# The chaotic history of the retrograde multi-planet system in K2-290A driven
by distant stars

Sergio Best Instituto de Astrofísica, Pontificia Universidad Católica de
Chile, Av. Vicuña Mackenna 4860, 782-0436 Macul, Santiago, Chile  Cristobal
Petrovich Instituto de Astrofísica, Pontificia Universidad Católica de Chile,
Av. Vicuña Mackenna 4860, 782-0436 Macul, Santiago, Chile  Millennium
Institute for Astrophysics, Chile

(Received November 2021; Accepted January 2022)

###### Abstract

The equator of star K2-290A was recently found to be inclined by $124\pm 6$
degrees relative to the orbits of both its known transiting planets. The
presence of a companion star B at $\sim 100$ au suggested that the birth
protoplanetary disk could have tilted, thus providing an explanation for the
peculiar retrograde state of this multi-planet system. In this work, we show
that a primordial misalignment is not required and that the observed
retrograde state is a natural consequence of the chaotic stellar obliquity
evolution driven by a wider-orbit companion C at $\gtrsim 2000$ au long after
the disk disperses. The star C drives eccentricity and/or inclination
oscillations on the inner binary orbit, leading to widespread chaos from the
periodic resonance passages between the stellar spin and planetary secular
modes. Based on a population synthesis study, we find that the observed
stellar obliquity is reached in $\sim 40-70\%$ of the systems, making this
mechanism a robust outcome of the secular dynamics, regardless of the spin-
down history of the central star. This work highlights the unusual role that
very distant companions can have on the orbits of close-in planets and the
host star’s spin evolution, connecting four orders of magnitude in distance
scale over billions of orbits. We finally comment on the application to other
exoplanet systems, including multi-planet systems in wide binaries.

##  1 Introduction

The K2-290A star (EPIC 249624646; an F8 star of 1.19 M⊙) was found to host two
transiting planets: a mini-Neptune at a $9-$day orbit and warm Jupiter with an
orbital period of 48 days (Hjorth et al., 2019). From adaptive optics imaging
and Gaia data the authors identify two M-dwarf companions at projected
separations of $\sim 110$ au and $\sim 2500$ au. You can find all the relevant
parameters in Figure 1.

Figure 1: Schematic depiction of the K2-290 system. The planets b and c are
nearly coplanar with an orbital plane inclined by $\sim 124^{\circ}$ relative
to the equator of star A. The distant M-dwarfs B and C have projected
distances of $\sim 100$ au and $\sim 2000$ au, respectively, and we define
their mutual inclination as $i_{\rm BC}$.

Furthermore, not only is K2-290 a triple stellar system hosting planets, but
it was later found in Hjorth et al. (2021) that both of its planets have
retrograde stellar obliquities111The stellar obliquity of planet c is well
determined, while that of b is less so, but still firmly retrograde. Since $b$
and $c$ are likely nearly coplanar, a single absolute obliquity of $124\pm 6$
degrees can be determined for both planets b and c. ($124\pm 6$ degrees
obliquity with respect to their host star spin axis). This retrograde state is
unique among the current sample of compact multi-planet systems with nearly a
dozen well-aligned ones and a handful of misaligned, still prograde, ones (see
the recent discussion in Wang et al. 2021).

The extreme orbital tilt and the unusual presence of two companions stars
suggest a peculiar dynamical evolution for K2-290. Other systems with
misaligned ($\psi\sim 40^{\circ}$) multi-planet systems include Kepler-56
(Huber et al., 2013; Otor et al., 2016) and Kepler-129 (Zhang et al., 2021),
and may be explained by interactions with the detected distant Jovian-sized
planets in the systems (Boué & Fabrycky, 2014; Li et al., 2014; Gratia &
Fabrycky, 2017). A more extreme example is the polar system HD3167 (Dalal et
al., 2019), which may still be accommodated by interplanetary interactions by
a yet undetected companion (Boué & Fabrycky, 2014; Petrovich et al., 2020).

Although retrograde systems do exist, these correspond to lonely short-period
planets—hot Jupiters and warm Netpunes. These systems likely acquired their
stellar obliquities via a different path, possibly by high-eccentricity
migration (e.g., Dawson & Johnson 2018), a process that is forbidden for
multi-planet systems like K2-290 (Mustill et al., 2015).

