Paper Title: The Bulk Densities of Small Solar System Bodies as a Probe of Planetesimal Formation.









Misako Tatsuuma
misako.tatsuuma@gmail.com

0000-0003-1844-5107]Misako Tatsuuma
RIKEN Interdisciplinary Theoretical and Mathematical Science Program (iTHEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

0000-0003-4562-4119]Akimasa Kataoka
Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

0000-0001-9659-658X]Hidekazu Tanaka
Astronomical Institute, Graduate School of Science, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai 980-8578, Japan

0000-0002-7188-8428]Tristan Guillot
Observatoire de la Côte d’Azur, Boulevard de l’Observatoire CS 34229, F 06304 NICE Cedex 4, France

§ ABSTRACT  

Constraining the formation processes of small solar system bodies is crucial for gaining insights into planetesimal formation.
Their bulk densities, determined by their compressive strengths, offer valuable information about their formation history.
In this paper, we utilize a formulation of the compressive strength of dust aggregates obtained from dust N-body simulations to establish the relation between bulk density and diameter.
We find that this relation can be effectively approximated by a polytrope with an index of 0.5, coupled with a formulation of the compressive strength of dust aggregates.
The lowest-density trans-Neptunian objects (TNOs) and main-belt asteroids (MBAs) are well reproduced by dust aggregates composed of 0.1-μm-sized grains.
However, most TNOs, MBAs, comets, and near-Earth asteroids (NEAs) exhibit higher densities, suggesting the influence of compaction mechanisms such as collision, dust grain disruption, sintering, or melting, leading to further growth.
We speculate that there are two potential formation paths for small solar system bodies: one involves the direct coagulation of primordial dust grains, resulting in the formation of first-generation planetesimals, including the lowest-density TNOs, MBAs, and parent bodies of comets and NEAs.
In this case, comets and NEAs are fragments or rubble piles of first-generation planetesimals, and objects themselves or rubbles are composed of 0.1-μm-sized grains.
The other path involves further potential fragmentation of first-generation planetesimals into compact dust aggregates observed in protoplanetary disks, resulting in the formation of second-generation planetesimals composed of compact dust aggregates, which may contribute to explaining another formation process of comets and NEAs.








§ INTRODUCTION
 


Constraining the formation processes of comets and asteroids in the solar system is crucial for understanding the formation of planetesimals, which are kilometer-sized building blocks of planets.
The insights into the formation history of comets and asteroids have been provided by some space explorations, such as Hayabusa <cit.>, Rosetta <cit.>, Hayabusa2 <cit.>, and Origins, Spectral Interpretation, Resource Identification, Security, Regolith Explorer (OSIRIS-REx) missions <cit.>.
There are several scenarios for the formation of comets and asteroids: remnants of planetesimals, fragments of planetesimals, or rubble piles, which are bodies made from piles of debris from the collisional destruction of planetesimals <cit.>.


One of the scenarios for planetesimal formation is the direct coagulation scenario of aggregates of submicrometer-sized dust grains <cit.>, but it is not consistent with observations of protoplanetary disks in terms of porosity.
Some numerical simulations have demonstrated that aggregation of dust grains results in highly porous structures, in other words, the porosity of dust aggregates exceeds ∼0.99 <cit.>.
However, some millimeter polarization observations by the Atacama Large Millimeter/submillimeter Array (ALMA) have indicated that dust aggregates in protoplanetary disks are relatively compact, which means that the porosity is 0.7–0.97 <cit.>.
Here, we refer to dust aggregates with porosity greater than 0.99 as highly porous dust aggregates, while those with porosity less than 0.99 as compact dust aggregates.


Another scenario for planetesimal formation is the streaming instability <cit.> and subsequent gravitational instability <cit.>, assuming millimeter–centimeter-sized compact pebbles are the initial condition.

Pebbles are also assumed in the pebble accretion scenario <cit.>, in which some seed planetesimals capture pebbles to form protoplanets before the dissipation of protoplanetary disk gas.

The SI/GI formation scenario is consistent with some exploration results.
For example, the low tensile strength of comet 67P/Churyumov–Gerasimenko <cit.> is consistent with loosely packed pebble aggregates <cit.>.

However, the formation process of compact pebbles from submicrometer-sized dust grains remains unknown.
Some experiments and numerical simulations have suggested that once dust aggregates become compact with porosity less than ∼ 0.99, they bounce and are compressed <cit.>, while highly porous dust aggregates with porosity greater than ∼ 0.99 do not bounce <cit.>.





With a scope of constraining the formation process of planetesimals, in this work, we theoretically derive the bulk density of objects made of single-sized grains called monomers and discuss whether it is consistent with the bulk density of small solar system bodies, such as asteroids, comets, and trans-Neptunian objects (TNOs).
We derive a relation between bulk density and diameter of dust aggregates by using polytropes and compressive strength of dust aggregates, which has been investigated numerically <cit.> and experimentally <cit.>.
We use the compressive strength formulated by <cit.>, which can be applied to dust aggregates with volume filling factors both lower and higher than 0.1, because the direct coagulation scenario results in highly porous structures and the subsequent compaction results in less porous structures.
Moreover, we clarify the dependence of the bulk density on the radius and composition of monomers to trace backward the formation process of planetesimals.
In this work, we assume the direct coagulation scenario resulting in highly porous dust aggregates, which can be considered as a lower limit of the bulk density.






This paper is organized as follows.
In Section <ref>, we derive a relation between bulk density and diameter of dust aggregates.
Then, we compare the relation for dust aggregates with bulk densities and equivalent sphere diameters of small solar system bodies in Section <ref>.
Based on our results, we discuss the formation processes of small solar system bodies and planetesimals in Sections <ref> and <ref>, respectively.
We also speculate about the possible origin of compact dust aggregates observed in protoplanetary disks in terms of porosity in Section <ref>.
In addition, we note some caveats about the other mechanisms that may affect the bulk density of dust aggregates in Section <ref>.
Finally, we conclude our work in Section <ref>.



§ METHODS
 


Here, we briefly introduce the compressive strength of dust aggregates derived from numerical simulations by <cit.> in Section <ref>.
Then, we derive the relation between bulk density and diameter in Section <ref>.
In Section <ref>, we explain the assumed material parameters of monomers.



 §.§ Compressive Strength of Dust Aggregates



We use a formulation of the compressive strength of dust aggregates with single-sized monomers by <cit.>, who performed numerical simulations of the compression of dust aggregates.
The compressive strength P_comp <cit.> is given by taking into account the rolling motion of aggregates for low volume filling factors and the closest packing of aggregates for high volume filling factors as

    P_comp   =   E_roll/r_m^3(1/ϕ-1/ϕ_max)^-3
       ≃    4.7×10^5 Pa(γ/100 mJ m^-2)(r_m/0.1 μ m)^-2
          ×(ξ_crit/8 Å)(1/ϕ-1/ϕ_max)^-3,

 where E_roll=6π^2γ r_mξ_crit is the energy needed for a monomer to roll a distance of (π/2)r_m, r_m is the monomer radius, γ is the surface energy of monomers, ξ_crit is the critical rolling displacement of monomers, ϕ is the volume filling factor of dust aggregates, and ϕ_max=√(2)π/6=0.74 is the volume filling factor of the closest packing.


Equation (<ref>) shows that dust aggregates are polytropes with a polytropic index n defined as P_comp∝ρ^(n+1)/n, where ρ=ρ_mϕ is the density inside dust aggregates and ρ_m is the material density.
Although the actual index is not constant, n=0.5, i.e., P_comp∝ρ^3∝ϕ^3, if the density is small enough, while n=0 if the density is large enough.
Polytropes with a polytropic index of 0.5 correspond to small dust aggregates, which have small central density and pressure.
On the other hand, polytropes with a polytropic index of 0 correspond to large dust aggregates, which have a uniform density and a large pressure.



 §.§ Relation between Bulk Density and Diameter




We derive a relation between bulk density ρ_bulk and diameter D based on polytropes with a polytropic index of 0.5 and the compressive strength of dust aggregates.
The diameter D of polytropes with a polytropic index of 0.5 is given by solving the Lane-Emden equation as

    D = 2[(0.5+1)P_c/4π Gρ_c^2]^1/2ξ_1
    = 2(3P_c/8π Gρ_c^2)^1/2ξ_1,

where P_c is the central pressure, ρ_c is the central density, G is the gravitational constant, and ξ_1=2.75 <cit.>.
In this case of a polytropic index of 0.5, P_c/ρ_c^3 = P_bulk/ρ_bulk^3, where P_bulk is the bulk pressure.
Therefore,

    P_c/ρ_c^2 = P_bulk/ρ_bulk^2ρ_c/ρ_bulk
    = 1.84P_bulk/ρ_bulk^2

because ρ_c/ρ_bulk=1.84.
Finally, we obtain

    D    =    2(3/8π GP_bulk/ρ_bulk^2·1.84)^1/2ξ_1 
       =    2[3/8π GE_roll/r_m^3(ρ_m/ρ_bulk-1/ϕ_max)^-31/ρ_bulk^2·1.84]^1/2ξ_1 
       ≃    5.28(E_roll/m_mGρ_m)^1/2(ρ_m/ρ_bulk-1/ϕ_max)^-3/2ρ_m/ρ_bulk,

where m_m=(4/3)π r_m^3ρ_m is the monomer mass, by assuming P_bulk=P_comp and using Equation (<ref>).

Equation (<ref>) is the combination of polytropes and dust aggregates because we use Equation (<ref>) instead of a polytropic equation of state.
For the bulk densities comparable to ρ_mϕ_max, the assumption of the polytropes with an index of 0.5 is not valid.
Therefore, we numerically calculate the internal density and pressure profiles of dust aggregate by solving the hydrostatic equilibrium and compare them with those of polytropes with a constant polytropic index of 0.5 in Appendix <ref>.
We confirm that the assumption of a constant index of 0.5 is not valid for high density.
We also confirm that the relation of Equation (<ref>) is a good approximation to that of the exact numerical solution.



 §.§ Material Parameters



We assume monomers have material parameters listed in Table <ref>, and additionally, we assume that each ice monomer has a silicate core whose mass equals the mass of an ice mantle.
The average material density of ice monomers ρ_m,ice+sil is given as

    ρ_m,ice+sil
    = 2ρ_m,iceρ_m,sil/ρ_m,ice+ρ_m,sil
    = 1.45 g cm^-3,

because the specific volume of the monomer is 1/ρ_m,ice+sil=0.5/ρ_m,ice+0.5/ρ_m,sil, where ρ_m,ice=1.0 g cm^-3 and ρ_m,sil=2.65 g cm^-3 are the material densities of ice and silicate.
Hereinafter, we use the material densities of 1.45 and 2.65 g cm^-3 for ice and silicate monomers, respectively.



lcc
Material Parameters of Ice and Silicate 

Parameter     Ice     Silicate

Material density ρ_m (g cm^-3)     1.0     2.65 

Surface energy γ (mJ m^-2)     100     20 

Poisson's ratio ν     0.25     0.17 

Young's modulus E (GPa)     7     54 

Critical rolling displacement ξ_crit (Å)     8     20 

We use ρ_m, γ, ν, E, and ξ_crit of ice of <cit.> and <cit.> and of silicate of <cit.>.




§ COMPARISON WITH SMALL SOLAR SYSTEM BODIES
 

In this section, we show our results on the bulk density and diameter of dust aggregates derived by using polytropes with a polytropic index of 0.5 and the compressive strength.
The methods of numerical calculation of solving the hydrostatic equilibrium and the comparison of numerical results with our analytical results are referred to in Appendix <ref>.

Figure <ref> shows the relation between bulk density and diameter of dust aggregates (Equation (<ref>)) in four cases: 0.1-μm-radius ice, 1.0-μm-radius ice, 0.1-μm-radius silicate, and 1.0-μm-radius silicate monomers.
Here, we refer to primordial dust aggregates as aggregates composed of 0.1–1.0-μ m-radius monomers.

We also plot the bulk density and equivalent sphere diameter of small solar system bodies in Figure <ref>.
The data of TNOs, comets, and asteroids investigated by sample return explorations are listed in Tables <ref>, <ref>, and <ref>, respectively.
We use the data of near-Earth asteroids (NEAs) and main-belt asteroids (MBAs) from Table 1 of <cit.> but exclude some data with large errors that exceed the density values for simplicity.
Small solar system bodies show trends in their bulk densities and diameters according to their classification.
First, TNOs have diameters of 100–1000 km, where bulk densities of dust aggregates converge.

Second, comets have diameters of 1–10 km and bulk densities of 0.1–1 g cm^-3.
Third, Itokawa, Bennu, Ryugu, and NEAs are smaller and denser than comets, and have diameters of 0.3–3 km and bulk densities of 1–2 g cm^-3.
Finally, some MBAs are similar to TNOs while MBAs have a large diversity of bulk density.


We also show further growth modes in Figure <ref>: fragmentation, melting, and monomer disruption.
Fragmentation keeps the bulk density constant but reduces the diameter, while melting and monomer disruption keep the mass constant but reduce the diameter.
We discuss the possibility of melting and monomer disruption in Section <ref> and Appendix <ref>.
Especially, monomer disruption cannot be neglected in the center of dust aggregates due to high pressure, and therefore we derive a relation between central pressure and diameter in Appendix <ref> and the upper limit of diameter below which our results are valid in Appendix <ref>.
We find that the upper limit is ∼ 200 km in the case of 0.1-μ m-radius monomers, which corresponds to the diameters of TNOs.



lccccccl
Bulk Density and Diameter of TNOs


Name     3cDensity (g cm^-3)     3cDiameter (km)     Reference 

2-4 5-7
          error     error          error     error     

(134340) Pluto     1.860     0.013     -0.013     2374     8     -8     <cit.> 

(136199) Eris     2.52     0.05     -0.05     2326     12     -12     <cit.> 

(134340) I Charon     1.702     0.021     -0.021     1212     6     -6     <cit.> 

(50000) Quaoar     1.99     0.46     -0.46     1110     5     -5     <cit.> 

(90482) Orcus     1.53     0.15     -0.13     958.4     22.9     -22.9     <cit.> 

(120347) Salacia     1.29     0.29     -0.23     901     45     -45     <cit.> 

(174567) Varda     1.27     0.41     -0.44     792     91     -84     <cit.> 

(55637) 2002 UX_25     0.82     0.11     -0.11     692     23     -23     <cit.> 

(47171) 1999 TC_36     0.64     0.15     -0.11     393.1     25.2     -26.8     <cit.> 

(79360) Sila-Nunam     0.73     0.28     -0.28     343     42     -42     <cit.> 

(148780) Altjira     0.30     0.50     -0.14     331     51     -187     <cit.> 

2001 QC_298     1.14     0.34     -0.30     303     27     -30     <cit.> 

(26308) 1998 SM_165     0.51     0.29     -0.14     279.8     29.7     -28.6     <cit.> 


(65489) Ceto     0.64     0.16     -0.13     281     11     -11     <cit.> 

(275809) 2001 QY_297     0.92     1.30     -0.27     229     22     -108     <cit.> 

2001 XR_254     1.00     0.96     -0.56     221     41     -71     <cit.> 

(88611) Teharonhiawako     0.60     0.36     -0.33     220     41     -44     <cit.> 

(66652) Borasisi     2.1     2.60     -1.2     163     32     -66     <cit.> 

(42355) Typhon     0.60     0.72     -0.29     157     34     -34     <cit.> 

This table is cited from Table A.1. of <cit.>.


lcccllcc

Bulk Density and Equivalent Sphere Diameter of Comets


Name     3cDensity (g cm^-3)     Method     Reference     2cDiameter (km) 

2-4 7-8
          lower limit     upper limit                    error

1P 
    0.6     0.2     1.5     NGF     <cit.>     11.0     


    1     0.03     4.9     NGF     <cit.>     11.0     


    0.5     0.2     0.8     NGF     <cit.>     11.0     

2P 
    0.8         1.6     NGF     <cit.>     4.8     

6P 
    0.15     0.1     0.2     NGF     <cit.>     3.2     

9P 
    0.4     0.2     1     Deep impact ejecta     <cit.>     6.0     0.2 


    0.45     0.2     0.7     NGF     <cit.>     6.0     0.2 


    0.2     0.1     0.3     NGF     <cit.>     6.0     0.2 


    0.47     0.24     1.25     Deep impact ejecta     <cit.>     6.0     0.2 

10P 
    0.7     0.3     1.1     NGF     <cit.>     10.6     

19P 
    0.49     0.29     0.83     NGF     <cit.>     4.8     

22P 
    0.2     0.1     0.3     NGF     <cit.>     3.0     0.4 

46P 
    0.4     0.1     0.7     NGF     <cit.>     1.2     

67P 
    0.535     0.5     0.57     In-situ mass and volume     <cit.>     4.0     


    0.532     0.525     0.539     In-situ mass and volume     <cit.>     4.0     

81P 
    0.3         0.8     NGF     <cit.>     4.0     

103P 
    0.22     0.14     0.52     Erosion rate     <cit.>     1.6     

We exclude the data without the preferred value of density in <cit.>.
The reference of diameters is <https://ssd.jpl.nasa.gov/sbdb_query.cgi>.
NGF denotes non-gravitational forces, which cause an acceleration of a comet's nucleus and change its orbital parameters, so that its mass can be estimated.


lcccccl
Bulk Density and Equivalent Sphere Diameter of Asteroids Investigated by Sample Return Explorations


Name     2cDensity (g cm^-3)     2cVolume (km^3)     Diameter (km)     Reference 

2-3 4-5
          error          error          

162173 Ryugu     1.19     0.02     0.377     0.0049     0.90     <cit.> 

101955 Bennu     1.19     0.013     0.0615     0.0001     0.49     <cit.> 

25143 Itokawa     1.9     0.13     0.0184     0.00092     0.33     <cit.> 

We derive their diameters from their volumes by assuming that they are spheres.






§ FORMATION OF SMALL SOLAR SYSTEM BODIES




In this section, we compare our results with those of TNOs, comets, and asteroids plotted in Figure <ref>, and discuss their formation processes in Sections <ref>, <ref>, and <ref>, respectively.
We assume some formation scenarios of small solar system bodies: primordial dust aggregates or further growth modes.
The further growth modes contain fragmentation and compaction; fragmentation means fragments or rubble piles of primordial dust aggregates and compaction means additionally compacted dust aggregates due to melting or monomer disruption.



 §.§ Formation Process of TNOs



We can explain TNOs, which have diameters of 100–1000 km and bulk densities of 0.3–2 g cm^-3, in terms of primordial dust aggregates and additionally compacted dust aggregates.

Figure <ref> shows that the lowest density TNO can be explained by primordial ice aggregates composed of 0.1-μ m-sized monomers.
The diversity of TNOs' bulk densities can be explained by the diversity of monomer radius, although its upper limit cannot be constrained because bulk densities of dust aggregates converge in the diameter range of TNOs.
The diversity of composition can also explain the diversity of TNOs' bulk densities.
For further discussion about the composition of TNOs, see other works, for example, <cit.>.

The additionally compacted dust aggregates due to melting or monomer disruption can explain the diversity of TNOs' bulk densities.
TNO-sized objects experience thermal evolution due to the decay energy of radionuclides (Section <ref>).
In addition, monomers in the center of TNO-sized objects can be broken due to high central pressure (Appendices <ref> and <ref>).



 §.§ Formation Process of Comets




We can explain comets, which have diameters of 1–10 km and bulk densities of 0.1–1 g cm^-3, in terms of fragments or rubble piles of primordial dust aggregates and additionally compacted dust aggregates.


Figure <ref> shows that even the lowest-density comet cannot be explained by primordial ice aggregates composed of 0.1–1.0 μ m-sized monomers.
If comets are primordial dust aggregates, the monomer radius should be 1.0 μ m<r_m≲10.0 μ m for ice and 0.3 μ m<r_m<3.0 μ m for silicate.
Especially, we estimate the monomer radius of comet 67P, whose bulk density is precisely estimated, by using

    r_m≃ 19.8(γξ_crit/ρ_m^2 D^2 G)^1/2(ρ_m/ρ_bulk-1/ϕ_max)^-3/2ρ_m/ρ_bulk

from Equation (<ref>).
If we assume that ρ_bulk=0.532 g cm^-3 <cit.> and D=4 km, we obtain r_m≃6.3 μ m for ice and r_m≃1.0 μ m for silicate by using Equation (<ref>).

However, the derived monomer radius is not consistent with the monomer radius that can explain comets' low tensile strength, and therefore we may have to consider other mechanisms that make strength low.
For example, <cit.> have shown that the tensile strength of dust aggregates composed of submicrometer-sized monomers is higher than the estimated tensile strength of comets.
Especially, they have suggested that comet 67P is composed of 3.3–220-μ m-sized silicate monomers, while our results suggest 1.0-μ m silicate monomers.
This inconsistency may be caused by other missing mechanisms to explain the low tensile strength of comets, such as sintering <cit.> and the dependence of tensile strength on the volume of dust aggregates <cit.>.
We note that the compressive and tensile strengths of pebble aggregates formed via SI/GI can be different from those of dust aggregates if comets are composed of compact pebbles <cit.>, but investigating the strength of pebble aggregates numerically is future work.


Figure <ref> suggests that the comet formation process needs further growth modes: fragmentation and compaction.
Fragmentation means fragments or rubble piles and needs parent dust aggregates.
If we do not assume compaction, we can estimate that the diameters of parent dust aggregates are ∼60 km or larger for 0.1-μ m-sized ice monomers, larger than 10 km for 0.1-μ m-sized silicate monomers, and ∼6 km or larger for 1.0-μ m-sized ice monomers.
Especially, we can estimate the diameter of the parent body of comet 67P.
If we assume that ρ_bulk=0.532 g cm^-3 <cit.>, we obtain D≃2.5×10^2 km for 0.1-μ m-sized ice monomers, D≃42 km for 0.1-μ m-sized silicate monomers, and D≃25 km for 1.0-μ m-sized ice monomers by using Equation (<ref>).

However, the diameters of parent dust aggregates are only approximate values, and some mechanisms may increase and decrease the bulk densities of comets.
Their bulk densities become lower than expected if comets are composed of surface materials of the parent dust aggregates, while their bulk densities become higher than expected if comets are composed of central materials of the parent dust aggregates.
Moreover, melting and monomer disruption inside the parent dust aggregates and the collisional compression make the bulk densities of comets higher, while voids among the rubble make the bulk densities of comets lower if comets are rubble piles.
We note that the tensile strength of dust aggregates is higher than that of comets, and therefore we need to consider sintering <cit.> and volume dependence <cit.> that lowers the tensile strength of dust aggregates if comets are single fragments, as well as possible pebble aggregates formed via SI/GI <cit.>.



 §.§ Formation Process of Asteroids




We can explain the majority of MBAs, which have diameters of ∼100 km and bulk densities of ≳ 1 g cm^-3, in terms of primordial dust aggregates and additionally compacted dust aggregates, while we can explain Itokawa, Bennu, Ryugu, and NEAs, which have diameters of 0.3–3 km and bulk densities of 1–2 g cm^-3, in terms of fragments or rubble piles of primordial dust aggregates and additionally compacted dust aggregates.

Figure <ref> shows that some MBAs can be explained by primordial silicate aggregates composed of 0.1–1.0 μ m-sized monomers.
The dispersion of MBAs' bulk densities is much larger than that due to different monomer radii, and therefore we need further growth modes.

The additionally compacted dust aggregates due to melting or monomer disruption can explain the dispersion of MBAs' bulk densities.
100-km-sized objects experience thermal evolution due to the decay energy of radionuclides, such as melting and differentiation (Section <ref>).
In addition, monomers in the center of objects whose diameter exceeds ≃ 200 km can be broken due to high central pressure (Appendices <ref> and <ref>).


In the case of Itokawa, Bennu, Ryugu, and NEAs, Figure <ref> shows that they cannot be explained by primordial silicate aggregates composed of 0.1–1.0 μ m-sized monomers.
If these asteroids are primordial silicate aggregates, the monomer radius should be r_m≳10.0 μ m, while the bulk density of Itokawa is too high to be explained by primordial silicate aggregates, which suggests its parent body has experienced melting to make it denser.
Especially, we estimate the monomer radius of Ryugu and Bennu by using their bulk densities and diameters in Table <ref>, and we obtain r_m≃18 μ m for Ryugu and r_m≃32 μ m for Bennu by using Equation (<ref>).


Figure <ref> suggests that the formation processes of Itokawa, Bennu, Ryugu, and NEAs need further growth modes: fragmentation and compaction.
If we do not assume compaction, we can estimate that the diameters of parent dust aggregates are ∼1.0×10^2 km or larger for 0.1-μ m-sized silicate monomers and ∼10 km or larger for 1.0-μ m-sized silicate monomers.
Especially, we can estimate the diameter of the parent bodies of Ryugu and Bennu, while we cannot estimate that of Itokawa because of possible melting.
If we assume that ρ_bulk=1.19 g cm^-3 for Ryugu and Bennu <cit.>, we obtain D≃1.6×10^2 km for 0.1-μ m-sized silicate monomers and D≃16 km for 1.0-μ m-sized silicate monomers by using Equation (<ref>).

The previously suggested size of the parent body of Ryugu, which is 100–160 km <cit.>, is consistent with our results of 0.1-μ m-sized silicate monomers.


The parent bodies of Ryugu are believed to be Eulalia and Polana from the results of the observed visible spectral type and orbital dynamics <cit.>.
Eulalia is a member of the Eulalia family, whose member asteroids are dynamically linked.
The Eulalia family is thought to be formed as a result of the breakup of a 100–160 km parent body <cit.>, which is consistent with our results that D≳1.6×10^2 km for 0.1-μ m-sized silicate monomers.
However, our results of the parent bodies' diameters have some uncertainty because of some mechanisms that increase and decrease the bulk densities of fragments (see Section <ref>), and therefore we cannot exclude the possibility of 1.0-μ m-sized silicate monomers.



§ PLANETESIMAL FORMATION AND ORIGIN OF OBSERVED COMPACT DUST AGGREGATES



In this section, we discuss the formation process of planetesimals and the origin of compact dust aggregates observed in protoplanetary disks.
First, we speculate the formation process of observed compact dust aggregates in terms of porosity in Section <ref>.
Then, we propose a unified formation scenario of planetesimals including TNOs, comets, asteroids, and observed compact dust aggregates in terms of direct coagulation in Section <ref>.



 §.§ Formation Process of Observed Compact Dust Aggregates




First, we focus on millimeter-sized compact dust aggregates observed in protoplanetary disks, whose porosity is 0.7–0.97 <cit.>.
The bulk density of compact dust aggregates is 0.044–0.44 g cm^-3 if we assume that they are composed of ice with silicate cores, whose material density is 1.45 g cm^-3.


Figure <ref> suggests that compact dust aggregates need their parent bodies to achieve the bulk density of observed compact dust aggregates and they are fragments of parent bodies.
Equation (<ref>) shows that their diameters are 30–180 km or larger for 0.1-μ m-sized ice monomers and 3–18 km or larger for 1.0-μ m-sized ice monomers.
Compression mechanisms other than self-gravitational compression cannot compress highly porous dust aggregates with porosity greater than 0.99, which is expected from direct coagulation, as <cit.> have shown that the bulk density of dust aggregates compressed by collisions and gas pressure is from 10^-5 g cm^-3 to 10^-3 g cm^-3, which is much smaller than that determined by self-gravitational compression.



 §.§ Speculation about Planetesimal Formation Process



Next, we discuss and propose a unified formation scenario of planetesimals including TNOs, comets, asteroids, and observed compact dust aggregates in terms of direct coagulation.
First, we summarize the possible formation process of TNOs, comets, asteroids, and observed compact dust aggregates in Table <ref> based on our results in Sections <ref>, <ref>, <ref>, and <ref>.


We can constrain the monomer radius as an initial condition of planet formation to 0.1 μ m for both ice and silicate by the discussion about the monomer radius of the lowest density TNO and the diameter of the parent body of Ryugu in Sections <ref> and <ref>, respectively.

Once we can constrain the monomer radius to 0.1 μ m and if we assume the same monomer radius as that of comets, there are two formation scenarios for NEAs and comets: direct coagulation and fragments of primordial dust aggregates, or aggregation of compact dust aggregates.


The first formation scenario needs primordial dust aggregates with a diameter of 30–180 km or larger, which we call the first-generation planetesimals, as shown in Figure <ref>.
The first-generation planetesimals can experience compaction due to melting or monomer disruption because of their diameter.
The diameter of the first-generation planetesimals is consistent with diameters of TNOs, MBAs, parent bodies of NEAs including Ryugu and Bennu, parent bodies of comets, and parent bodies of observed compact dust aggregates as shown in Table <ref>.
In this case, NEAs and comets are fragments or rubble piles, and observed compact dust aggregates are fragments of planetesimals.



Our results suggest that there are planetesimals in protoplanetary disks where compact dust aggregates are observed.


The second formation scenario needs planetesimals composed of aggregates of compact dust aggregates, which we call the second-generation planetesimals, because there is a contradiction between the expected tensile strength of dust aggregates composed of 0.1-μ m-radius monomers and the estimated tensile strength of comets as discussed in Section <ref>.
One of the possible explanations for the low tensile strength of comets is aggregates of compact dust aggregates.
Our results suggest that there is a possibility that NEAs and comets are second-generation planetesimals whose diameter is 0.1–10 km as shown in Figure <ref>.










lcccccc


Possible Formation Process of TNOs, Comets, Asteroids, and Observed compact dust aggregates


Objects     2cPrimordial dust aggregates     4cFragments or rubble piles 

2-3 4-7
     Ice     Silicate     2cIce     2cSilicate 

4-5 6-7
             r_m=0.1 μ m     r_m=1.0 μ m     r_m=0.1 μ m     r_m=1.0 μ m


TNOs     r_m≳0.1 μ m                     

Comets     1.0 μ m<r_m≲10.0 μ m     0.3 μ m<r_m<3.0 μ m     D≳60 km     D≳6 km     D>10 km     

67P     r_m≃6.3 μ m     r_m≃1.0 μ m     D≃2.5×10^2 km     D≃25 km     D≃42 km     

MBAs         r_m≳0.1 μ m                 

NEAs         r_m≳10.0 μ m             D≳1.0×10^2 km     D≳10 km 

Ryugu         r_m≃18 μ m             D≃1.6×10^2 km     D≃16 km 

Bennu         r_m≃32 μ m             D≃1.6×10^2 km     D≃16 km 

Observed

compact             D≳30–180 km     D≳3–18 km         

dust aggregates
r_m is the monomer radius and D is the diameter of the parent body.






§ CAVEATS



In this section, we discuss some caveats about the other mechanisms that may affect the bulk density of dust aggregates: thermal evolution of km-sized objects, uncertainty of material parameters, and pebble aggregates in Sections <ref>, <ref>, and <ref>, respectively.






 §.§ Thermal Evolution of km-sized Objects



We discuss the possibility of melting of monomers by the head-producing short-lived radionuclides, such as ^26Al <cit.>.
If the temperature exceeds ∼1400 K, silicate would start melting <cit.> and our calculations are no longer valid.
Moreover, if the temperature exceeds ∼300 K, ice would start melting although the temperature depends on the pressure.
Here, we roughly discuss at which diameter the maximum temperature may reach these limits and monomers cannot keep their shape. 

The previous calculations of temperature increase inside planetesimals have normally assumed fixed diameter, porosity, thermal conductivity, and formation timing, which determine the amount of ^26Al included <cit.>.
For example, <cit.> estimated the maximum temperature that can be reached inside planetesimals.
If the porosity is 0 and the formation time is the same as that of calcium-aluminium-rich inclusions (CAIs), which is the most heated case, the planetesimals with a diameter of ≃16 km reach 1500 K, and those with a diameter of ≃4 km reach 400 K while smaller bodies do not reach such temperatures.
If the porosity is 0.5, the planetesimals with a diameter of ≃2 km reach 1500 K, and those with a diameter of sub-km reach 400K.
We note that the small bodies we treat in this paper can experience melting, although the porosity dependence of the maximum temperature is not fully understood.



 §.§ Uncertainty of Material Parameters



Equations (<ref>) and (<ref>) show the relation among the object's diameter, object's bulk density, and monomer radius, and demonstrate that they depend on some material parameters, which may have uncertainty.
For example, the critical rolling displacement of monomers ξ_crit in E_roll has a range from 2 Å of theory <cit.> to 32 Å of experiments <cit.>.
The surface energy of monomers γ in E_roll also has uncertainty <cit.> depending on materials and temperatures.
Moreover, the material density has uncertainty depending on the mass ratio between an ice mantle and a silicate core, although we assume 1:1 in this work.
If we assume other materials, such as carbon, the material density, surface energy, and other parameters can be different from the values of ice and silicate.
However, the object's diameter and bulk density of other materials should be between those of ice and silicate if the material parameters have values between those of ice and silicate.



 §.§ Bulk Density of Pebble Aggregates



Pebble aggregates may have a different relation between their bulk density and diameter, which is beyond the scope of this work.
Their compressive strength, which determines their bulk density, may be different from that of dust aggregates in this work and can be more fragile than them <cit.>.
However, the compressive strength of pebble aggregates has not been investigated by numerical simulations.
The bulk density of pebble aggregates can be higher if their compressive strength is lower than that of dust aggregates, while the bulk density of pebble aggregates tends to be lower because they have voids both inside and between pebbles.



§ CONCLUSIONS
 

We derived the relation between bulk density and diameter of dust aggregates by using polytropes and compressive strength formulated by <cit.>.
Then, we compared our results with those of small solar system bodies to discuss their formation process as well as the planetesimal formation process.

Our main findings are as follows.





  * The relation between bulk density and diameter of dust aggregates is given by Equation (<ref>) by using polytropes with a polytropic index of 0.5 and the compressive strength of dust aggregates (Equation (<ref>)).


  * We discussed the general formation process of TNOs.
The lowest density TNO can be explained by primordial ice aggregates composed of 0.1-μ m-sized monomers, where primordial refers to those that have not undergone additional compaction.
The diversity of TNOs' bulk densities can be explained by the diversity of monomer radius and composition and by additional compaction due to melting or monomer disruption.


  * We also discussed the formation process of comets.
Even the lowest-density comet cannot be explained by primordial ice aggregates composed of 0.1–1.0 μ m-sized monomers.
If comets are primordial dust aggregates, the monomer radius should be 1.0–10.0 μ m for ice and 0.3–3.0 μ m for silicate.
The comet formation process needs further growth modes: fragmentation and compaction.
In the case of fragmentation, which means comets are fragments or rubble piles of their parent dust aggregates, the diameters of parent bodies are ∼60 km or larger for 0.1-μ m-sized ice monomers, larger than 10 km for 0.1-μ m-sized silicate monomers, and ∼6 km or larger for 1.0-μ m-sized ice monomers.


  * We also discussed the formation process of asteroids.
MBAs can be explained by primordial silicate aggregates composed of 0.1–1.0 μ m-sized monomers.
The dispersion of MBAs' bulk densities can be explained by compaction due to melting or monomer disruption.
NEAs including Itokawa, Bennu, and Ryugu cannot be explained by primordial silicate aggregates composed of 0.1–1.0 μ m-sized monomers.
If NEAs are primordial silicate aggregates, the monomer radius should be ≳10.0 μ m, while Itokawa needs additional compaction.
The NEA formation process needs further growth modes: fragmentation and compaction.
In the case of fragmentation, the diameters of parent bodies are ∼1.0×10^2 km or larger for 0.1-μ m-sized silicate monomers and ∼10 km for 1.0-μ m-sized silicate monomers.
The previously suggested diameter of the parent body of Ryugu, which is 100–160 km <cit.>, is consistent with our results of 0.1-μ m-sized silicate monomers.


  * We discussed the formation process of observed compact dust aggregates.
They can be explained by fragments of parent dust aggregates whose diameters are 30–180 km or larger for 0.1-μ m-sized ice monomers and 3–18 km or larger for 1.0-μ m-sized ice monomers.



  * We proposed a unified scenario to explain the formation process of planetesimals including TNOs, comets, asteroids, and observed compact dust aggregates in terms of direct coagulation as follows and shown in Figure <ref>.
First, 0.1-μ m-sized dust grains coagulate and grow into dust aggregates, and then the first-generation planetesimals composed of dust aggregates form.
The first-generation planetesimals include TNOs, MBAs, parent bodies of NEAs including Ryugu and Bennu, and parent bodies of comets.
In this case, NEAs and comets are fragments or rubble piles of the first-generation planetesimals.
Second, a part of the first-generation planetesimals fragment into compact dust aggregates, they coagulate and grow into aggregates of compact dust aggregates, and then the second-generation planetesimals composed of aggregates of compact dust aggregates form.
The second-generation planetesimals include NEAs and comets.





We appreciate the discussion about monomer disruption with Sota Arakawa.
This work was supported by JSPS KAKENHI grant Nos. JP19J20351, JP22J00260, JP22KJ1292, and JP22K03680.
This work has made use of NASA’s Astrophysics Data System.
This work has made use of adstex (https://github.com/yymao/adstex).







§ ACCURACY OF THE ANALYTICAL RELATION BETWEEN BULK DENSITY AND DIAMETER




In this section, we numerically calculate the internal density and pressure profiles of dust aggregates by solving the hydrostatic equilibrium, and then we calculate their bulk density and compare it with the analytical relation (Equation (<ref>)).
The numerical calculation is necessary because the pressure, i.e., compressive strength, inside dust aggregates shows their polytropic index is not constant although the analytical relation is based on polytropes with a polytropic index of 0.5.

First, we numerically solve the internal density and pressure profiles of dust aggregates.
The hydrostatic equilibrium is given as

    d/dr(r^2/ρdP/dρdρ/dr) = -4π Gρ r^2,

where r is the radius from the center of the object and P is the pressure. 
Here, the pressure is equivalent to the compressive strength of dust aggregates, and therefore we derive dP/dρ from Equation (<ref>) as

    dP/dρ = dP_comp/dρ = 3E_rollρ_m/r_m^31/ρ^2(ρ_m/ρ-1/ϕ_max)^-4.


After numerically solving Equations (<ref>) and (<ref>) by integrating from the center where ρ=ρ_c to the surface where ρ=0, we calculate the bulk density by dividing the total mass by its volume.

Figure <ref> shows the numerically calculated internal density and pressure profiles of ice dust aggregates that have similar sizes to TNOs and comets.
The monomer radius is 0.1 μ m and the other parameters but the material density are listed in Table <ref>.
Here, we assume the central densities ρ_c=0.35 and 0.002 g cm^-3.
In these cases, the bulk densities ≃ 0.23 and 0.0011  g cm^-3, and the objects' radii ≃ 44 and 2.1 km, respectively (see the blue dashed line in Figure <ref>).

For comparison, we also plot the internal density and pressure profiles of polytropes with a constant polytropic index of 0.5 in Figure <ref>.
In the case of ρ_c=0.002 g cm^-3, the polytrope with a constant index agrees well with the numerical results because the density is low enough, while in the case of ρ_c=0.35 g cm^-3, the polytrope with a constant index deviates from the numerical results.
However, even in the latter case, the bulk density obtained from Equation (<ref>) agrees well with the numerically calculated bulk density.

Figure <ref> shows the relative error to the analytical relation (Equation (<ref>)), which is based on polytropes with an index of 0.5 and the compressive strength of dust aggregates, of numerically calculated bulk density.
The relative error is below 5% for all cases, and therefore we conclude that the analytical relation corresponds to numerical calculated results.







§ CENTRAL PROPERTIES OF DUST AGGREGATES



In this section, we derive relations for central density and central pressure against diameter to discuss the validity of our results in terms of central properties.





First, we derive a relation between central density ρ_c and diameter D by using Equation (<ref>) as

    D    =    2(3P_c/8π Gρ_c^2)^1/2ξ_1 
       =    2[3/8π GE_roll/r_m^3(ρ_m/ρ_c-1/ϕ_max)^-31/ρ_c^2]^1/2ξ_1 
       ≃    3.89(E_roll/m_mGρ_m)^1/2(ρ_m/ρ_c-1/ϕ_max)^-3/2ρ_m/ρ_c.



Next, we derive a relation between central pressure P_c and diameter D.
By using Equation (<ref>), we obtain

    P_c   =   E_roll/r_m^3(ρ_m/ρ_c-1/ϕ_max)^-3,
    ρ_m/ρ_c   =   (E_roll/P_cr_m^3)^1/3 + 1/ϕ_max.

We obtain the diameter D by using Equations (<ref>) and (<ref>) as

    D    =    2(3P_c/8π Gρ_c^2)^1/2ξ_1 
       =    2{3P_c/8π G1/ρ_m^2[(E_roll/P_cr_m^3)^1/3 + 1/ϕ_max]^2}^1/2ξ_1 
       ≃   1.90(P_c/Gρ_m^2)^1/2[(E_roll/P_cr_m^3)^1/3 + 1/ϕ_max].


Figure <ref> shows the relations for the central density (Equation (<ref>)) and the central pressure (Equation (<ref>)) in four cases: 0.1-μm-radius ice, 1.0-μm-radius ice, 0.1-μm-radius silicate, and 1.0-μm-radius silicate monomers.





§ MONOMER DISRUPTION




We discuss the validity of our results in terms of the pressure that monomers can be broken <cit.>.
If the central pressure is too high, monomers can be broken, and then the bulk density and diameter can become different from our results.
The pressure at which materials can be broken P_dis has been investigated in the context of material science.
For example, ice can be broken at 5–25 MPa when the temperature is from -10^∘C to -20^∘C <cit.>.
Silica glasses, on the other hand, can be broken at ∼5 GPa at room temperature <cit.>.
Here, we assume P_dis=10 MPa and 1 GPa for ice and silicate, respectively.
We note that these pressures are not for submicrometer–micrometer-sized spheres, so P_dis and further discussion can be different from our estimation.



We estimate the upper limit of diameter D_ul below which our results are valid according to the discussion in Section 4.2 in <cit.>.
By considering the stress applied to the contact surface between monomers, we obtain the upper limit as

    P_ul   ≃    0.014P_dis(γ/100 mJ m^-2)^2/3
          ×(r_m/0.1 μ m)^-2/3(E^∗/3.7 GPa)^-2/3,
    
    E^∗   =   E/2(1-ν^2),



    D_ul≃ 1.90(P_ul/Gρ_m^2)^1/2[(E_roll/P_ulr_m^3)^1/3 + 1/ϕ_max],

where P_ul is the upper limit of pressure, E is the Young's modulus, E^∗ is the reduced Young's modulus, and ν is the Poisson's ratio, by using Equation (<ref>).
We plot the upper limit of both pressure (Equation (<ref>)) and diameter (Equation (<ref>)) as a function of monomer radius in Figure <ref>.
We find that the upper limit below which monomer disruption can be neglected is D_ul∼ 200 km in the case of 0.1-μ m-radius monomers, which corresponds to the diameters of TNOs.




aasjournal




 Paper Title: Schumann Resonances as a tool to constrain the depth of Titan's buried water ocean: Re-assessment of Huygens observations and preparation of the EFIELD/Dragonfly experiment.


1
.001


    


  


mode = title]Schumann Resonances as a tool to constrain the depth of Titan’s buried water ocean: 
Re-assessment of Huygens observations and preparation of the EFIELD/Dragonfly experiment 


inst1]mailto:paul.lagouanelle@latmos.ipsl.frPaul Lagouanelle
 inst1]mailto:alice.legall@latmos.ipsl.frAlice Le Gall

[inst1]organization=LATMOS/IPSL,
            addressline=Université Paris-Saclay, UVSQ, Sorbonne Université, CNRS, 
            postcode=78280,
            city=Guyancourt,
            country=France


§ ABSTRACT 
Among the lines of evidence for a buried ocean on Titan is the possible detection, in 2005, by the Permittivity, Wave and Altimetry (PWA) analyzer on board the ESA Huygens probe of Schumann-like Resonances (SR). SR are Extremely Low Frequency electromagnetic waves resonating between two electrically conductive layers. On Titan, it has been proposed that they propagate between the moon’s ionosphere and a salty subsurface water ocean. Their characterization by electric field sensors can provide constraints on Titan’s cavity characteristics and in particular on the depth of Titan’s ocean which is key to better assess Titan’s habitability. For this work we have developed a numerical model of Titan’s electromagnetic cavity as well as a surrogate model to conduct simulations and sensitivity analyses at a low computational cost. This surrogate model is used both to re-assess PWA/Huygens measurements and to predict the future performance of the EFIELD experiment on board the NASA Dragonfly mission. We demonstrate that the PWA/Huygens measurements, in particular due to their low spectral resolution, do not bring any meaningful constraint on Titan’s ocean depth. On the other hand, the finer resolution of the EFIELD experiment and its ability to capture several harmonics of SR should provide more robust constraints on Titan’s internal structure, especially if the electrical properties of the ice crust and the atmosphere can be better constrained.










  * Surrogate modeling of Titan's planetary cavity for Schumann resonances

  * Re-assessment of Huygens observations to constrain the thickness of Titan's ice crust

  * Potential performances of the EFIELD experiment on board Dragonfly for estimating the thickness of the ice crust






Titan, interior Instrumentation Ionospheres Radio observations Titan, atmosphere


[
    [
    Received 2 August 2023 / Accepted 7 May 2024
================================================







§ INTRODUCTION



Several lines of evidence point to the presence of a global water ice ocean in Titan’s interior. The strongest evidence arises from the investigation of the tidal variations of Titan’s gravity fields inferred from Cassini flybys of the satellite <cit.>. Indeed, the tidal Love number k_2 of 0.62 derived by <cit.> is compatible with a high-density ocean while a recent re-assessment of k_2 (0.375) points to a low-density water or ammonia ocean <cit.>. Titan’s measured obliquity of ∼ 0.3^∘ <cit.> is also significantly larger than the value expected for an entirely solid object and therefore suggests a decoupling between the outer ice shell and the interior of Titan <cit.>. Based on both the values of <cit.>’s k_2 and of the obliquity, <cit.> estimate that the outer ice shell of Titan is at least 40 and at most 170 thick consistent with the results published in <cit.> which test a wide range of internal structure models for Titan including thermal considerations. 

Another, possible evidence for an internal ocean on Titan is the detection of ELF (Extremely Low Frequency) waves by the PWA/HASI (Permittivity, Waves and Altimetry analyzer, part of the Huygens Atmospheric Structure Instrument) experiment on board the Huygens interpreted as Schumann Resonances <cit.>. Schumann resonances (SR) are a set of ELF electromagnetic propagation modes that can develop in a planetary cavity excited with a broadband electromagnetic source <cit.>. On Earth, these modes are generated by lightning discharges and propagate between the ionosphere and the surface. Theoretically, SR could be observed on other planets and serve as a tool to obtain information on the planetary cavities, in particular on their dimensions <cit.>.

If SR exist on Titan, there are probably not triggered by lightning as such activity was never observed and is not expected to be common on Titan <cit.>. In addition, the surface of Titan being very poorly conductive <cit.>, it cannot act as the lower boundary of the resonant cavity which instead must be an internal electrically conductive layer. <cit.> propose that SR on Titan could be excited by interactions with Saturn’s magnetosphere and that the signal detected at ∼36 by PWA/HASI is the second harmonic of a SR propagating between Titan’s ionized atmospheric layer (at ∼60-70 altitude) and a buried salty ocean lying at a depth encompassed between 40 and 80. However, this interpretation is still debated as the 36 line may actually be due to mechanical oscillations of the booms on which the PWA/HASI electrodes were installed or of other parts of the Huygens probe <cit.>. 

Nevertheless, if SR occur on Titan, their detection and characterization would place new and more robust constraints on the buried ocean. In particular, knowing more accurately its depth is key to estimate the likeliness of exchange between the ocean and the surface and therefore to assess Titan’s habitability and astrobiological potential. That is the reason why the forthcoming mission to Titan, Dragonfly (NASA), will embark sensors to measure the time-varying electrical field, namely the EFIELD experiment which is part of the DraGMet (Dragonfly Geophysics and Meteorology) package <cit.>. Using two spherical electrodes mounted at different locations on the body of the Dragonfly drone, EFIELD will passively record the AC electrical field between ∼5 and 100 with a much finer spectral resolution than PWA/HASI.

In this paper, we describe the numerical model we have developed to simulate Titan’s electromagnetic cavity and predict its resonant frequencies and associated quality factors (section 2). This model is used to build a much less computationally expensive surrogate model which allows to perform an accurate sensitivity analysis of the SR characteristics to the cavity parameters. The surrogate model is then used to re-examine the PWA/HASI measurements leading to results very different from the ones published in Béghin et al. (2012) (section 3). In section 4, it is used to investigate the expected performance of the EFIELD/DraGMet experiment. Lastly, we conclude and discuss the implications of this work in section 5.

 






§ MODELING TITAN'S RESONANT CAVITY



Following <cit.> who developed cavity models for Titan, Venus and other planetary environments, we used the COMSOL Multiphysics© tool to build a numerical model of electromagnetic wave propagation in the cavity of Titan. We then used the numerical model to construct a surrogate model of the propagation of SR on Titan in order to conduct eigenfrequency analysis at a low computing and memory cost.



 §.§ Numerical model




  §.§.§ Cavity description and parameters


The numerical code solves the Maxwell's equations in a spherical structure made of discrete slabs. Figure <ref> displays the simplified structure of Titan’s cavity we considered; it consists of three concentric layers:


    
  * the atmosphere/ionosphere layer for which an analytic conductivity profile is given to the model, namely the one proposed by <cit.> or, more recently, by <cit.>  displayed in figure <ref>. Both conductivity models include an ionized layer at an altitude of about 60-70   on which ELF waves are reflected. They rely on Huygens measurements of the electron-density performed from an altitude of 140 down to the surface <cit.>. In <cit.>, the conductivity profile is interpolated from 140 to 750 and better respect the upper limit of the near-surface conductivity imposed by Huygens Relaxation Probe measurements.
    
  * the ice crust layer in which ELF waves are refracted following Fresnel’s laws. This layer is assumed uniform in terms of electrical properties with a very small conductivity that allows ELF waves to propagates over a very long path (i.e., the skin depth is >1000).
    
  * the salty ocean layer which is assumed to be a perfect electric conductor and therefore on which ELF waves are fully reflected.


The parameters of the cavity model considered for parametric analysis are the followings:

    
  * the thickness of the ice crust z_c : Based on Cassini and Huygens observations as well as on gravity and thermal modeling (see section <ref>), we consider that z_c can vary over a wide range of values from 5 to 200, and most likely between 40 and 170.

    
  * the real part of the ice crust relative permittivity ε_c: The relative permittivity of water ice at Titan’s temperatures and ELF frequencies is ∼3 (e.g., <cit.>). However, the crust permittivity also depends on its porosity and on the presence of impurities or contaminants such as ammonia. We therefore test values in the range 2-4 which encompasses the value measured at the Huygens landing site by the permittivity probe PWA-MIP/HASI, namely 2.5±0.3 <cit.>.

    
  * the electrical conductivity of the ice crust σ_c: Through expected to be small, the conductivity of the ice crust of Titan remains uncertain. It is especially sensitive to the possible presence of ionic contaminants. <cit.> investigates the 1-4  range while <cit.> found a conductivity of 1.2±0.6  at the Huygens landing site. To account for this measurement, we consider values in the range 0.6-4 .




As outputs, the numerical model computes the eigenfrequencies of the first three modes of the resonant cavity (f_1,f_2,f_3) along with their corresponding Q-factors (quality factors: Q_1,Q_2,Q_3) which describe wave attenuation in the cavity. More specifically, the model provides the complex frequencies of the different eigenmodes from which the Q-factor is computed as followed:

    Q_n=Re(f_n)/2Im(f_n)≈f_n/Δ f_n

where Re and Im are respectively the real and imaginary parts of the complex eigenfrequency, f_n is the peak power frequency of mode n, and Δ f_n is the width at half-power.

For a given uncertainty δ f_n on f_n and Δ f_n, which in practice is mainly dictated by the measurement spectral resolution, the uncertainty δ Q_n on the corresponding quality factor Q_n can be derived by logarithmic differentiation of equation <ref>:

    δ Q_n ≈ Q_n (1+Q_n)δ f_n/f_n




  §.§.§ Numerical approach


The numerical model uses the Finite Element Method <cit.> for solving Maxwell’s equations with the boundary conditions and layers properties as described above. Since layers properties are only functions of the radial distance, the resonant cavity problem can be solved in a 2D axi-symmetric configuration. We nevertheless validate our 2D model with comparison to a 3D model and results from the numerical model from <cit.>.





An example of a 3D model of Titan’s cavity is displayed on figure <ref> as well as a 2D cut of the mesh. Due to the level of discretization needed to accurately reproduce the behavior of the electric field in Titan's atmosphere, the complete 3D mesh consists of 774,258 domain elements, 181,100 boundary elements, and 2,588 edge elements. A single resolution of Maxwell's equations using such a mesh takes ∼2 (on an Intel Core i5-12500H, 2.5, 32 GB of RAM). In contrast, the design of a 2D axi-symmetric model with a mesh composed of 202,993 domain elements and 3,652 boundary elements requires ∼30 of computation time which is much more reasonable for the optic of using the model to perform an accurate parametric inversion.

Table <ref> reports the results from the 3D model and the 2D approximation for two different cavities. The 2D approximation is accurate enough so that the results from the 3D model are reproduced with relative errors smaller than 5% for both the Q-factor and the resonant frequency. Table <ref> reports the results from the 2D approximation and the model from <cit.> for a lossless atmosphere showing that the 2D approximation is able to accurately reproduce the cavity behavior in the case of study case with a relative error on the first three resonant frequencies smaller than 2%. Thus, for the remainder of the paper, we only consider the 2D axi-symmetrical model of Titan's planetary cavity.






 §.§ Surrogate model



Even with the 2D axi-symmetrical approximation, a complete sensitivity analysis or data inversion using Titan's cavity numerical model would require to compute several thousand of datapoints. To avoid this, we built a surrogate model (also called "metamodel") which provides an approximate description of the behavior of Titan’s resonant cavity by analytical functions. The resulting metamodel can then be used, instead of the numerical model, to perform various analysis (e.g. optimization, sensitivity analysis) at a low computational cost <cit.>.



  §.§.§ Description


The surrogate model used here is an exact interpolator; the Polynomial-Chaos based Kriging (PCK) <cit.> metamodeling combines both Polynomial Chaos Expansion (PCE) and Kriging to predict the variations of a given model ℳ(X). Kriging is used to interpolate the local variations of the output model while PCE is useful for the global approximation. A PCK metamodel is defined by:

    ℳ(X)= ∑_α∈𝒜y_αψ_α(X) + σ^2 Z(X,ω)

where ∑_α∈𝒜y_αψ_α(X) is a weighted sum of orthonormal polynomials describing the trend of the PCK model and σ^2 Z(X,ω) is a zero-mean stationary Gaussian process with a variance of σ^2. The computation of the metamodel parameters is performed by the UQLab framework available on Matlab <cit.>.

The costly part of the metamodelling process is the training time which can be greatly reduced by using sequential sampling instead of classical space-filling approaches. In this work an adaptive sampling algorithm combined with PCK has been used to build the metamodel. This algorithm has already been proven useful for various electromagnetic problems <cit.>.



  §.§.§ Metamodel accuracy



Once built, the surrogate model is used to predict the behavior of the cavity outside of the training data. For such a purpose, a proper metric is crucial to quantify the accuracy of these predictions. A classical approach consists in using a validation dataset outside of the training dataset and computing the mean squared error (MSE) of the metamodel prediction compared to the real input values. However, the resulting MSE is biased by the use of only one dataset and could vary greatly from one validation dataset to another. Moreover, this approach requires additional calls of the expensive computational numerical model for building the training dataset, which ultimately increases the computation time.
Thus, a better metric was chosen : the Leave-one-out cross-validation error (LOO), which does not require additional computations. 

Let us consider a set {(X_1,Y_1),…,(X_N,Y_N)} of N input samples. Using this set, one can build a PCK metamodel ℳ and evaluate the LOO as follows:

    LOO= 1/N∑_i=1^N ( ||ℳ_/i(X_i)-Y_i||||Y_i|| ) ^2

where ℳ_/i is the mean predictor that was trained using all (X, Y) except (X_i, Y_i).
For a given datapoint (X_i,Y_i), a metamodel is built with all datapoints except datapoint i, which gives N-1 training datapoints.  This metamodel is then used to predict the value Y_i at the remaining datapoint i, where the difference is classified with a MSE. The process is repeated for every datapoint which, after average, provides the LOO. 
The use of N different validation sets of one datapoint guarantees that the LOO is much less biased than a classical MSE and reduces the probability of overestimating the validation error <cit.>. In this work, we therefore consider the LOO  as our accuracy metric. A LOO  close to 1 (or 100%) implies that the surrogate model does not provide a good approximation of the system. On the other hand, the smaller the LOO, the more accurate the surrogate model.
 



  §.§.§ Sensitivity indices



Since the resulting metamodel consists of an analytical function, calling the metamodel is extremely cheap in terms of computation time. Thus, sensitivity analyses, which are usually performed by Monte-Carlo analyses over the parameter spaces, are now feasible at a low computation cost. 

The sensitivity analysis we conduct relies on Sobol' indices which are scalars between 0 and 1 describing the influence of a set of inputs on a model output <cit.>. The most commonly used Sobol' indices is the first-order Sobol' index defined, for a given parameter P_i, as:

    S_i=Var_P_i(𝔼_ X_/i( Y | X_i))/Var( Y)

S_i is a measure of the fraction of the output variance caused by the variance of a given input parameter. In other words, it describes the impact of a parameter P_i alone on the output model compared to other parameters. The closer to 1, the bigger impact P_i has on the model output.

However,  parameters are usually not independent and their relative effects cannot be separated from each other. This leads to the definition of higher-order Sobol' indices as, for a subset of parameters (P_i_1,...,P_i_s):

    S_i_1,...,i_s=Var_(P_i_1,...,P_i_s)(𝔼_ X_/i_1...i_s( Y | X_i_1,...,X_i_s))/Var( Y)

which describes the sensitivity of the model to the variations of several input parameters simultaneously.

For high dimensional output models, the interpretation of all orders Sobol' indices can be difficult due to the high number of possible combinations. Therefore, for an input parameter P_i, a total-effect index (or total Sobol' index) S_i^T is defined by summing all the Sobol' indices as follows:

    S_i^T= ∑_{ u,  u⊆ 1,d  and  i∈ u } S_ u

S_i^T is the most suited sensitivity tracker for our study and will be refereed to as S_i in the remainder of the paper. When using PCK metamodels, the computation of the various Sobol' indices of the surrogate model can be easily extracted from the polynomial decomposition (see equation <ref>). Therefore, no additional computation of the surrogate model are required to perform the sensitivity analysis based on the total Sobol' indices.



§ RE-ASSESSMENT OF PWA/HASI/HUYGENS OBSERVATIONS



<cit.> first reported the detection of a narrow power line at ∼36 in the ELF spectrum measured by PWA/HASI during Huygens’ descent in Titan’s atmosphere in January 2005, from an altitude of 140 down to the surface. The magnitude of this signal is especially enhanced just after the deployment of the stabilizer parachute, at an altitude of ∼110. <cit.> proposed different scenarios, both natural and artificial, to explain the 36 signal. In <cit.>, a natural scenario is preferred: the signal and associated Q-factor of about ∼6 would be the second harmonic of a SR propagating between Titan’s ionosphere and ocean and triggered by interactions with Saturn’s magnetosphere. Using an approximate analytical model of Titan’s cavity, <cit.> further derive constraints on the physical parameters of the cavity from PWA/HASI measurements. More specifically, they conclude that the measured f_2=36±3 and Q_2∼6 are indicative of a water-ammonia ocean lying at a depth of 40-80.

In this section, we re-asses the PWA/HASI data using the surrogate model we have developed (see section <ref>) to investigate, in a more accurate fashion, the constraints Huygens measurements bring on the thickness of the ice crust (i.e., the depth of the ocean) z_c. As a starting point, we adopt exactly the same hypotheses as in <cit.> that is the same conductivity profile in the atmosphere and ranges of variation for ε_c (2-4) and σ_c (1-4 ).

The metamodelling process estimates consistently (LOO≈3.9%, n_samples=1584) the second harmonic of SR (f_2, Q_2). Using the resulting surrogate model, two regular 3D grids of f_2 and Q_2 values along all three input parameters z_c, ε_c and σ_c can be computed at a low computation cost: 50×50×50=125,000 values, which would have taken 43 days of computation time using directly the numerical model instead of only 13 hours for training the metamodel. 








 

By analyzing the 3D grids, every combination of z_c, ε_c and σ_c which gives f_2∈[33-39] and Q_2∈[3-9] can be classified as a potential solution explaining Huygens measurements. Figure <ref> displays all the potential inversion of Huygens measurements in the plane (z_c,ε_c) where all the grids in the direction σ_c have been stacked. When accounting for 3 spectral resolution, the range of possible values for z_c covers all the parameter space (i.e. 5-200) which means that no constraint can be deduced on z_c from the PWA/HASI dataset (blue zone). On the other hand, if the signal characteristics were known with a 1 resolution, only a narrow range of z_c values could explain the observations (pink zone). This hypothetical inversion would restrain the thickness of the ice crust to z_c∈ [165-200] .

Our results are in contraction with <cit.> which can be explained by the various analytical approximations they consider to solve the wave propagation equation. Furthermore, since the metamodel is sufficiently accurate (LOO≈3.9%), it can be used to conduct an accurate Sobol'-based sensitivity analysis. The following total Sobol' indices are found: S_z_c=0.73, S_ε_c=0.28 and S_σ_c=0.74. These indexes indicate that the ice crust thickness is the parameter that has the most significant impact on the SR characteristics. However, since all indexes are of the same order of magnitude, no parameter can be regarded as having a negligible impact. This is a further guarantee that the metamodel accuracy is correctly estimated with the LOO and that our metamodel is highly accurate. Therefore the differences from    <cit.> 's results and the present work  cannot be ascribed to an incorrect estimate of the errors from the metamodel.

The surrogate model also demonstrates that solutions (in terms of combinations of z_c, ε_c and σ_c values) in which the detected signal is not the second harmonic but the fundamental are possible (figure <ref>). If so, the constraints on z_c derived from PWA/HASI would be different (namely >80, see figure <ref>).

Lastly, figure <ref> shows how the constraints on z_c are modified if the numerical model from which the surrogate model was built rather uses <cit.>'s conductivity profile and values of ε_c and σ_c from <cit.>. We will keep these hypotheses for the remainder of the paper and, in particular, to assess EFIELD/DraGMet future performance in the frame of the Dragonfly mission. The associated sensitivity analysis provides the following total Sobol' indices: S_z_c=0.78, S_ε_c=0.34 and S_σ_c=0.51. Again and notably, z_c is the  parameter that have the most impact on SR characteristics. Nevertheless, the effects of ε_c and σ_c cannot be neglected as their respective total Sobol' index are of the same order of magnitude. This further implies that their accurate knowledge would greatly reduce the uncertainty on the inversion of Huygens measurements as well as be very valuable for the analysis of future EFIELD data. In particular, if the real part of the permittivity of the ice crust were measured as ε_c=2.5±0.1 instead of 2.5±0.3, it would help discriminating between the two distinctive domains in figure <ref>: z_c∈[5-40] and z_c∈[150-200]. This is further discussed below.






§ ANTICIPATED EFIELD/DRAGMET/DRAGONFLY PERFORMANCE




This section investigates the performance and possible outcome from the forthcoming electric-field experiment on Titan.



 §.§ The EFIELD experiment on board Dragonfly
 

In June 2019, NASA selected the Dragonfly mission project for its New Frontiers program <cit.>. The primary goal of the Dragonfly mission is to investigate the chemistry and habitability of Titan. Starting operations in mid-2030s, the Dragonfly quadcopter drone will visit a variety of sites, from a dune field to the rim of a young impact crater, and sample materials in different geologic settings. Dragonfly embarks a Geophysical and Meteorological package (DraGMet) which is a suite of sensors designed to measure e.g. the temperature, pressure, methane humidity, wind speed and direction, ground dielectric constant, thermal properties and level of seismic activity at each Dragonfly landing site. 

Among these sensors is the EFIELD experiment which consists of two independent spherical electrodes (∼5 in diameter) accommodated at The End of ∼25 long stalks pointing away from the drone body. From two different locations on the drone, these electrodes will passively record the time-varying electrical field at low frequencies (∼5-100 ) with the main goal of detecting SR, if any. As a secondary objective, the EFIELD probes will detect and characterize near-surface wind-blown charged grains flying in their vicinity <cit.>. 

The EFIELD experiment offer many advantages over the PWA/HASI one. It will operate during an extended period of time (several times a Titan day for the 3.3 years of the nominal mission), from a stable and much mechanically-quieter platform than the Huygens probe. Further, the EFIELD design should guarantee a spectral resolution of 1 (against, at best, 3 for PWA/HASI) and the capture of the first three harmonics of the SR. Figure <ref> demonstrates how valuable a finer spectral resolution would be to bring more robust constraints on the depth of the buried ocean z_c. The benefit of detecting more than one SR harmonics is investigated below.



 §.§ Multi-modal analysis


The re-assessment of Huygens data presented in section <ref> relies on the measurement of only one mode of the SR; it concludes that a wide range of values are possible for the thickness of the ice crust (z_c∈[5-200] ). The multi-modal analysis of SR enabled by EFIELD should drastically reduce this range because, in addition to a better spectral resolution, the three first modes of the Schumann resonances will be associated with three different domains of possible z_c values wh intersection may be narrow.


As an illustration, we numerically simulate Titan’s cavity for the following parameter values: z_c=60,ε_c=2.5,σ_c=1.2 and the up-to-date conductivity profile from <cit.>. The numerical model provides the following outputs: f_1=28.4  ,  f_2=44.9 ,  f_3=62.4  along with their corresponding quality factors: Q_1=3.25, Q_2=3.58 and Q_3=3.81. Using the metamodel, we further compute the variations of the resonant frequencies (f_1,f_2,f_3) and of the quality factors (Q_1,Q_2,Q_3) as a function of the thickness of the ice crust z_c, assuming fixed values for ε_c=2.5 and σ_c=1.2. These variations are displayed on figure <ref>.




 
Assuming a 1 resolution, each harmonic can be inverted separately (using the method described in section 3), resulting in three different possible domains for z_c:

    
  * f_1=28.4±1 z_c∈[19.7-21.3]∪[56.0-73.0]∪[81.4-85.8]
    
  * f_2=44.9±1 z_c∈[20.9-22.4]∪[54.8-72.3]∪[83.4-87.8]
    
  * f_3=62.4±1 z_c∈[21.7-23.5]∪[33.2-46.3]∪[56.7-63.2]

The intersection of these domains is: z_c∈[56.7-63.2] which corresponds to an uncertainty of 6% with respect to the input value of 60 km. A similar inversion is performed on the quality factors separately:

    
  * Q_1=3.25± 0.49  z_c∈[13.8-89.4]
    
  * Q_2=3.58± 0.36 z_c>11.9
    
  * Q_3=3.81± 0.29 z_c>14.9

In the case considered here, no restriction can be further obtained on z_c from the Q-factor values (see figure <ref>). Nevertheless, this example well illustrates the values of measuring several modes of the SR.



 §.§ Inversion uncertainty


The example above assumes fixed values for both ε_c and σ_c. To take into account our imperfect knowledge of the electrical properties of Titan’s ice crust, the uncertainties on these parameters have to be propagated through the inversion to estimate their effects on the derivation of z_c.  As another illustration, we numerically simulate Titan’s cavity with a thickness of the ice crust fixed at z_c=60. A multi-modal analysis is then performed for every point (z_c,ε_c,σ_c) in a regular parameter grid with ε_c and σ_c respectively varying in 2.5±0.3 and 1.2±0.6  as found at the Huygens landing site by <cit.>. The parameter grid consists of 17×25×25=15,300 datapoints in the 3D parameter space {z_c}×{ε_c}×{σ_c}. Every datapoint on the grid which returns the desired values for the first three modes considering their measurement uncertainty of 1 is a possible inversion of EFIELD. 



This allows us to compute the probability density function of the returned values of z_c assuming that the three first harmonic of the SR are detected with a resolution of 1, as figure <ref> displays. Interestingly, 90% of the inversion cases fall in the range z_c∈[51-69]  while ≈10% of the cases return z_c values in the ranges [20-40]  or [70-80]  (blue bars). Assuming a normal distribution of z_c centered in 60, this corresponds to a standard deviation (STD) of 6.8. Figure <ref> further shows that reducing the uncertainty on σ_c from 1.2±0.6 to 1.2±0.2 reduces the STD to the value of 5.7 (orange bars).



Such exercise was repeated for a set of z_c values in the [10-85]  range to produce figure <ref> which displays the STD (in km and %) of the distribution of the inferred z_c as a function of the z_c value for two assumptions on the range of variation of σ_c. In most cases, the STD is smaller than 10. Moreover, it significantly decreases as z_c increases, especially after 60 where it becomes smaller than 10%. In contrast, for small values of z_c (<25), the theoretical relative error on z_c can reach almost 100%. This can be partially explained by the relative coarse mesh of the numerical model; indeed the dimensions of the cells have been imposed greater than 5 in order to reduce computation time. Further developments are required to build a surrogate model more appropriate to small thicknesses (i.e., with a finer mesh) but this is out of the scope of this paper.  

The same analysis was conducted assuming a more constrained knowledge of σ_c (namely σ_c=1.2±0.2) resulting in smaller STD for almost all cases (red lines in figure <ref>). In particular, the uncertainty at estimating small thicknesses drops from ≈80% to ≈40%. This further shows the need for a more accurate knowledge of the electrical properties of the crust. This point and others are discussed in the following section.




§ DISCUSSION AND CONCLUSION


For this work we have developed a numerical model of Titan’s cavity to then build a less computationally expensive surrogate model able to describe how the cavity characteristics (i.e., eigenfrequency and Q-factors) vary with the main cavity parameters (i.e., Titan’s ice crust thickness and electrical properties). This model (and its use for data inversion) is a powerful tool for the analysis of electric field measurements on Titan (and elsewhere). It was used to re-assess Huygens observations leading to the conclusion that the 2005 detection of a line at ∼36, if indeed due to SR, does not provide any specific constraint on the depth of Titan’s ocean in the range 5-200  contrary to what is advanced in <cit.>. 

The surrogate model was also used to estimate the possible outcomes from the EFIELD/DraGMet/Dragonfly experiment. EFIELD is designed to detect several modes of SR with a fine spectral resolution; we have demonstrated that it has the ability to put a meaningful constraint on the thickness of the ice crust. Considering the electromagnetic properties varying in the ranges specified by <cit.>, the various sensitivity analysis presented throughout this work, reach the same conclusion: although the thickness of the ice crust is the most influential parameter on SR characteristics, the resonant frequencies are dependent in the same order of magnitude on the electrical properties of Titan’s crust. Therefore, in order to reduce the uncertainty at estimating Titan's crust thickness, it is crucial to reduce the uncertainty on the crust electrical properties. 

Another experiment on board the Dragonfly drone will contribute to this task: the DIEL/DraGMet experiment. Acting as a mutual-impedance probe with a pair of electrodes mounted on each skid of the drone, DIEL will measure the complex permittivity (which includes the electrical conductivity) of the ground at several low frequencies (<10) thus providing insights into the composition, moisture and porosity of the near-subsurface of Titan as well as on the spatial and temporal variations of such properties. Though all measured permittivity values may not be representative of the crust, values measured on the ejecta blanket of the geologically-young Selk crater (the final destination of Dragonfly) may be. In addition, variations of the measured complex permittivity along the Dragonfly journey to Selk and its possible correlation with otherwise inferred vicinity of the water ice bedrock in the near subsurface will provide further constraint on the ice crust electrical properties. 

Nevertheless, as highlighted in <cit.>, one of the main sources of uncertainty is and will remain our limited knowledge of the lower ionosphere conductivity structure. Unfortunately, no improvement is to be expected from forthcoming observations as Dragonfly will not perform measurements during its descent in Titan’s atmosphere. As a consequence, only theoretical developments can provide further insights on the atmosphere conductivity profile and its expected variations with the local hour, solar activity and the position of Titan in Saturn’s magnetosphere. 

Lastly, future investigations will include the simulation of the actual EFIELD electrodes accommodated on the Dragonfly (conductive) body as well as the study of the effect on measurements of the location and polarization of the possible sources of SR. In the mid-2030s, when the Dragonfly drone will be on Titan, methane-storms are expected at the South Pole which will be the first hypothesis to be tested. Given the different orientation and altitude on the drone of the two EFIELD electrodes, at least two components of the electrical field will be measured. The third component could be captured by rotating the drone and future study will also explore the value of measuring the full electrical field vector.



§ ACKNOWLEDGEMENTS


The authors wish to thank Ralph Lorenz (APL, PI of DraGMet/Dragonfly) and Jean-Jacques Berthelier, Audrey Chatain and  Franck Montmessin from LATMOS for useful discussions on this project. The authors are also grateful to Région Ile-de-France (DIM-ACAV+ program) for funding this research.



cas-model2-names


35
natexlab#1#1









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 Paper Title: The frequency of transiting planetary systems around polluted white dwarfs.



firstpage–lastpage
Ge-based Clinopyroxene series: first principles and  experimental local probe study
    João P. Araújo
    August 5, 2024
===================================================================================




§ ABSTRACT 
This paper investigates the frequency of transiting planetary systems around metal-polluted white dwarfs using high-cadence photometry from ULTRACAM and ULTRASPEC on the ground, and space-based observations with TESS.  Within a sample of 313 metal-polluted white dwarfs with available TESS light curves, two systems known to have irregular transits are blindly recovered by box-least-squares and Lomb-Scargle analyses, with no new detections, yielding a transit fraction of 0.8_-0.4^+0.6 per cent. Planet detection sensitivities are determined using simulated transit injection and recovery for all light curves, producing upper limit occurrences over radii from dwarf to Kronian planets, with periods from 1 h to 27 d.  The dearth of short-period, transiting planets orbiting polluted white dwarfs is consistent with engulfment during the giant phases of stellar evolution, and modestly constrains dynamical re-injection of planets to the shortest orbital periods. Based on simple predictions of transit probability, where (R_* + R_ p)/a≃0.01, the findings here are nominally consistent with a model where 100 per cent of polluted white dwarfs have circumstellar debris near the Roche limit;  however, the small sample size precludes statistical confidence in this result.
Single transits are also ruled out in all light curves using a search for correlated outliers,  providing weak constraints on the role of Oort-like comet clouds in white dwarf pollution.



    circumstellar matter – 
    planetary systems – 
    white dwarfs










§ INTRODUCTION


Transit photometry is the leading method of detecting extrasolar planets, and is responsible for several thousand discoveries that constitute the bulk of confirmed detections <cit.>. From a geometric standpoint, the transit method favours the detection of planets on orbits close to their host stars, whereas sensitivity favours larger transit depths, such as those caused by giant and dwarf gas planets.

Relative to main-sequence stars, white dwarfs provide two potential advantages in transit searches: their small radii and the ubiquitous presence of closely orbiting planetary debris. Compared to the sun, while a typical white dwarf radius R_*=0.013 R_⊙ decreases the nominal transit probability by two orders of magnitude, it produces a 50 per cent transit depth for an Earth-size planet, and thus for similar photometric precision yields a sensitivity gain of nearly four orders of magnitude. Unlike main-sequence stars, there is comprehensive evidence of planetesimal debris discs orbiting within roughly 1 R_⊙ of metal-polluted white dwarfs <cit.>. At face value, this predicts around 1 per cent of polluted white dwarfs may exhibit transits. 

At least 25 per cent of white dwarfs exhibit signatures of planetary systems, which include emission from circumstellar dust and gas <cit.>, photospheric metals that require ongoing replenishment <cit.>, and transits <cit.>.  In support of these observations, there are theoretical predictions for planets closely orbiting white dwarfs, possibly resulting from debris disc evolution <cit.>, or post-main sequence dynamics <cit.>.