An explanation provided by Hjorth et al. (2021) for the retrograde orbits of
K2-290b and c is that the birth protoplanetary disk tilted relative to the
host star’s equator (Batygin, 2012). Here, an inclined binary companion $B$
would tilt the disk, while the disk dispersal leads to a secular resonance
that tilts the host star spin’s axis sometimes reaching retrograde states
(Lai, 2014; Spalding & Batygin, 2014). Although this may be a reasonable
possibility and calculations show that it can reproduce the system’s obliquity
for typical parameters, there is a fair amount of uncertainty, including the
disk size and dispersal history, planet migration, star-disk coupling, and so
on, that may inhibit the occurrence of the resonance (Zanazzi & Lai, 2018).
Moreover, the questions of whether the stellar obliquity in the disk phase
becomes the observed obliquity after billions of years and the role of the
third star C remain unaddressed.

Motivated by these questions and by the striking fact that all the relevant
precession timescales in the system are comparable, we study the long-term
evolution of K2-290’s stellar obliquity. Our paper is organized as follows. In
§2 we describe our methods. In §3 we show our results, a population synthesis
and a physical description of the mechanism that drives obliquity excitation.
We discuss our results in §4 and present our conclusions in §5.

##  2 Procedure

We simulate the long-term evolution of the system using a secular set of
equations described in Appendix A. This allows us to quickly evolve the system
for up to 5.6 Gyrs (the upper limit on the age estimate for the central star).

As some parameters are not well constrained (e.g. the mass of planet b,
distance to the companion stars, initial rotational period of the central
star, etc) we explored this parameter space and see which combinations
reproduce the desired outcome.

###  2.1 Simplifying the problem

The K2-290 system is composed of fives bodies (three stars, two planets),
which would mean 10 pair-wise interaction terms. However, only a fraction
dominates the dynamics and this allows us to simplify the problem.

The most important simplification is to consider the two planets effectively
locked together (i.e. sharing their orbital planes). This coupling is
justified using equation 12 from Lai & Pu (2016):

$\epsilon_{\rm b,c}\approx\left(\frac{M_{\rm B}}{10^{3}m_{\rm
c}}\right)\left(\frac{10a_{\rm c}}{a_{\rm B}}\right)^{3}\left[\dfrac{3a_{\rm
b}/a_{\rm c}}{b^{(1)}_{3/2}(a_{\rm b}/a_{\rm c})}\right]\dfrac{(a_{\rm
c}/a_{\rm b})^{3/2}-1}{1+(L_{\rm b}/L_{\rm c})},$ (1)

which results in $\epsilon_{\rm b,c}\sim 2\times 10^{-5}$ for the values in
Figure 1, assuming Mb = 7.6 M⊕, this means that the two planets are strongly
coupled together and their mutual inclination could only reach values in the
same order when perturbed by star B. A similar conclusion is reached comparing
with stellar $J_{2}$. Also, we assume their orbits remain circular 222In
Hjorth et al. (2019) there is the possibility for a nonzero eccentricity of
planet c, although reported values are for circular orbits and to 3$\sigma$
they report an upper limit on the eccentricity of 0.241. as the precession
rate for planet c due to its inner companion is higher than the precession
induced by star B even for high values of $e_{B}\sim 0.99$ (equation 10 from
Denham et al. 2018).

Further simplifications include planets b and c interacting with star C only
indirectly via star B, justified since the torque from C is weaker than that
of B by a factor of $\left(b_{\rm B}/b_{\rm C}\right)^{3}\sim 10^{-4}$. We
also assume that the planets do not torque stars B and C.

###  2.2 Precession frequencies

From the equations of motion in Appendix A, we can evolve the stellar spin
axis and planetary angular momentum vectors as