Transits provide a distinct view of white dwarf planetary systems, with a modest but growing number of discoveries <cit.>. These have revealed new empirical insights, and confirmed inferences based on prior work.  The rings of debris can have semimajor axes significantly larger than a solar radius, and thus potentially be highly eccentric <cit.>, where some exhibit line of sight gas absorption <cit.>.  The debris discs are highly dynamic, with large scale heights <cit.>, and therefore distinct from the canonical picture based on the rings of Saturn <cit.>. White dwarfs with transits generally do not have infrared excesses, suggesting their detection results from favourable viewing geometry and not enhanced dust production <cit.>. 

Currently, white dwarf transits fall into two categories: those with regular transits from self-gravitating bodies <cit.>, and irregular transits consistent with circumstellar debris fields, where metals have been detected when high-resolution spectroscopy is available <cit.>.  While all white dwarfs with infrared dust emission exhibit photospheric metals <cit.>, some that exhibit transits do not, or currently lack decisive data <cit.>.

Given that white dwarf pollution requires the ongoing or recent deposition of circumstellar debris, it may be inevitable that some exhibit transits. However, despite key forerunner investigations of transiting planets around white dwarfs <cit.>, the transit frequency toward polluted white dwarfs is currently unconstrained, and in particular for transiting debris discs. This work aims to constrain the frequency of planetary systems around metal-polluted white dwarfs, and utilizes standard methodology of box-least-squares and simulated injection and recovery for high-cadence photometric observations of 33 targets from the ground, and 313 objects from space.  The observations are described in Section 2, and the transit signal search and sensitivity are detailed in Section 3.  The results are discussed in Section 4, including a determination of transit frequency and comparison to related surveys, with a brief summary and outlook in Section 5.




§ OBSERVATIONS AND DATA


The data used in this paper are based on two programs targeting polluted white dwarfs: the first is a ground-based transit search for a sample of 33 stars using the high-speed photometry cameras ULTRACAM and ULTRASPEC <cit.>, and the second analyzes Transiting Exoplanet Survey Satellite (TESS; ) pdcsap light curves for 313 objects. 

The latter targets are a bright subsample that represent around 20 per cent of all known polluted white dwarfs, and were identified by cross-matching literature sources with the TESS Input Catalogue (TIC v8.2; ).






 §.§ ULTRACAM and ULTRASPEC


High-speed photometry for 19 polluted white dwarfs was obtained with ULTRACAM, a triple-beam imaging camera that enables simultaneous acquisition of data in wavelength regions covering the u, g, and riz bands, and facilitates rapid capture of the field of view via frame transfer technology. The observations were carried out using super-SDSS filters u_ sg_ sr_ s (with higher throughput compared to the conventional filters; ), between 2017 Nov 17 and 2021 Aug 20 at La Silla Observatory, where ULTRACAM is mounted on the 3.5-m New Technology Telescope.

These data were reduced using the HiPERCAM pipeline software <cit.>.  Differential photometry was performed on the source and at least one comparison star using a variable aperture (the number of field stars available for normalisation ranged from one to five depending on the field). The frames were corrected for bias and flat-fielded using normalized sky flats obtained in a continuous spiral (to remove stars) during twilight. To maximise the signal-to-noise ratio and the sampling cadence in the final light curves used here for transit detection, only the g and r bands are used in the subsequent analysis. These data were then co-added to further improve the fidelity of the final light curves, with a median scatter in the individual g and r bands of 1.2 and 1.7 per cent, respectively.

A further 14 polluted white dwarfs were observed with ULTRASPEC, which is mounted on the 2.4-m Thai National Telescope at Doi Inthanon. ULTRASPEC operates similarly to ULTRACAM, but instead utilises a single channel.  The data were obtained with a Schott KG5 filter to maximise light throughput, where the transmission approximates the wavelengths spanned by ugr combined. The ULTRASPEC observations were taken between 2015 Nov 30 and 2020 Jan 09, and the data were reduced as for ULTRACAM. The median scatter in ULTRASPEC target star light curves is 2.5 per cent. The time stamps for light curves from both instruments were converted to Barycentric Julian Day using Barycentric Dynamical Time.  All ground-based target observations are summarized in Table <ref>.




 §.§ TESS Data


TESS has surveyed the entire sky for transiting planets since its launch in 2018. During science operations, its four cameras take nearly continuous exposures, from which full-frame images and target pixel files of selected targets are produced by a dedicated data pipeline. All available light curves from Sectors 1–65 with 120 s-cadence data were downloaded via the Science Processing Operations Center pipeline. In addition, 214 of these stars had 20 s-cadence light curves, which were downloaded and analyzed in the same fashion as the 120 s data. This sample had mean TESS light curve baselines of 2.34 and 1.72 sectors for the 120 and 20 s cadences, respectively.  When downloading the light curves, only data with pipeline quality flags of 0 or 512 (impulsive outlier flag) were accepted. Despite typically indicating anomalous data, the impulsive outlier flag was accepted, based on the short durations expected for white dwarf transits (less than 2 min for a transiting Earth analogue, and thus only one or two in-transit measurements for the 120 s-cadence data). 

The source photometry obtained from TESS is susceptible to crowding owing to the 21 arcsec size and response function of the camera pixels. For exoplanet transits, there will be a reduction in the observed depths in blended light curves, leading to underestimated planet radii. This problem is worse for white dwarfs as they are intrinsically faint, where the TESS estimation of crowding, encoded by the crowdsap parameter, ranges from 0.011 to 0.998 (median and standard deviation 0.53±0.32) for stars in this study.  Despite the correction of any flux dilution in the pdcsap light curves <cit.>, crowding often results in false positive detections of variability and transit-like signals (see Section <ref>).

All long-term trends were removed from the light curves using a Tukey bi-weight slider from the wōtan package <cit.>, where a kernel size of 0.2 d was used for most of the data; this is approximately a factor of three larger than the maximum expected transit duration for a planet with radius 2 R_ Jup and period 27 d.  In special cases where the light curve displayed rapid variability (owing to contamination from nearby sources), a smaller, tailored kernel size was adopted.  The de-trended TESS light curves have significantly larger scatter than for the ground-based observations, with a median of 12.5 per cent, illustrated in Fig. <ref>.



§ METHODS AND DATA ANALYSIS


The following section details the algorithms used to search for new transiting systems, and the procedures 
to determine (upper limit) occurrences for planetary systems, in the case of either real or null detections. First, a transit search was executed for all light curves using the box-least-squares algorithm <cit.>, which identifies exoplanet transit signals by fitting boxcar functions to light curves phase folded on trial periods. Second, transit injection and recovery was used to determine the sensitivities of the respective instruments to planetary transit signals. 








 §.§ The transit search algorithm


For the ULTRACAM and ULTRASPEC light curves, where the median light curve duration is 4.1 h, BLS was used to search for transit signals in a period range spanning 15 min to 7 h, with a grid spacing of 3.6 s, where the upper period bound is slightly less than twice the median, ground-based observation baseline.  The simulated period range for the ground-based data encompasses orbits from the inner edge of a debris disc to just outside the Roche limit. For the TESS sample, periods from 1 h to 27 d, spaced by 60 s, were searched for both the 120 and 20 s cadence light curves. The lower bound of 1 h is interior to the Roche limit for a white dwarf, and the upper bound of 27 d corresponds to the approximate baseline of a single TESS sector. In both cases, the period grid is sufficiently fine to detect short-duration transits, while also enabling calculation of the BLS periodogram for each light curve at reasonable computational expense. From the periodogram, three candidate periods that pass the detection threshold are selected. The first candidate period corresponds to the signal with the highest power in the BLS spectrum, and likewise the second and third candidate periods correspond to the next two respective maxima in the periodogram, with the condition that they are neither harmonics of the best candidate period nor of each other. 



Owing to the irregular dimming events in the light curves of white dwarfs with transiting debris, the Lomb-Scargle algorithm was used in addition to BLS, to facilitate the detection of any possible periodic signals. Both algorithms were used to perform a blind search in both the ground-based and TESS samples. For all light curves, the Lomb-Scargle periodogram was computed for the raw data, and BLS was run after de-trending. 

The detection thresholds for all light curves are individually determined using a bootstrap randomisation technique <cit.>. In short, for a given light curve, the fluxes are randomly shuffled while preserving the original time stamps of the observations. The periodogram is subsequently re-computed for each randomised light curve to extract the signal with the highest peak in the power spectrum. This randomisation process is repeated a total of 1000 times per light curve, from which the 99.9^ th percentile power is calculated, and this value is adopted as the detection threshold for that light curve.  The median signal detection efficiencies (defined in eq. 6 of ) and standard deviations of the detection thresholds are 10.8±2.9 for the TESS light curves, and 12.1±3.7 for the ground-based data. 

All TESS targets were inspected for contamination owing to aperture crowding using tess-localize <cit.>, which determines the source of variability in the sky for a given target pixel file and set of frequencies. The input frequencies were informed by the output of the Lomb-Scargle periodogram for each light curve. All periodograms and phase-folded light curves for each candidate period were visually examined for each target, with particular attention given to stars that were determined to be variable by tess-localize. Visual inspection may introduce uncertainties in the characterisation of light curve variability, but is unlikely to have a significant effect on the overall results. There were 32 instances where signals in the Lomb-Scargle periodogram passed the associated false alarm threshold, but that were subsequently determined to be spurious by tess-localize. This corresponds to a 9.6 per cent contaminant rate for variability detections in the space-based sample. 




 §.§ Transit injection and recovery


In an ideal scenario, the planetary occurrence is simply the ratio of the number of detected systems to the total sample size N. However, only a proportion Np_ transit = N · (R_* + R_ p)/a of planetary systems transit the stellar disc, and within this reduced subset, only a fraction S(R_ p, P) – the detection sensitivity – can be found with a given instrument. Both of these factors hinder the detection of transiting systems in a photometric survey, and in the case of a null detection, yield only upper limits on their actual frequency. To account for both of these, an effective sample size N' can be used instead <cit.>, defined as 

    N'(R_ p, P) = N · p_ transit(R_ p, P) · S(R_ p, P)   ,

and this approach is used to determine all transit occurrences hereafter. 

The detection sensitivity encodes the probability that an instrument can detect planets of a given period and radius, and can be computed via the method of injection and recovery. This process injects a simulated transit signal into a light curve, for which a BLS periodogram is computed using the injected data. Because the period and radius of the synthetic transit are known quantities, if the correct period or a harmonic is identified by BLS, then a real signal (i.e. regular transits from a self-gravitating body or irregular dimming events from debris) of the same period and radius should have also been detected. The sensitivity is then simply the ratio of the number of recovered injections to the total number of simulations. The Python software package batman <cit.> was used to inject transit signals into the de-trended light curves where no pre-existing transit features were present (32 ground-based, 311 and 213 TESS 120 s- and 20 s-cadence light curves, respectively). Because transits were injected following light curve de-trending, it may be possible that the resulting detection sensitivities are marginally higher than otherwise. All sample stars were assumed to have R_*=0.013 R_⊙ and M_*=0.65 M_⊙, with uniform limb darkening, and the impact parameter of the synthetic planet was drawn from a uniform distribution spanning 0 to (R_ p + R_*)/R_* (i.e. grazing transits are permitted). The injected transits had zero eccentricity: on the one hand, tides are theorised to circularise orbits within a few Myr for planets that survive until the white dwarf phase <cit.>; on the other hand, the data provide no actual constraints on eccentricity in the absence of detections. 

The model integration time was chosen to match the exposure times for the ground-based observations and TESS cadences, respectively. 

To initialise the injection and recovery process, a period-radius grid was defined using eight logarithmically-spaced radii from 0.0625 to 8.00 R_⊕ (corresponding to dwarf planets like Ceres up to the radius of Saturn). The injected period range for each sample was identical to the period grid used in the respective transit searches in Section <ref>. For each injection, a light curve was randomly selected from the sample, and values for the period and radius of the synthetic transit signal were randomly drawn from a uniform distribution. A synthetic transit signal was considered to be recovered by the search algorithm if the best-fit and injected periods were within a tolerance of 2 per cent. The simulations were run until the recovered sensitivity converged to within 0.5 per cent of the local median value within each period-radius cell, with a total of 37 800 injections performed for each of the ground-based and TESS 120 s cadence light curves, and 20 400 injections for the TESS 20 s subset (fewer in the latter case, to reduce computational expense). 




 §.§ Occurrence constraints


If transits are detected, the occurrence in radius-period phase space is the number of detections n divided by the effective sample size N' <cit.>. In the case that no transits are found, upper limit occurrences can be calculated as a function of radius and period. Assuming a binomial distribution describes the probability of finding n planetary systems in an effective sample of N' white dwarfs <cit.>, the rate of occurrence for a null detection is given by <cit.>


    f_ max = 1 - (1-C)^1/1+N' ,


where f_ max is the maximum occurrence for a null detection at an arbitrary confidence level C. 




The uncertainty in effective sample size is 

    σ_N' = √((p_ transit× N ×σ_S)^2 + (S × N ×σ_ transit)^2) ,


where σ_S and σ_ transit are the uncertainties on the detection sensitivity and transit probability, respectively. The uncertainty on the maximum occurrence f_ max for a null detection is therefore

    σ_f_ max = | ∂ f_ max/∂ N' σ_N'| = | ln(1-C) · (1-C)^1+N'/(1+N')^2 σ_N'| .

As a compromise between under- and over-predicting the transiting planet occurrence, a value of C = 0.95 (i.e. a 2 confidence interval) is adopted for all rates and uncertainties presented and discussed hereafter. Because the 20 s-cadence data have a higher detection sensitivity than the 120 s data, the optimum effective sample size is a combination of data from both cadences. The combined uncertainty can be computed via error propagation, and both of these steps were executed following eqs. 8 and 9 in <cit.>.




 §.§ Single transit detection


The non-periodic nature of single transits requires an alternative detection strategy. To search for such signals in the TESS light curves, the data were combed for outlier flux points, using a sliding filter of size 8 h (a factor of five greater than the maximum expected duration for a Jupiter-size planet with a 200 d period). Fluxes more than four standard deviations below the filter median were flagged as candidate single transits. This threshold represents a good balance between sensitivity to shallow transits and a manageable number of spurious detections. These were subsequently filtered by only accepting correlated changes in flux, with a minimum number of four and 12 data points in the event for 120 and 20 s cadence light curves, respectively. This minimum-case signal was chosen to be representative of a transiting Mars-size planet on a 45 d orbit. 

Adjacent arrays were then compared to reduce the possibility of long-term systematics or correlated noise being flagged as candidate events. This detection routine was tested on a sample of 200 TESS light curves with injected single transit signals, and produced a 90 per cent true positive rate with zero false negatives for (synthetic) transit depths larger than 4 per cent. The single transit detection algorithm was then run for the ground-based data and TESS light curves. No single transits, consistent with a planetary body, were confidently detected in either the ground-based or TESS data. However, two false positives, WD 0145+234 and WD 2253+803, were flagged in this search, and are shown in Fig. <ref>. The double-dip feature present in the Sector 43 light curve of WD 0145+234 is caused by a spurious increase in the sky estimate in the pdcsap pipeline aperture photometry, which is likely caused by a passing asteroid. As is the case with WD 2230–125 (see Fig. <ref>), the estimated speed of the passing objects are consistent with the orbits of asteroids, which are slower than the earth (25 km/s) owing to wider orbits (although the relative speed to the TESS satellite can be slightly smaller or larger). The speeds of these objects are not consistent with satellites. The single light curve dip apparent for WD 2253+803 in Sector 58 was caused by an instrumental pointing excursion lasting approximately 30 min, during which fluxes were re-distributed into adjacent pixels, resulting in corresponding dimming and brightening events in different parts of the array. Although these features are not true dimming events in the target star light curves, their detection underscores the ability of this method to recover single transits.








§ RESULTS AND DISCUSSION


This section outlines the results of the transit search, and derives upper limits for transiting planet occurrence based on their null detection. The TESS results are compared with related studies and simple predictions based on transit probability, applied to polluted white dwarfs.



 §.§ Recovered transiting and variable systems


The search blindly recovered two of three targets with known irregular transits from debris clouds via their TESS light curves; ZTF J0328–1219 <cit.> and WD 1054–226 <cit.>, but not WD 1145+017 <cit.>.  Normalized BLS periodograms computed for the pdcsap light curves of both stars with detected transits are shown in Fig. <ref>. Of the remaining, metal-rich white dwarfs with known transiting debris, SDSS J0107+2107, ZTF J0139+5245 and SBSS 1232+563 are too faint to be included in the TIC and are not part of the target sample. WD 1856+534, which has regular transits from a candidate giant planet <cit.>, does not meet the sample selection criteria because it is not metal-polluted. Thus, a nominal fraction of 0.8_-0.4^+0.6 per cent of polluted white dwarfs exhibit detectable transits in TESS, where the median value and limits (16^ th and 84^ th percentiles) are derived using a beta distribution with a flat prior and a binomial likelihood.

The non-detection of WD 1145+017 is notable, as it has been observed to exhibit transit depths of 40 to 60 per cent in observations spanning at least a few years <cit.>.  Despite having T=17.3 mag, it has an excellent crowdsap of 0.93, hence variations of several per cent should be detectable in TESS data (; see Fig. <ref>). Nevertheless, neither BLS nor Lomb-Scargle reveal any significant periodicities, and the null detection suggests a change in the circumstellar activity of the prototype transiting white dwarf. This is corroborated by recently published, ground-based observations that show the transits are greatly diminished in depth and frequency in 2022 and 2023, and notably correlated with long-term, flux-calibrated photometry <cit.>.  Had WD 1145+017 been detected, the transiting fraction would have been 1.2^+0.7_-0.5 per cent, and thus not substantially changed.


The Lomb-Scargle search recovered two known ZZ Ceti stars that passed the tess-localize validation process: ZZ Psc (G29-38; ) and KX Dra (PG 1541+651; ). Notably, the BLS algorithm did not detect either of these pulsators, where the light curve variations are increases from the baseline flux.  Fig. <ref> plots the detected amplitudes of the variable polluted stars (both transiting and pulsating), together with the detection thresholds for non-variable pdcsap light curves, each computed using Lomb-Scargle. Only stars with crowdsap >0.5 are included in the plot, because photometric contamination from nearby field stars can superficially augment the variability threshold.  For the 180 stars plotted, the figure demonstrates excellent sensitivity to variable signals with amplitudes smaller than 1 per cent in most cases.








 §.§ Results of the injection and recovery


The resulting detection sensitivities, transit occurrences and upper limits for all samples are displayed in Fig. <ref>. The errors for the detection sensitivities were calculated using bootstrap re-sampling (this procedure is outlined in Section <ref>). The results of the injection and recoveries were re-sampled a total of 10 000 times per period-radius bin, and the standard deviation of the re-sampled distribution encodes the error that arises owing to a finite number of injections.  The ground- and space-based results have some common trends: for synthetic planets with larger radii and shorter orbital periods, the transit depths are larger and there are more transits in a given light curve, thus enabling higher detection sensitivity, while the opposite is true for simulated planets with smaller radii and longer periods. Because transit probability is also higher for closer orbits, Eq. <ref> provides stronger constraints for such planets. The shorter light curve duration (median 4.1 h) and sample size for the ground-based observations result in weak upper limits for transits with periods longer than around 2 h.  This effect can be seen by comparing the strongest occurrence constraints for the ground-based data (31.7 per cent for P<1 h) with a similar period-radius bin for the TESS sample (10.6 per cent for 1 h < P < 3 h). The visual inspection process was not applied to the synthetic transits, hence there may be potential imperfections in the recovery, and thus detection sensitivity. Because self-gravitating objects such as planets and brown dwarfs cannot have orbits interior to the Roche limit, the ground-based results presented in Fig. <ref> are better suited to describe the occurrence of circumstellar debris around polluted white dwarfs. However, for orbits longer than several hours, the space-based results are applicable to the occurrence of debris clouds and self-gravitating planetary bodies.






 §.§ Comparison with prior work


The first effort to constrain the occurrence of transiting planets around white dwarfs used data from the Wide-Angle Search for Planets survey (WASP) where the analysis consisted of 194 white dwarfs with 12 mag <V<15 mag that were selected without regard to spectral type, mass, or multiplicity <cit.>.  Assuming the sample are all products of single star evolution, the WASP data most strongly constrain the frequency of brown dwarf and giant planetary companions to no more than 10 per cent for periods less than 0.2 d. The results presented here for TESS provide similar, focused limits for polluted white dwarfs in corresponding period-radius bins, but also provide modest constraints for planets at greater distances, and significantly smaller radii.


A search for planetary transits around white dwarfs was also performed using data from the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; ). The size of the white dwarf sample in this study is significantly uncertain, with 661 strong candidates identified by reduced proper motion, but the bulk of the 1718 candidates selected using color-color cuts that likely include contaminants.  Nevertheless, the study found no transit candidates for light curves with a few 10^3 epochs per star, with four to eight consecutive 240 s exposures per night.  The study reports an upper limit of 0.4 per cent for transiting planets with R_ p≳ 2  R_⊕ and semimajor axes smaller than 0.02 au. Keeping in mind that this upper limit frequency may be significantly underestimated based on the uncertain sample size of bona fide, single white dwarfs, TESS compares more favorably in the bulk of the parameter space in Fig. <ref>, especially for planets smaller than Earth at any orbital distance.

A search for transiting substellar objects around 1148 high-probability white dwarfs in K2 recovered nine eclipsing binaries, and the transits towards WD 1145+017 <cit.>. An upper limit frequency of 82 per cent was placed for planets larger R_ p ≈ 0.5  R_⊕ with orbital periods less than a few days, and is comparable to 84 per cent found with the TESS sample of single, polluted white dwarfs.  However, K2 is more sensitive to planets with P < 15.6 h and R_ p >  R_⊕), assuming a significant fraction of the sample white dwarfs are intrinsically single. With the crude assumption that the values reported in each K2 period-radius bin are representative of the entire bin range, then orbits between 10 and 27 d are roughly equally-well constrained by TESS, with weak constraints on smaller planetary radii than K2.   


Using both forward modelling and hierarchical Bayesian inference, it has recently been predicted that transiting rocky planets around white dwarfs are exceedingly rare <cit.>. While consistent with the results presented here for transits of self-gravitating bodies, the prediction does not seem to account for engulfment, which is primarily responsible for their absence.  In fact, direct engulfment is expected to clear all planetary bodies out to several au following both the first-ascent and asymptotic giant branches <cit.>. Thus, the candidate transiting planet WD 1856+534b is indeed a surprise, but likely accounted for by Lidov-Kozai migration <cit.>.  Furthermore, the prediction does not capture the necessarily frequent orbital injection of rocky planetesimals that are required for the observed debris discs and white dwarf photospheric pollution. This planetary material must have an origin further out to avoid engulfment during prior stellar evolution.  If the two stars with irregular transits recovered in this work are indicative of a 100 per cent occurrence of closely-orbiting planetary material, then a similar fraction of polluted white dwarfs are host to distant planets that are dynamically responsible for the star-grazing orbits required to approach the Roche limit.



 §.§ Debris transits consistent with frequent occurrence


Because all white dwarfs in this study are metal-polluted, the accretion of planetary material is either ongoing or recent, and thus most or all targets are expected to have circumstellar debris, where two stars were detected to have associated transits. If all polluted white dwarfs have debris discs orbiting near their Roche limits (1.2 R_⊙), and these are sufficiently opaque and vertically extended to marginally result in detectable transits (p_ transit=1.4 per cent), then 4.4 events are expected in a sample of 313.  These predictions are within the error bars of the 0.8^+0.6_-0.4 per cent transit fraction and 2.0 events recovered in this study.  The results are broadly compatible with a model where all polluted white dwarfs have debris orbiting near the Roche limit, and thus their detections are the result of favourable viewing geometry.

The bulk of occurrences presented in Fig. <ref> are upper limits relevant to null detections, with the exception of the two bins where transiting debris is identified (outlined in white). Owing to the modest detection sensitivity and small transit probability, both bins with detected transits have n/N' that saturates at 100 per cent. Alternatively, using a modified Eq. 2 to compute the upper limit occurrence given a single detection also yields unity. Despite a small effective sample size at these radii and periods, these occurrence values are consistent with circumstellar material orbiting all polluted white dwarfs, and in agreement with the prediction of transit probability above.




 §.§ Constraints on exocomets


There are several models that have considered Oort-like comet clouds to account for metal pollution in white dwarfs, where the first preceded the discovery of their debris discs <cit.>.  All studies agree that a modest fraction of distant comets should remain bound, even in the case of asymmetric mass loss during stellar evolution <cit.>. Given the potential numbers, exocomets might significantly contribute to metal pollution, where simulations demonstrate they can be perturbed from several thousand au into the Roche limit of a white dwarf, either by a natal kick <cit.>, or from Galactic tides <cit.>. Extrasolar comets are apparent toward young stars like  Pic via time-series spectroscopy <cit.>, and more recently via transits in that system <cit.>, and others <cit.>.  It thus seems plausible and worthwhile to consider their survival and possible transits near periastron around polluted white dwarfs, particularly because cometary comae are typically Earth-sized or larger, and so their transits should be straightforward to detect in white dwarf light curves.  

The probability that a transit is detected within a sample of N=313 white dwarfs is 


    p_ detect = p_ transit× N × m × f_ orbit× f_ coma,


where m is the number of exocomets hosted by each star, f_ orbit is the fraction of the orbital period observed (e.g. by TESS), and f_ coma is the fraction of the orbit over which a coma (i.e. outgassing) is present and detectable. To estimate this overall probability, the following parameters are adopted: a semimajor axis of 3000 au <cit.>, periastron at 1 R_⊙, p_ transit = (R_* + R_⊕)/a/(1-e^2) for Earth-sized comae, f_ orbit = 8.5 × 10^-7 for the median observing coverage of 2.3 TESS sectors in this study, and f_ coma = 5.5 × 10^-8 for activity that begins around 150 K (i.e. interior to 0.15 au) for a white dwarf luminosity of 2×10^-3 L_⊙. Under the assumption that each exocomet orbit approaches the Roche limit, m = 2 × 10^12 comets are needed to achieve p_ detect=0.5, and thus a reasonable chance of detecting one in transit.  This ad hoc scenario would require all polluted white dwarfs to have bountiful Oort-like clouds, each dominated by star-grazing orbits. While unrealistic, this calculation nevertheless provides a detection benchmark, where it is notable that a recent model predicts distant planetesimal belts can be excited to high eccentricities following white dwarf natal kicks <cit.>.

There are models that specifically calculate the rate at which Oort-cloud analogs are delivered to the Roche limit of white dwarfs, including the effects of Galactic tides, as well as the mitigating influences of companions and stellar evolution <cit.>.  The most optimistic of these predict that comets may reach the innermost regions at rates as high as ∼2×10^4 Myr^-1, in the absence of a companion and without substantial loss of comets during post-main sequence evolution (each of which can reduce this number by an order of magnitude).  During the median observing coverage of 2.3 TESS sectors for the 313 white dwarfs in the sample, this rate predicts each star could have up to 0.0035 comets on such steeply infalling orbits, where a fraction of 0.011 of these would transit.  Overall, there would have been a 1 per cent chance at best to detect an exocomet transit in the space-based sample, and this probability is likely less favourable based on the aforementioned model considerations. Thus, the lack of a cometary transit here does not significantly constrain the prevalence of extrasolar Oort clouds, but future samples that are orders of magnitude larger should be more informative.




§ CONCLUSION AND OUTLOOK


This study searched for transiting planetary systems around polluted white dwarfs using ground-based observations with ULTRACAM and ULTRASPEC, and space-based data from TESS. Both the BLS and Lomb-Scargle algorithms were used to search for periodic signals in the target light curves, and two systems known to have periodic dimming events associated with debris discs were blindly recovered. The calculated detection sensitivities provide upper limits for the occurrence of planets and planetesimals ranging from dwarf to Kronian planetary radii.  The nominal fraction of polluted white dwarfs with detectable transits is 0.8_-0.4^+0.6 per cent.

The 40 pc sample of white dwarfs contains 1076 stars with 99.3 per cent spectroscopic completeness <cit.>. Assuming a pollution fraction of 30 per cent <cit.>, and the transit incidence found here, then 2.6_-1.6^+2.0 metal-rich white dwarfs within this volume are expected to exhibit transits. Currently, WD 1054–226 is the only polluted white dwarf with transits within 40 pc (ZTF J0328–1219 is just outside this volume), and it is therefore possible that one or two additional nearby, polluted white dwarfs will have transiting debris discs. 

While models for white dwarf pollution require major planets to dynamically perturb asteroids to the Roche limit, these are expected to have semimajor axes of several to tens of au <cit.>.  However, planets might be born of recycled disc material, as a result of viscous spreading beyond the Roche limit for sufficiently massive debris discs <cit.>, or achieve temporary, star-grazing orbits due to unpacking of previously stable, multi-planet systems <cit.>. While none are detected here, the planetary constraints are modest outside of orbital periods of a few days.

There is a basic necessity for planets in polluted white dwarf systems, as the tidally-disrupted planetesimals responsible for debris discs and pollution require dynamical perturbations.  Moreover, these orbits have to be sufficiently distant to escape engulfment during stellar evolution, and will likely have expanded further owing to stellar mass loss <cit.>.  Such widely-bound planets may be directly imaged by JWST if they are at least as massive as Jupiter at favorable angular separations <cit.>.  However, the requisite number of observations to achieve a detection will be challenging. 

It is likely that the most promising avenue for discovering planets orbiting (polluted) white dwarfs will be astrometric detection with Gaia. The fraction of white dwarfs that host planets detectable with Gaia astrometry is predicted to be as high as 0.1 per cent <cit.>, yielding an order of ten planets on au-scale orbits. There are currently three white dwarfs with candidate planets that have been inferred through a combination of Gaia and Hipparcos astrometry as proper motion anomalies <cit.>. Furthermore, Gaia DR4 should provide epoch photometry for millions of stars, and increase the number of white dwarfs with space-based light curves by over two orders of magnitude compared to TESS.  While the cadence and photometric precision of Gaia will remain inferior to TESS, the sheer numbers should result in new transiting systems.  Rubin-LSST will obtain light curves for up to 10^8 white dwarfs, but the observing cadence and baseline may complicate immediate characterization of any transits.  Nevertheless, if the 0.8 per cent fraction determined here is correct to an order of magnitude, then several thousands of transiting systems await detection. Even if only a tiny fraction of these are sufficiently bright for follow up, this can transform the field from the study of individual stars to statistical analyses of a population.



§ ACKNOWLEDGEMENTS


The authors thank E. M. Bryant for useful discussions about injection and recovery simulations.  A. Robert acknowledges support from a Science and Technology Facilities Council studentship. This work was supported by Naresuan University (NU), and National Science, Research and Innovation Fund (NSRF) (grant No. R2566B021). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101020057).The authors acknowledge the European Southern Observatory for the award of telescope time via programs 0100.C-0546, 0102.C-0700, 0103.C-0247, and 105.209J. This work has made use of data obtained at the Thai National Observatory on Doi Inthanon, operated by NARIT.  ULTRACAM and ULTRASPEC are funded by the Science and Technology Facilities Council (grant ST/V000853/1). This paper utilizes data collected by the TESS mission, which is funded by the NASA Explorer Program, including targets from Guest Investigator programs: G011113, G011179, G011203, G011288, G011294, G022017, G022057, G022065, G022077, G022158, G022176, G022217, G03046, G03124, G03126, G03207, G03221, G03252, G03274, G04036, G04056, G04091, G04106, G04123, G04137, G04171, G04177, G04200, G04209, G04211, G04219, G04236, G04240, G05003, G05057, G05071, G05081, G05084, G05115, G05148. 





§ DATA AVAILABILITY

ULTRACAM and ULTRASPEC data are available on reasonable request to the authors. TESS data are available through the Mikulski Archive for Space Telescopes. 





mnras






















§ GROUND-BASED OBSERVATION SUMMARY











 Paper Title: Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem.



Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem]Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem













1]L. B. Putra

[2]I. Nurul Hudaibnu.nurul.huda@brin.go.id

1]H. S. Ramadhan

1,2]M. B. Saputra

3]T. Hidayat

[1]Departemen Fisika, FMIPA, Universitas Indonesia,  Depok, 16424,  Indonesia

*[2]Research Center for Computing, National Research and Innovation Agency,  Bogor,  Indonesia

[3]Department of Astronomy and Bosscha Observatory, Institut Teknologi Bandung, Bandung, Indonesia





In this paper, we intend to investigate the dynamics of the Circular Restricted Three-Body Problem. Here we assumed the primaries as the source of radiation and have variable mass. The gravitational perturbation from disk-like structure are also considered in this study. There exist five equilibrium points in this system. By considering the combined effect from disk-like structure and the mass transfer, we found that the classical collinear equilibrium points depart from x-axis. Meanwhile, this combined effect also breaks the symmetry of tringular equlibrium point positions. We noted that the quasi-equilibrium points are unstable whereas the triangular equilibrium points are stable if the mass ratio μ smaller than critical mass μ_c. It shows that the stability of triangular equilibrium points depends on time.






















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§ INTRODUCTION



Circular Restricted Three Body Problem (CRTBP) consists of the movement of the third body with respect to the two primaries. The primaries move in a circular orbit and the third body is influenced by but not influences the primaries. In the classical case, the primaries and the third body are assumed as the point mass <cit.>. There exist five equilibrium points which are divided into two categories named collinear equilibrium points L_1, L_2, and L_3 and triangular equilibrium points L_4 and L_5. The collinear equilibrium points are always unstable while the triangular equilibrium points are stable if the mass ratio μ < μ_c = 0.038520896504551.

The complexity of nature has made the CRTBP not suitable for some cases. Therefore, some authors have tried to develop the CRTBP by incorporating various effects. For instance, <cit.> and <cit.> have considered the effect of photogravitation in the CRTBP for mimicking the stellar objects . More recently, the influence of disk-like structure has been incorporated in the CRTBP <cit.>. There are several studies that combined various additional effects in CRTBP. For instance, <cit.> has studied CRTBP by assuming both primaries are radiating and oblate bodies, together with the effect of disk-like structure. <cit.> combined the effect of photogravitational and disk-like structure, with addition of oblateness and finite-straight segment for the primaries, to study the stability of equlibrium points in CRTBP.

Several close binary star systems have been discovered <cit.>. It is already studied that some of them have planets that have mass much less compared to the binary <cit.>. Meanwhile, previous studies also suggest that there is also possibility that an asteroid belt like structure also exist in the binary system <cit.>. In certain cases, the transfer of mass between binary stars is unavoidable <cit.>. However, accurately predicting how mass moves between close-orbiting stars is still a major challenge. In the case of CRTBP, the transfer mass between star in binary sytem can be modelled by the variability of mass of each primary. The study of variable mass in the restricted three-body problem was done by <cit.> in 1930s. More recently, <cit.> studied the CRTBP system which the primaries have variable masses but the sum of their masses remains constant. <cit.> consider the variation of mass of primaries in accordance with the combined Meshcherskii law. Several studies also consider the variable mass of the third body <cit.>.



In this study, we investigate the possible movement of the infinitesimal mass in the close binary star system. We used a framework of CRTBP where the binaries are primaries. We assumed that the stars emit radiation and there is a mass transfer between primaries. We also considered a disk-like structure surrounding this three-body system, mimicking the Kuiper or asteroid belt structure. 

This paper is outlined as follows. In the Section <ref>, we give a detail about the equation of motion of the system. The detail about the equilibrium points is given in Section <ref>. Section <ref> describes the stability of the system. Finally, the conclusion is given in Section <ref>. Here we used Mathematica software to conduct a numerical calculation or algebraic manipulation.




§ EQUATION OF MOTION



Let the mass of the first and second primaries are m_1 and m_2 respectively. The mass ratio between primaries is represented by μ = m_2/(m_1 + m_2), where 0 < μ < 1. Hence we represent the mass of the primaries by 1-μ and μ. For simplifying the problem, we consider the system in a two dimensional rotational coordinate Oxy and the primaries always lays on x-axis. The origin of the coordinate system is located in the positon of m_1. We take the distance between primaries as the unit of length and the unit of time is chosen in such a way so that the gravitational constant is unity. Let (x, y) be the position of the third body. We follow <cit.> for describing the CRTBP where the primaries have variable mass. However, we also consider the effect of the radiation pressure on the primaries and the disk-like structure surrounding the three-body system. 

The radiation force (F_p) has an opposite direction with respect to the gravitational force (F_g). In order to consider the radiation pressure in the CRTBP, we defined the mass reduction factor q = 1 - (F_p/F_g), where 0 < 1 - q ≪ 1. Meanwhile, the disk-like structure effect can be modelled by following <cit.>. The potential of disk-like structure for planar version is given as

    V(x,y) = M_b/√(r^2 + T^2)

where M_b is the total mass of the disk-like structure and r^2 = x^2 + y^2 is the radial distance of the infinitesimal mass. The mass parameter of the disk-like structure is M_b and is assumed to be small compared to the total mass of the primaries (M_b<<1). Here T = a + b means the dust belt's density profile, where a and b are the flatness and core parameters of the disc respectively. 



Figure <ref> shows a graphic representation of the system. We follow <cit.> to model the mass transfer between the primaries. It has a center of inertia (A) as its center. This center is a static point. There is also center of mass (D) that moves around point A. It has to be noted that in our case the distance between A and D is so small. For simplicity, we shall consider conservative linear mass transfer law between the primaries,

    μ(t)=m_2(t)/m_1(t)+m_2(t)=kt.

Here t means time and 0<t<1/k. It has to be noted that the sum of mass m_1(t) and m_2(t) is constant. We assume that the rate of transfer k is much slower compared to the orbital period of the primaries, i.e. k<<1/n where n is the mean motion of the two body system,

    n^2=1+2M_br_c/(r_c^2+T^2)^3/2.

The reference radius of the disk-like structure is given by r_c^2=1-μ+μ^2 as in <cit.>. Assuming that the transfer mass between primaries is very slow and that the dominant order in the expansion is the first order, we have

    μ (t)≈μ (t_0)+μ(t_0)(t-t_0),

where we could define μ(t_0)≡ k and μ(t_0)≡μ_0, so that

    μ (t)=μ_0+k(t-t_0).

Note that for μ_0=kt_0, <ref> reverts back to <ref>. In the case of μ_0=kt_0, the domain for t is (t_0-μ_0/k)<t<(t_0+1-μ_0/k).

The equation of motion of the system is given as follows

    x   -2ny=W_x, 
    y   +2nx=W_y,

where W_x and W_y mean the partial derivative of W with respect to x and y respectively. The pseudo potential is given by

    W=1/2n^2(x^2+y^2)-μ n^2 x+(1-μ)q_1/r_1+μ q_2/r_2+M_b/(ℛ^2+T^2)^1/2.

The first derivatives of the pseudo potential with respect to the third body position is given by

    W_x   =n^2x-μ n^2-(1-μ)q_1x/r_1^3-μ q_2(x-1)/r_2^3-M_b(x-μ)/(ℛ^2+T^2)^3/2, 
    
        W_y   =n^2y-(1-μ)q_1y/r_1^3-μ q_2y/r_2^3-M_b(y-2k/n)/(ℛ^2+T^2)^3/2.

Here q_1 and q_2 are the radiation pressure factor for m_1 and m_2. If we do not consider the radiation and disk-like structure effects, <ref> will be similar to the equation of motion in <cit.>. We consider the same coordinate system in <cit.> where the origin of the rotational coordinate is the position of m_1, hence

    r_1^2   =x^2+y^2, 
    
        r_2^2   =(x-1)^2+y^2.

Since the center of the disk-like structure is the point C, i.e. the point around which the primaries barycenter orbits, the radial distance of the infinitesimal mass becomes

    ℛ^2=(x-μ)^2+(y-2k/n)^2.





§ EQUILIBRIUM POINTS





 §.§ Quasi-Collinear Points


The collinear points L_1, L_2, and L_3 are the solution located in the interval 1 < x < ∞, 0 < x < 1, and -∞ < x < 0, respectively. The position of collinear points are found by considering x = y = ẋ = ẏ = y = 0 into <ref>. We have

    (x-μ)(n^2-M_b/((x-μ)^2+4k^2/n^2+T^2)^3/2)-(1-μ)q_1x/x^3-μ q_2(x-1)/(x-1)^3=0, 
    2M_bk/n((x-μ)^2+4k^2/n^2+T^2)^3/2=0.

It is clear from <ref> that the collinear equilibrium points only exist if M_b = 0 or k = 0. Nevertheless, we searched a possible equilibrium points near x-axis when M_b ≠ 0 and k ≠ 0 by calculating the equilibrium points numerically. The numerical values are obtained using a numerical algorithm in Mathematica. Here we consider μ_0=0.3 and t_0=0. The resulting time dependence graph can be seen in <ref>. As expected, we found that the equilibrium points are quasi-collinear. They shifted slightly towards the +y axis, due to the existence of mass variation and the disk like structure, which has the point C (that is not on the barycenter, nor is it anywhere in the x axis) as its center.

According to <ref>, the position of L_1, L_2, L_3 have affected by M_b. Higher M_b makes L_1, L_2, and L_3 position further away from x-axis. Meanwhile, higher k means that the position of L_1, L_2, and L_3 are shifted higher with respect to the original position as time increase (see <ref>). In contrast, as shown in <ref> the value of q_1 and q_2 has contributed lower compared to k in determining the shift of L_1, L_2, and L_3 as time increase.





 §.§ Triangular Points


In order to find the position of equilibrium points, we have to solve <ref> by considering x = y = ẋ = ẏ = 0. The position of  triangular equilibrium points can be calculated by considering y ≠ 0. Hence we get

    (x-μ)[n^2-M_b/(ℛ^2+T^2)^3/2]-(1-μ)q_1x/r_1^3-μ q_2(x-1)/r_2^3 = 0, 
    
        y [n^2-M_b/(ℛ^2+T^2)^3/2]   -(1-μ)q_1y/r_1^3-μ q_2y/r_2^3+2kM_b/n(ℛ^2+T^2)^3/2 = 0.

We assume that the position of triangular points in the modified CRTBP is the perturbed version of classical case (r_1 = 1; r_2 = 1), i.e.

    r_1   =1+ϵ_1, 
    
        r_2   =1+ϵ_2,

where ϵ_1,2≪ 1. By substituting <ref> to <ref>, neglecting higher order of ϵ_1,2, and solving for x and y, we have position of triangular equilibrium points as follows

    x    = 1/2+ϵ_1-ϵ_2, 
    
            y    = √(3/4+ϵ_1+ϵ_2),

Following <cit.>, we consider <ref> and <ref> into <ref>. Hence, with additional expansion of k to the first order, we obtain

    ϵ_1   =-1-q_1/3+M_b(1-2r_c)/3(r_c^2+T^2)^3/2, 
    ϵ_2   =-1-q_2/3+M_b(1-2r_c)/3(r_c^2+T^2)^3/2.

Substituting <ref> to <ref> yields the triangular points L_4 and L_5

    x=1/2-q_2-q_1/3

and

    y=±√(3)/2(1-2/9(2-q_1-q_2)+4/9M_b(1-2r_c)/(r_c^2+T^2)^3/2).

It can be seen that the triangular points for this system are identical (to the first order) with the constant primary mass counterpart, albeit with a (slow) time dependence.

<Ref> show the effect of M_b in the position of triangular points. We observe that the position of L_4 and L_5 is not symmetric due to the combination of disk-like structure and mass transfer. This asymmetric is larger when M_b is higher. We noted also that the position of L_4 and L5 is closer to the primaries  with increasing M_b. In <Ref>, it is clear that the decreasing value of k makes L_4 closer to the primaries, in contrast with L_5. From <Ref> we observe that radiation pressure has the impact on the location of triangular equilibrium points. The location of L_4 and L_5 are closer to the source of radiation pressure when the radiation pressure getting stronger, either for m_1 or m_2. 


















§ LINEAR STABILITY






The stability of equilibrium points are studied by introducing the perturbation in the equilibrium point (x_0,y_0), hence we define

    x   =x_0+α, 
    
        y   =y_0+β,

where α and β is small displacements with respect to the equilibrium points. By substituting <ref> to <ref> and expand the equation, we get

    α   -2nβ=W_xx^0α+W_xy^0β, 
    β   +2nα=W_yx^0α+W_yy^0β,

where

    W_xx   =n^2 + (1-μ)q_1/r_1^3(-1+3x^2/r_1^2) + μ q_2/r_2^3(-1+3(x-1)^2/r_2^2) + M_b/(R^2+T^2)^3/2(-1+3(x-μ)^2/(R^2+T^2)), 
    
        W_yy   =n^2 + (1-μ)q_1/r_1^3(-1+3y^2/r_1^2) + μ q_2/r_2^3(-1+3y^2/r_2^2) + M_b/(R^2+T^2)^3/2(-1+3(y-2k/n)^2/(R^2+T^2)), 
    
        W_xy   =W_yx=3(1-μ)q_1xy/r_1^5 + 3μ q_2(x-1)y/r_2^5 + 3M_b(x-μ)(y-2k/n)/(R^2+T^2)^5/2.

The characteristic equation is given by

    λ^4+(4n^2-W_xx^0-W_yy^0)λ^2+W_xx^0W_yy^0-(W_xy^0)^2=0.

The solution of this equation is given as follows 

    λ_i = ±√((-b ±√(b^2 - 4c))/2);    i=1,2,3,4.

where b = 4n^2-W_xx^0-W_yy^0 and c = W_xx^0W_yy^0-(W_xy^0)^2. The stability of equilibrium points can be achieved when all λ_i are purely imaginary, otherwise we have an unstable equilibrium point. 

Table <ref> shows the characteristic roots (λ_i) of the collinear equilibrium points by considering several configuration of perturbing parameters. All λ_1 have the form real which signify instability. In the range of mass parameter 0 < μ < 1 we found that b^2-4c > 0 for L_1, L_2, and L_3. Consequently we have at least one positive real for the solution of characteristic equation. Hence, the collinear equilibrium points are always unstable.

In the case of triangular equilibrium points, the stability is achieved when 0<μ<μ_c, where μ_c means the critical mass. Following <cit.>, the critical mass is given as follows

    μ_c   =1/2(1-√(23/27))-22-q_1-q_2/27√(69)
       +(3/2+(76-8r_c)(r_c^2+T^2)/27√(69)-83+12r_c^2/6√(69))M_b/(r_c^2+T^2)^5/2.

Table <ref> shows the examples of characteristic roots in the stability of triangular equilibrium points. All case have stable equilibrium points during t=0 and unstable in t=0.2 and t=0.3. We noted that λ_1 and λ_3 in L_4 are similar to L_5 for the case 1 and case 2. However, due to the combination of disk-like structure and mass transfer effects, this similarity is not sound for case 3 and case 4.

Since, in our case, μ depends on time, besides μ_c there exists also a so called critical time (t_c) as follows

    t_c=t_0+(μ_c-μ_0)/k,

where t>t_c means unstable. <ref> shows the effect of perturbing parameters M_b, q_1, and T on t_c for the case of k=0.1, μ_0=0.3, and t_0=3. We noted that t_c becomes shorter when M_b and 1-q_1 increase. In contrast, t_c is longer if T increases.













§ CONCLUSIONS


We have studied the influence of mass transfer, disk-like structure, and radiation pressure, on the position and stability of CRTBP equilibrium points. In this system, we found there are five equilibrium points, where two of them are triangular equilibrium points, and the others are quasi-collinear equlibrium points. Unlike the classical collinear equilibrium points, we noted that L1, L2, and L3 are slightly departed from x-axis since there exist the effects from disk-like structure and mass transfer. Moreover, the symmetry of L_4 and L_5 is broken when we consider the mass transfer and disk-like structure together. Furthermore, we found that the quasi-collinear equilibrium points remain unstable. The stability of triangular points depend on the initial mass parameter μ_0 as well as the time. We found there exist critical time for achieving the stability of triangular points.



§ ACKNOWLEDGMENT

We thank the reviewers for their insightful comments and suggestions on the manuscript.This research has been supported by RIIM LPDP-BRIN 2023-2025 and UI research grant No. PKS-026/UN2.F3.D/PPM.00.02/2023. 








 Paper Title: TESS discovery of two super-Earths orbiting the M-dwarf stars TOI-6002 and TOI-5713 near the radius valley.

 






Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium 
Oukaimeden Observatory, High Energy Physics and Astrophysics Laboratory, Cadi Ayyad University, Marrakech, Morocco 
Department of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 
Dpto. Física Teórica y del Cosmos, Universidad de Granada, 18071, Granada, Spain 
Avature Machine Learning, Spain
Carl Sagan Institute at Cornell University, Space Science Building, Ithaca, NY 14850, USA 
School of Physics & Astronomy, University of Birmingham, Edgbaston, Birmimgham B15 2TT, UK 
Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía s/n, 18008 Granada, Spain 
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 
Department of Astronomy & Astrophysics, University of California San Diego, La Jolla, CA 92093, USA 
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38200, La Laguna, Tenerife, Spain 
Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain 
Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA 
Space Sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Allé du 6 Août 19C, B-4000 Liège, Belgium 
NASA Ames Research Center, Moffett Field, CA 94035, USA 
Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 
Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 
Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 
Royal Astronomical Society, Burlington House, Piccadilly, London W1J 0BQ, UK 
NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA 
Crow Observatory, Portalegre, Portugal 
Bryant Space Science Center, Department of Astronomy, University
of Florida, Gainesville, FL 32611, USA 
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 USA 
NASA Exoplanet Science Institute, IPAC, California Institute of Technology, Pasadena, CA 91125 USA 
Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA 
Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA 
Department of Physics, Center for Astrophysics & Space Sciences, University of California San Diego, La Jolla, CA 92093, USA
Department of Physics and Astronomy, University of Notre Dame, South Bend, IN 46556, USA
AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, F91191 Gif-sur-Yvette, France 
Paris Region Fellow, Marie Sklodowska-Curie Action
Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, UK 







 discovery of two super-Earths orbiting the M-dwarf stars TOI-6002 and TOI-5713 near the radius valley Ghachoui et al.



We present the validation of two  super-Earth candidates transiting the mid-M dwarfs TOI-6002 and TOI-5713 every 10.90 and 10.44 days, respectively. The first star (TOI-6002) is located 32.038±0.019 pc away, with a radius of 0.2409^+0.0066_-0.0065 , a mass of 0.2105^+0.0049_-0.0048  and an effective temperature of 3229^+77_-57 K. The second star (TOI-5713) is located 40.946±0.032 pc away, with a radius of 0.2985^+0.0073_-0.0072 , a mass of 0.2653±0.0061  and an effective temperature of 3225^+41_-40 K. We validated the planets using  data, ground-based multi-wavelength photometry from many ground-based facilities, as well as high-resolution AO observations from Keck/NIRC2. TOI-6002 b has a radius of 1.65^+0.22_-0.19  and receives 1.77^+0.16_-0.11 S_⊕. TOI-5713 b has a radius of 1.77_-0.11^+0.13 and receives 2.42±0.11 S_⊕. Both planets are located near the radius valley and near the inner edge of the habitable zone of their host stars, which makes them intriguing targets for future studies to understand the formation and evolution of small planets around M-dwarf stars.    




 discovery of two super-Earths orbiting the M-dwarf stars TOI-6002 and TOI-5713 near the radius valley
    M. Ghachoui      <ref>,<ref>
B.V. Rackham        <ref>, <ref>                      
M. Dévora-Pajares <ref>, <ref>                            
J. Chouqar          <ref> 
M. Timmermans       <ref>                                          
L. Kaltenegger      <ref>                                        
D. Sebastian        <ref>                                          
F.J. Pozuelos       <ref> 
J.D. Eastman        <ref>                            
A.J. Burgasser      <ref>                                           
F. Murgas           <ref>,<ref>                     
K.G. Stassun        <ref>                                          
M. Gillon           <ref>                                          
Z. Benkhaldoun      <ref>                                     
E. Palle            <ref>,<ref>                     
L. Delrez           <ref>,<ref>                         
J.M. Jenkins        <ref>                                      
K. Barkaoui         <ref>,<ref>,<ref>                 
N. Narita           <ref>,<ref>,<ref> 
J. P. de Leon       <ref>                                  
M. Mori             <ref>,<ref>                       
A. Shporer          <ref>                                      
P. Rowden           <ref>                                     
V. Kostov           <ref>                                      
G. Fűrész <ref>                                      
K.A. Collins        <ref>                            
R.P. Schwarz        <ref>                            
D. Charbonneau      <ref>                            
N.M. Guerrero       <ref>,<ref>                             
G. Ricker           <ref>,<ref>                         
E. Jehin            <ref>                                     
A. Fukui            <ref>,<ref>                   
Y. Kawai            <ref>                                  
Y. Hayashi           <ref>                                  
E. Esparza-Borges   <ref>,<ref>                     
H. Parviainen       <ref>,<ref>                     
C.A. Clark          <ref>,<ref>                               
D.R. Ciardi         <ref>                                         
A.S. Polanski       <ref>                                
J. Schleider        <ref>                                      
E.A. Gilbert       <ref>                                            
I. J.M. Crossfield <ref>                             
T. Barclay          <ref>                                   
C.D. Dressing       <ref>                                         
P.R. Karpoor        <ref>      
E. Softich          <ref>      
R. Gerasimov        <ref>,<ref>    
F. Davoudi          <ref>                                          

    August 5, 2024
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§ INTRODUCTION


The Transiting Exoplanets Survey Satellite <cit.> has significantly increased the sample of exoplanets smaller than Neptune and larger than Earth (broadly called "super-Earths" and "sub-Neptunes") known to orbit nearby stars.
The absence of planets in this size range in our own solar system limits us to understand how these planets form and evolve, making them compelling targets to investigate. Although it is arguably important to focus on a wide range of stellar types to study exoplanets, M dwarfs provide appealing targets for our study of super-Earths due to their abundance in our galaxy <cit.> and the significantly more favorable planet-to-star size ratio compared to Sun-analog stars. Especially planets at the inner edge of the Habitable Zone (HZ) provide compelling targets because of the transition from hot, potentially habitable rocky worlds to Venus-like worlds <cit.>. 
Especially planets close to the inner edge of the HZ of M dwarfs, which could arguably show extended secondary atmospheres resulting from a runaway greenhouse effect, thus provide very interesting targets for observations <cit.>

The abundance and the distribution of planets with radii below 4  for orbits shorter than 100 days allows the identification of specific features that can link their abundance to their formation and evolution like the radius valley <cit.> and the density valley <cit.>, based on their radii and densities distributions. The radius valley features a lack in planets ranging from 1.5 to 2 . In this regard, some studies argue that rocky super-Earths are a byproduct of atmospheric escape experienced by gaseous sub-Neptunes driven by the energy radiated by the host stars <cit.> and/or the energy released by their cooling planetary cores <cit.>. Other studies argue that some planets are formed as rocky super-Earths in a gas-depleted <cit.> or gas-poor environment with thin gaseous envelopes <cit.>.
Additional theoretical studies predict that the sub-population of sub-Neptunes consists typically of water-rich worlds formed beyond the water ice-line and migrate toward their host stars <cit.>.
The density valley features three peaks in the density distribution defining rocky super-Earths, water-rich worlds, and puffy sub-Neptunes. <cit.> suggest that the pebble accretion is the main mechanism forming these planets with super-Earths formed within the ice line and water-rich world formed outside it <cit.>. For now the mechanisms forming super-Earths remain unclear, making observations of planets in this mass-radius regime a critical part to understand it.     

 is increasing the number of small exoplanets amenable to mass and atmospheric characterization <cit.>. Detailed studies of such exoplanets will allow insights into the origin of the radius/density valley in the near future. Such analyses should shed further light on the mechanisms dominating the formation and evolution of super-Earths. In addition the analysis of exoplanets with incident irradiation between Earth and Venus will allow key insights into the limits of the HZ <cit.>.

This paper reports the validation of two super-Earths, TOI-6002 b (1.65^+0.22_-0.19 ) and TOI-5713 b (1.77_-0.11^+0.13 ) orbiting two mid- type M-dwarf stars, TOI-6002 and TOI-5713. They were first detected by , and validated in this study using various ground-based observations (see Sect. <ref>). The masses of both planets are left unconstrained by our observations, but their radii suggest rocky or water-rich composition. Note that precise mass measurements should be possible with state-of-the art Doppler spectographs to derive their masses, as discussed in more detail in Sect. <ref>.

The paper is structured as follows. Sect. <ref> introduces the data from  and all ground-based observations. Stellar characterization, validation of the planetary nature of the two  candidates, and global data analyses are covered in Sect. <ref>. Results and potential for mass measurements and atmospheric characterization are discussed for both planets in Sect. <ref>. Sect. <ref> summarizes our findings.



§ OBSERVATIONS
 

This section presents all observations of TOI-6002 and TOI-5713 obtained from space and ground-based instruments. In Table <ref>, we summarize all the ground-based, photometric time-series observations, as well as the observation parameters. We note that the differential photometry was performed using uncontaminated circular apertures for all ground-based observations. 






 §.§  photometry
 

TOI-6002 was observed by  in sectors 14 (18 July to 14 August 2019), 41 (24 July to 20 August 2021), and 54–55 (9 July to 1 September 2022). TOI-5713 was observed by  in sectors 16 (12 September to 06 October 2019), 22–23 (19 February to 15 April 2020), and 49 (26 February to 25 March 2022). Note that this target was also observed in sectors 74 and 75 from 03 January to 26 February 2024. These two sectors were not included in our analyses here (see Sect. <ref>), but the data appears consistent with our results and will be used as part of future research on these planets. 

The  time-series observations were processed by the  Science Processing Operations Center (SPOC) pipeline <cit.>. SPOC calibrates the image data, carries out quality control, labels bad data, and performs simple aperture photometry (SAP) for each star in the field of view. Then, it corrects for instrumental systematic offsets using a Presearch Data Conditioning module <cit.> and produces pre-search data conditioned light curves. Fig. <ref> shows the target pixel files (TPFs) and the SAP apertures used for photometry, along with the overlaid locations of nearby   DR2 <cit.> sources. 

The SPOC pipeline executed a transit search of the combined light curve from sectors 14–55 on 11 September 2022 for TOI-6002 and from sectors 16–49 on 05 May 2020 for TOI-5713 <cit.>, yielding a threshold crossing event (TCE, a statistically significant periodic, transit-like signal on the target star) with a 10.9 day period for the first target and a TCE with a 10.44 day period for the second. Initial limb-darkened transit models were fitted <cit.> and a suite of diagnostic tests were performed to prove or refute the planetary origin of the signals <cit.> for both TCEs. The difference image centroiding tests conducted for the sectors 14-55 search of TOI-6002 and the sectors 16-49 search of TOI-5713 constrained the location of the target star to within 4.6±4.1 and 3.4±3.2 of the transit source, respectively.  The two transit signatures passed all diagnostic tests presented in the SPOC data validation reports to discard different possible false positives, mainly eclipsing binaries and blended eclipsing binaries. The  Science Office (TSO) evaluated the vetting information and issued an alert for TOI-6002 b on 01 December 2022 and for TOI-5713 b on 8 August 2022.

For our analyses, we downloaded the 2-minute PDC-SAP light curves from the Mikulski Archive for Space Telescopes (MAST[<https://archive.stsci.edu/>]). Using the "QUALITY" flag, we identified all the bad-quality data points that were removed from the data files. The year two data of  were subject to sky background overestimation in the SPOC pipeline for dim and/or crowded targets, affecting the transit depth and, thus, the planet radius. TOI-6002 is a crowded-field target and was subjected to this effect in sector 14. We corrected the PDC-SAP light curves of sector 14 using equation 2 described in section 4.2 of the data release notes of sector 27[<https://tasoc.dk/docs/release_notes/tess_sector_27_drn38_v02.pdf>] (see section 4.2, equation 2). This effect was negligible for TOI-5713 due to the low level of crowding. The light curve of TOI-5713 showed some clear flare-like signals, which we removed using an upper 3σ clipping.
We used  <cit.> to detrend the light curves for stellar variability, utilizing a bi-weight filter with a window 2.5 times longer than each planets' transit duration. The obtained lightcurves are presented in Fig. <ref>, with the locations of the transit signals highlighted with different colors.








 §.§ Ground-based photometry




  §.§.§ LCOGT-1m0 photometry


We used the 4096×4096 LCOGT SINISTRO CCD camera on the 1.0-m robotic telescope of the Las Cumbres Observatory Global Telescope (LCOGT; ) network node at Teide Observatory (Tenerife, Spain) to observe two full transits of TOI-6002 b on UT 2023 May 13 and 24.
Table <ref> provides the observation parameters. 
We used the standard LCOGT BANZAI pipeline <cit.> to calibrate the science images, and we used  <cit.> for the photometric extraction.



  §.§.§ LCOGT MuSCAT3 photometry


We used the MuSCAT3 multi-band imager <cit.> on the LCOGT 2m Faulkes Telescope North at Haleakala Observatory on Maui (Hawaii) to observe one full transit of TOI-6002 b on UTC August 19, 2023 (see Table <ref> for more details).
The multi-band observation covered g, r, i, and z_ s bands simultaneously. 
The data reduction and aperture photometry were performed in a similar way as described in the previous section.



  §.§.§ TRAPPIST-North photometry


We used the CCD camera on TRAPPIST-North, a 0.6-m robotic telescope at Oukaimeden Observatory in Morocco (), to observe a full transit of TOI-5713 b on UTC June 17, 2023 (see Table <ref>).
We observed with the non-standard I+z' filter.
The data reduction and photometry were performed using AstroImageJ.



  §.§.§ MuSCAT2 photometry


For TOI-5713 b, we observed one full transit on UTC July 8, 2023 (see Table <ref>), using the multi-color imager (g, r, i, and z_ s bands) MuSCAT2 <cit.>, mounted on the 1.52-m Telescopio Carlos Sánchez (TCS) at Teide Observatory, Spain. 
We used the g band images for guiding, as the star is faint in the blue optical (B ∼ 17 mag). 
We calibrated and reduced the raw data using the code and procedure of <cit.>. 
Due to the short exposure time and the faintness of the target, the MuSCAT2 g band photometry presented a much larger scatter than in the other bands, and it was not used in the analysis presented below.

We also observed a full transit of TOI-6002 b with MuSCAT2 on UTC October 01, 2023.
However, this observation was affected by clouds around the mid-transit time, and thus was omitted in the global analyses.



 §.§ Spectroscopic observations




  §.§.§ IRTF/SpeX


We used the SpeX spectrograph <cit.> on the 3.2-m NASA Infrared Telescope Facility (IRTF) to collect a  near-infrared spectrum of TOI-5713 on 2023 May 16 (UT) and TOI-6002 on 2023 Oct 31 (UT).
On both nights, conditions were clear with seeing of 06–07.
We used the short-wavelength cross-dispersed (SXD) mode, and aligned the 03 × 15” slit to the parallactic angle.
The resulting spectra span 0.80–2.42 μm and have a resolving power of R∼2000.
We collected six integrations of 139.9 s on TOI-5713 and 299.8 s on TOI-6002, using an ABBA nod pattern.
Immediately after the science observations, we gathered a standard set of flat-field and arc-lamp calibrations and observed A0 V standard stars.
We reduced the data with Spextool v4.1 <cit.>, following the standard instructions from the Spextool User's Manual[Available at <http://irtfweb.ifa.hawaii.edu/ spex/observer/>.].
The final spectrum of TOI-5713 has a median per-pixel SNR of 92 with peak values in the J, H, and K bands of of 116, 131, and 121, respectively.
For TOI-6002, the final spectrum has a median SNR per pixel of 156 with peak values of 195, 219, and 205 in the J, H, and K bands, respectively.



  §.§.§ Shane Telescope/Kast Double Spectrograph Observations


We observed TOI-5713 and TOI-6002 with the Kast Double Spectrograph <cit.> on the 3-m Shane Telescope at Lick Observatory. 
TOI-5713 was observed on 2023 July 7 (UT) in clear conditions with 1.1 seeing; 
TOI-6002 was observed on 2023 November 5 (UT) in partly cloudy conditions with 12 seeing.
We used both the red and blue channels of Kast split by the D57 dichroic at 5700 Å and the 1.5-wide slit for both observations. 
In the red channel, we used the 600/7500 grating providing 5900–9000 Å wavelength coverage at an average resolution of λ/Δλ = 1450.
In the blue channel, we used the 600/4310 grism providing 3600–5600 Å wavelength coverage at an average resolution of λ/Δλ = 1100. 
For TOI-5713, we obtained one blue exposure of 1200 s and two red exposures of 600 s each at an average airmass of 1.07; 
for TOI-6002, we obtained one blue exposure of 900 s and two red exposures of 450 s each at an average airmass of 1.03. 
We observed a nearby G2 V star immediately after each source for calibration of telluric absorption, and spectrophotomeric flux standards PG 1708+60 and BD +28 4211 <cit.> during each night for flux calibration.
Flat-field lamps, HeNeAr arc lamps, and HgCd arc lamps were also observed at the start of each night for pixel response and wavelength calibration. 

All data were reduced using the kastredux package[kastredux: <https://github.com/aburgasser/kastredux>.], which included image calibration (pixel response correction, conversion to electrons/second, bad pixel masking), boxcar extraction with median background subtraction, wavelength calibration, flux calibration, and correction of telluric absorption in the red channel.
The final spectra have median signals-to-noise of 140 in both red channels (near 7450 Å) and 45 in the blue channel of TOI-6002 (near 5100 Å). Due to calibration issues, we discarded the blue observations of TOI-5713.



 §.§ High-Resolution Imaging
 

To constrain possible contamination by bound or line-of-sight companions <cit.>, We observed TOI-5713 and TOI-6002 with near-infrared adaptive optics (AO) imaging at the Keck (2023-Jun-10 UT) and at the Palomar (2023-Jun-08 UT) Observatory. 

TOI-5713 was observed with the NIRC2 instrument on Keck-II (10 m) using the natural guide star AO system <cit.> utilizing a standard 3-point dither pattern. Each position in the dither pattern is offset from the previous position by 3. The 3-point dither pattern was performed 3 times, each pattern offset from the previous pattern by 0.5, resulting in a total of 9 on-target exposures. NIRC2 was used in the narrow-angle mode with the full field of view (∼10) and a pixel scale of roughly 0.0099442 per pixel. Observations were made in the Kcont filter (λ_o = 2.2706; Δλ = 0.0296 μm) with an integration time of 15 seconds per dither position, yielding a total of 135 seconds on-source time.
	
We observed TOI-6002 with the PHARO instrument <cit.> on the Palomar Hale (5m) using the P3K natural guide star AO system <cit.> in the narrowband Br-γ filter (λ_o = 2.1686; Δλ = 0.0326 μm). The PHARO pixel scale is 0.025 per pixel. We twice repeated a standard 5-point quincunx dither pattern with steps of 5, separating each repeat by 0.5. We used a single frame integration time of 15 sec, yielding a total integration of 225 seconds.
    
Flat fields were taken on-sky, dark-subtracted, and median averaged. We generated sky frames using the median average of the dithered science frames. We then sky-subtracted and flat-fielded each science frame and combined the calibrated science frames into a single mosaiced image.  The final resolutions of the combined dithers, determined  from the full-width half-maximum (FWHM) of the point spread functions, were 0.057 and 0.108 for TOI-5713 and TOI-6002, respectively.  
	
We assessed the sensitivities of the final combined AO image by injecting simulated sources azimuthally around the primary target, following the approach of <cit.>.
Fig <ref> shows the resulting sensitivity curves. 
We do not detect close-in (≲ 1) stellar companions for either star. 
The Keck and Palomar data to have sensitivities of δ mag ≈ 7.5 mag and 6.9 mag at 0.5, respectively. 
This corresponds to a spatial limit ≲ 32 au for TOI-6002 and ≲ 40.94  au for TOI-5713. 



 §.§  Assessment



In addition to high-resolution imaging, we used  to pinpoint any widely separated, bound stellar companions.
The  catalog does not identify any nearby sources that have the same distance and proper motion as TOI-5713 or TOI-6002 <cit.>.  
There is a 4 neighbor source to TOI-6002, but it is a line-of-sight source at >2000 pc and unrelated to TOI-6002.
     
We also used the  renormalised unit weight error (RUWE) to assess the possibility of any close-in companions.
RUWE values ≳ 1.4 indicate excess astrometric noise, which can potentially be caused by a bound companion <cit.>. 
The  DR3 RUWE values for TOI-5713 and TOI-6002 are 1.3 and 1.1, respectively, indicating astrometric solutions that are consistent with single-star models. 





§ ANALYSES
 



 §.§ Stellar characterization
 



  §.§.§ Spectroscopic analysis


Fig. <ref> shows the SpeX SXD spectra of TOI-5713 and TOI-6002.
To assign spectral types, we compared the spectra to single-star spectral standards in the IRTF Spectral Library <cit.> using the SpeX Prism Library Analysis Toolkit <cit.>.
For both stars, we find the single best match to the M3.5 dwarf GJ 273, with M3 and M4 dwarf standards providing the next closest matches.
Accordingly, we adopt an infrared spectral type of M3.5 ± 0.5 for both targets.
Following <cit.>, we used the <cit.> relation between the equivalent widths of the K-band Na i and Ca i doublets, the H2O–K2 index <cit.>, and stellar metallicity to estimate an iron abundance of [Fe/H] = -0.03 ± 0.13 for TOI-5713 and [Fe/H] = -0.18 ± 0.12 for TOI-6002.





The Shane/Kast optical spectra of TOI-5713 and TOI-6002 are shown in Fig. <ref>. Comparison to SDSS M dwarf templates from <cit.> shows a best-match to the M4 template in both cases, 
with index-based classifications from <cit.>; <cit.>; and <cit.> indicating equivalent classifications of M3.5–M4, also consistent with the near-infrared classifications.
We detect Hα emission in TOI-5713 with an equivalent width of 1.94±0.06 Å, corresponding to log_10L_Hα/L_bol = -4.16±0.07 based on the χ factor relation of <cit.>. For TOI-6002, we see no evidence of H I emission, with an Hα equivalent width limit of 0.6 Å. These emission signatures indicate ages less than and greater than ∼4.5 Gyr, respectively <cit.>. 
We measured metallicity-sensitive ζ indices <cit.> of 1.07±0.01 and 1.02±0.01 for TOI-5713 and TOI-6002, respectively, consistent with near-solar metallicities of [Fe/H] = +0.08±0.20 and [Fe/H] = +0.04±0.20 <cit.>, again formally consistent with the metallicities inferred from the near-infrared spectra. 






  §.§.§ SED and empirical relations within the global modeling



We performed a Spectral Energy Distribution (SED) fit jointly with transit modeling (see Sect. <ref>) using the latest version of [<https://github.com/jdeast/EXOFASTv2/branches>] <cit.> software package. The latest version of  
comes with the possibility of simultaneously utilizing the empirical relations of <cit.> to determine the stellar radius R_*, and of <cit.> to determine the stellar mass M_*. We activated this feature using the two corresponding flags,  and . The R_* and M_* values obtained by  are optimized and self-consistent compromises between the constraints from the SED, empirical relations, and transits (via stellar density). The stellar parameters obtained in this case are more precise than when using the MIST stellar evolution models <cit.>, which was disabled by setting the  flag as recommended in <cit.>. For the SED fit, we used the JHK_S magnitudes from 2MASS, the W1–W4 magnitudes from WISE <cit.>, the G_ BP G_ RP magnitudes from  DR3 <cit.>, and the Pan-STARRS <cit.> yz magnitudes (see Table <ref>). We applied a Gaussian prior on the  EDR3 parallax, which we corrected for systematics by subtracting -0.044991895 mas from the nominal value according to <cit.>. We used an upper limit of A_V = 8.1537 for TOI-6002 and A_V = 0.0477 for TOI-5713 on the extinctions from the dust maps of <cit.> and a Gaussian prior on the stellar metallicity from  IRTF/SpeX (see Table <ref>). Also, a Gaussian prior was applied on the 2MASS m_K band to be used by <cit.> and <cit.> relations. We run the fit with R_*, M_*, and the effective temperature T_eff as free parameters. The SED fit results are reported in Table <ref>.





  §.§.§ Galactic kinematics



We used the  DR3 data for each target, including coordinates, proper motions, parallaxes, and radial velocities <cit.>, to infer their UVW galactic velocities in the local standard of rest (LSR) frame.
We used  <cit.> to convert the  data to UVW velocities, utilizing the correction of <cit.> for the barycentric velocity relative to the LSR.
The resulting velocities are summarized in <ref>.
Following Equations 6 and 7 of <cit.>, we find these UVW velocities correspond to thin-disk, thick-disk, and halo membership probabilities of {63.9 ± 1.3%, 36.1 ± 1.3%, < 0.1%} and {97.2 ± 0.1%, 2.9 ± 0.1%, < 0.1%} for TOI-6002 and TOI-5713, respectively.
The likely thin-disk membership of both stars implies ages of ≲ 8.2 Gyr <cit.>.










 §.§ Discarding possible false-positive scenarios


Two primary cases of astrophysical false positives can masquerade as exoplanet transits in  data. The first case is background/foreground eclipsing binaries (EBs), especially those in grazing configurations. For both targets, we searched for archival imaging, retrieving images from POSS I/DSS <cit.> and POSS II/DSS <cit.>. These images, in addition to those taken from our recent observations from MusCAT3 (for TOI-6002) and MusCAT2 (for TOI-5713) in 2023, span more than 70 years, as shown in Figure <ref>. TOI-6002 has a high proper motion of ∼344.42 mas/yr and has moved by ∼24.4 from 1955 to 2023. This allowed us to confirm that there was no bright star in the current position of TOI-6002. On the other hand, TOI-5713 has a relatively low proper motion (∼110.24 mas/yr) and has moved by ∼7.7 from 1953 to 2023, making it challenging to confirm the absence of any unresolved companion in its current position. However, high-resolution adaptive optics imaging from Keck-II (for TOI-5713) and Palomar (for TOI-6002) confirm that no stellar companions were detected close to (≲ 1) the targets as discussed in Sect. <ref>. Additionally, the renormalized unit weight error (RUWE) from  DR3 is below 1.4 for both stars, indicating that the stars are singles (see Sect. <ref>).

The second astrophysical false positive case is that of bound eclipsing binaries. Many findings argue against this scenario for both stars: (1) the spectral energy distribution is fit well by single-star models (see Fig. <ref>); (2) the spectra collected by SpeX SXD and Shane/Kast, for both stars, show no secondary spectrum (see Sect. <ref>), and Fig. <ref> and <ref>; (3) the achromatic nature of the transits, where their depths in all bands agree within 1σ for each planet (see Tables <ref> and <ref>); (4) the stellar densities determined from the transits alone are in good agreement with the those determined from the the basic parameters of each star (see Table <ref>).
Considering these arguments, we discard these false-positive scenarios.





 §.§ Statistical validation


In addition to ruling out false-positives via the arguments in the previous section, we used the Tool for Rating Interesting Candidate Exoplanets and Reliability Analysis of Transits Originating from Proximate Stars <cit.> to statistically validate TOI-6002 b and TOI-5713 b. 
TRICERATOPS simulates potential astrophysical false positives due to bound stellar companions, nearby stars, and line-of-sight stars.
To assess the planetary nature of the transit signal, it calculates two quantities:
the false positive probability (FPP), which quantifies the probability that the signal originates from other astrophysical sources; 
and the nearby false positive probability (NFPP), which quantifies the probability that the transit signal originates from a resolved nearby star rather than the target star. 
The thresholds for statistically validating a planet candidate are FPP < 0.01 and NFPP < 0.001. 

We ran  on  data supplied with contrast curves from NIRC2 speckle imaging for TOI-5713 and from Palomar for TOI-6002. As our ground-based photometry confirmed the transit events were on target, removing the possibility of a nearby false positive, we set the NFPP to 0. The inferred FPP is 0.0024±0.001 for TOI-5713 b 
and 0.00783±0.00016 for TOI-6002 b. Independently, we also ran  on the data obtained with MuSCAT3 (for TOI-6002 b) and MuSCAT2 (for TOI-5713 b) because they present a  higher photometric precision. Also supplied with contrast curves, this led to FPP and NFPP lower than 6×10^-4 for both candidates. Given these results using both  and  ground-based data independently, we consider the two candidates as statistically validated 
planets.