$\displaystyle\frac{d{\bf s}}{dt}$ $\displaystyle=$
$\displaystyle\dot{\Omega}_{\rm s,bc}~{}\left({\bf s}\times\hat{{\bf j}}_{\rm
bc}\right),$ (2) $\displaystyle\frac{d\hat{\bf j}_{\rm bc}}{dt}$
$\displaystyle=$ $\displaystyle\dot{\Omega}_{\rm bc,s}~{}\left(\hat{\bf
j}_{\rm bc}\times{\bf s}\right)+\dot{\Omega}_{\rm bc,B}~{}\left(\hat{\bf
j}_{\rm bc}\times{\bf j}_{\rm B}\right),$ (3)

where $\hat{\bf{j}}_{\rm bc}$ is the unit vector of the angular momentum of
both planets and $\dot{\Omega}$ are precession frequencies. The magnitudes of
angular momenta are $L_{\rm bc}=m_{\rm b}\sqrt{\mathcal{G}M_{\rm A}a_{\rm
b}}+m_{\rm c}\sqrt{\mathcal{G}M_{\rm A}a_{\rm c}}$ and that of the spin is
$S=k_{q}M_{\rm A}R_{\rm A}^{2}\Omega_{\star}$.

The precession frequencies are approximated by

$\displaystyle\dot{\Omega}_{\rm bc,s}$ $\displaystyle=$
$\displaystyle\frac{3J_{2}(GM_{\rm A})^{1/2}R_{\rm A}^{2}}{2(m_{\rm
b}\sqrt{a_{\rm b}}+m_{\rm c}\sqrt{a_{\rm c}})}\left(\frac{m_{\rm b}}{a_{\rm
b}^{3}}+\frac{m_{\rm c}}{a_{\rm c}^{3}}\right)\cos(i_{\rm s,bc})$ (4)
$\displaystyle\simeq$ $\displaystyle
0.28\mbox{Myr}^{-1}\left[0.5+0.45\left(\frac{33m_{\rm b}}{m_{\rm
c}}\right)\right]$ $\displaystyle\times\left(\frac{J_{2}}{3\times
10^{-6}}\right)\cos(i_{\rm s,bc})$

where we have used the quadrupole moment of the central star
$J_{2}\approx\frac{1}{3}k_{2}\left(\Omega_{\star}^{2}R_{\rm A}^{3}/GM_{\rm
A}\right)$ (Ward et al., 1976) with $k_{2}$ its stellar Love number and

$\displaystyle\dot{\Omega}_{\rm s,bc}=\dot{\Omega}_{\rm
bc,s}\left(\frac{S}{L}\right)\simeq 0.45\left(\frac{6.63\mbox{
d}}{P_{\star}}\right)\dot{\Omega}_{\rm bc,s},$ (5)

where $P_{\star}$ is the period of the central star and we have assumed that
$k_{q}=0.06$.

In turn, for the planets and star B we get:

$\displaystyle\dot{\Omega}_{\rm bc,B}$ $\displaystyle=$
$\displaystyle\frac{3G^{1/2}M_{\rm B}a_{\rm c}^{3/2}}{4a^{3}_{\rm B}M_{\rm
A}^{1/2}}\frac{\cos(i_{\rm bc,B})}{(1-e_{\rm B}^{2})^{3/2}}$ (6)
$\displaystyle\simeq$ $\displaystyle
0.23\mbox{Myr}^{-1}~{}\left(\frac{100\mbox{ au}}{a_{\rm
B}}\right)^{3}\frac{\cos(i_{\rm bc,B})}{(1-e_{\rm B}^{2})^{3/2}},$

where we have assumed that the star B only couples to the outer planet c. This
is reasonable as the outer planet will precess faster due to B by a factor of
$(a_{\rm c}/a_{\rm b})^{3/2}=5.25$ compared to planet b.

Finally, the A-B binary undergoes nodal precession and/or
eccentricity/inclination oscillations due to star C in a von Zeipel-Lidov-
Kozai (ZLK) oscillations timescale (von Zeipel, 1910; Kozai, 1962; Lidov,
1962):

$\displaystyle\tau_{\rm ZKL}$ $\displaystyle=$ $\displaystyle\frac{2P_{\rm
B}}{3\pi}\left(\frac{b_{\rm C}}{a_{\rm B}}\right)^{3}\frac{(M_{\rm A}+M_{\rm
B})}{M_{\rm C}}$ $\displaystyle\simeq 1.55$ Myr
$\displaystyle\left(\frac{a_{\rm B}}{100\mbox{
au}}\right)^{3/2}\left(\frac{b_{\rm C}}{20a_{\rm B}}\right)^{3}\frac{(M_{\rm
A}+M_{\rm B})}{M_{\rm C}}.$ (7)