 §.§ Planet search and detection limits


As mentioned in Section <ref>, TOI-5713 and TOI-6002 have both been observed in multiple sectors of  — respectively, sectors 16, 22, 23, 46, and sectors 14, 41, 54, 55. Since both planets were detected by the  pipelines with a low signal-to-noise ratio, it is possible additional planets remained unnoticed below the threshold set to trigger an alert (Multiple Event Statistic = 7.1σ) <cit.>. 
In this context, we performed a search for additional planet candidates in the  data with the [<https://github.com/franpoz/SHERLOCK>] pipeline <cit.> and assessed the detection limits with injection and recovery tests with the [<https://github.com/PlanetHunters/tkmatrix>] ToolKit <cit.>. 

The  pipeline is an open-source package that provides six different modules to find and validate transit signals in photometric time-series observations: (1) acquiring automatically the data from an online database, such as MAST in the case of  data; (2) searching for planetary transit signals; (3) performing a vetting of the detected signals; (4) conducting a statistical validation of the vetted signals; (5) modeling them to retrieve their ephemerides; and (6) finding the upcoming transit windows observable from ground-based observatories. See <cit.> and <cit.> for recent applications and further details.  

 obtains the PDC-SAP fluxes from the MAST archive, and applies a bi-weight filter using the  package <cit.> to detrend the light curve with various window sizes. The transit search is then performed over the nominal light curve as well as the detrended ones using the  (TLS) package <cit.>. This strategy maximizes the signal-to-noise ratio (S/N) and the signal detection efficiency (SDE) of the transit signals found. In the case of TOI-6002 and TOI-5713, we downloaded the short-cadence data from  and applied seven different window sizes between 0.2 and 1.0 days to detrend the data. As the transit search is performed in an iterative way, if a signal is detected with a S/N⩾5, it is masked, and a new search is implemented. The operation is repeated until the algorithm no longer finds signals above the threshold or reaches the number of loops (so-called “runs”) set by the user. For all our analyses, we kept the number of detrends to 7, set the maximum number of runs to 5, and also kept the S/N threshold to 5. 

We performed a first test to see if the first planet is recovered well by  for both systems with a period grid between 1 and 15 days. For TOI-6002, the signal was not recovered in the first instance. We then applied a Savitzky-Golay digital filter <cit.> and repeated the test, where we indeed recovered the planet. TOI-6002 b was recovered in the second run and in seven of the detrended light curves with a maximum S/N of 12.8 and an SDE of 13.0. Similarly, TOI-5713 b was recovered in the first run and in six of the detrended light curves and, at best, with an S/N of 11.8 and an SDE of 23.1. In order to assess the presence of additional candidates, we performed a search for the period grid between 0.5 and 25 days on 7 light curves detrended by a bi-weight filter with window sizes from 0.2 and 1.2 days, allowing a maximum of 5 runs. Neither of these analyses yielded positive planetary-like signals; all signals could be attributed to noise, variability, or systematics. 

The lack of additional transiting planet candidates in the  data for both systems does not necessarily exclude the existence of extra planets. Several scenarios could explain the lack of interesting signals in our transit search with : (1) the systems only host one planet; (2) if other planets exist, they are not transiting; (3) the data we examined do not have the photometric precision required to detect the additional transiting planets, or they have a longer period than the period grid searched in our analyses <cit.>. In the case of scenario (2), radial velocity measurements may be able to detect them, as discussed in Section <ref>. In the case of scenario (3), a ground-based high-precision photometric follow-up could possibly detect more planets, such as was the case for LP 890-9 c <cit.>. However, to test scenario (3) against the  data, we performed injection and recovery tests over the available short-cadence data for both systems.

 is an injection and recovery software specifically built to establish the retrieval thresholds of hypothetical transiting exoplanet signals in space-based data for a given star. It is built on top of the same core features as  so that it can reproduce the same algorithms and techniques implemented in the pipeline, and their results can be compared. That is,  sets detection limits to , and by extension, to any similar pipeline using preliminary detrending and phase-folded least squares search techniques. To do so, three-dimensional grids (orbital period, planet size, orbital phase) are built to create synthetic scenarios. For each of these scenarios,  runs the pre-processing of the curve (smooth, noise-masks, detrending) and searches for the injected planet either using TLS or Box Least Squares <cit.>. 




As both systems' planets are very similar in orbital periods and sizes, we planned the same scenarios for both of them. We wanted to explore the recoverability of planets with sizes between 0.5 and 3 R_⊕ and orbital periods from 1 up to 25 days. Therefore, the final injection and recovery test cases included a grid of 25 periods (from 1 to 25 days), 15 planet radii (from 0.5 to 3 R_⊕), and 3 orbital phases, summing a total of 1125 independent injected synthetic transiting planets for each target. 

Our tests show that TOI-5713 b and TOI-6002 b lie at the detectability limits found for each of their systems (see Figure <ref>). This is consistent with the low S/N obtained for both planets, either in  (S/N_TOI-5713 b=11.8, S/N_TOI-6002 b=12.8) or  (S/N_TOI-5713 b=7.7, S/N_TOI-6002 b=8.1), and the low number of  detrends where they were found. It is essential to recognize that the S/N reported by the SHERLOCK algorithm is derived from the TLS package, which employs a straightforward estimation approach that is not directly comparable to the metrics produced by alternative pipelines, such as SPOC. Instead, SHERLOCK utilizes these values internally to evaluate the detected signals and identify the most significant ones among them.


From these results, we rule out the existence of transiting planets larger than 2.3 R_⊕ with orbital periods up to 25 days around TOI-6002, whilst the same statement applies for planets larger than 3 R_⊕ in TOI-5713. If additional transiting exoplanets were present in the analyzed period range, they would probably be within the limits or below the super-Earth radius regime in TOI-6002. On the other side, there could also be undetected sub-Neptune transiting exoplanets in TOI-5713, especially for outermost orbits above 14 days, where planets with a radius between 2 R_⊕ to 2.75 R_⊕ could remain undetected with detection rates between 33–66%.




 §.§ Stellar activity and rotation period


If the host star is magnetically active and has an inhomogeneous photosphere, its rotation can pose a challenge for accurately measuring an exoplanet's mass via radial velocity (RV) observations, especially if the rotation time scale is similar to the orbital period of the planet. Therefore, we examined the effect of stellar activity on RV observations. 

We first visually inspected the PDC-SAP data light curves of the two targets. We observed no rotational modulations nor flares for TOI-6002, while we found clear flares in the four observed sectors of TOI-5713 in addition to hints of rotational modulation. 

We then used the Systematics-Insensitive Periodogram
([<https://github.com/christinahedges/TESS-SIP>]) <cit.> to build a periodogram using the whole  photometry (4 sectors) for each of the two targets. This tool simultaneously creates a Lomb-Scargle periodogram and detrends  systematics as described in <cit.>. The SPOC pipeline is known to attenuate rotational periods, thus  reproduces simple aperture photometry (SAP) light curves from the TPFs and the SPOC apertures. We applied this procedure on all four sectors observed by . Searching for periods between 2 and 15 days, we applied  on the target and on all the background pixels around the SPOC apertures. We found no significant rotational signal for TOI-6002; however, we found a significant rotational period at 9.32±0.20 days for TOI-5713. Thus, we report the SIP power spectrum for only TOI-5713 in Figure <ref>. We also found that this signal remains on the PDC-SAP light curves of each individual sector, which underscores the robustness of the signal. Beyond 15 days, we did not find any rotational signal.  

Given these rotation-period constraints, we can estimate the ages of the stars using the mass-dependent spindown relation from <cit.>.
The “jump” times at which fully convective M dwarfs transition from rapidly to slowly rotating are 2.19 ± 0.10 Gyr and 1.81 ± 0.10 Gyr for stars with the masses of TOI-6002 and TOI-5713, respectively <cit.>.
Assuming that our non-detection of rotational modulations for TOI-6002 owes to a long rotational period (> 10 d), we can constrain its age to > 2.19 ± 10 Gyr.
By contrast, the relatively rapid rotation of TOI-5713 implies an age of < 1.813 ± 0.10 Gyr.
These estimates are consistent with the age estimates we derive from their Hα emission (for TOI-5713) or lack of it (for TOI-6002) as well as the likely thin-disk membership of both stars.





 §.§ Global modeling



For each planet, we used the latest version of  <cit.> software package to globally analyze the  and ground-based data described in Sect. <ref>, jointly with the star's SED fit and the empirical relations <cit.> and <cit.> to characterize the host stars (see description in Sect. <ref>).

In the priors file, we set starting values of M_*, R_* and T_eff. We set lower and upper limits priors on the orbital period (P ± 0.1P d) and the transit epoch (T_c± 0.33P) from the values determined by SPOC and reported in ExoFOP. For the LD parameters, we disabled the priors based on the Claret tables <cit.> by setting the  flag given the low mass of our stars where these tables are not reliable <cit.>. Instead, we fitted for quadratic LD parameters assuming Gaussian priors of variance 0.1, computed with the  code <cit.> for each passband (see Table <ref>). Regarding the  data, we used the detrended PDC-SAP light curves (see Sect. <ref>) which are corrected for the contribution of neighboring stars and subsequently corrected for sky background bias for TOI-6002 in sector 14 (see Sect. <ref>), but we still fit for a possible dilution by applying a Gaussian prior of 0±10% of the contamination ratio reported in ExoFOP. This is to propagate any uncertainty in the correction. Also, Gaussian priors were applied on the metallicities from IRTF/SpeX (see Table <ref>). For each planet, we fitted for circular and eccentric orbits to check for orbital eccentricity evidences using only photometric data. The two fits were run until convergence. We found the difference in the loglikelihood is Δln Z = ln Z_eccentric - ln Z_circular⩾ 2 for both TOI-6002 b and TOI-5713 b, which suggests that eccentric orbits are marginally favored for both planets. Tables <ref> and <ref> present the fits' results for both the circular and eccentric cases. 


We also re-fitted the data for a circular orbit and with the same priors as explained above but without any Gaussian prior on the stellar mass. This is to check for consistency between the stellar densities determined from R_* and M_* (from SED + empirical relations) and those determined from the transits via the a/R_* (the semi-major axis over the stellar radius) ratio. We found  ρ_* = 23.4^+1.4_-3.2 g cm^-3   for TOI-6002 b and 11.4_-6^+7 g cm^-3 for TOI-5713 b, which are consistent with the previously determined stellar densities (see Table <ref>). This provides additional assurance that the transit signals for both candidates originate from transiting planets and not astrophysical false positives. 




















§ RESULTS AND DISCUSSION
 



 §.§ Radius valley



The radius valley, discovered by <cit.> from 1.5 to 2  for solar-type stars and by <cit.> from 1.4 to 1.7  for low-mass stars (i.e., M ≲ 0.65 ), has attracted attention because it is important to understand planet formation and evolution. There are several theories to explain the origin of the radius valley: (1) thermally-driven mass loss, where planets lose their atmospheric envelopes because of stellar high energetic X-ray and ultra-violet (UV) radiation <cit.> and of the energy emerging from the cooling planetary cores <cit.>; (2) gas-poor formation, where some low-mass planetary cores accrete a thin gaseous envelope <cit.>. These planets can still experience thermally-driven mass loss; and (3) gas-depleted formation, where the population of super-Earths is formed late in a gas-depleted environment <cit.>; (4) Other studies based on coupled formation and evolution models predict that the sub-Neptunes are rich in water formed outside the ice-line and migrate toward the host stars <cit.>.  

Observational and theoretical studies showed that the position of the radius valley is period-dependent (see Figure <ref>). Its location has been found trending to smaller planet radii for long periods in both thermally-driven mass loss and gas-poor formation models for solar FGK stars <cit.>. For M dwarfs, its location has been found trending to larger radii for long periods in the gas-depleted theory <cit.>.
Conversely, other studies on small planets around M dwarfs found a negative slope <cit.>, something similar to that of FGK stars. 

As shown in Figure <ref>, a wedge of opposing predictions emerged because of the difference in the slopes of the radius valley. Planets in this region are known as "keystone planets". They are predicted to be purely rocky by thermally-driven mass loss and gas-poor formation, and potentially rocky with small gaseous envelopes by the gas-depleted formation model <cit.>. Only a few planets with measured densities are in this wedge. Having a large sample of keystone planets is important to understand the models dominating the formation and evolution of small planets. TOI-5713 b (R_p=1.75_-0.16^+0.14  and P = 10.441989^+0.000015_-0.000014 d) is located within the wedge of keystone planets with the possibility of being at its upper edge, given the uncertainty on the planetary radius. TOI-6002 b (R_p=1.63^+0.21_-0.19 and P = 10.904821^+0.000021_-0.000018 d) is located at the lower edge of this wedge with the possibility of being inside it, given the uncertainty on the planetary radius. When their masses are determined with RV observations (see Sect.<ref>), these planets will join the small sample of keystone planets with density measurements. This sample will, in the future, help disentangle which model is dominating the formation of small (i.e., R_p ≲ 4) planets. 

In addition, for M dwarfs, a later study of <cit.> on small planets (with mass and radius precision better than 25% and 8%, respectively)  identified instead a density valley separating rocky planets, with density peaked at ρ=0.94±0.13 ρ_⊕, and water-rich worlds, with density peaked at ρ=0.47±0.05 ρ_⊕. A third peak was identified at ρ=0.24±0.04 ρ_⊕, attributed to gas-rich planets. This study favored the pebble accretion model as the main mechanism that forms these populations of planets. In this model, water-rich planets are formed outside the ice-line and migrate toward the host stars. This finding was theoretically predicted by <cit.>, and newly supported by <cit.> and <cit.> using coupled formation and evolution models. Conversely, <cit.> stand against the existence of distinct population of water-rich planets around M dwarf stars. However, in all previous studies on radius/density valley, strong conclusions require larger sample of precisely characterized exoplanets.

Given their predicted masses estimated from their radii using the relations of <cit.>, these planets are predicted to have densities of 0.855^+0.345_-0.218 ρ_⊕ for TOI-6002 b, and 0.782^+0.2911_-0.182 ρ_⊕ for TOI-5713 b. We also used the recent [<https://github.com/hpparvi/spright>] python package of <cit.>, where we found median values of the masses (and then densities) consistent with those obtained using <cit.> relations. To further guess their compositions, we referred to the radius space of the study of <cit.> and the corresponding density regimes in the density space. Given their radii, TOI-5713 b falls slightly towards the range of water-rich planets, and TOI-6002 b falls in the region where we find both rocky and water-rich planets. Therefore, it is uncertain whether the two planets could be rocky or water-rich worlds. However, even if they are not of intermediate composition, they might then be either some of the largest rocky planets or some of the smallest water-rich worlds known today. As for the radius valley, both planets straddle near the opposing side of the region of "keystone planets", where it is difficult to guess whether they lost their gaseous envelopes or not. When their masses are determined, these two planets can be added to the sample of "keystone planets," which will be used to conduct in-depth statistical inferences on the radius/density valley. Such studies will help to refine the relative dominance of the various mechanisms proposed to be responsible for the formation and evolution of super-Earths and sub-Neptunes around M-dwarf stars. 






 §.§ Prospects for a radial velocity follow-up
 

In Tables <ref> and <ref>, we show the predicted semi-amplitudes for TOI-6002 b  and TOI-5713 b for both circular and eccentric orbits. With both planets inducing reflex motions to their host-stars with semi-amplitudes on the order of 3 m s^-1, high-resolution, well-stabilized spectrographs are needed for their measurement. 
Due to the relatively faint apparent magnitudes of both stars, reaching such precision will be challenging for instruments on 4-m class telescopes, such as HARPS-N <cit.> in the optical. This is also true for NIR instruments such as CARMENES <cit.> or SPIRou <cit.> which would allow to reach a RV precision in the order of  10 m s^-1 or less.
The MAROON-X instrument <cit.> at the 8.1-m Gemini North telescope offers the necessary stability and throughput to successfully measure the reflex motions of both stars. Assuming exposure times of 900 s, the instrument's red arm will achieve a maximum signal-to-noise-ratio larger than 95 and 65 for TOI-6002 and TOI-5713, respectively. 

Following <cit.>, we can expect negligible line broadening and a low activity level for TOI-6002 due to the age of this star. Assuming an additional noise term of 1 m s^-1 accounting for stellar activity, we expect that the reflex motion can be detected at a 5σ level with less than 11h telescope time using MAROON-X. This would not only allow us to measure the mass of TOI-6002 b, but also to derive the eccentricity of the planet's orbit.

In Sect. <ref>, we determine the stellar rotation period of TOI-5713 from photometry. This translates to a maximum rotational line broadening (v sin i_*) on the order of 1.6  km s^-1, which is below the resolution limit for most high-resolution spectrographs, and thus does not affect the radial-velocity precision. Nevertheless, we observe moderate activity and a photometric rotational variability close to the orbital period of TOI-5713 b, making activity mitigation challenging. Assuming an optimistic noise term of 3 m s^-1 accounting for this stellar activity after successful mitigation, a 5σ detection of the radial velocity signal due to TOI-5713 b would need at least 16 hr telescope time with MAROON-X.



 §.§ Potential for atmospheric characterization


TOI-6002b and TOI-5713b, with radii of 1.63 R_⊕ and 1.75 R_⊕ and incident stellar fluxes of 1.77 and 2.42 times that of Earth respectively, could be classified as 'super-Venus' planets if they have rocky compositions. Venus, in comparison, receives about 1.9 times the flux of Earth and is 95% the size of Earth. According to <cit.>, Venus analogs are defined as predominantly rocky planets within the 'Venus-Zone (VZ)', receiving insolation fluxes between approximately 0.95 and 25 times that of Earth. This classification suggests that both TOI-6002b and TOI-5713b fall within this category. 
With incident flux slightly lower (TOI-6002b) and slightly higher (TOI-5713b) than modern Venus, these two exoplanets are intriguing candidates to expand our understanding of the evolution of rocky planets in the state from a temperate Earth-like to a hot Venus-like planet. The incident flux levels indicate that both planets might experience or might have experienced strong runaway greenhouse effects, depending on factors like the initial water inventory, planetary evolution and atmospheric escape rates <cit.>. Combined with the estimated ages of the stars, this could indicate atmospheric conditions similar to modern Venus. However, studying planets with incident irradiation higher than modern Earth to modern Venus levels and beyond, that may be experiencing strong greenhouse effects or may have evolved into a post-runaway greenhouse state is crucial, as it can provide insights into the factors that cause some rocky planets to resemble Venus-analogs while others may maintain Earth-analog characteristics <cit.>.

To investigate two cases of potential atmospheric characteristics of these planets, we constructed models simulating a) cloudless and 10 mbar cloudy, 92-bar Venus-like, and b) 1-bar Earth-like atmospheres. 
We link the one-dimensional VULCAN chemical kinetic code <cit.> to the one-dimensional radiative transfer model PETITRADTRANS to predict the transmission spectra for these clear atmospheres <cit.>. The radiative–convective temperature–pressure profiles were computed with the HELIOS code <cit.>. The species considered for our atmospheric models include N_2, O_2, O_3 H_2O, CO_2, CO, HCN, CH_4, NO, NH_3, and C_2H_2 for the Earth-like atmosphere (rich in N_2). Additionally, Rayleigh scattering for N_2 and O_2, as well as collision-induced absorption for the N_2–N_2 and O_2–O_2 pairs, are considered. For the Venus-like atmosphere (rich in CO_2), we include H_2S, SO_2, OCS, H_2O, CO_2, CO, CH_4, HCN, and C_2H_2 as well as CO_2–CO_2 collision-induced absorption, as opacity sources in the atmosphere. Rayleigh scattering is included for CO_2.

<cit.> showed that during Venus's 2012 transit, its transmission spectra indicated that the cloud and haze layers obstruct probing below an altitude of 80 km. <cit.> modeled the transmission spectra of the L 98-59 planets, a benchmark system for studying the VZ planets <cit.>. They showed that if the L 98-59 planets possess a clear Venus-like atmosphere, NIRSpec could detect CO_2 within 26 transits for each planet. However, the presence of H_2SO_4 clouds would significantly suppress CO_2 absorption. 
A new study by <cit.> shows that there is a wide range of possible atmospheres for a rocky planet experiencing a runaway greenhouse effect, which changes the atmosphere and observable features significantly. Model transmission spectra of the planet LP 890-9c <cit.>, another benchmark system for studying the VZ planets <cit.> show that depending on the state of the planet, JWST observations could infer evidence of H_2O with 3 transit (at 3σ confidence) for a full runaway greenhouse scenario for LP 890-9c, and CO_2-dominated atmospheres resembling Venus without high-altitude terminator clouds with 8 transits 
<cit.>. A 3D Venus model for the same planet with higher, thick cloud coverage increases the amount of transits needed to be able to explore the atmospheric composition. Note that all predictions could be complicated by the impact of clouds and/or unocculted starspots, especially for cool host stars. 

These results show that observing these exoplanets is of critical importance to understanding their evolutionary stage and the effect of increasing solar irradiation on rocky worlds. Additionally, planetary systems like the TRAPPIST-1 system could also allow insight into the effect of the increase of irradiation on the environment on a planet (e.g. <cit.>). However, even for a specific orbital distance and stellar irradiation, the observable spectral features in transit for an exoplanet depend on their specific atmospheric evolution and cloud coverage (e.g., <cit.>), requiring a wider range of observations to explore the characteristic changes of rocky worlds with stellar irradiation. TOI-6002b and TOI-5713b provide two important datapoints in that exploration. 

For TOI-6002b and TOI-5713b, the synthetic transmission spectra generated from our models (Earth-like surface temperatures are 524 K and 580 K, respectively; Venus-like surface temperatures are 1053 K and 1110 K, respectively) show key differences between Earth-like and Venus-like atmospheres (see Fig. <ref>). 

Models of TOI-6002b and TOI-5713b for clear hot Earth-like atmospheric compositions are dominated by water vapor. Water vapor is a strong absorber across the infrared for Earth-like atmospheric compositions, with amplitudes of about 300 ppm in the mid-infrared (see Fig. <ref>, top panel). Note that Fig. <ref> (top panel) also shows a notable CO_2 feature at 4–5 μ with a depth of 400 ppm. However, we would assume the atmospheric CO_2 concentration to decrease during a runaway greenhouse stage, reducing that feature strength. Because we are not modeling the possible evolution of the planet's atmosphere, we have not changed the CO_2 concentration here, but note that the features could be much smaller than shown.

Models of TOI-6002b and TOI-5713b using clear Venus-like atmospheric composition are primarily characterized by CO_2 opacity, with a depth of about 250 ppm in the mid-infrared, and a prominent SO_2 feature at 4 μ and between 7 and 10 μ, with depths above ≈ 100 ppm. With a cloud deck at 10 mbar, the CO_2 feature at approximately 4.2 μm remains discernible, exhibiting a depth greater than 200 ppm.


These models show that TOI-6002b and TOI-5713b represent promising targets for atmospheric characterization with JWST, as they may provide valuable insights into the diverse evolutionary pathways and climate states of hot rocky or water-rich exoplanets. Note that we assume rocky or water-rich compositions in our models based on the radius of the planets. Mass measurements could help provide better constraints on the viability of atmospheric studies and would be possible with the MAROON-X instrument at the 8.1-m Gemini North telescope.   






§ CONCLUSION
 

In this paper, we reported the  discovery and initial characterization of TOI-6002 b and TOI-5713 b, two super-Earth planets orbiting the mid-M-dwarf stars TOI-6002 (V = 14.6) and TOI-5713 (V = 15.355). We used the 2-min-cadence TESS observations from four sectors for each target, ground-based photometry, high-angular resolution imaging, and spectroscopic observations to validate the planetary nature of the detected transit events. We jointly analyzed the transit light curves observed by  and ground-based facilities to characterize the planets. 

TOI-6002 b has a radius of 1.65^+0.22_-0.19, an orbital period of 10.904821^+0.000021_-0.000018 days, and receives 1.77^+0.16_-0.11 S_⊕. TOI-5713 b has a radius of 1.77_-0.11^+0.13, an orbital period of 10.441989^+0.000015_-0.000014 days, and receives 2.42±0.11 S_⊕. Both planets are located just outside but near the inner edge of the habitable zone around their host stars, making them interesting targets for future studies to explore the evolution of exoplanets from hot, potentially habitable to Venus-like worlds. These two planets provide compelling additions to the sample of planets critical for understanding the formation and evolution of small worlds on the basis of the radius/density valley. 


This research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).

This article is based on observations made with the MuSCAT2 instrument, developed by ABC, at Telescopio Carlos Sánchez operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide.

TRAPPIST-North is a project funded by the University of Liege (Belgium), in collaboration with Cadi Ayyad University of Marrakesh (Morocco).

We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center.

This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF.

This research has made use of the Exoplanet Follow-up Observation Program (ExoFOP; DOI: 10.26134/ExoFOP5) website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

We acknowledge the use of public  data from pipelines at the  Science Office and at the  Science Processing Operations Center.

This paper includes data collected by the TESS mission that are publicly available from the Mikulski Archive for Space Telescopes (MAST).

Funding for the  mission is provided by NASA's Science Mission Directorate. KAC acknowledges support from the  mission via subaward s3449 from MIT.

Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.

This paper is based on observations made with the MuSCAT3 instrument, developed by the Astrobiology Center and under financial supports by JSPS KAKENHI (JP18H05439) and JST PRESTO (JPMJPR1775), at Faulkes Telescope North on Maui, HI, operated by the Las Cumbres Observatory. 

This work was partially supported by a grant from the Erasmus+ International Credit Mobility program (M. Ghachoui).

This publication benefits from the support of the French Community of Belgium in the context of the FRIA Doctoral Grant awarded to MT.
MG is F.R.S-FNRS Research Director.
The postdoctoral fellowship of KB is funded by F.R.S.-FNRS grant T.0109.20 and by the Francqui Foundation.
Author F.J.P acknowledges financial support from the Severo Ochoa grant CEX2021-001131-S funded by MCIN/AEI/10.13039/501100011033 and 
Ministerio de Ciencia e Innovación through the project PID2022-137241NB-C43. 
H.P. acknowledges support by the Spanish Ministry of Science and Innovation with the Ramon y Cajal fellowship number RYC2021-031798-I.
This work is partly supported by JSPS KAKENHI Grant Number JP18H05439 and JST CREST Grant Number JPMJCR1761.

E.E-B. acknowledges financial support from the European Union and the
State Agency of Investigation of the Spanish Ministry of Science and
Innovation (MICINN) under the grant PRE2020-093107 of the Pre-Doc
Program for the Training of Doctors (FPI-SO) through FSE funds

We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 and the ERDF “A way of making Europe” through project PID2021-125627OB-C32, and from the Centre of Excellence “Severo Ochoa” award to the Instituto de Astrofisica de Canarias.

This material is based upon work supported by the National Aeronautics and Space Administration under Agreement No. 80NSSC21K0593 for the program “Alien Earths”.
The results reported herein benefitted from collaborations and/or information exchange within NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate.

This research was supported by Wallonia-Brussels International


aa





 Paper Title: TESS Giants Transiting Giants. VI. Newly Discovered Hot Jupiters Provide Evidence for Efficient Obliquity Damping after the Main Sequence.






0000-0003-2657-3889]Nicholas Saunders
NSF Graduate Research Fellow
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0003-4976-9980]Samuel K. Grunblatt
Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA

0000-0003-2657-3889]Ashley Chontos
Henry Norris Russell Fellow
Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA

0000-0003-2657-3889]Fei Dai
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0001-8832-4488]Daniel Huber
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

UTF8gbsn
0000-0002-2696-2406]Jingwen Zhang (张婧雯)
NASA FINESST Fellow
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0001-7409-5688]Guðmundur Stefánsson
Anton Pannekoek Institute for Astronomy, 904 Science Park, University of Amsterdam, Amsterdam, 1098 XH

0000-0002-4284-8638]Jennifer L. van Saders
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0002-4265-047X]Joshua N. Winn
Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA

0000-0003-3244-5357]Daniel Hey
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA



0000-0001-8638-0320]Andrew W. Howard
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0003-3504-5316]Benjamin Fulton
NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E. California Boulevard, Pasadena, CA 91125, USA

0000-0002-0531-1073]Howard Isaacson
Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA
Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD, Australia

0000-0001-7708-2364]Corey Beard
NASA FINESST Fellow
Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA

0000-0002-8965-3969]Steven Giacalone
NSF Astronomy and Astrophysics Postdoctoral Fellow
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0002-4290-6826]Judah van Zandt
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA

0000-0001-8898-8284]Joseph M. Akana Murphey
NSF Graduate Research Fellow
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA

0000-0002-7670-670X]Malena Rice
Department of Astronomy, Yale University, New Haven, CT 06511, USA

0000-0002-3199-2888]Sarah Blunt
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
CIERA, Northwestern University, Evanston IL 60201, USA

0000-0002-1845-2617]Emma Turtelboom
Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA

0000-0002-4297-5506]Paul A. Dalba
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA

0000-0001-8342-7736]Jack Lubin
Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA

0000-0002-4480-310X]Casey Brinkman
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0003-3179-5320]Emma M. Louden
Department of Astronomy, Yale University, New Haven, CT 06511, USA

0000-0002-3221-3874]Emma Page
Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015, USA




0000-0001-8621-6731]Cristilyn N. Watkins
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0001-6588-9574]Karen A. Collins
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0003-2163-1437]Chris Stockdale
Hazelwood Observatory, Australia


0000-0001-5603-6895]Thiam-Guan Tan
Perth Exoplanet Survey Telescope, Perth, Western Australia


0000-0001-8227-1020]Richard P. Schwarz
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0001-8879-7138]Bob Massey
Villa '39 Observatory, Landers, CA 92285, USA

0000-0002-2532-2853]Steve B. Howell
NASA Ames Research Center, Moffett Field, CA 94035, USA

0000-0001-7246-5438]Andrew Vanderburg
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA


Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA

0000-0002-4715-9460]Jon M. Jenkins
NASA Ames Research Center, Moffett Field, CA, 94035








0000-0002-6892-6948]Sara Seager
Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA

0000-0002-8035-4778]Jessie L. Christiansen
NASA Exoplanet Science Institute, IPAC, MS 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA

0000-0002-6939-9211]Tansu Daylan
Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA





Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA



0009-0008-9808-0411]Max Brodheim
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0009-0004-4454-6053]Steven R. Gibson
Caltech Optical Observatories, Pasadena, CA, 91125, USA

0000-0002-7648-9119]Grant M. Hill
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0000-0002-6153-3076]Bradford Holden
UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA

0000-0002-5812-3236]Aaron Householder
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA


Caltech Optical Observatories, California Institute of Technology, Pasadena, CA 91125, USA

0000-0003-2451-5482]Russ R. Laher
NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E California Blvd, Pasadena, CA 91125, USA

0009-0004-0592-1850]Kyle Lanclos
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0000-0003-0967-2893]Erik A. Petigura
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA

0000-0001-8127-5775]Arpita Roy
Astrophysics & Space Institute, Schmidt Sciences, New York, NY, 10011, USA

0000-0003-3856-3143]Ryan A. Rubenzahl
NSF Graduate Research Fellow
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0002-4046-987X]Christian Schwab
School of Mathematical and Physical Sciences, Macquarie University, Balaclava Road, North Ryde, NSW 2109, Australia

0000-0003-3133-6837]Abby P. Shaum
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0009-0007-8555-8060]Martin M. Sirk
Space Sciences Laboratory, University of California, 7 Gauss Way, Berkeley, CA 94720, USA


Space Sciences Laboratory, University of California, 7 Gauss Way, Berkeley, CA 94720, USA

0000-0002-6092-8295]Josh Walawender
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0000-0002-4037-3114]Sherry Yeh
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743








§ ABSTRACT 

    The degree of alignment between a star's spin axis and the orbital plane of its planets (the stellar obliquity) is related to interesting and poorly understood processes that occur during planet formation and evolution. Hot Jupiters orbiting hot stars (≳6250 K) display a wide range of obliquities, while similar planets orbiting cool stars are preferentially aligned. Tidal dissipation is expected to be more rapid in stars with thick convective envelopes, potentially explaining this trend. Evolved stars provide an opportunity to test the damping hypothesis, particularly stars that were hot on the main sequence and have since cooled and developed deep convective envelopes. We present the first systematic study of the obliquities of hot Jupiters orbiting subgiants that recently developed convective envelopes using Rossiter-McLaughlin observations. Our sample includes two newly discovered systems in the Giants Transiting Giants Survey (, ). We find that the orbits of hot Jupiters orbiting subgiants that have cooled below ∼6250 K are aligned or nearly aligned with the spin-axis of their host stars, indicating rapid tidal realignment after the emergence of a stellar convective envelope. We place an upper limit for the timescale of realignment for hot Jupiters orbiting subgiants at ∼500 Myr. Comparison with a simplified tidal evolution model shows that obliquity damping needs to be ∼4 orders of magnitude more efficient than orbital period decay to damp the obliquity without destroying the planet, which is consistent with recent predictions for tidal dissipation from inertial waves excited by hot Jupiters on misaligned orbits.






§ INTRODUCTION
 





Giant planets on close-in orbits (hot Jupiters) have been observed in a wide variety of stellar environments, but it remains unclear how they arrive at such small orbital separations from their host stars <cit.>. Two popular scenarios are disk-driven migration and high-eccentricity migration <cit.>. In both theories, giant planets form at larger semi-major axes, before losing angular momentum to either the protoplanetary disk (in disk-driven migration; e.g. ) or to another planet or star (in high-eccentricity migration; e.g. ). In the latter case, the planet ends up on an eccentric orbit that gradually shrinks and circularizes as the eccentricity is damped by tidal interactions with the star. 

The stellar obliquity—the angle between the spin axis of the star and the planet's orbital axis—is a key piece of evidence to distinguish between these pathways as each scenario produces a distinct distribution of spin-orbit angles. Disk migration results in aligned populations at early times, while high-eccentricity migration predicts early scattering events that might alter the obliquity <cit.>. Observations of hot Jupiters around main-sequence stars have shown that the stellar obliquity distribution has a strong dependence on the star's effective temperature <cit.>, with hot hosts displaying a wide range of obliquities, and cool hosts being preferentially aligned. The critical separation is approximately at the Kraft break <cit.>, around 6250 K, corresponding to the break between stars with primarily radiative envelopes (>6250 K) and stars with thick convective envelopes (<6250 K). The distinct distributions of obliquities separated by the Kraft Break suggest two possible scenarios:

    
  * Hot Jupiters have unique formation pathways around hot and cool stars, resulting in aligned orbits around cool stars and a broad range of obliquities around hot stars.
    
  * Hot Jupiters are formed by the same mechanism around hot and cool stars, and the spin-orbit angle is realigned during the main sequence by tidal dissipation in stars with convective envelopes.


The correlation between spin-orbit alignment and the presence of a convective envelope has driven theories for realignment through the tidal damping of obliquity <cit.>. During their main sequence lifetimes, low mass stars (≲1.2 M_⊙) have surface convection zones that deepen as the star cools and evolves into a subgiant. However, it is challenging to constrain the timescales of damping mechanisms using a main sequence population in part due to uncertain age estimates. More massive stars ≳1.2 M_⊙) have radiative envelopes on the main sequence, which cool and become convective as the stars drop below the Kraft break during the subgiant or red giant branch phase. As these stars only develop substantial convective envelopes after leaving the main sequence, they provide a unique avenue to test the timescales of tidal damping. If convection zones efficiently damp obliquities, as required by high-eccentricity migration, subgiants that were formerly 'hot' and had a broad range of obliquities should be found to have low obliquities due to rapid obliquity damping.

Studies have suggested that the presence of a convective envelope drives efficient damping of orbital period, eccentricity, and obliquity <cit.>. Diminishing orbital period has been unambiguously observed in only a small handful of systems <cit.>, though these measurements provide useful benchmarks for the timescales and efficiencies of mechanisms for orbital evolution. Orbital decay is driven by the transfer of orbital angular momentum to stellar rotation <cit.>. As an orbiting planet raises a tidal bulge in the outer envelope of its host star, turbulent viscosity <cit.> and the excitation of waves through the stellar interior <cit.> sap energy from the orbit. This orbital evolution can take place over human timescales <cit.>, and the changing orbital period can be observed. 











The sky-projected obliquity can be constrained by obtaining high-resolution spectra during a planet's transit and modeling the Rossiter-McLaughlin (RM) effect <cit.>. As a planet crosses in front of its host star, it blocks regions of the stellar surface that are red-shifted or blue-shifted relative to the observer due to the stellar rotation. These red- and blue-shifts imprint an additional signal onto the traditional radial velocity curve caused by the planet's orbit, the morphology and amplitude of which depend on the relative alignment of the planet's orbital path across the rotating star, and can be used to constrain the sky-projected obliquity of the system. The nearly all-sky coverage of NASA's Transiting Exoplanet Survey Satellite (; ) has enabled the discovery of a large sample of hot Jupiters orbiting bright subgiants which are amenable to measurements of the RM effect <cit.>, allowing us measure spin-orbit alignment and understand late-stage planetary orbit evolution. 

In this paper we present observations of the RM effect for three hot Jupiters orbiting subgiants observed by : (R_ p = R_ J, P_ orb = days, R_⋆ = 1.89±0.18 R_⊙, T_ eff = 6106±100 K), (R_ p = R_ J, P_ orb = days, R_⋆ = 2.23±0.05 R_⊙, T_ eff = 6223±100 K), and (R_ p = R_ J, P_ orb = days, R_⋆ = 1.82±0.25 R_⊙, T_ eff = 6020±100 K). is a previously confirmed system <cit.>, while and are newly-identified planets confirmed by our work as part of the Giants Transiting Giants Survey. We find that all three systems have a low sky-projected obliquity (|λ| < 15^∘), indicating that tidal realignment is efficient after the post-main-sequence emergence of a convective envelope. We then test models for realignment, and find that enhanced efficiency of obliquity damping after the main sequence can lead to realignment without inspiral and engulfment. 















§ OBSERVATIONS
 



 §.§ Photometry
 



and were discovered as part of our Giants Transiting Giants survey (GTG; ). data are available for in 30-minute cadence Full-Frame Images (FFIs) for Sectors 17, 18, and 24, spanning from October 7, 2019 - May 13, 2020, and in 200-second cadence FFIs for Sector 58 from October 29 - November 26, 2022. It was flagged by our survey as a planet candidate in 2021, and was selected for 2-minute cadence observations in Sector 58. data are available for in 30-minute FFIs in Sector 12 (April 22 - May 21, 2019), 10-minute cadence in Sector 39 (May 26 - June 24, 2021), and 200-second cadence in Sector 66 (June 2  - July 1, 2023). We identified as a planet candidate in 2021, and it received 2-minute cadence observations in Sectors 39 and 66. 

was first identified by the MIT Quick Look Pipeline (QLP; ) in 2019, and data are available in 30-minute cadence FFIs for Sectors 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, and 26, spanning from July 18, 2019 - July 4, 2020. data are available in 10-minute cadence FFIs for Sectors 40, 41, 47, 48, 49, 50, 51, 52, 53, 54, 55, spanning from June 24, 2021 - September 1, 2022. In 200-second cadence FFIs, it has so far been observed in Sectors 56, 57, 58, 59, and 60, covering September 1, 2022 - January 18, 2023. was scheduled for 2-minute cadence observations beginning on April 16, 2020 with Sector 24, and was selected for 2-minute cadence data through Sector 60. The planet was confirmed by <cit.>, and was followed up as part of the -Keck Survey (TKS; ). 

All photometry data used are the Presearch Data Conditioning simple aperture photometry (PDC_SAP; ) 2-minute light curves produced by the Science Processing Operations Center (SPOC; ) at NASA Ames Research Center downloaded from the Mikulski Archive for Space Telescopes (MAST).

We searched for additional transiting planets within these systems in the photometry by masking the known transits and performing a Box-fitting Least Squares (BLS) periodogram analysis with the  implementation of the  BLS tools. We did not identify additional transiting planet signals.

The Science Processing Operations Center (SPOC; ) performed multi-transiting planet search in the subsequent 2-min cadence observations and detected the signatures of TOIs 1181 b, 6029 b and 4379 b using a noise-compensating matched filter <cit.>. The transit signatures were fitted with an initial limb-darkened transit model <cit.> and were subjected to a number of diagnostic tests <cit.>, including the difference image centroiding test, which located the host stars within 0.409 ± 2.6, 0.78 ± 2.5, and 2.6 ± 2.6. The SPOC searches likewise failed to identify additional transiting planet signals.



 §.§ Radial Velocity Follow-up
 

High precision RVs were taken using the High Resolution Echelle Spectrograph (HIRES) on the 10-meter Keck I telescope on Maunakea, Hawaiʻi <cit.>. For , 87 RV measurements were taken between December 2, 2019 and May 16, 2022. For , 46 RV measurements were taken between December 25, 2020 and October 6, 2022. For , 12 RV measurements were taken between September 21, 2021 and July 16, 2022. RV observations were taken in the HIRESr configuration with an iodine cell in the light path for wavelength calibration. The C2 decker was used for all observations, providing a spectral resolution of R≈60,000. RVs were reduced with the California Planet Search (CPS; ) pipeline. Table <ref> in Appendix <ref> contains all RV observations and corresponding uncertainties used in this analysis.



 §.§ Rossiter-McLaughlin Observations
 

In addition to RVs over the full phase of the planets' orbits, we obtained RV observations during each planet's transit to measure the Rossiter-McLaughlin (RM) effect. We obtained high resolution spectra during transit events with HIRES for and , and with the Keck Planet Finder (KPF; ) for . 

The HIRES observing configuration during the RM observations matches that described in <ref>. We obtained 40 HIRES RV observations during the transit of on September 8, 2020. The typical counts for each ∼10-minute exposure were 20,000 e^- with an average RV uncertainty of ∼3 m s^-1. We obtained full coverage of the transit of , with ∼1 hour before ingress and ∼1.5 hours after egress to establish an RV baseline. For , we obtained 34 HIRES RVs on September 18, 2022 with typical counts of 5,000 e^- over ∼13-minute exposures and average RV uncertainty of ∼5 m s^-1. Poor weather prevented observation of the full transit egress, and we were unable to continue the observations after the transit event concluded.

We obtained 30 observations of on June 6, 2023 using KPF, with 9-minute exposures. KPF is a fiber-fed spectrograph with a resolution of R≈98,000. Observations were taken with simultaneous wavelength calibration from an etalon lamp fed through a dedicated calibration fiber. In addition to simultaneous calibrations, we took dedicated etalon calibration exposures (so-called “slew-cals") each hour to monitor RV stability. Due to its southerly declination (-32^∘), the window to observe the transit of was tight, and the star was not visible above the Keck telescope dome during ingress. RVs were measured using two distinct approaches—first, using a cross-correlation function (CCF) on a set of spectral lines produced by the public KPF data reduction pipeline[https://github.com/Keck-DataReductionPipelines/KPF-Pipeline/https://github.com/Keck-DataReductionPipelines/KPF-Pipeline/]. This reduction method resulted in a typical RV uncertainty of ∼7 m s^-1. We also reduced the RVs by analyzing the spectra using an adapted version of the SpEctrum Radial Velocity Analyser code (SERVAL; ), which measures shifts in the observed spectra relative to a matched template spectrum, but found that the CCF reduction resulted in RVs with higher precision and smaller scatter. The KPF data reduction pipeline is being actively developed, and future improvements to the measured RV precision are likely to be implemented.



 §.§ Ground-based Imaging
 

We obtained high-contrast imaging of and to search for close stellar companions which might have biased the measurement of the amplitude of the transit signal observed by . was observed by the NIRC2 near-infrared imager in the Kp bandpass on the 10-meter Keck II telescope on June 26, 2023.

We first pre-processed the data using the Vortex Imaging Processing (VIP) software package <cit.>. We performed flat-fielding, bad-pixel removal, and correct for geometric distortions by applying the solution in <cit.>. Since the data were taken in vertical angle mode, we de-rotated each image according to the parallactic angles and then stacked the individual images into a combined image. To register the eight frames, we identify the star's position by fitting a 2D Gaussian to the stellar point spread function (PSF) in each frame.

A companion to was identified with a separation of 763 ± 20 milliarcseconds at a position angle of 110.2 ± 0.5^∘. We measured a difference in magnitude in the Ks bandpass of Δ m=4.44 ± 0.05. The reduced and derotated image can be found in Figure <ref> in Appendix <ref>.

After obtaining the pre-processed cubes, we extracted the astrometry of the companion applying a least-squares minimization code. Specifically, we fit the PSF of the companion using the unsaturated PSF of the primary star in the same images (see details in ).

was observed by the 8-meter Gemini South telescope on Cerro Pachon using the speckle interferometric instrument Zorro <cit.> on May 18, 2022 UT. Three sets of 1000 × 60 ms images were obtained simultaneously at 562 and 832 nm, the data was reduced using our standard software pipeline <cit.> and the limiting magnitude contrast was computed for each filter as a function of the angular separation from TOI-4379. The 5-σ contrast curves reached depths of 5 to 6.5 magnitudes in the two filters over the angular range of 0.1 to 1.2 arcsec. Within these contrast levels and angular limits, no close companion was detected. The raw and reduced data, including the contrast curves and reconstructed images at each wavelength, are available on the  website[exofop.ipac.caltech.edu/tess/target.php?id=97766057exofop.ipac.caltech.edu/tess/target.php?id=97766057].



 §.§ Ground-based Photometry


The TESS pixel scale is ∼ 21 pixel^-1, often causing multiple stars to blend in the TESS photometric aperture. To confirm the true source of the TESS detection, we acquired ground-based follow-up photometry of the fields around TOI-6029 and TOI-4379 as part of the TESS Follow-up Observing Program <cit.>[tess.mit.edu/followup]. We used the , which is a customized version of the  software package <cit.>, to schedule our transit observations.



  §.§.§ LCOGT


We observed an ingress window of in Sloan i' band on UTC 2023 August 03 from the Las Cumbres Observatory Global Telescope <cit.> 1 m network node at McDonald Observatory near Fort Davis, Texas, United States (McD). The 1 m telescopes are equipped with 4096×4096 SINISTRO cameras having an image scale of 0389 per pixel, resulting in a 26×26 field of view. The images were calibrated by the standard LCOGT  pipeline <cit.>, and differential photometric data were extracted using  <cit.>. The transit is detected within the 35 follow-up photometric aperture which excludes the flux from the nearest known neighbor in the Gaia DR3 catalog (Gaia DR3 430269136521931776) that is ∼17 north of .

We also observed a full transit window of in Sloan i' band on May 15, 2022 (UTC) from the LCOGT 1 m network node at Cerro Tololo Inter-American Observatory in Chile (CTIO). The transit is detected within the 27 follow-up photometric aperture, which excludes most of the flux from the nearest known neighbor in the Gaia DR3 catalog (Gaia DR3 6027376203516633728), which is ∼3 west of . 



  §.§.§ PEST

We observed a full transit window of in Sloan g' band on May 27, 2023 (UTC) using the Perth Exoplanet Survey Telescope (PEST), located near Perth, Australia. The 0.3 m telescope is equipped with a 5544×3694 QHY183M camera with an image scale of 07 pixel^-1, resulting in a 32×21 field of view. A custom pipeline based on [c-munipack.sourceforge.net] was used to calibrate the images and extract the differential photometry. The transit is detected within the 64 follow-up photometric aperture, which includes the flux from the nearest known neighbor in the Gaia DR3 catalog. Figure <ref> shows the ground-based transit observations of and with best-fit transit models. All light curve data are available on the  website[exofop.ipac.caltech.edu/tess/target.php?id=419523962exofop.ipac.caltech.edu/tess/target.php?id=419523962], but are not included in the joint model of this work. 



§ HOST STAR CHARACTERIZATION
 



 §.§ High-resolution Spectroscopy
 

We obtained out-of-transit, iodine-free spectra with Keck/HIRES for each of our host stars to extract precise stellar parameters. Stellar properties were computed using SpecMatch empirical (SpecMatch-Emp; ) and synthetic (SpecMatch-Synth; ) methodologies. All of our host stars have effective temperatures T_ eff > 5000 K, and we therefore adopt values from SpecMatch-Synth. Our adopted values can be found in Table <ref>.



 §.§ Isochrone Fitting
 

We used the [https://github.com/danxhuber/isoclassifyhttps://github.com/danxhuber/isoclassify] Python package <cit.> to compute additional stellar properties. We ran the code in grid mode where we pass input observables taken from SpecMatch-Synth and position and parallax distance from Gaia DR3 <cit.> and K-band magnitude adopted from the Input Catalog (TIC;  v8.2.  uses the stellar model grid to compute stellar mass (M_⋆), stellar radius (R_⋆), surface gravity (logg), distance (d), luminosity (L), and stellar age. We used the Combined19 allsky dust map from  <cit.>. 

We account for the companion to described in <ref> by running  in binary mode, including the correction for the measured companion magnitude. For the companion to , we obtain a mass of 0.44±0.02 M_⊙, radius of 0.45±0.02 R_⊙, and temperature of 3528±40 K.

The  outputs and associated uncertainties can be found in Table <ref>. All host stars are found to be evolved (∼2 R_⊙) stars with masses between 1.2 and 1.6 M_⊙ and metallicities between 0.1 and 0.4 dex.



 §.§ Asteroseismology
 

Finally, to improve the characterisation of the host stars, we searched for signatures of stellar p-mode oscillations in the light curve.  To do this, we first remove transits and apply a simple high pass-filter to remove further low-frequency variability. We use the PySYD pipeline <cit.> to search for variability in the host stars. We find no evidence of oscillations in TOI-6029 and TOI-4379. In TOI-1181, however, we identify a weak signal associated with the global oscillation power excess, commonly referred to as ν_ max, at 737±69 μHz at an SNR of 4.55. This value is slightly lower than the expected ν_ max of 930 μHz based on asteroseismic scaling relations (cf. ). As a result of this, and non-detections in the other host stars, we do not use asteroseismology to constrain the stellar parameters.







§ PLANET MODELING
 



 §.§ Joint Transit and RV Fit
 

We used the  Python package <cit.> to simultaneously model the planet transit, radial velocity curve, and RM effect. All photometry data used are 2-minute cadence observations reduced by the Science Processing Operations Center (SPOC; ) pipeline. We used the 2-minute cadence data as it provides the best constraint on the ingress and egress times, which are important for characterizing the shape of the Rossiter-McLaughlin signal. These data are also reduced with a uniform approach which accounts for contamination by nearby stars, providing the most accurate constraint on the transit depth and planet radius. To enable longer sampling within reasonable computation time (≲ 24 hours), we select only one Sector of 2-minute cadence data for each target (Sectors 73, 58, 66 for , , and , respectively). Including additional Sectors did not significantly impact the measured transit depth, period, or epoch.

We fit for orbital period, P, transit epoch, t_0, ratio of planetary to stellar radius, R_p/R_⋆, cosine of orbital inclination, cosi, RV semi-amplitude, K, sky-projected spin-orbit angle, λ, and the eccentricity and argument of periastron, parameterized by the expressions √(e)cosω and √(e)sinω. For each instrument, we also fit for two quadratic limb darkening coefficients, described by q_1, TESS, q_2, TESS, q_1, HIRES, q_2, HIRES, q_1, KPF, and q_2, KPF. We include an offset parameter for the RV instruments, which captures small global offsets for all RVs taken by a given instrument. 



For each target, we tested (a) constant instrumental offset shared between all observations for a given instrument and (b) instrumental offset with a linear slope to capture long term trends in the RV signal. The linear offset improved the fit by reducing structure in the residuals for , while the fits to and preferred the constant intrumental offset. We predicted the RV signal due to the resolved companion for to test whether it could cause the observed trend, assuming the companion was observed at maximum separation on a circular orbit. We found that the maximum slope caused by a companion of mass M=0.44 M_⊙ at the observed angular separation is 0.001 m s^-1 day^-1, a factor of 10 smaller than the observed slope of ∼0.01 m s^-1 day^-1. We conclude that the outer companion is unlikely to be the source of the RV slope and may indicate the presence of an unresolved stellar or planetary-mass companion. Further monitoring of RV and astrometry for this system is required to determine the source of the RV trend.



We also included the RM effect in our  model (see  for details). The amplitude of the RM effect depends on the ratio of the planet's radius (R_ p) to the star's radius (R_⋆), the projected rotation velocity of the star (vsini), and impact parameter (b), given by <cit.> as:

    A_ RM≈2/3(R_ p/R_⋆)^2 vsini√(1-b^2).

As stars evolve off of the main sequence, they become larger, more luminous, and slower in their rotation. These changes have three primary effects that make RM observations challenging—first, the larger radius causes the transit duration to increase, limiting available observation windows; second, the increased stellar radius also causes the ratio R_ p/R_⋆ to decrease, reducing the RM amplitude; finally, the slowed stellar rotation (vsini_⋆) also reduces the RM amplitude. Additionally, granulation for subgiants can introduce RV jitter with an amplitude of ∼5 m s^-1 on timescales of hours <cit.>. 

By modeling the RM signal jointly with the photometric and RV observations, the constraints on the obliquity are strengthened. The RVs provide additional information about the out-of-transit RV baseline, and the deviation from that baseline informs the spin-orbit angle. The photometric transit gives a constraint on the planet radius, which plays a strong role in the amplitude of the RM signal. In turn, the amplitude of the RM signal sheds additional light on the planet radius. Because we have an independent constraint on the planet radius, the projected stellar rotation velocity, vsini_⋆, is also well-constrained by the amplitude of the RM signal. 



All priors and inferred values for our model parameters can be found in Table <ref>. For each planet, we additionally derive planet mass, M_p, planet radius, R_p, and eccentricity, e from the outputs of our model. Figures <ref>, <ref>, and <ref> show the transit, RV, and RM fits, respectively. For , a significant long-term RV trend was detected, shown in Figure <ref>. 



For each target, we sample the posterior distribution using affine-invariant Markov Chain Monte Carlo (MCMC). We initiate 100 walkers with 50,000 sampling steps and 5,000 burn-in steps per walker, ensuring that the chains ran at least 30 times the autocorrelation length and successfully converged. The residual scatter in the RVs is consistent with what we would expect for a subgiant (10-15 m s^-1), and there is no remaining structure in the residual photometry.













 §.§ Doppler Shadow
 

As an independent measure of the obliquity, we analyzed the spectral timeseries for each of our systems, and identified a Doppler shadow signal in the HIRES observations for and (Figures <ref> and <ref>, respectively). We selected lines in the non-Iodine portion of the HIRES spectra (4000-5000 Å) to search for variations in the line profile during transit. We performed 5-σ outlier clipping and removed the continuum and blaze function with a polynomial fit. Each spectrum was then cross-correlated with the SpecMatch template spectrum and the globabally-averaged line profile was subtracted off to isolate line profile variations. Our approach follows <cit.>. Analysis of the KPF data for did not result in the Doppler shadow detection, due to low signal-to-noise ratio of the observations, and the fact that the line profile is dominated by turbulence rather than rotational modulation.







§ RESULTS
 



 §.§ TOI-1181
 

is an intermediate-mass (M_p= M_J) hot Jupiter on a circular orbit. The planet radius is typical for a hot Jupiter of this mass (R_p= R_J). At a T_ eff of 6054 K, its subgiant host star has cooled after its main sequence lifetime and is likely to have developed a non-negligible surface convective zone. The RM effect of has the highest amplitude in our sample, due in part to the high projected rotation velocity of its host star (vsini_⋆ = 10.6±1.0 km s^-1), providing the most precise measurement of the spin-orbit angle. The posterior from our model gives the constraint on the sky-projected obliquity of TOI-1181 of λ = ^∘. For , we measure an obliquity from the Doppler shadow analysis of 7.5±17.2^∘, which is consistent with the measurement from the RM analysis. 



 §.§ TOI-6029
 

is the most massive planet in our sample, with M_p= M_J, as well as the smallest (R_p= R_J), indicating a high density relative to other hot Jupiters. Its host star is the hottest, most massive, and most luminous in the sample. TOI-6029 sits near the Kraft break at T_ eff = 6223 K, indicating that the system has had a substantial surface convective envelope (≳10% of stellar radius) for only a relatively brief window of its lifetime. The out-of-transit residuals to our RV model have larger scatter than those of the other two systems, and we searched for additional non-transiting planets. We performed a Lomb-Scargle periodogram analysis on the residuals as well as attempting a two-planet fit to the observations, neither of which returned conclusive evidence of additional planetary companions. The lack of evidence for additional non-transiting planets indicates that the residual scatter is likely caused by stellar jitter. The spin-orbit alignment of the TOI-6029 system is similar to that measured for TOI-1181, with λ = ^∘. The Doppler shadow measurement for provides an obliquity constraint of 22.3±8.6^∘, which is consistent with the degree of misalignment provided by the RM measurement. We note that the sign of the obliquity from the different methods is flipped for due to the degeneracy of the orientation of the transit chord.



 §.§ TOI-4379
 

is similar to Jupiter in mass (M_p= M_J) but has a larger radius (R_p= R_J). Its host star has the lowest projected stellar surface rotation velocity in our sample (vsini_⋆ = 5.9±1.0 km s^-1). The lack of sufficient RV baseline observations caused the measured sky-projected obliquity of the system to have larger uncertainty than the previously discussed measurements. The spin-orbit angle of the TOI-4379 system is consistent with being fully aligned (λ = ^∘).



 §.§ Population Comparison
 

We now compare the radii, masses, and incident flux of our planet sample to the known population of exoplanets. Literature values were downloaded from the NASA Exoplanet Science Institute (NExScI) Exoplanet Archive[https://exoplanetarchive.ipac.caltech.eduexoplanetarchive.ipac.caltech.edu] on January 3, 2024. We limited the sample to exoplanets with both measured radii and masses, and removed planets for which the fractional uncertainty on either mass or radius exceeded 20%. Figure <ref> shows the well established relationship between incident flux, planet radius, and planet mass. We estimate incident flux with the relation from <cit.> given by

    F_ p/F_⊕ = (R_⋆/R_⊕)^2 ( T_ eff,⋆/ T_ eff,⊙)^4 (a/ AU)^-2.

, , and show inflation that is consistent with known planets of similar mass. The planets presented in this work constitute a small representative sample of hot Jupiters. We refer to <cit.> for a population-level comparison of masses and radii for planets orbiting main-sequence and subgiant stars.


















§ TIDAL REALIGNMENT
 



 §.§ Comparison to Main Sequence Obliquities
 

Figure <ref> shows a Hertzsprung-Russell (H-R) diagram of host stars for systems with measured sky-projected obliquities. To populate this figure, we downloaded obliquity measurements from the  catalog <cit.> and included additional measurements reported in <cit.>. We then performed cuts to select hot Jupiters systems (M_P>0.3 M_J, P_ orb < 10 days, a/R_⋆≤8) and removed obliquity measurements with uncertainties >30^∘.



To estimate the evolution of our host stars, we fit evolutionary tracks from  Isochrones & Stellar Tracks (; ). We adopt the  mass and interpolate between values for metallicity in order to identify isochrone models that pass through our host stars near our stellar age estimates. The resulting tracks are shown as dotted lines in Figure <ref>. The evolutionary tracks show that these stars have indeed dropped below 6250 K during their post-main sequence evolution. In Figure <ref> we highlight two additional hot Jupiter systems for which the host star is likely somewhat evolved: WASP-71 b at 6180 K <cit.> and HAT-P-7 b at 6310 K <cit.>.

Near the estimated main-sequence positions of our targets, hot Jupiter hosts are observed with a wide range of obliquities. We test the likelihood that our three aligned systems are randomly drawn from the previously-aligned population in the observed distribution using a bootstrapping technique: we select a range of effective temperature between 6300 K and 6700 K which encompasses the zero-age main sequence (ZAMS) positions of our host stars. Next, we perform 100,000 random draws of three measured obliquities and calculate the fraction of draws in which all three systems were aligned (<20^∘), and find that we retrieve the observed measurements ∼10% of the time. If we include the literature measurement of WASP-71 b, the fraction drops to ∼4%. This indicates that it is unlikely for our measured obliquities to be randomly sampled, however a larger sample would improve the confidence of this constraint.

Using a sample of subgiants which have decreased in temperature and crossed the Kraft break, we can test whether systems that were part of a frequently misaligned population become realigned after developing a deep surface convective envelope. Main sequence stars hotter than the Kraft break—similar to the H-R positions of our sample at the ZAMS—show a wide range of obliquities. If spin-orbit misalignment persists over the full lifetime of planetary systems, hot Jupiters orbiting stars which are hotter than the Kraft break should still be misaligned. In contrast, if the presence of a convective envelope causes spin-orbit realignment, the population of stars which gained deep convective envelopes after the main sequence should host hot Jupiters which are aligned or in the process of being aligned.







 §.§ Evolution of the Convective Envelope
 

Figure <ref> qualitatively suggests that misaligned hot Jupiters can be realigned as their host stars cross the Kraft break on the subgiant branch, implying efficient obliquity damping due to a convective envelope. To quantify whether obliquity damping can indeed act over subgiant evolution timescales, we examine the evolution of the surface convective envelope. We performed stellar modeling with the Modules for Experiments in Stellar Astrophysics code (; ) to trace the emergence of surface convective envelopes and estimate the time each star spent cooler than the Kraft break. Our models used initial elemental abundances from <cit.> and an atmospheric temperature structure following an Eddington T() relation with fixed opacity. We assume no diffusion and no core or envelope overshoot. 



We used the best-fit stellar properties for each star (given in Table <ref>) as initial properties, and evolved our stellar models from the pre-main sequence until the base of the red giant branch. In Figure <ref>, we show the depth of the surface convection zone boundary in units of fractional stellar radius R_ CZ/R_⋆ (where 0 is the center and 1 is the surface) as a function of stellar age.





Near the ZAMS, the convective boundary is near R_ CZ/R_⋆=1, indicating a thin surface convection zone. As the stars evolve into subgiants, the convective boundary moves deeper into the stellar interior. While the mass fraction (M_ CZ/M_⋆) contained in the convective envelope remains small (≲0.01%), the deepening surface convection zone may lead to enhanced tidal damping. We note that at the age when these stars drop below 6250 K, the outer ∼10% of the stellar envelope is convective, but there is no clear feature to indicate a substantial deepening of the convective envelope over a short timescale. However, the observed change in the distribution of obliquities corresponds to this temperature, and we therefore select 6250 K as the threshold for enhanced damping. Realignment may require a minimum threshold of material for efficient damping, or it may correspond to the disappearance of the convective core. Further stellar modeling will be required to examine this threshold with more detail in the future.

If we assume that efficient realignment begins when a star decreases in temperature and drops below the Kraft break, we can roughly estimate the timescale for realignment. Because all systems in our sample are already aligned, we can only estimate an upper limit to this timescale by calculating the time between when the host star crossed the Kraft break and the estimated age at which it is observed today. This results in an upper limit on the realignment timescale of ∼1.5 Gyr for , and ∼0.5 Gyr for and . Using and as references, we can place a global upper limit on the timescale for realignment at ∼ 500 Myr. 



 §.§ Realignment Model
 

Stellar tides have been suggested to effectively damp out the obliquity of hot Jupiter systems, however they also damp orbital periods on similar or shorter timescales <cit.>. This leads us to an important question—how can a system realign without the planet facing runaway inspiral and engulfment? 

To test this, we model the evolution of orbital period and obliquity under the influence of tidal damping. We construct a toy model for orbital evolution, using a single stellar evolution model that is characteristic of stars in our sample, with M_⋆=1.4 M_⊙, [Fe/H]=0.3 dex, and M_p=1.2 M_J. For simplicity, we assume a circular orbit for the planet. We use our toy model to explore the damping efficiencies required to reproduce a post-main sequence distribution similar to what we observe. 

We assume exponential decay prescribed by ȧ = - a / _a for the semi-major axis, a, and λ̇ = - sin(λ) / _λ for the obliquity, λ.








The coefficients _a and _λ are the damping timescales, defined by <cit.> as

    _a ≈ 1.28 ( Q'_⋆/10^7) (M_⋆/10^3 M_ p) (ρ̅_⋆/ρ̅_⊙)^5/3(P_ orb/1 d)^13/3 Gyr

and 

    _λ≈ 1.13 _a (κ/0.1) (M_⋆/10^3 M_ p) (ρ̅_⊙/ρ̅_⋆)^2/3(10 d/P_ s) (1 d/P_ orb)^1/3

where Q'_⋆ is the tidal quality factor of the star, ρ̅_⋆ is the mean stellar density, and κ is related to the stellar moment of inertia I_⋆ by I_⋆ = κ M R^2.

The calculation of _λ in Equation <ref> requires knowledge of the stellar rotation period. We therefore produced our stellar models with a modified version of which calculates and stores parameters necessary to trace the rotational evolution of the star (see  for details). We assume standard Skumanich-like spindown over the main sequence lifetime. This is a reasonable assumption for stars in this mass range, as they are not expected to undergo significant magnetic braking <cit.>. 

The choice of tidal quality factor Q'_⋆ has a significant effect on the dynamical evolution. However, values for Q'_⋆ range over multiple orders of magnitude for subgiants <cit.>. In Figure <ref>, we show the impact of this choice on our dynamical models. If Q'_⋆ is assumed to be the same for the damping of P and λ, the planet enters a phase of runaway inspiral and is engulfed before it can realign (panel a). If we instead define the damping timescales _a and _λ with different effective tidal quality factors, it is possible to produce a model in which the planet realigns before it can be engulfed. Because the obliquity distribution appears distinct on either side of the Kraft break, we assume that _a and _λ share the same tidal quality factor until the model drops below 6250 K, at which point we invoke a deviation in Q'_⋆, described by the order of magnitude difference δlogQ'_⋆. Figure <ref>b shows that a difference in effective tidal quality factor δlogQ'_⋆≳4 is required to efficiently realign hot Jupiters on orbits between ∼3-8 days without entering a phase of runaway inspiral. 



Figure <ref> shows the sample from Figure <ref> but instead plotting obliquity versus effective temperature, highlighting the correlation between alignment and stellar effective temperature. To examine the change in λ as the host star evolves, we take the model in Figure <ref> b and plot it in λ versus T_ eff space. We observe that the toy model indeed reproduces an evolutionary track in which a deepening convective envelope during subgiant evolution causes obliquity damping.



The toy model shown in Figures <ref> and <ref> make several assumptions. First, we assume that the tidal quality factor Q'_⋆ does not evolve smoothly in time, and instead drops precipitously in the case of λ-damping after the star crosses the Kraft break. It is unlikely that this fully captures the behavior seen in nature, as Q'_⋆ depends on the interior stellar structure, which evolves smoothly over main sequence lifetimes. However, main sequence internal structure changes are small, and we choose to fix the values of Q'_⋆ on the main sequence, as well as after the threshold is crossed. We also assume that the value of Q'_⋆ applied to the damping of period is held fixed for the entire evolution of the star. As the stellar structure evolves after the star leaves the main sequence, its Q'_⋆ should change, though the relationship between Q'_⋆ and stellar radius and density is uncertain. We again choose the simple case of holding the value fixed, though future studies of how Q'_⋆ relates to stellar interior evolution may allow us to improve this model in the future. There are promising pathways to better characterize Q'_⋆ through eclipsing binary system evolution <cit.> and asteroseismology <cit.>.



We now consider what physical mechanisms could lead to a large difference in the efficiencies of the orbital period and obliquity damping timescales. <cit.> invoke sustained core-envelope decoupling as a potential source of enhanced obliquity damping. In this scenario, the planet is able to exert a torque on the outer envelope of the star without rapidly losing orbital angular momentum. They note, however, that this explanation is challenged by evidence that the core and envelope of main sequence stars appear to be well-coupled <cit.>. It is unclear whether this expectation applies to subgiant stars, and the theory of sustained core-envelope decoupling could be tested with asteroseismic measurements of radial differential rotation or core-envelope misalignments in evolved stars <cit.>.

Another explanation, posed by <cit.>, suggests that in the presence of a convective envelope, a hot Jupiter on a misaligned orbit can excite inertial waves which dissipate angular momentum as they propagate through the stellar interior. <cit.> examined tidal dissipation in stars with convective envelopes in the context of hydrodynamical theory, and found that these inertial waves can result in the value of Q'_⋆ being reduced by up to 4 orders of magnitude. Because these inertial waves are static in the inertial frame, they do not contribute to orbital decay <cit.>, and only impact the efficiency of obliquity damping. This suggested order of magnitude difference is consistent with our toy model in Figures <ref> and <ref> following the prescription for tidal damping laid out in <cit.>.

A recent study by <cit.> showed that hot Jupiter orbits can resonantly lock to gravity mode pulsations in the radiative interior of cool stars and efficiently damp the system's obliquity (similar to semi-major axis damping described in ). This mechanism is able to damp the stellar obliquity without causing the planet to inspiral into its host star. The interior structure of our subgiant hosts is amenable to gravity mode pulsations, providing another possible mechanism for the observed obliquity damping.





 §.§ Predicting Orbital Evolution of Main Sequence Systems
 

To investigate period and obliquity damping over a range of parameters, we calculate our toy model over a grid of main sequence orbital period, P_ ZAMS, planet mass, M_ p, and δlogQ'_⋆. We consider a system “engulfed” if the orbital period is reduced to zero before the star reaches the base of the red giant branch, “realigned” if the obliquity drops below 10^∘ and the orbital period is greater than zero for the full track, and “not aligned” if the planet is not engulfed but retains an obliquity greater than 10^∘. This threshold corresponds to the typical uncertainty on a sky-projected obliquity measurement (see Figure <ref>).

Figure <ref> shows the parameter space where realignment is expected for a range of values for δlogQ'_⋆ as a function of main sequence orbital period, P_ ZAMS, and planet mass, M_ p. The regions inside the contours are expected to realign without being engulfed before the star reaches the base of the red giant branch. As the orbital period increases, a larger M_ p is required to efficiently damp the obliquity. However, if the planet mass is too high, the damping of orbital period will be efficient enough to catalyze runaway inspiral. Figure <ref> can be used as a rough guide to predict whether a hot Jupiter orbiting a hot main sequence star will realign after the main sequence.







§ CONCLUSIONS


The obliquities of hot Jupiter systems have been observed to show a strong dependence on the effective temperature of the host star, with cool stars (<6250 K) hosting mostly aligned planets and hot stars (≥ 6250 K) hosting mostly misaligned planets. Theories have suggested that a surface convective envelope can damp the obliquity of a hot Jupiter's orbit.

To test this theory, we have measured the sky-projected spin-orbit angles for three subgiants hosting hot Jupiters. Each of the host stars in our sample crossed the Kraft break and gained deep surface convective envelopes during their post-main sequence evolution. Our evolved hot Jupiter systems have low sky-projected obliquities, indicating efficient realignment as stars develop convective envelopes. 

We summarize our main conclusions as follows:

    
  * We confirm two newly identified hot Jupiters orbiting subgiant stars: (R_ p = R_ J, M_ p = M_ J, P_ orb = days), and (R_ p = R_ J, M_ p = M_ J, P_ orb = days).
    
  * We measured the RM effect for three hot Jupiters orbiting subgiants that have recently developed convective envelopes. All three are aligned (|λ| < 15^∘), which is consistent with the picture that convective envelopes can efficiently damp obliquities. We place an upper limit on the timescale for tidal realignment of ∼500 Myr.
    
  * A simple model incorporating tidal realignment to explain these observations requires the effective tidal quality factor for obliquity damping to be at least four orders of magnitude lower than that for orbital decay (δlog Q'_⋆≳ 4). This is consistent with angular momentum dissipation through the driving and damping of inertial waves in the stellar convective envelope or coupling to gravity modes in the radiative interior.



















Further work is required to place tighter constraints on the timescales for obliquity damping. A larger sample of obliquity measurements for planets orbiting subgiant stars similar to those reported here would provide additional benchmarks for the dynamical evolution, and potentially distinguish between mechanisms for enhanced λ-damping efficiency. One approach to better understand the relationship between λ and T_ eff would be to measure the stellar obliquity for massive subgiants which have not yet crossed the Kraft break. By observing systems at a variety of evolutionary stages, we will better constrain the efficiency of obliquity damping to test high-eccentricity migration as the mechanism for hot Jupiter formation.



§ ACKNOWLEDGEMENTS


N.S. acknowledges support by the National Science Foundation Graduate Research Fellowship Program under Grant Numbers 1842402 & 2236415. A.C. & D.H. acknowledge support from the National Aeronautics and Space Administration (80NSSC21K0652). D.H. acknowledges support from the Alfred P. Sloan Foundation and the Australian Research Council (FT200100871). M.R. is supported by Heising-Simons grant #2023-4478. K.A.C. and C.N.W. acknowledge support from the mission via subaward s3449 from MIT. T.D. acknowledges support by the McDonnell Center for the Space Sciences at Washington University in St. Louis. J.M.A.M. is supported by the National Science Foundation (NSF) Graduate Research Fellowship Program (GRFP) under Grant No. DGE-1842400. 



This paper made use of data collected by the mission and are publicly available from the Mikulski Archive for Space Telescopes (MAST) operated by the Space Telescope Science Institute (STScI). Funding for the mission is provided by NASA's Science Mission Directorate. We acknowledge the use of public data from pipelines at the Science Office and at the Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. Funding for the mission is provided by NASA's Science Mission Directorate. 

Some of the data presented herein were obtained at Keck Observatory, which is a private 501(c)3 non-profit organization operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. 

The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.



This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF.

Some of the observations in this paper made use of the High-Resolution Imaging instrument Zorro and were obtained under Gemini LLP Proposal Number: GN/S-2021A-LP-105. Zorro was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Zorro was mounted on the Gemini South telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).



This research has made use of the Exoplanet Follow-up Observation Program (ExoFOP; DOI: 10.26134/ExoFOP5) website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

 <cit.>,  <cit.>,  <cit.>

tange2018

aasjournal





§ GROUND-BASED IMAGING FIGURES
 

Figure <ref> shows the high-contrast image of obtained with the Keck/NIRC2 near-infrared imager (for details, see <ref>). The corrected and derotated image, taken in the Kp bandpass, shows a companion with a separation of 763±20 milliarcseconds at a position angle of 110.2±0.5^∘.

Figure <ref> shows photometric time-series observations of the transits of and . These data were obtained with the ground-based LCOGT and PEST telescopes (for details, see <ref>), and they display transit depths and ephemerides consistent with our predictions. 



































§ RADIAL VELOCITY MEASUREMENTS
 

Table <ref> lists RV measurements obtained with Keck/HIRES and Keck/KPF for each of our targets. A machine-readable version of this table containing all observations is available.


l l l l l
    Target     Instrument     Time (BJD)     RV (m/s)     RV Uncertainty (m/s) 
 
    TOI-1181     Keck/HIRES     2459000.040402     -67.993     3.735 

TOI-1181     Keck/HIRES     2459003.045446     14.927     3.907 

TOI-1181     Keck/HIRES     2459004.050001     72.699     3.539 

TOI-1181     Keck/HIRES     2459026.093511     -50.794     4.016 

TOI-1181     Keck/HIRES     2459029.024667     118.585     3.540 



















































































⋮                

TOI-6029     Keck/HIRES     2459423.093788     84.436     6.597 

TOI-6029     Keck/HIRES     2459442.102513     -136.860     6.772 

TOI-6029     Keck/HIRES     2459447.07665     -26.309     7.122 

TOI-6029     Keck/HIRES     2459449.108516     -30.814     6.492 

TOI-6029     Keck/HIRES     2459457.094269     111.598     6.563 










































⋮                

TOI-4379     Keck/HIRES     2459478.753368     -38.193     8.845 

TOI-4379     Keck/HIRES     2459482.716898     104.904     5.204 

TOI-4379     Keck/HIRES     2459633.119397     63.349     4.963 

TOI-4379     Keck/HIRES     2459694.041773     79.915     5.086 

TOI-4379     Keck/HIRES     2459710.004739     24.236     4.605 

⋮                








TOI-4379     Keck/KPF     2460101.830527187     68.210     15.086 

TOI-4379     Keck/KPF     2460101.837867083     49.998     9.299 

TOI-4379     Keck/KPF     2460101.844402326     37.009     7.578 

TOI-4379     Keck/KPF     2460101.85159022     40.815     9.384 

TOI-4379     Keck/KPF     2460101.858644097     48.221     7.549 


























⋮                

Radial velocity measurements for TOI-1181, TOI-6029, and TOI-4379 used in our analysis. A full machine-readable table can be accessed online.
    