From these estimates we note a crucial result from this paper that all
timescales are in the same order of magnitude for K2-290. Secular
commensurabilities can occur for reasonable values of the system and when they
do not, we will show that ZKL cycles can modulate $e_{\rm B}$ so that
$\dot{\Omega}_{\rm bc,B}\propto 1/(1-e_{\rm B}^{2})^{3/2}$ periodically
333Strictly speaking, $\dot{\Omega}_{\rm bc,B}$ can cross the other
frequencies at points other than the ZKL eccentricity peaks, but these cause
the most dramatic changes to the system. crosses $\dot{\Omega}_{\rm bc,s}$
and/or $\dot{\Omega}_{\rm s,bc}$.

###  2.3 Spin down

Even though the current value of the period for star A is known to be 6.63
days, presumably the star started with a shorter period ($P_{\star,0}$) and
then, due to magnetic braking it slowed down 444Although, given its short
period, this effect was probably not significant for star A. (e.g. Anderson &
Lai 2018). To study the sensitivity of our results on the initial period of
the star we assume the Skumanich law for the spin-down (Skumanich, 1972;
Bouvier, 2013):

$\Omega_{\star}(t)=\dfrac{\Omega_{\star,0}}{\sqrt{1+\alpha_{\rm
MB}\Omega_{\star,0}^{2}t}},$ (8)

where $\Omega_{\star,0}$ is the initial angular velocity of star A and the
constant $\alpha_{\rm MB}$ was chosen so that the period at the current age of
the star (about 4 Gyr) coincides with the current period of star A.

###  2.4 Planet b mass

Hjorth et al. (2019) give a mass estimation of 7.6 M⊕ using the mass-radius
relation from Weiss & Marcy (2014). It is worth noting that in Hjorth et al.
(2019) this mass could not be observationally constrained and was only found
to be 5.8 $\pm$ 5.1 M⊕. Therefore, we shall vary mb in our population
synthesis in a wide range of 1-15 M⊕.

Using the relation from Chen & Kipping (2016) we get a mass of 9.8 M⊕ which is
close to the previously cited value and within the observational constrain. As
it will later be shown in Figure 5d, our results do not depend much on this
parameter and so we adopt the mass from Hjorth et al. (2019) as our fiducial
value.

Figure 2: Evolution of the inclination of the stellar spin axis ($i_{\rm
s}=\cos^{-1}[\hat{\bf s}\cdot\hat{\bf z}]$) and the stellar obliquity ($i_{\rm
bc,s}=\cos^{-1}[\hat{\bf s}\cdot\hat{\bf j}_{\rm bc}]$) for two examples using
the fiducial values given in Figure 1 and final $J_{2}=10^{-5.6}$. Panel a
shows the case with spin-down of the host star from $P_{0}=2$ days to its
current value of 6.63 days, while panel b shows the same case without spin-
down. In panel c, we show the ZKL cycles of the AB binary and observe that
sudden and chaotic changes in the obliquities coincide with the instances when
the eccentricity peaks are reached.

##  3 Results

We show two examples of successful simulations (i.e. achieving obliquities of
$124^{\circ}$) in Figure 2 for our fiducial parameters including the von
Zeipel-Lidov-Kozai (ZLK) oscillations (von Zeipel, 1910; Kozai, 1962; Lidov,
1962) to emphasise that this is what kicks the system out of the equatorial
plane and decouples the inner planets from their star in most cases.