 Paper Title: Chemical abundance gradients of organic molecules within a protostellar disk.




















































Levi G. Walls
levigwal@umich.edu

0000-0001-6253-1630]Levi G. Walls
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107

0000-0002-2555-9869]Merel L.R. van 't Hoff
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107



0000-0003-4179-6394]Edwin A. Bergin
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107















§ ABSTRACT 
Observations of low-mass protostellar systems show evidence of rich complex organic chemistry. Their low luminosity, however, makes determining abundance distributions of complex organic molecules (COMs) within the water snowline challenging. However, the excitation conditions sampled by  differing molecular distributions may  produce substantive changes in the resulting emission.  Thus, molecular excitation may recover spatial information from spatially unresolved data. By analyzing spatially-unresolved NOrthern Extended Millimeter Array (NOEMA) observations of CH_3OH and CH_3CN, we aim to determine if CH_3OH and CH_3CN are distributed differently in the protostellar disk around HOPS-370, a highly-luminous intermediate mass protostar. Rotational diagram analysis of CH_3OH and CH_3CN yields rotational temperatures of 198 ± 1.2 K and 448 ± 19 K, respectively, suggesting the two molecules have different spatial distributions. Source-specific 3D LTE radiative transfer models are used to constrain the spatial distribution of CH_3OH and CH_3CN within the disk. A uniform distribution with an abundance of 4×10^-8 reproduces the CH_3OH observations. In contrast, the spatial distribution of CH_3CN needs to be either more compact (within ∼120 au versus ∼240 au for CH_3OH) or exhibiting a factor of ≳ 15 increase in abundance in the inner ∼55 au.  A possible explanation for the difference in spatial abundance distributions of CH_3OH and CH_3CN is carbon-grain sublimation.























 



§ INTRODUCTION

Complex organic molecules (COMs) are carbon-bearing molecules with 6 or more atoms <cit.>. Many COMs are formed in ice mantles on dust grains. For example, hydrogenation of CO ice can lead to the formation of simple oxygen-bearing COMs, such as methanol (CH_3OH), up to biologically relevant molecules like glycerol <cit.>. Whether a COM will remain in the ice or transitions into the gas depends on energetics and the local temperature <cit.>. The binding energies of many COMs are similar to that of water, resulting in the transition between gas and ice to occur roughly around the water snowline (∼ 150 K for pressures P ∼ 10^-12 bar).  For example, the binding energy is 5640 K for amorphous solid water <cit.>, 5728 K for methanol <cit.>, and 4954 K for methyl cyanide  <cit.>, as measured for carbonaceous grains.   

Based on this picture, the spatial distribution of COMs in circumstellar material is expected to be uniform post-sublimation within the snowline in the absence of gas phase chemical processing. However, for some species, for example, the nitrogen-bearing COM methyl cyanide, CH_3CN, (additional) gas phase formation routes have been proposed <cit.>. In addition, chemical processing in hot (≳300-500 K) gas may be possible, potentially following the destruction of carbonaceous grains <cit.> or through high-temperature pathways that become active <cit.>. 

For low-mass sources, the snowline lies within a few au from the star in disks up to a few tens of au from the star in protostellar envelopes, making resolving the snowline location technically challenging. As such, most studies of low-mass protostellar sources have focused on the detection of COMs <cit.> or the chemical inventory of individual sources <cit.>, rather than their spatial distribution (see reviews by e.g., ). 

Evidence for chemical changes in hot gas have been found toward high-mass protostars. <cit.> is one of the first studies of the high mass star forming region Orion KL to indicate the observed kinematic and spatial difference in N-COM and O-COM abundances could be due to hot gas phase chemistry. More recent studies on the Orion KL region also show abundance enhancements of N-COMs relative to O-COMs <cit.>.  
<cit.> finds the ratio of column densities in hot gas relative to warm gas for N-COMs is approximately 5 times higher than for O-COMs. While all of these studies toward high-mass systems indicate resolved chemical segregation <cit.> no systematic study of the spatial distribution of COMs within the snowline of low-mass sources has been conducted. 


The disk around the highly-luminous protostar HOPS-370 <cit.> also displays evidence of hot-gas chemistry based on observations with the NOrthern Extended Millimeter Array (NOEMA; van 't Hoff et al. accepted). Being an intermediate mass protostar <cit.> with a luminosity of ∼314 L_⊙ <cit.>, this system is warm, which pushes the water snowline out to ∼225 au. This source has been well-studied and well-modeled: ALMA observations resolved a disk structure in the dust continuum emission (of 62 au) and rotation in multiple molecular lines perpendicular to the outflow direction, suggesting the presence of a near edge-on (inclination of 74) gas disk <cit.> that subtends a radial extent of ∼ 05. As such, HOPS-370 is a good target to study the spatial distribution of COMs within the snowline.

In this paper, we use spatially unresolved NOEMA observations of  and  in the disk around HOPS-370 to probe their spatial abundance distributions within the snowline using molecular excitation and source-specific radiative transfer modeling.
In  <ref>, we show how  can be and is used as a temperature probe. 
In  <ref> and  <ref>, we present the  and  observations toward HOPS-370 and our analysis that indicates differences in the spatial distributions of these species within the NOEMA beam.  These results are discussed in  <ref> and the main conclusions are summarized in  <ref>.




§  AS A TEMPERATURE PROBE
 
Symmetric-top molecules like  are excellent probes for the gas temperature. These molecules break symmetry along the rotational axis in one dimension.  Thus their energy states are characterized by the two quantum numbers J (total angular momentum) and K (the projection of J onto the axis of symmetry).  
Radiative transitions for symmetric-tops have ΔJ = 1 and ΔK = 0, while ΔK≠ 1 transitions are forbidden. 
The selection rules for dipole transitions imply collisions dominate over radiative processes when relative populations are explored across K ladders (same J - different K), and thus probe the gas temperature <cit.>.  To further enhance this capability, the rotational emissions of symmetric top molecules for transitions with the same J but different K are found closely spaced in frequency. Thus a single observation might capture multiple energy states at once.


In general, the excitation temperature, T_exc, defines the relative population of upper level u to lower level l in the Boltzmann distribution:


    N_u/N_l = g_u/g_lexp( -E_u - E_l/k_B T_exc).


If molecular emission is in LTE then the excitation temperature is equal to the kinetic temperature (T_k).  For complex molecules, with multiple emission lines sampling a range of excitation states, it is common to use the rotation diagram method <cit.> to measure the rotational temperature, T_rot.  Here the rotational temperature is a relative measure of the excitation temperature.  Below we explore the relation of T_rot and T_k for the J=14-13 transition of CH_3CN using a series of statistical equilibrium calculations to lay the ground work for our subsequent observational analysis.  

The methodology laid out by <cit.> states

    ln( N_u/g_u) = ln( N_T/Q(T_rot)) - E_u/T_rot,

where N_u is the upper level column density of the molecule, g_u the statistical weight of that level and E_u is the upper energy level in units of K. The total column density of the molecular gas is given by N_T and the partition function is Q(T_rot). 

For optically thin transitions, ln (N_u / g_u) is related to the observed total integrated flux, W in units of K km s^-1, via

    N_u/g_u = 8π k_B/h c^3ν^2 W/A_ul g_u,

where k_B is the Boltzmann constant, G is the gravitational constant, h is Planck's constant, c is the speed of light, A_ul is the Einstein coefficient for spontaneous emission, and ν is the transition frequency.  

With Eq. <ref> presented as such, a linear regression can be fit in semi-log space to the data, ln(N_u/g_u), as a function of upper energy level, E_u. The intercept is then proportional to the total column density and the negative slope yields the rotational temperature. For a fixed abundance, lowering the rotational temperature, T_rot, steepens the slope.

We use   slab model radiative transfer simulations <cit.> of  J=14_K - 13_K, K=0-12 transitions for different kinetic (gas) temperatures, H_2 density, and total CH_3CN column densities.  The spectroscopic data and collisional rate coefficients we adopt in these simulations are provided by <cit.>. For a given set of physical conditions  calculates the line emission, from which upper level column densities and rotational temperatures are derived using the method outlined above.  Fig. <ref> displays the resulting rotational temperatures versus gas density for different kinetic (gas) temperatures, and total   column densities.   Based on this analysis, LTE (in this case T_rot = T_k) is reached near densities of n_H_2∼ 10^6 cm^-3.  Furthermore, LTE conditions are retrieved (within a factor of 2) when n_H_2 < 10^6 cm^-3 provided N_T >10^16 cm^-2 and the emission is optically thin. 

Additionally, a statistical equilibrium argument illustrates that LTE is appropriate for realistic disk temperatures and densities such as those described above (see  <ref> for our disk model). Ignoring radiative processes, a system is in LTE when densities exceed the critical density, i.e. n_H_2≫ n_cr≡ A_ul/C_ul, where C_ul is the collisional rate coefficient from the upper level to the lower level. <cit.> show updated collisional rate coefficients for E- (K = 3n ± 1 for n ≥ 0) which are lower by at most a factor of 5 as compared with the rates from <cit.>; A- (K = 3n for n ≥ 0) is even closer to agreement. We thus approximate the most-up-to-date E- collisional rate cofficients of <cit.> at 140 K (roughly corresponding to the lowest temperature in our disk model). Calculating the critical density in this most conservative case, we find values do not exceed ∼3.4×10^7 cm^-3 (occurring for K≤ 2 transitions). This is approximately an order of magnitude lower than 2.2×10^8 cm^-3 (corresponding to the lowest gas number density in our disk model). Other locations in the disk are increasingly dense and hot and, thus, would only further solidify the fact the disk is in LTE. This effectively validates the use of CH_3CN to probe the gas temperature as demonstrated previously <cit.>. 


In our observational analysis we will explore, and contrast, the rotational temperatures of CH_3OH to CH_3CN.  
CH_3OH is an asymmetric top molecule and its rotational temperature relationship to the gas kinetic temperature is more complex.  However, if its emission is in LTE then a comparison between measured rotational temperatures may be revealing.
For analysis of observations in the PErseus ALMA CHEmistry Survey (PEACHES), <cit.> performs 1D, isothermal non-LTE radiative transfer calculations of . They find that for the ( 5_1,  4) A^- - ( 4_1,  3) A^-  transition (ν_t = 0) at (243915.788 MHz) transition, the excitation temperature approached the kinetic temperature to within 10-30% for gas density estimates appropriate for this type of sources (between ∼4×10^9 and 10^12 cm^-3). A similar analysis is done by <cit.> in their study of IRAS 16293, albeit for a different frequency range, but with a similar conclusion. Furthermore, models of <cit.> show LTE is appropriate for  at densities around ∼10^9 cm^-3, albeit for a low kinetic temperature, T_k = 50 K. We target the same frequency range as the PEACHES sources and the density for the HOPS-370 disk falls within the same range (see  <ref>), suggesting LTE is a valid assumption for our modeling of the optically thin emission spectrum of .    




 

§ OBSERVATIONS
 
HOPS-370 was observed by the Northern Extended Millimeter Array (NOEMA) as a part of the Chemistry And Temperature: Searching with NOEMA for Destruction Of carbon GrainS project (CATS-N-DOGS; HOPS-370 project code: W20AF) four times from December 2020 to March 2021 (van 't Hoff et al. accepted). In this work, we use the first three observations, which were at 1 mm spanning the frequency ranges of 240.9–248.6 GHz and 256.4–264.1 GHz each with 2 MHz (Δ v = 2.3 km s^-1 spectral resolution) and spent 3.0, 1.9, and 2.6 hours, on source in configuration 11C, 11C, and 10A, respectively.  

The data are calibrated using the CLIC package of the  software[<https://www.iram.fr/IRAMFR/GILDAS>], using the most recent  version available at the time of data reduction (July 2021). For each observation the bandpass calibrator was 3C84. The phase calibration was done using J0542-0913 and J0458-020 for each observation. The flux was calibrated using LKHA101 for all observations except for the second, in which MWC349 was used. 

Imaging and cleaning was done using the  package of . Due to the line richness of the spectrum, the data was imaged without continuum subtraction and no self-calibration was performed. The resulting beam size is  3.37× 0.6 (9^∘ ). The gas disk of HOPS-370 is ∼0.5in size (∼200 au in diameter at ∼400 pc; ), and thus unresolved in these observations. The noise level has been measured in a region off source in the image cube and is 16 mK in 2 MHz channels.

After extracting spectra toward the continuum peak position, the continuum was subtracted using a statistical method as, for example, used by <cit.> for the line-rich spectrum of the protostellar binary IRAS16293. This method entails the formation of a density distribution of flux values for the spectrum of each sideband. In the case of only continuum emission, the density distribution would represent a symmetric Gaussian centered at the continuum level. In the case of continuum and line emission, the Gaussian will contain an exponential tail toward higher values. The continuum level is then determined by fitting a skewed Gaussian to the distribution, and subtracted from the spectrum. The continuum-subtracted   J = 14_K - 13_K spectrum is shown in Fig. <ref>. 

We securely detect 12 transitions of the 14_K - 13_K ladder of  shown in Fig. <ref> (see also Table <ref>). The 16 mK noise level results in a weak detection of the K=11  transition. There are multiple other weak lines present near the K=12 frequency, which makes a detection of the K=12 transition very uncertain. We therefore do not report a detection of the K=12 transition. The K=5  transition at 257403.585 MHz coincides with the (18_3, 16) A^+ - (18_2, 17) A^-  transition (ν_t = 0) at 257402.086 MHz. We thus report a detection of the K=5  transition, but do not include it (nor the  transition) in the observed rotation diagram analysis. 






§ ANALYSIS
 
In this section, we model the emission of  and  via full 3D  radiative transfer calculations. The disk around HOPS-370 has a well-characterized physical structure <cit.>. We use this structure for 3D radiative transfer modeling with the LIne Modeling Engine <cit.> which includes both gas and dust emission and their respective optical depth effects. We then convolve the predicted emission to the NOEMA beam (at the source distance) to match the spatially unresolved observations. Using the strategy outlined in Sect. <ref>, we derive the rotational temperature for each molecular species.  A focus of our efforts will be to use the observed molecular excitation to set constraints on the overall spatial distribution of  and . In  <ref> we first describe the physical model in detail, and in  <ref> and  <ref> - <ref> we present the results of our modeling efforts for  and , respectively.


  



 §.§ Physical structure and model setup
  
Following the approach of <cit.>, we adopt the gas disk structure  from their modelling results of  emission, where 
the gas surface density, Σ_g, of the disk is assumed to be a power-law with an exponential cut-off <cit.>

    Σ_g(r) = Σ_0 ( r/r_c)^-γexp[-( r/r_c)^2 - γ],

where Σ_0 = 51.25 g cm^-2 is the gas surface density normalization, r is the cylindrical radius, r_c = 110 au is the disk critical radius, and γ = 0.89 is the surface density power-law index. 

The vertical structure of the disk is assumed to be in hydrostatic equilibrium, such that:

    h_g(r) = (k_B r^3 T_g(r)/G M_* μ_m m_H)^1/2

where k_B is the Boltzmann constant, G is the gravitational constant, M_* = 2.37 M_⊙ is the mass of the central protostar, μ_m = 2.37 is the mean molecular weight <cit.>, and m_H is the mass of the hydrogen atom. A radial power-law is adopted for the temperature profile, T_g(r), 

    T_g(r) = T_0 ( r/au)^-q

where T_0 = 998.7 K is the temperature at 1 au <cit.>. The temperature power-law index q=0.35 was chosen based on results for other protostellar disks, e.g., L1527 <cit.>. 
With this temperature profile, as shown in the right panel of Fig. <ref>, the 150 K water snowline is located at 225 au. 
See  <ref> for more discussion on the effect temperature profile choice has on our modelling work. 
 

The gas (mass) density then is found by integrating Eq. <ref>:

    ρ_g(r) = 1/√(2π)Σ_g(r)/h_g(r)exp[-1/2(z/h_g(r))^2],

where Σ_g(r) is given by Eq. <ref>, h_g(r) is given by Eq. <ref>, and z is the cylindrical height above the midplane. The gas number density, n_H_2, shown in the left panel of Fig. <ref> is found by dividing the gas mass density (Eq. <ref>) by the mean molecular weight.

The dust density is assumed to be 1% of the gas density <cit.>. Furthermore, the dust size distribution is assumed to follow n(a) ∝ a^-3.5 <cit.>, where a is the radius of the dust, which is assumed to be between 0.005 and 1 μm. Dust grains are assumed to be mixtures of water ice (mass fraction: 1.19×10^-3), iron  (mass fraction: 2.53×10^-4), olivine (mass fraction: 2.51×10^-3), refractory and volatile organics (mass fraction: 4.13×10^-3), orthopyroxene (mass fraction: 7.33×10^-4), and troilite (mass fraction: 5.69×10^-4). Following the results of <cit.>, we model the mass-weighted dust opacities presented in <cit.> using  <cit.>.  






The main focus of this work is investigating the effects of varying the abundance distribution (e.g., constant abundance uniformly distributed over the entire disk) of  and . The physical structure outlined above is used to perform LTE radiative transfer calculations for different transitions of  and  with  <cit.>.  solves the radiative transfer equation on a grid of Voronoi cells constructed from the Delauney triangulation of randomly chosen points, whose acceptance can be weighted based on, for example, a provided molecular abundance distribution. The result is an image cube containing the intensity for a prescribed spectral resolution. 

For both  and  we provide as input a Keplerian velocity profile and thermal Doppler broadening parameter in addition to the gas density (e.g. Eq. <ref>) and temperature structure (e.g. Eq. <ref>) as modeled by <cit.>. Additionally we adopt in  the distance <cit.> and inclination angle <cit.> for the disk around HOPS-370. The molecular abundance and its distribution for each molecular species is also provided and is the main quantity we vary.  

The spectroscopic information for both  and  (i.e. ν, g_u, E_u, A_ul) is present in  <cit.> and provided in a file conforming to the  format <cit.>. The spectroscopic data for  is provided 
by <cit.> (see also Table <ref>). 

Since  calculates the partition function internally using the energy levels in the provided molecular data file, we calculate the partition function in the same way for the analysis of the observations using 

    Q(T) = ∑_i g_i exp(-E_i/k_B T),

where g_i and E_i is the statistical weight and energy of level, i, respectively. In calculating the total column density for each species, the partition function is evaluated at the rotational temperature. This is justified given LTE conditions have been established within the disk (see  <ref>).

Given the temperatures considered here, however, higher  vibrational states can contribute to its partition function. We determine the multiplicative vibrational correction factor at the rotational temperature (a factor ≲2) by interpolating the correction factors provided in CDMS. Importantly, the derived rotational temperature does not depend on the partition function.

We include in our models the K=0-12 lines of the J=14_K-13_K transition, each at a velocity resolution of 0.16 km s^-1. While we include the  K=0 and 1 lines in our observational and model rotation diagrams, we do not include them when performing the linear regression as these transitions are blended (see Table <ref> for more information on what transitions are used in constructing the rotation diagrams in Figs. <ref>, <ref>, and <ref>).

The spectroscopic data for  is provided by <cit.> (see also Table <ref>).
From the  entry for , which consists of  torsionally-excited ν_t ≤ 2 states up to J=40 with A- and E-symmetry, we construct a molecular data file containing 38,643 total transitions up to 3 THz. The Einstein A_ul values are calculated as detailed by <cit.>, and we calculate the partition function from the line list as detailed above. Each line is modeled with a velocity resolution of 1 km s^-1.  


Each model yields an image cube containing the intensity in Jy/pixel for each line considered per molecule. For each line, the continuum is determined as the mean intensity in the first three velocity channels devoid of line emission. This is then subtracted from the rest of the velocity channels per line per molecule. This data is then converted to brightness temperature via the Rayleigh-Jeans law <cit.>:

    T_b = 1.222×10^3I/ν^2 θ_min θ_max,

where I is the spectral intensity in mJy, ν is the line frequency in GHz, and θ_min = 06 and θ_max=337 are the beam minor and major axes, respectively. After extracting the spectrum from the data cube, these modeled fluxes are then convolved with a 1D Gaussian assuming the velocity resolution of the NOEMA observations, i.e. Δ v = 2.3 km s^-1 and integrated with respect to velocity. The models, in which abundances and spatial distributions for  and  are varied, are analyzed with rotation diagram analysis and, in this way, compared with observations. 



 §.§ 
 
Fig. <ref> shows the rotation diagram for the  observations of HOPS-370 (see also van 't Hoff et al. accepted). A fit to the data using least-squares minimization yields a rotational temperature of T_rot = 198 ± 1.2 K and a total column density of N_T = (3.42 ± 0.04)×10^16 cm^-2. Given the higher number of  transitions with E_u ≳ 400 K relative to lower energy transitions, it is the higher energy transitions which strongly influence the derived rotational temperature. 

To model the  emission, we adopt a constant abundance of X(CH_3OH) = 4×10^-8 distributed uniformly throughout the 240 au disk, based on the results by <cit.> towards HOPS-370. This model is also shown in Fig. <ref> and results in a rotational temperature of T_rot = 197 K. 
Fitting the ν_t = 0 and 1 torsional states or the A- and E-symmetries separately results in very similar results. This is in agreement with the observed value.







 §.§ : constant abundance model
 
The rotation diagram for the observed K-ladder of J=14_K-13_K   transition is shown in Fig. <ref>. A least squares minimization results in an observed  rotational temperature of T_rot = 448 ± 19 K and a total column density of N_T = (8.28 ± 0.13)×10^14 cm^-2, given the discussion in  <ref>. The temperature, which is unaffected by our treatment of the partition function, is approximately 250 K higher than observed for  and well above the estimated uncertainties. This difference likely indicates different spatial distributions for both COMs. 

To study the spatial distribution of , we first explore a constant abundance distributed uniformly throughout the 240 au disk, as done for . The  abundance is varied to find the abundance that reproduces the observed peak K=4 flux to within 1σ (16 mK). We choose the K=4 line as it has an intermediate upper-level energy (E_u = 207 K) and large signal-to-noise (S/N ∼ 70). The K= 0 - 2 lines show some degree of blending, obfuscating the true fluxes of each line and the K=3 line (and K=6, 9, ...), while having high signal-to-noise, has a different statistical weight  due to the threefold symmetry of the methyl group and nuclear spin statistics. The K=5 transition is blended with a  line. As such, the line K=4 has the highest signal-to-noise of unblended transitions, is unaffected by statistical weights, and has an upper energy level that is readily observable.

The rotation diagram for the full J=14-13 ladder using the resulting constant abundance X(CH_3CN) = 5.2×10^-10 is shown in Fig. <ref>. This model severely underpredicts high K fluxes. This results in a rotational temperature of only T_rot = 225 K, which is ∼220 K cooler than observed. To obtain a higher rotational temperature, a higher  abundance is required in the warmer gas closer to the star. However, simply increasing the  abundance, results in low-K transitions being overpredicted.
Thus, a constant abundance  distribution is unable to reproduce the NOEMA observations, in contrast to . 






 §.§ : compact abundance models
 
The simplest way to increase the calculated rotational temperature while ensuring simulated fluxes agree with observations is by limiting the distribution of  to warmer gas. In these models, we assume a constant  abundance inside a given radius, r_out, and zero outside. Beginning with a radius, r_out∼ 25 au, we first systematically increase r_out in steps of ∼ 50 au. Based on the effect increasing r_out (that is, increasing the relative amount of warm gas we are introducing) has on the T_rot and K=4 line flux, we refine the step size such that a better agreement between model and observations is found. At each step, we determine whether the model and observed rotation diagram agree. This is accomplished when the K=4 line flux agrees with observations to within 1σ (16 mK) and the rotational temperature agrees to within 1σ (19.0 K). We limit the  abundance to be below 10^-4, which corresponds to the canonical CO abundance. Table <ref> provides the results of a set of models that met one or both of the above listed criteria.


cccccccc 

0pt 
Summary of compact model results with varying outer extent of emission

 r_out [au]    T_g(r_out) [K]     X_in(CH_3CN)     Peak T_b,K=4     Fits?^a     T_rot [K]     Fits?^b    N_T×10^14 [cm^-2]  

110     193     6.4× 10^-9     1.727     √     531                 12.15
 
110     193     4.0× 10^-9     1.518                 444     √     8.56 
 
115     190     4.2× 10^-9     1.726     √     436     √     9.39 
 
120     187     5.0× 10^-9     1.985                 452     √     11.17
 
120     187     3.1× 10^-9     1.713     √     386                 8.02 
 
167     167     9.0× 10^-10     1.731     √     257                5.13 
 
The observed values are: Peak T_b,K=4 = 1.725 ± 0.016 K, T_rot = 448 ± 19, N_T = (8.28 ± 0.13)×10^14 cm^-2. 
aIf a checkmark is present, this model reproduces the observed peak K=4 flux to within 1σ (16 mK).
bIf a checkmark is present, this model reproduces the observed rotational temperature to within 1σ (19.0 K).


When r_out≲ 25 au, the emitting area is too small, regardless of abundance; even with an abundance of 10^-4, the peak K=4 flux is not reached. Systematically increasing r_out, and thereby the emitting area, the smallest r_out for which the K=4 line flux can be reproduced is ∼ 100 au. However, for these models, the rotational temperature is generally ≳ 1σ larger than observed. 
A model with r_out = 110 au and abundance of X(CH_3CN) = 4.0×10^-9 (see Fig. <ref>, panel a) is able to reproduce the observed rotational temperature, but yields a K=4 line flux that is >1σ lower than observed. Increasing the abundance slightly to X(CH_3CN) = 6.4×10^-9 reproduces the K=4 flux, but results in a too high rotational temperature, as was the case for the model with r_out = 100 au. 
The smallest r_out that matches both criteria is 115 au (Fig. <ref>, panel b). This model has an abundance of X(CH_3CN) = 4.2×10^-9.  
Thus, the inner boundary for a compact model is ∼110 au.

Moving r_out further out quickly produces results comparable to the models with r_out = 110 au. As shown in Fig. <ref> (panel c), when r_out = 120 au, a model with X(CH_3CN) = 3.1×10^-9 is able to reproduce the observed K=4 line flux, but not the rotational temperature. On the other hand, a model with a higher abundance of X(CH_3CN) = 5.0×10^-9 reproduces the rotational temperature (to within 1σ), but does not reproduce the K=4 line flux. The outer boundary for a compact model is thus ∼120 au. In summary,  can be distributed as a compact abundance inside ∼ 110 - 120 au, while the observed rotational temperature and K=4 line flux cannot be simultaneously reproduced (to within 1σ) for models outside this radial range.








 §.§ : jump abundance models
 
The compact models illustrate the sensitivity of  to warm gas or more strictly the need for its emission to have significant contributions from gas closer to the star.  However, these models are somewhat inconsistent with chemical expectations.  That is, nearly the entire disk lies within the  sublimation point.  Thus  should have an abundance that is distributed throughout the disk, albeit with a strong emission component that samples the hot gas closer to the star, but also the warm gas that lies just inside the water snowline. Another way to increase the contribution of hot gas compared to a constant or compact abundance distribution is by including a non-zero abundance outside r_out that is lower than the inner abundance. This is accomplished with a jump abundance model <cit.>. 

We systematically increase r_jump and the relative inner and outer  abundances such that again the K=4 flux and rotational temperature are within 1σ of observed values.  As before, we only consider abundances less than the canonical CO abundance; that is, X_in(CH_3CN) < 10^-4. 
For realistic lower limits on the  abundance in the outer disk, we adopt an abundance ratio X(CH_3CN)/X(CH_3OH) of 0.1% based on the weighted-average column density ratio presented by <cit.> using observations of high-mass protostars in the ALMAGAL sample and literature values toward low- and intermediate-mass protostars. This ratio is ∼10 times lower than found by, for example, <cit.> (CALYPSO survey) and <cit.> (PEACHES survey), so the lowest outer abundance we consider (X_out(CH_3CN) ∼ 10^-11) may be conservative. 



Table <ref> shows the result of varying the jump location and corresponding inner/outer abundances required to reproduce the observations. In each case, there is a  enhancement of ≳ 15 inside the jump radius relative to outside. Fig. <ref> shows rotation diagrams for a range of these models.

ccccccc 

0pt 

Summary of jump-abundance models that can reproduce the observations.

 r_jump [au]     T_g(r_jump) [K]     X_in(CH_3CN)     X_out(CH_3CN)     Peak T_b,K=4 [K]     T_rot [K]     N_T×10^14 [cm^-2]

35      288     9.60× 10^-7     4.90× 10^-10     1.721     444     8.00 

55      246     1.90× 10^-8     4.60× 10^-10     1.727     447     8.64 

90      207     6.10× 10^-9     2.70× 10^-10     1.726     445     9.66 

100     199     5.30× 10^-9     1.70× 10^-10     1.712     448     9.70 

105     196     4.95× 10^-9     1.15× 10^-10     1.715     448     9.72 

The observed values are: Peak T_b,K=4 = 1.725 ± 0.016 K, T_rot = 448 ± 19, N_T = (8.28 ± 0.13)×10^14 cm^-2. 





While not shown in Fig. <ref>, when the jump location is located inside 35 au (where the disk temperature is ≳ 290 K), levels with K≤4 are generally well-populated, while higher K-levels are not. For K ≥ 6, peak line fluxes are ∼45% lower than observed on average. This ultimately results in a derived rotational temperature about 200 K lower than observed, even with an inner  abundance in excess of the canonical CO abundance.
This is primarily due to having too small an emitting area as well as optically thick dust obscuring the emission from the inner most region.  However, a jump at 35 au with an abundances of X_in(CH_3CN) = 9.60× 10^-7  and X_out(CH_3CN) = 4.90× 10^-10 provides a good match to the  observations. 

Good agreement between model and observation is found for models with jump radii out to 105 au (Fig. <ref> and Table <ref>). As r_jump is increased we also see that while inner/outer abundances decrease (as expected and also seen in the compact abundance distributions), the inner abundance is consistently ≳15 larger than the outer abundance.  Furthermore, we see that despite retrieving the observed rotational temperature and K=4 peak flux, the total column density increases as the jump radius is moved outward. This is expected since more material is being added as the radius increases. Our most concentrated jump model, with r_jump = 35 au has a total column ∼2σ lower than observed. In contrast, our best-fit compact abundance model with r_out=115 au, results in a total column that is ≳ 11σ larger than observed. 
This is to say, the observed total column necessitates a jump abundance model with  being distributed inside ∼55 au, much more concentrated than the best-fit compact abundance model. 

When the jump location is shifted outward of 105 au (where the disk gas temperature is <196 K), the jump model stops being as effective as, for example, the compact emission model with r_out=110 au at reproducing observed values. Outside 105 au, the contribution from the warm rather than hot gas begins to dominate, and it becomes difficult to reproduce emission from the high K-levels. 

To summarize, models with 35 < r_jump < 105 au are able to simultaneously reproduce the observed rotational temperature as well as the peak K=4 flux. For models with r_jump < 35 au, the emitting area is too small, and, as such, the rotational temperature cannot be reproduced despite using an abundance of 10^-4. For models where r_jump is larger than 105 au, we cannot distinguish whether  is distributed as a jump model or compact abundance. Furthermore, when the total column density is considered, models indicate  needs to have a compact component with the jump radius being r_jump≲ 55 au.   




§ DISCUSSION
 



We observe  to have a rotational temperature of T_rot = 448 ± 19 K, which is approximately 250 K warmer than observed for . Models of their spatial distribution indicate that  must indeed be different from , with  shown to be enhanced in the warm gas relative to . Nominally this requires a  production mechanism in the hot gas or a destruction mechanism preferentially diminishing  in the warm gas. Below we explore how detailed excitation might influence our results and discuss potential mechanisms that could enhance  in the hot gas or deplete it in the warm gas.




 §.§ Effect of temperature profile choice
 
The choice of temperature structure could have a significant impact on the outcome of our modelling work <cit.>. A potentially influential choice in our model is the assumption of a vertically isothermal temperature profile. <cit.> and <cit.> illustrated that the temperature structure for an evolved disk is not simply vertically isothermal. However, for deeply embedded protostars, the addition of an envelope (and its opacity) acts to intercept and reprocess the incident starlight <cit.>. Ultimately, this results in only a weak vertical temperature gradient compared to more evolved disks. This is confirmed for HOPS-370 specifically by 2D  calculations <cit.>, as shown in Fig. <ref>; the temperature does not vary greatly as a function of polar angle (height), especially between 1 and 100 au (demarcated by the gray arrows). In other words, the assumption of a vertically isothermal structure is reasonable for HOPS-370. 


Assuming the temperature is vertically isothermal implies that a 1D temperature profile can be used to describe the entire 2D disk temperature. We adopt the power-law temperature profile from <cit.>, who model the molecular line emission and dust continuum, albeit separately. This is done because calculating the dust temperature self-consistently using the radiative transfer calculations of  is computationally expensive. This quickly becomes intractable when full dust thermal calculations are to be done for each model with a different spatial distribution for multiple molecular species. To assess the validity of this simplified temperature profile, Fig. <ref> shows the difference between the dust temperature calculated self-consistently with  and the power-law temperature profile adopted in  <ref> based on the NOEMA  observations.


A power-law fit to the 2D  data yields:

    T_g+d(r) = 1470K (r/au)^-0.47

whereas in  <ref> we find that the observed  rotation diagram can be described with the following power law as modelled by <cit.>:

    T_g(r) = 998.7K (r/au)^-0.35.

With a larger T_0 and steeper power-law index, q, the fit to the  calculated temperature profile is hotter at 1 au but colder further out in the disk (> 10 au) than the power-law we assume (i.e. Eq. <ref>). This could affect our constraint on the jump location. However, even at 240 au, where there is the largest deviation between the two temperature profiles, we see there is at most a difference of ∼35 K between models. Any difference between the temperature structures where the jumps were placed is less than a factor of 2, which is less than ∼50 K. Even though a slightly different radial temperature profile may influence the exact location of the outer radius in the compact models or the jump radius in the jump-abundance models, it will not affect the conclusion that  is differently distributed than  within the snowline of HOPS-370.


    






 §.§ Pebble drift
  
One potential explanation for the existence of an increase in  in the inner disk could be pebble drift. From the perspective of dynamics, dust of different sizes interacts differently with the gas. The small dust is dynamically well-coupled to the gas, while large dust experiences maximum inward radial drift. The presence of a jump abundance distribution could in principle be explained by the inward drift of large dust, causing an enhanced abundance in the inner parts of the disk where the ice sublimates <cit.>. 
 


However, if radial drift were occurring and giving rise to the jump abundance seen in , we would also expect to a jump abundance distribution for  unless there is some additional ice chemistry serving to keep  enhanced outside the  outer radius or jump location in the outer disk. Since the simpler constant abundance model reproduces the  fluxes observed by NOEMA, a jump model is unnecessary. Thus, radial drift is unlikely to be the reason for the difference in abundance distributions between  and .




 §.§ Destroying CH_3CN in warm gas
 
Models of their spatial distribution indicate inner  abundances are heightened by a factor of ∼ 15 relative to outer within ∼105 au, where warm gas (150 K≲ T_g ≲ 300 K) resides. This enhancement could be due to a chemical mechanism reducing the  in the warm gas. 

Two such destruction mechanisms could be gas-phase reactions between He^+ or H_3^+ for both  and . The reaction between  He^+ and  occurs on a timescale similar to that for He^+ and .  However,  the reaction between  H_3^+ and  is approximately a factor of three faster than for  <cit.>. After ice sublimation this would lead to quicker re-equilibration towards chemical equilibrium for  with a timescale that is a factor of three shorter.   This could lead to a shift in the abundance of  relative to  in the outer disk gas, but any model would have to be fine-tuned in timing to capture this change within the short period of time as timescales are < 10^5 yrs at these high densities <cit.>.
Thus, the elevated  abundance is difficult to explain by a gas-phase destruction mechanism. 






 §.§ Carbon-grain sublimation
 
We see  is characterized by hot gas, where there is a relatively concentrated  abundance enhanced relative to . This could indicate the presence of hot gas phase chemistry taking place, which could be the result of carbon grain sublimation. Carbon-grain sublimation is the process by which molecules in refractory carbon grains are released to the gas. This occurs at ≳ 300 K; the so-called “soot line" <cit.>. 


When dust grains are destroyed at the soot line, more carbon and nitrogen will be observed in the hot gas relative to oxygen, since the cometary carbon and nitrogen budget is higher for dust grains than for ices, while the reverse is true for the cometary oxygen budget <cit.>.  After the initial release of carbon- and nitrogen-bearing species in the hot gas,  could then form there via reactions between CH_3^+ and HCN, the product of which, CH_3CNH^+, rapidly equilibrates to  via electronic dissociative recombination <cit.>. The conditions in HOPS-370 could facilitate these gas phase reactions. 


<cit.> provide three potential signatures of carbon grain sublimation and top-down chemistry to occur within young disks: 


  * An increased abundance of  N-COMs e.g.,  inside the soot line,

  * N-COMs being distributed within a smaller spatial area than O-COMs e.g., , and therefore

  * N-COMs exhibiting higher excitation temperatures than O-COMs. 

Potential evidence for carbon-grain sublimation has been observed by <cit.> for high mass protostars in the ALMAGAL survey. However, we also see potential evidence in HOPS-370.


The excitation temperature for the 1 mm NOEMA observations of  (see also van 't Hoff et al. accepted) is T_rot = 448 ± 19 K, approximately 250 K warmer than observed for  (T_rot = 198 ± 1.2 K). 

While we find a compact  abundance out to 115 au is able to reproduce the observations, so do several jump abundance models. All of these jump abundance models have the jump occurring between 35 ≲ r_jump≲ 105 au, corresponding to 196 < T_g < 288 K. Models with smaller jump radii (e.g. r_jump≲ 55 au) are favored based on the total column density calculated being most similar to observations. 
While the 300 K line is expected to correspond to the soot line, our analysis is agnostic of this.  However, we clearly see an enhanced  abundance inside of the expected soot line location. Thus, criteria 1 and 3 are satisfied.