Figure 3: Nodal precession frequencies $\dot{\Omega}_{\rm s,bc}$ (red lines),
$\dot{\Omega}_{\rm bc,s}$ (blue lines) and $\dot{\Omega}_{\rm bc,B}$ (green
lines and shaded region) for the example in Figure 2b comparing two cases:
mutual inclination $i_{\rm B,C}=10^{\circ}$ (no ZKL cycles; dashed lines) and
$i_{\rm B,C}=85^{\circ}$ (ZKL cycles, solid lines). Panel a: dependence on the
initial mutual inclination $i_{\rm B,C}$ with the shaded green region
displaying the whole possible range for $\dot{\Omega}_{\rm
bc,B}\propto\left[1-e_{\rm B}^{2}\right]^{3/2}$; resonances are only
encountered for $i_{\rm B,C}\gtrsim 50^{\circ}$. Panel b: time evolution
showing the resonance crossings that occur only for initial $i_{\rm
B,C}=85^{\circ}$ (solid lines). Panel c: evolution of the stellar obliquity
showing that the sudden changes coincide with resonance crossings. The case
with initial $i_{\rm B,C}=10^{\circ}$ without resonance crossing have only
small periodic changes.  Figure 4: Fraction of the time that the system spends
with an inclination of 124$\pm$ 6 within the system’s age (3.2-5.6 Gyr).
Parameters where set according to Figure 1, while varying the stellar
inclinations (vertical axes) and $J_{2}$ in a broad range. The red vertical
lines indicate the current values of $J_{2}$ for Love numbers $k_{2}$ of 0.01
and 0.02. Panel a: Case without star C. Panels b and c: case with star C with
fractions as a function of the initial values of $i_{\rm bc,B}$ and $i_{\rm
B,C}$, respectively. The white lines correspond to the limiting condition to
cross the resonances in Equation (10) and the horizontal ones correspond to
the limit in which the system is destroyed because star B approaches too much
the inner planets ($a_{B}[1-e_{B}]\approx a_{c}$).

###  3.1 Secular chaos: conditions and available phase-space

When star C is sufficiently inclined relative to AB, the eccentricity of star
B (eB) reaches high values via ZKL oscillations which modulate the precession
frequency of the inner planets as peaks in eccentricity translate into peaks
in the nodal precession rate ($\dot{\Omega}_{\rm bc,B}\propto 1/(1-e_{\rm
B}^{2})^{3/2}$). This can be seen in Figure 3 which compares a case of low
mutual inclination (iBC) to a higher one reaching higher eccentricities. This
modulation of the potential can knock over the orbit of the planets out of the
equatorial plane or decouple them from their star. We can calculate an
analytical maximum value of this precession rate by knowing the maximum
eccentricity of the cycles and using equation (6). It is worth noting that the
angular momentum of stars B and C are comparable, which is why the effect does
not peak at 90 degrees as it would for a test particle system.

Also eC plays a role here but is assumed zero to compare with the analytical
equation given in Naoz et al. (2013) (we later consider the case of e${}_{\rm
C}\neq 0$ for the population synthesis):

$\displaystyle\gamma^{2}e_{\rm B,max}^{4}+\left[3+4\gamma\cos i_{\rm
BC}+\frac{\gamma^{2}}{4}\right]e_{\rm B,max}^{2}$ (9)
$\displaystyle+\gamma\cos i_{\rm BC}-3+5\cos^{2}i_{\rm BC}=0$

Where $\gamma=L_{\rm B}/L_{\rm C}$. We can see in Figure 3b that for an
inclination of 10 degrees, $\Omega_{\rm bc,B}$ does not cross the other
precession rates, but when we move to 85 degrees and start the ZKL cycles they
do intersect which then knocks the system over.

In Figure 4 we run a simplified population synthesis fixing various parameters
($a_{B}$, $a_{C}$, $m_{b}$, $e_{\rm B}=e_{\rm C}=0$, no spin-down) to gauge
the role of the star C. We can see that when star C is added (panels b and c),
there is a wider parameter space in which we get to the desired obliquity,
which appears to be fairly isotropic with isB, compared to the case without
star C in which we need values of ibc,B close to 85 degrees. It is also worth
noting that for our estimation of the value of J2, using a Love number of 0.01
and 0.02 555As star A should be entering the sub-giant branch and so its
convective zone should be deepening, this range in $k_{2}$ covers all the way
from fully radiative to fully convective stars (Batygin & Adams, 2013)).  and
using the star parameters given in Figure 1, we cannot get the retrograde
obliquity without star C, at least when $e_{B}=0$.

In panel c of Figure 4 we observe empty regions marked by white lines where
extreme obliquities are never reached. First, the horizontal band around
$i_{\rm BC}\sim 95^{\circ}$ is empty simply because the star B achieves
eccentricities close to 0.99 and the system is destroyed as star B approaches
too much to the planets ($a_{B}(1-e_{B})\approx a_{c}$).

Second, the envelopes at large $J_{2}$ indicate the empty regions where the
precession rate of the planets due to star A’s $J_{2}$ is greater than the one
due to star B ($\dot{\Omega}_{\rm bc,s}>\dot{\Omega}_{\rm bc,B}$). Equating
the precession rates from Equations (4) and (6) at the maximum eccentricity
$e_{\rm B,max}$, we get the following condition

$\displaystyle\frac{J_{2}}{1.9\times 10^{-5}}=\frac{1}{\left[1-e_{\rm
B,max}^{2}\left(i_{\rm BC}\right)\right]^{3/2}}.$ (10)

We see this envelope explains very well the observed results from our
simulations.

###  3.2 Population Synthesis

In Figure 5 we explore a parameter space neighbouring the observed values in
Figure 1 and a range of spin-down and $J_{2}$ histories (the $J_{2}$ parameter
in Figure 5b refers to the current value for star A, assuming a period of 6.63
days) with $\sim$ 50 000 trials.

As in all previous simulations, the planetary system is initially aligned with
the stellar equator ($i_{\rm bc,s}=0$). On the other hand, the stellar orbits
are drawn following the following properties:

  * •

The inclinations are isotropically oriented $\cos i\sim\mbox{U}(-1,1)$, with
uniformly distributed longitude of periapsis and longitudes of the ascending
nodes.

  * •

As we had no prior knowledge of the current $J_{2}$ or initial period of the
central star, we assumed a log-Uniform and uniform distribution respectively.

  * •

The eccentricities of stars B and C were drawn according to the distributions
in Tokovinin & Kiyaeva (2015), considering that star C has an inner binary:

$\displaystyle f(e_{\rm B})$ $\displaystyle=$ $\displaystyle 1.2e_{\rm
B}+0.4,~{}\left<e_{\rm B}\right>=0.6,$ (11) $\displaystyle f(e_{\rm C})$
$\displaystyle=$ $\displaystyle 0.24e_{\rm C}+0.88,~{}\left<e_{\rm
C}\right>=0.52.$ (12)

  * •

The semi-major axes $a_{\rm B}$ and $a_{\rm C}$ were drawn from a log-Uniform
distribution, rejecting those that do not satisfy the stability criterion by
Mardling & Aarseth (2001):

$\frac{a_{\rm C}(1-e_{\rm C})}{a_{\rm B}}>2.8\bigg{[}\left(1+\frac{M_{\rm
C}}{M_{\rm B}+M_{\rm A}}\right)\frac{(1+e_{\rm C})}{(1-e_{\rm
C})^{1/2}}\bigg{]}^{2/5}.$ (13)

Finally, we also reject the systems whose orbit-averaged projected distance

$D_{\rm
proj}=\frac{a(1-e^{2})^{5/2}}{2\pi}\int_{0}^{2\pi}\frac{df\left[1-\sin^{2}(\omega+f)\sin^{2}I\right]^{1/2}}{\left[1+e\cos
f\right]^{3}}$ (14)

do not fall within the observed $1\sigma$ error bars of $D_{\rm proj,B}=113\pm
2$ au and $D_{\rm proj,C}=2467^{+177}_{-155}$ au (Hjorth et al., 2019).

We present these results in Figure 5 as the fraction of total simulations
achieving the observed obliquity within the estimated age of the system of
$4.0^{+1.6}_{-0.8}$Gyr and find that $56\%$ of the systems could explain
K2-290.

Looking into each panel, we see no dependence on initial period (panel a). On
panel b we see the expected trend for $J_{2}$ as for lower values, the planets
are less coupled to their star and are more easily knocked over. This trend
was already hinted by the white envelope in Figure 4c, where for larger
$J_{2}$ the inclinations needs to be larger to cross the resonances (Eq. 10).

We also observe that the mutual inclination distribution has a dip at
i${}_{\rm BC}\sim 100^{\circ}$ (panel c) due to the large eccentricities that
disrupt the planetary system, while it reaches two maxima just outside this
dip that allow for large $e_{\rm max}$ exploring a wider range of nodal
frequencies $\dot{\Omega}_{\rm bc,B}\propto(1-e_{\rm B}^{2})^{-3/2}$ to drive
chaos.

In turn, panel d shows a slightly higher success rate for lower masses of the
inner planet. As this planet represents very little of the angular momentum of
the bc system this is only due to the coupling between bc and the central star
where both planets contribute on the same order. Finally for panels e and f we
see there is a preferred distance for the outer stars. More noticeable for
star B.