The rotation diagram analysis we perform significantly constrains the location where any hot gas chemistry might be taking place, despite the size of the NOEMA beam. Regardless, future observations with high angular resolution could spatially resolve the location of the water snowline (∼225 au or 095) and maybe even the putative soot line (∼31 au or 013 assuming the soot line occurs at 300 K for HOPS-370).   Regardless, there is clear evidence for chemical shifts in this system that hint at changes that can be resolved.




§ CONCLUSION
 
With spatially-unresolved 1 mm NOEMA observations of  and  in the protostellar disk around HOPS-370, 
we aim to determine if  and  are distributed differently. 
We use the radiative transfer modelling software  in combination with a source-specific physical model to calculate spectra for each of these molecules for different parameterized spatial distributions. Rotation diagrams are used to analyze the observed and modeled spectra. Our main findings are as follows:


  *  has an observed rotational temperature of T_rot = 198 ± 1.2 K, while  is observed to be significantly warmer with T_rot = 448 ± 19 K.

  * The  observations can be reproduced with a constant abundance model out to 240 au.

  * The  observations cannot be reproduced by a uniform distribution similar to . Instead, the  emission is either described by a more compact abundance distribution within ∼115 au or by a jump-abundance distribution with the increase in abundance happening for radii between ∼35 and ∼105 au. In the inner region, the  is enhanced by a factor of ∼15 compared to the outer region.  






These changes suggest chemical gradients are present inside the water snowline within hot gas.
HOPS-370 is a unique laboratory for characterization of chemistry in hot gas which is needed to understand the formation of inner solar/stellar system planets like our own.  

L.G.W. acknowledges support from the Rackham Merit Fellowship, under which a majority of this work was funded. M.L.R.H. acknowledges support from the Michigan Society of Fellows. 




§ MOLECULAR DATA

Tables <ref> and <ref> provide spectroscopic data for observed and modelled transitions of  and , respectively (see also  <ref>). For , we provide in Fig. <ref> the observed transitions (with quantum numbers as they appear in the  database), along with the resultant Gaussian fits calculated with .

cccccccc 

0pt 
Spectroscopic data for  obtained from 
Upper level     Lower level     Frequency     E_u     g_u     A_ul     Observed flux     Flux error 
 
(J_K_a, K_c)      (J_K_a, K_c)     [MHz]     [K]          [s^-1]     [K km s^-1]     [K km s^-1] 
 

            v_t = 0                  


(25_3, 22) A^-     (25_2, 23) A^+     241588.758     803.7     204     8.19e-05     7.4304     0.56  
 
( 5_0,  5) A^+     ( 4_0,  4) A^+     241791.352      34.8      44     6.05e-05     25.436     1.17  
 
(24_3, 21) A^-     (24_2, 22) A^+     242490.245     745.7     196     8.19e-05     9.3002     1.33  
 
(23_3, 20) A^-     (23_2, 21) A^+     243412.610     690.1     188     8.20e-05      9.499     0.193 
 
( 5_1,  4) A^-     ( 4_1,  3) A^-     243915.788      49.7      44     5.97e-05     26.813     0.714 
 
(21_3, 18) A^-     (21_2, 19) A^+     245223.019     585.7     172     8.23e-05     13.751     0.977 
 
(20_3, 17) A^-     (20_2, 18) A^+     246074.605     537.0     164     8.25e-05     17.744     0.978 
 
(19_3, 16) A^-     (19_2, 17) A^+     246873.301     490.7     156     8.27e-05     17.737     0.979 
 
( 4_2,  2) A^+     ( 5_1,  5) A^+     247228.587      60.9      36     2.12e-05     21.876     1.44  
 
(18_3, 15) A^-     (18_2, 16) A^+     247610.918     446.6     148     8.29e-05     19.583     1.39  
 
(17_3, 14) A^-     (17_2, 15) A^+     248282.424     404.8     140     8.30e-05     20.338     0.258 
 
(19_3, 17) A^+     (19_2, 18) A^-     258780.248     490.6     156     9.01e-05     13.311     0.451 
 
(20_3, 18) A^+     (20_2, 19) A^-     260381.463     536.9     164     9.14e-05     15.384     1.22  
 
(21_3, 19) A^+     (21_2, 20) A^-     262223.872     585.6     172     9.28e-05     11.785     0.879 
 
 
(14_1, 14) E_2     (13_2, 11) E_2     242446.084     248.9     116     2.29e-05     21.325     1.32  
 
(16_2, 15) E_1     (15_3, 13) E_1     247161.950     338.1     132     2.57e-05     19.189     1.45  
 
(23_1, 22) E_1     (23_0, 23) E_1     247968.119     661.4     188     4.44e-05     8.6042     0.934 
 
(24_1, 23) E_1     (24_0, 24) E_1     259581.398     717.0     196     4.91e-05     6.8224     1.38  
 
(21_4, 18) E_2     (20_5, 16) E_2     261061.320     623.6     172     3.02e-05     7.3807     1.09  
 
( 2_1,  1) E_1     ( 1_0,  1) E_1     261805.675      28.0      20     5.57e-05     19.046     0.315 
 
            v_t = 1                  


( 5_0,  5) A^+     ( 4_0,  4) A^+     241267.862     458.4      44     6.00e-05     10.786     0.621 

( 5_1,  4) A^-     ( 4_1,  3) A^-     241441.270     360.0      44     5.79e-05     13.471     0.643 

(17_2, 15) A^-     (16_1, 15) A^-     259273.686     652.7     140     5.58e-05     8.2852     0.74  

 
( 5_1,  4) E_2     ( 4_1,  3) E_2     241238.144     448.1      44     5.75e-05     11.515     0.654 
 
(12_2, 11) E_2     (13_3, 11) E_2     247840.050     545.1     100     6.28e-05     12.673     0.696 
 
The quantum numbers (QNs) for each level of the transition are listed as (J_K_a, K_c), where J is the total rotational angular momentum, K_a is the projection of J onto the A and C inertial axes, respectively. Following the parentheses in each entry is the symmetry A or E. For transitions with A-symmetry, the parity is denoted with + or -. For transitions with E-symmetry, the subscript 1 indicates K_a ≥ 0, while the subscript 2 indicates K_a < 0 



    


cccrrccc 

0pt 
Spectroscopic data for  obtained from LAMDA
Upper level     Lower level     Frequency     E_u     g_u^a     A_ul      Observed flux      Flux error  

J_K     J_K     [MHz]     [K]          [s^-1]     [K km s^-1]     [K km s^-1]

                v_t = 0             

14_12     13_12     256817.163     1118.7     29.0     3.89e-4     —        —   
 
14_11     13_11     256930.140      955.2     14.5     5.62e-4     1.5713     0.266 
 
14_10     13_10     257033.444      805.8     14.5     7.20e-4     4.6887     1.05  
 
14_9     13_9      257127.036      670.5     29.0     8.63e-4     9.5756     0.78  
 
14_8     13_8      257210.878      549.4     14.5     9.91e-4     7.317      1.2   
 
14_7     13_7      257284.935      442.4     14.5     1.11e-3     9.6442     1.2   
 
14_6     13_6      257349.180      349.7     29.0     1.20e-3     21.106     1.2   
 
14_5     13_5      257403.585      271.2     14.5     1.29e-3     —        —   
 
14_4     13_4      257448.128      207.0     14.5     1.36e-3     21.196     0.301 
 
14_3     13_3      257482.792      157.0     29.0     1.41e-3     31.866     0.269 
 
14_2     13_2      257507.562      121.3     14.5     1.45e-3     28.774     0.305 
 
14_1     13_1      257522.428       99.8     14.5     1.47e-3     30.202     0.198 
 
14_0     13_0      257527.384       92.7     14.5     1.48e-3     23.013     0.338 
 
aThe statistical weights presented in the  entry for  have the contribution from the nuclear spin factored out. However, those present in the  file (and used in our calculation of the total column density) have the nuclear spin factor included. See, for example, <cit.> and <cit.> for discussion of the statistical weights for molecules with C_3ν symmetry, like .
While not used in constructing the observed rotation diagram, we include the K=5 and 12 lines in our models and the linear regressions. In both observed and model rotation diagrams, we do not include the K=0 and 1 lines in the rotation diagram analysis as these transitions are blended. 



 
 

 
  
 
 

 
 














NOEMA






 <cit.>,
         <cit.>, 
         <cit.>,
         <cit.>,
        , <cit.>,
        astropy <cit.>, <cit.>, 
        SciPy <cit.>,
        Matplotlib <cit.>
          











aasjournal












 Paper Title: Dynamical Viability Assessment for Habitable Worlds Observatory Targets.





0000-0002-7084-0529]Stephen R. Kane
Department of Earth and Planetary Sciences, University of
  California, Riverside, CA 92521, USA
skane@ucr.edu

0000-0002-4860-7667]Zhexing Li
Department of Earth and Planetary Sciences, University of
  California, Riverside, CA 92521, USA

0000-0002-0569-1643]Margaret C. Turnbull
SETI Institute, Carl Sagan Center for the Study of Life
  in the Universe, Off-Site: 2613 Waunona Way, Madison, WI 53713, USA

0000-0001-8189-0233]Courtney D. Dressing
Department of Astronomy, 501 Campbell Hall #3411,
  University of California, Berkeley, CA 94720, USA

0000-0001-5737-1687]Caleb K. Harada
NSF Graduate Research Fellow
Department of Astronomy, 501 Campbell Hall #3411,
  University of California, Berkeley, CA 94720, USA



§ ABSTRACT 

Exoplanetary science is increasingly prioritizing efforts toward
direct imaging of planetary systems, with emphasis on those that may
enable the detection and characterization of potentially habitable
exoplanets. The recent 2020 Astronomy and Astrophysics decadal survey
recommended the development of a space-based direct imaging mission
that has subsequently been referred to as the Habitable Worlds
Observatory (HWO). A fundamental challenge in the preparatory work for
the HWO search for exo-Earths is the selection of suitable stellar
targets. Much of the prior efforts regarding the HWO targets has
occurred within the context of exoplanet surveys that have
characterized the stellar properties for the nearest stars. The
preliminary input catalog for HWO consists of 164 stars, of which 30
are known exoplanet hosts to 70 planets. Here, we provide a dynamical
analysis for these 30 systems, injecting a terrestrial planet mass
into the Habitable Zone (HZ) and determining the constraints on stable
orbit locations due to the influence of the known planets. For each
system, we calculate the percentage of the HZ that is dynamically
viable for the potential presence of a terrestrial planet, providing
an additional metric for inclusion of the stars within the HWO target
list. Our analysis shows that, for 11 of the systems, less than 50%
of the HZ is dynamically viable, primarily due to the presence of
giant planets whose orbits pass near or through the HZ. These results
demonstrate the impact that known system architectures can have on
direct imaging target selection and overall system habitability.









§ INTRODUCTION



Although thousands of exoplanets have now been discovered, a full
census and exploration of nearby planetary systems has yet to be
realized. One of the primary reasons for incompleteness regarding
nearby planetary architectures are the observational biases inherent
within the two detection techniques responsible for the vast majority
of current exoplanet discoveries: the radial velocity (RV) and transit
methods
<cit.>. Both
of these methods are heavily biased toward large, close-in planets,
with strong dependencies on survey design, data precision, observing
strategy, and the time baseline of observations
<cit.>. However,
RV surveys that have been operating for multiple decades have been
able to probe beyond the snowline, providing insight into the
prevalence of giant planets, even at relatively large separations from
the host star <cit.>. The
architectures of these systems are informative for their orbital
dynamics and evolution, that can also be used to constrain the
possible presence of additional planets within those systems
<cit.>.
Of particular interest is the potential for harboring planets within
the Habitable Zone (HZ) of the system
<cit.>.
Such HZ planets are expected to be the target of extensive follow-up
observations, especially those nearby systems whose host star
brightness and angular separation of their planets may enable direct
imaging as a means to characterize the planetary atmospheres
<cit.>.

Among the recommendations of the 2020 Astronomy and Astrophysics
decadal survey (hereafter Astro2020) was the prioritization of a
space-based mission with the capability to directly image terrestrial
planets within the
HZ [https://www.nationalacademies.org/our-work/decadal-survey-on-astronomy-and-astrophysics-2020-astro2020]. The
currently adopted name for this recommended mission is the Habitable
Worlds Observatory (HWO), and considerable efforts are being
undertaken to define the mission designs that would maximize the
science yield for terrestrial planet characterization
<cit.>. Work is also being carried out to both
define and characterize the target stars that are best suited for HWO
observations. An initial list of 164 HWO target stars has been
provided to the broader community for assessment and discussion
<cit.>, the stellar properties of which have been further
refined by <cit.> and examined in the context of yield
simulations by <cit.>. Of these 164 nearby stars, all of
which lie within 24 pcs, 30 are currently known to host exoplanets,
with a total inventory of 70 known exoplanets. These exoplanet
detections have primarily been enabled through long-term RV
observations, revealing the Keplerian orbits and architectures down to
the RV precision of the corresponding surveys. Though the
architectures for these planetary systems are likely incomplete, even
partial knowledge of the planets present enables a dynamical analysis
that can inform the likelihood that each system has a HZ that contains
dynamically viable orbital locations for terrestrial planets. Given
the need to determine a high quality list of HWO targets,
deprioritizing those systems that are dynamically unsuitable is a
crucial step in the target assessment process.

In this work, we present the results of a dynamical study for the
known exoplanet hosts within the current proposed HWO target list
provided by <cit.>, that specifically provides an
assessment for the dynamical stability within the HZ of those
systems. Section <ref> provides a description of the HWO
stellar sample, with a particular emphasis on the known exoplanetary
systems within that list. The description includes a tabulation of the
stellar and planetary properties for the known exoplanet systems, and
the methodology for the calculation of the system HZ boundaries. The
structure and results from the dynamical simulations are described in
Section <ref>, with a quantification of the available HZ that
is dynamically stable. We discuss the implications and limitations of
our work in Section <ref>, and provide concluding remarks
and suggestions for future work in Section <ref>.





§ THE HABITABLE WORLDS OBSERVATORY SAMPLE



Here, we provide the details of the potential HWO target list that
harbor known exoplanets, and discuss the calculation of the HZ for
these systems.





 §.§ Known Planetary Systems



From the 164 stars identified by <cit.>, we extracted the
30 stars that are currently known to host exoplanets
<cit.>. The full list of planets for all of the included
systems, including their relevant stellar and planetary properties,
are provided in Table <ref>. The selection of relevant
parameters means those that are needed for the calculation of the HZ
and/or the dynamical simulations, described in
Section <ref>. Note that the HH Peg (HD 206860) system was
not included in our analysis since the companion is a brown dwarf at
very wide separation from the host star, detected using the direct
imaging technique <cit.>. The sources for the
stellar/planetary parameters used in all cases are included in the
final column of Table <ref>, where the source selection is
based upon the most complete data source.

The stellar multiplicity of a planetary system can have a significant
dynamical effect on HZ planets
<cit.>. Of the 30 known exoplanet
hosts in our sample, 9 of the stars are known to have stellar or brown
dwarf companions. The HD 3651 system has a brown dwarf with a
projected separation of 476 AU away the primary
<cit.>. The HD 9826 system contains a red dwarf star with
a projected separation of 750 AU from the primary <cit.>.
HD 26965 is a triple system, whereby the B and C components are a
white dwarf and red dwarf star, respectively, and orbit each other at
a distance of ∼400 AU from the primary <cit.>. The
HD 75732 system contains a red dwarf star with a projected separation
of 1100 AU from the primary <cit.>. The HD 102365 includes
a red dwarf star with a projected separation of 211 AU from the
primary <cit.>. HD 115404A is in a binary orbit with a
red dwarf star that has projected separation of 289 AU
<cit.>. HD 147513 is in a binary orbit with a
white dwarf separated by ∼5300 AU <cit.>. The
HD 190360 system includes a red dwarf star that is separated by at
least 1266 AU from the primary <cit.> and is probably at a
much larger distance <cit.>. The HD 209100 system contains a
brown dwarf binary pair that is separated from the primary by 1500 AU
<cit.>. Given the wide separations
for all of these binary/multiple systems, and the considerable
uncertainties in their orbital parameters, the stellar or brown dwarf
companions were not included in our subsequent dynamical analysis.

There are several further important notes regarding the included
systems. The vast majority of the 70 planets are not known to transit
their host star, and so the planetary masses used are minimum masses
in most cases, the implications of which are discussed in
Section <ref>. Exceptions to this include HD 39091 c
<cit.>, HD 75732 e
<cit.>, HD 136352 b, c,
and d <cit.>, and HD 219134 b and c
<cit.>, all of which are known to transit their host
stars. There are also planets in our list for which orbital
inclination measurements have been made. For HD 39091, planet b has an
inclination of ∼50^∘, resulting in a mutual inclination of
∼40^∘ with respect to planet c <cit.>, and we
assume planet d is coplanar with planet b. For HD 115404A, planet c
has an inclination of ∼25^∘, and we assume planet b is
coplanar <cit.>. For HD 140901, planet c has an inclination
of ∼170^∘, and we assume planet b is coplanar
<cit.>. For HD 160691, astrometry places an upper limit on
the inclination of ∼60^∘ for the known planets
<cit.>. For HD 209100 b, <cit.> provides an
inclination of ∼64^∘, thus establishing the actual
planetary mass. Planet masses reported in the literature as Earth mass
units (M_⊕) have been converted to Jupiter mass units (M_J)
in Table <ref> for consistency. For cases where the
eccentricity is reported as zero, the argument of periastron is set to
90^∘, corresponding to the location of inferior
conjunction. Finally, note that the existence of the planet orbiting
HD 26965 <cit.> has been disputed from RV follow-up data
<cit.>, but we elected to err on the side of caution and
included the planet in our list.



The exoplanet population included in our sample is represented in
Figure <ref>, showing the distributions of semi-major axes and
eccentricities. The ranges of these values within the sample are
0.015–11.6 AU and 0.0–0.64 for the semi-major axes and
eccentricities, respectively. The data point sizes are logarithmically
proportional to the planet mass, with a range of 0.0055 M_J (tau
Ceti g) to 12.6 M_J (pi Men b). There are several features to
consider regarding the parameter distribution shown in
Figure <ref>. There is a clear trend toward higher mass
planets at larger separations. For example, the mean planet mass for
semi-major axes less than 1 AU is 0.19 M_J, whereas the mean planet
mass beyond 1 AU is 1.19 M_J. A significant factor in this planetary
mass trend is the observational bias inherent in the RV method,
whereby the RV amplitude decreases with increasing semi-major axis
<cit.>. A similar bias impacts the observed
eccentricity distribution, since such planets produce higher RV
amplitudes, albeit during a limited period of the orbital phase
<cit.>. Moreover, for
those long-period planets with insufficient phase coverage, the
relatively slow apastron passage may result in eccentricity
misclassifications
<cit.>. Even so, the
combination of relatively high masses and eccentricities produces
numerous planetary architectures that exhibit significant dynamical
consequences within the HZ of their systems.

lrrrlrrrrl
  10
  0pc
   Stellar and planetary parameters
    for HWO known systems.
  
    Star    
    NP^†    
    M_⋆    
    T_eff    
    Planet    
    M_p^    
    a    
    e    
    ω    
    Source 

        
        
    (M_⊙)    
    (K)    
        
    (M_J)    
    (AU)    
        
    (^∘)    
    
  
  HD 3651 (54 Piscium)     1     0.799     5221     b     0.228     0.295     0.645     243     <cit.>

HD 9826 (ups And)     3     1.29     6156     b     0.675     0.05914     0.0069     0     <cit.>

                    c     1.965     0.8265     0.266     58.2     

                    d     4.1     2.517     0.294     73.8     

HD 10647 (q1 Eridani)     1     1.11     6218     b     0.94     2.015     0.15     212     <cit.>

HD 10700 (tau Ceti)     4     0.78     5333     g     0.0055     0.133     0.06     395.34     <cit.>

                    h     0.0058     0.243     0.23     7.45     

                    e     0.0124     0.538     0.18     22.35     

                    f     0.0124     1.334     0.16     119.75     

HD 17051 (iota Hor)     1     1.34     6167     b     2.27     0.92     0.14     309     <cit.>

HD 20794 (82 Eridani)     4     0.813     5401     b     0.0089     0.127     0.27     386.17     <cit.>

                    c     0.0079     0.225     0.17     31.51     

                    d     0.0111     0.364     0.25     252.67     

                    e     0.015     0.509     0.29     268.14     

HD 22049 (eps Eri)     1     0.81     5020     b     0.651     3.5     0.044     350     <cit.>

HD 26965 (40 Eri)     1     0.78     5072     b     0.0267     0.219     0.04     148.969     <cit.>

HD 33564 (kappa Cam)     1     1.25     6250     b     9.1     1.1     0.34     205     <cit.>

HD 39091 (pi Men)     3     1.07     5998     c     0.0114     0.06805     0     90     <cit.>

                    d     0.0421     0.499     0.22     323     

                    b     12.6     3.2826     0.6396     331.03     

HD 69830     3     0.86     5385     b     0.0321     0.0785     0.1     340     <cit.>

                    c     0.0371     0.186     0.13     221     

                    d     0.057     0.63     0.07     224     

HD 75732 (55 Cancri)     5     0.905     5172     e     0.0251     0.01544     0.05     86     <cit.>

                    b     0.8036     0.1134     0     90     

                    c     0.1611     0.2373     0.03     2.4     

                    f     0.1503     0.7708     0.08     262.4     

                    d     3.12     5.957     0.13     290.9     

HD 95128 (47 UMa)     3     1.06     5872     b     2.53     2.1     0.032     334     <cit.>

                    c     0.54     3.6     0.098     295     

                    d     1.64     11.6     0.16     110     

HD 95735     2     0.3899     3712     b     0.0085     0.07879     0.063     330     <cit.>

                    c     0.0428     2.94     0.132     63     

HD 102365     1     0.85     5630     b     0.0503     0.46     0.34     105     <cit.>

HD 114613     1     1.27     5641     b     0.357     5.34     0.458     196     <cit.>

HD 115404A     2     0.83     5019     b     0.097     0.088     0.232     259     <cit.>

                    c     10.319     11.364     0.211     312     

HD 115617 (61 Vir)     3     0.942     5577     b     0.016     0.050201     0.12     105     <cit.>

                    c     0.057     0.2175     0.14     341     

                    d     0.072     0.476     0.35     314     

HD 136352 (nu2 Lupi)     3     0.87     5664     b     0.0149     0.0964     0     90     <cit.>

                    c     0.0354     0.1721     0     90     

                    d     0.0278     0.425     0     90     

HD 141004 (lambda Ser)     1     1.05     5885     b     0.0428     0.1238     0.16     19.48     <cit.>

HD 140901     2     0.99     5586     b     0.0503     0.085     0.472     108     <cit.>

                    c     6.284     7.421     0.607     298     

HD 143761 (rho CrB)     4     0.95     5817     e     0.0119     0.1061     0.126     359.4     <cit.>

                    b     1.093     0.2245     0.038     269.64     

                    c     0.0887     0.4206     0.096     9.7     

                    d     0.068     0.827     0     90     

HD 147513 (62 G. Sco)     1     1.11     5883     b     1.21     1.32     0.26     282     <cit.>

HD 160691 (mu Ara)     4     1.13     5773     d     0.033     0.0923     0.16     197     <cit.>

                    e     0.439     0.9296     0.091     193     

                    b     1.665     1.5224     0.036     39     

                    c     1.873     5.0937     0.022     84     

HD 189567     2     0.83     5726     b     0.0267     0.111     0     90     <cit.>

                    c     0.022     0.197     0.16     219     

HD 190360     2     0.99     5537     c     0.0677     0.1294     0.165     322     <cit.>

                    b     1.492     3.955     0.3274     13.9     

HD 192310     2     0.8     5166     b     0.0532     0.32     0.13     173     <cit.>

                    c     0.076     1.18     0.32     110     

HD 209100 (eps Indi A)     1     0.754     4611     b     3.25     11.55     0.26     77.83     <cit.>

HD 217987     2     0.489     3688     b     0.0132     0.068     0     90     <cit.>

                    c     0.0239     0.12     0     90     

HD 219134     6     0.794     4913     b     0.012     0.038474     0     90     <cit.>

                    c     0.011     0.064816     0     90     

                    f     0.028     0.14574     0     90     

                    d     0.067     0.23508     0     90     

                    g     0.0346     0.3753     0     90     

                    h     0.34     3.11     0.06     215     

  †Number of planets.
  Minimum mass, except for 55 Cancri e and the HD 136352 planets.






 §.§ System Habitable Zones







The extent of the HZ for each system was calculated using the stellar
parameters from the sources provided in
Table <ref>. Specifically, we adopt the methodology
provided by <cit.>, in which the
boundaries of the HZ are derived from 1D Earth-based climate models
that calculate the radiative balance for which surface liquid water is
retained. This approach divides the HZ into two primary regions: the
conservative HZ (CHZ) and the optimistic HZ (OHZ). The CHZ is bounded
by the runaway greenhouse criteria at the inner edge, and the maximum
CO_2 greenhouse criteria at the outer edge <cit.>. The
OHZ is an empirical extension to the CHZ based upon assumptions
regarding retention of surface liquid water in the Venusian and
Martian evolutionary histories
<cit.>. The
CHZ and OHZ boundaries for each system are provided in
Table <ref>.

Figure <ref> and Figure <ref> show top-down views for
8 of the planetary systems, that demonstrate the diversity of
architectures in our known exoplanet host sample. The extent of the
CHZ and OHZ are shown in light green and dark green, respectively. In
most of these cases, the planetary orbit that is present within the HZ
is a giant planet with significant perturbation potential. The cases
where eccentric orbits lie within and/or cross the HZ, such as
HD 39091 (pi Men), HD 114613, HD 147513, and HD 192310, are
particularly devastating to orbital stability in that region due to
angular momentum transfer that excites other planetary orbits into
high eccentricity regimes <cit.>. The others cases shown
exhibit planetary orbits that are more circular, and can often allow
regions of orbital stability at the edges of the HZ. These scenarios
may be tested with dynamical simulations that explore particle
injection throughout the HZ region.





§ ORBITAL DYNAMICS







 §.§ Simulation Design



We conducted an exhaustive series of dynamical simulations to test the
viability of terrestrial planets orbiting within the HZ of the target
systems. The simulations were conducted using the REBOUND N-body
integrator package <cit.> that applies the symplectic
integrator WHFast <cit.>. Our simulations explored the
effect and orbital evolution of an additional Earth-mass planet in a
circular orbit that is coplanar with the other planets within each
system, where 10 equally spaced mean anomalies were chosen for the
injected planet. For the case of HD 39091 (pi Men), where there is a
substantial mutual inclination between the planets, we injected the
additional planet in an orbit coplanar with planet b, since that is
the planet that dominates the gravitational influence on the HZ for
that system (see Section <ref> and Figure <ref>). The
semi-major axes explored spanned 20% less than the inner edge of the
OHZ to 20% larger than the outer edge of the OHZ (see
Section <ref>). This region within each system was divided into
1000 equally spaced semi-major axis starting locations for the
injected planet to ensure sufficiently dynamical sampling of the
HZ. Each of these semi-major axis and mean anomaly starting locations
were integrated for 10^7 years, resulting in 10,000 simulations
performed for each system. The time step for each integration was
selected based upon the orbital period for the innermost body within
the system divided by 20, where the innermost body is either a known
planet or the injected planet.

At each of the simulation starting locations, our simulations provided
the percentage of the total simulation duration for which the planet
was able to survive, where non-survival means that the injected planet
was captured by the gravitational well of the host star or ejected
from the system. From these results, we calculated the dynamically
viable HZ (DVHZ), which is the percentage of the OHZ locations for
which the injected planet survived the full 10^7 year simulation.





 §.§ Habitable Zone Stability Results



The methodology described in Section <ref> was applied to each of
the systems shown in Table <ref>. The time required to
complete the suite of simulations for each system varied enormously,
depending upon the number of planets in the system, their locations,
and the extent of the instability induced within the HZ. For example,
the 55 Cancri system (bottom-left system depicted in
Figure <ref>) contains 5 known planets, including one with a
0.74 day orbit, resulting in the need for high time resolution within
the dynamical simulation. By contrast, systems with a single,
eccentric giant planet with high scattering potential (such as the
HD 147513 system) concluded the simulation suite relatively quickly.





The simulation results for 8 of the systems are represented in
Figure <ref> and Figure <ref>, corresponding to
the same systems shown in Figure <ref> and
Figure <ref>, respectively. The plots provide the percentage
of the full 10^7 year simulation for which the injected terrestrial
planet was able to maintain a stable orbit for each semi-major axis
location. As for Figure <ref> and Figure <ref>, the
CHZ and OHZ are shown as light green and dark green shaded regions,
respectively. In all of the 8 cases shown, there are significant areas
of instability within the HZ resulting from the known planets in the
system. Most notable is the case of pi Men (second panel;
Figure <ref>) in which the entire region explored is
unstable due to the incursion of the highly eccentric b planet into
the HZ. The HD 147513 system (second panel; Figure <ref>) is
similarly affected, although stable regions exist just outside of the
HZ. The remaining examples shown have varying amounts of stable
locations within the HZ, including regions of stability/instability
resulting from a combination of mean motion resonance (MMR) effects
and the numerical randomness intrinsic to the simulations
<cit.>. For example, the
results for HD 10647 (first panel; Figure <ref>) exhibits
numerous stability islands induced by the single giant planet in the
system. The 3:2 MMR location at 1.53 AU creates an island of stability
whereby a planet located there has the potential to avoid
gravitational perturbing interactions with the known giant planet,
similar to the 2:3 MMR of Neptune with Pluto <cit.>. Of
particular note is the 5:2 MMR location at 1.09 AU that produces a
region of instability that coincides with the inner boundaries of the
OHZ and CHZ. Multi-planet systems can exhibit significantly
complicated stability patterns due to the multitude of MMR locations
cascading throughout the system, as shown in the case of HD 75732 (55
Cancri; third panel; Figure <ref>).

lrrrrr
  6
  0pc
   HZ boundaries and DVHZ results.
  
    Star    
    OHZ_in    
    CHZ_in    
    CHZ_out    
    OHZ_out    
    DVHZ 

        
    (AU)    
    (AU)    
    (AU)    
    (AU)    
    (%)
  
  HD 3651        0.56     0.71     1.27     1.34     92.41 

HD 9826        1.35     1.71     3.00     3.16     0.00 

HD 10647       0.87     1.10     1.92     2.02     38.65 

HD 10700       0.55     0.69     1.24     1.31     20.75 

HD 17051       0.94     1.20     2.09     2.20     55.02 

HD 20794       0.72     0.91     1.63     1.72     0.00 

HD 22049       0.45     0.57     1.03     1.09     100.00 

HD 26965       0.52     0.66     1.20     1.26     100.00 

HD 33564       1.58     2.00     3.49     3.68     61.50 

HD 39091       0.95     1.20     2.11     2.23     0.00 

HD 69830       0.60     0.75     1.35     1.42     84.48 

HD 75732       0.59     0.74     1.33     1.41     56.99 

HD 95128       0.94     1.19     2.09     2.20     27.59 

HD 95735       0.13     0.16     0.30     0.32     100.00 

HD 102365      0.57     0.72     1.28     1.35     78.21 

HD 114613      1.54     1.96     3.46     3.65     33.43 

HD 115404A     0.43     0.54     0.99     1.04     100.00 

HD 115617      0.68     0.86     1.53     1.61     44.93 

HD 136352      0.77     0.97     1.72     1.82     100.00 

HD 140901      0.69     0.87     1.54     1.62     89.06 

HD 141004      1.08     1.36     2.40     2.53     100.00 

HD 143761      1.01     1.28     2.25     2.38     88.17 

HD 147513      0.74     0.93     1.64     1.73     0.00 

HD 160691      1.00     1.26     2.23     2.35     5.39 

HD 189567      1.09     1.38     2.45     2.58     100.00 

HD 190360      0.82     1.04     1.84     1.94     85.67 

HD 192310      0.48     0.61     1.10     1.16     19.14 

HD 209100      0.38     0.48     0.87     0.92     100.00 

HD 217987      0.16     0.20     0.38     0.40     100.00 

HD 219134      0.41     0.52     0.95     1.00     99.26 

  

To quantify the dynamical simulation results on a system-by-system
basis, we calculate and provide the DVHZ in each case, as described in
Section <ref>. These calculations are shown in
Table <ref>, along with the inner and outer boundaries for
the CHZ and OHZ, described in Section <ref>. For 11 of the systems,
DVHZ is less than 50%, and there are 4 cases where the DVHZ is 0%,
including the previously discussed systems of HD 39091 (pi Men) and
HD 147513. In general, the architectures for the 11 cases where DVHZ
is less than 50% are dominated by a giant planet in an eccentric
orbit that passes near or through the HZ.

On the other hand, there are 9 cases for which the DVHZ is
100%. Cross-referencing these systems with Table <ref>
reveals that their architectures are dominated by low-mass,
short-period planets with limited gravitational influence within the
HZ. Shown in Figure <ref> is a histogram of the DVHZ values
that summarizes the findings of our dynamical simulations. This shows
that half of the studied systems have DVHZ values that are greater
than 80%, ensuring that there is substantial dynamically viable space
within the HZ of those systems based on the currently known
architectures, and therefore suitable for potential direct imaging
follow-up observations.







§ DISCUSSION



Given the number of known exoplanetary systems, including those
resulting from numerous decades of RV observations around bright
stars, it is little surprise that the HWO target list contains known
exoplanet systems. The description of these 30 systems in
Section <ref> shows that the architectures of these systems is
diverse, including low-mass multi-planet systems and high-mass
single-planets in eccentric orbits. The latter category includes giant
planets beyond the snow line, the frequency of which appears to be
relatively rare
<cit.>,
and can greatly influence the volatile delivery within the system
<cit.>. The high
eccentricity cases may be the result of planet-planet scattering
events <cit.> and can dynamically eliminate the
possibility of further planets being present in the system
<cit.>. The growing understanding of such dynamically
volatile systems, including their formation and prevalence, will
greatly inform our ability to infer the possible presence of HZ
exoplanets <cit.>.

As described in Section <ref>, the integration time for each of
our simulations was 10^7 years. The integration time was chosen to
minimize the effects of orbital element error propagation
<cit.>, and to create a feasible pathway to
explore the full parameter space for all 30 targets, each of which
required thousands of simulations. However, chaotic systems are often
the result of divergent orbital eccentricities that can require
significant time to develop. It is thus possible that, even for
locations identified as stable in our simulations, instability occurs
beyond the 10^7 year integration time
<cit.>. Although our criteria for stability is a
first-order approach that necessitates all planets survive the
duration of the dynamical simulation, this criteria has successfully
been validated against chaos indicators <cit.> and applied
to numerous exoplanetary systems
<cit.>.

As noted in Section <ref>, the preponderance of planets in our
sample are not known to transit their host stars, whereby the true
mass of the planet may be realized. Thus, the planetary masses used
are generally minimum masses. A consequence of this knowledge
limitation is that the dynamical influence of the planets on each
other within a system, and their effects on the injected planet, may
be larger than that calculated and presented here. For that reason,
the results presented in Section <ref> can be considered a
lower limit on the HZ orbital instability. Furthermore, the
observational bias in the RV and transit methods mentioned in
Section <ref> means that the architectures of the known systems
are likely incomplete, with additional planets yet to be discovered in
these systems. However, it is worth repeating here that we assumed
  coplanar orbits for the injected planets in our simulations
  (coplanar with planet b in the case of pi Men). A full exploration
  of terrestrial HZ planets whose orbits are mutually inclined from
  the known planets may yet reveal stable locations that are otherwise
  unviable.





§ CONCLUSIONS



The effective transition of the exoplanet community to prioritizing
direct imaging techniques requires that the data for the expected
stellar targets be sufficiently leveraged toward a robust target
list. Of the 164 targets provided by <cit.> for direct
imaging with HWO, only ∼20% are presently known to harbor
exoplanets. Most, if not all, of the remaining stars likely also have
planetary companions, but in most cases the current database of RV
observations may be sufficient to rule out the presence of giant
planets in or near the HZ. For those that are known exoplanet hosts,
our results show that 11 of the systems have less than 50% of the HZ
that is dynamically viable (DVHZ < 50%), including 4 cases where
there is no dynamically possible orbit (ups And, eps Eri, pi Men, and
62 G. Sco). These systems should therefore be treated with extreme
caution when considered as possible HWO targets. However, over half of
the tested systems have DVHZ values greater than 80%, making them
suitable HWO targets from the perspective of orbital
dynamics. Moreover, our overall results are not necessarily
representative of what may be expected for the broader list of HWO
targets where no exoplanets have yet been detected. Giant planets are
relatively rare among the exoplanet population, and the detection
methods utilized are biased toward detecting those kinds of planets
more quickly than lower-mass planetary counterparts. This means that
current exoplanet surveys may have already detected many of the nearby
systems that contain significant perturbing agents (giant planets) for
possible terrestrial HZ planets.

With respect to exoplanet architectures, the refinement of the HWO
target sample will greatly benefit from the continued search for
exoplanets. The era of extreme precision RVs <cit.>
provides opportunities to detect previously undiscovered exoplanets
around the nearby stars. These exoplanet detections will include
low-mass terrestrial exoplanets and high-mass planets with
low-inclinations relative to the plane of the sky. Additional planets
are also likely to be found in the known systems, further constraining
the stability profiles for the HZ region, as described in this
work. Results from the Gaia mission <cit.> will aid in the
detection of long-period giant planets and allow the true mass of
known RV giant planets to be determined. All of these precursor
observational efforts will continue to form an essential component for
the success of the Astro2020 recommendations, and maximize the yield
of HWO and the characterization of HZ terrestrial planets.





§ ACKNOWLEDGEMENTS


We acknowledge support from the NASA Astrophysics Decadal Survey
Precursor Science (ADSPS) program under grant
No. 80NSSC23K1476. C.K.H. acknowledges support from the National
Science Foundation (NSF) Graduate Research Fellowship Program (GRFP)
under Grant No. DGE 2146752. This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program. This research
has also made use of the Habitable Zone Gallery at hzgallery.org. The
results reported herein benefited from collaborations and/or
information exchange within NASA's Nexus for Exoplanet System Science
(NExSS) research coordination network sponsored by NASA's Science
Mission Directorate.



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