Figure 5: Fraction of the systems that reach the observed angle of
$124^{\circ}$ in our population synthesis study described in §3.2. Overall,
the success rate is $56\%$, while in $17\%$ of the systems the planets are
destroyed and $27\%$ never reach $\geq 124^{\circ}$ (if we only consider the
systems which are not destroyed, the success rate goes up to 67%). From panels
a to f we show the fraction dependence on: initial period of the star, final
value of J2, mutual inclination of B and C, mass of planet b, semi-major axis
of stars B and C. The red marks in panel b correspond to a case without star C
for comparison where the success rate is only $12\%$ and the initial
distribution of $e_{\rm B}$ (not shown) is largely skewed to very large values
with a mean of $0.86$.

##  4 Discussion

In this work, we have studied the inclination history of the multi-planet
system orbiting K2-290A perturbed by two distant stars and the rotationally-
induced stellar quadrupole. We show that the stellar obliquity increases
chaotically, often reaching retrograde configurations and reproducing the
observed value of $124\pm 6^{\circ}$.

A striking property of K2-290 is that all nodal precession frequencies for the
planets and the star’s spin axis are comparable ($\sim 0.2-0.3\mbox{
Myr}^{-1}$; equations 4-6). Thus, secular resonances are often crossed,
triggering secular chaos, a process that is assisted by the ZKL oscillations
of the AB binary driven by star C.

Other results include:

  * •

The parameter space where retrograde obliquities are attained grows by
including the effect of the distant star C. Without C we cannot reproduce the
observed obliquity for our estimated current value of $J_{2}$ (Panel b of
Figure 4), unless $e_{\rm B}$ is initially very large. A population synthesis
without star C shows a success rate of only $12\%$ (compared to $56\%$ with C)
and the successful systems have $\left<e_{\rm B}\right>\simeq 0.86$ (Figure
5).

  * •

We show that a necessary condition for secular chaos is that $i_{\rm BC}$ is
sufficiently large to drive larger values of $e_{\rm B,max}$ (see equation 10
and envelope in panel c of Figure 4). As such, we expect that B and C are
highly inclined relative to each other (see panel c in Figure 5).

  * •

We see some dependence on the final value of $J_{2}$ (panel b, in Figure 5),
but no dependence on initial period of the star (panel a). In other words, the
mechanism is independent on the initial value of $J_{2}$. As for the current
value, the somewhat evolved star with radius of $1.51R_{\odot}$ would have had
a radius closer to $1.25R_{\odot}$ during its main sequence; a factor of $\sim
1.8$ lower in the value of $J_{2}$.

In what follows, we discuss how our main findings depend on the presence of
another planet and how they fit in the bigger picture of obliquity excitation.

###  4.1 Presence of other planets in K2-290

Even though only two planets (K2-290b and K2-290c) have been discovered around
K2-290A, there is still the possibility that more planets could be discovered
in the future.

Using RV data from HARPS-N, Hjorth et al. (2021) find a radial acceleration
for star A of $\dot{\gamma}=9\pm 5\mbox{ m}\mbox{ s}^{-1}\mbox{ yr}^{-1}$.
This acceleration, although compatible with zero within 2$\sigma$ and with the
presence of star B (which would cause an acceleration on the order of $5\mbox{
m}\mbox{ s}^{-1}\mbox{ yr}^{-1}$) within 1$\sigma$, could also be caused by
other planets orbiting K2-290A with a longer period.

The main effect of this undiscovered planets would be to increase the coupling
between the planetary system and star B (increase $\dot{\Omega}_{\rm bc,B}$).
Thus, in order to encounter the resonance $\dot{\Omega}_{\rm
s,bc}\sim\dot{\Omega}_{\rm bc,B}$, it would demand a larger value of $J_{2}$,
possibly earlier in the evolution of the star.

###  4.2 Link to retrograde hot Jupiters

The chaotic spin-orbit behavior observed in this work is reminiscent of the
work by Storch et al. (2014); Storch & Lai (2015) in the context of Jovians
undergoing ZKL cycles and high-eccentricity migration to hot Jupiter orbits.
In their work, the Jovian itself plays the role of both the star B undergoing
eccentricity oscillations and the planetary system that torques the star.

A qualitative difference occurs when the planet(s) do not have enough angular
momentum to torque the star ($|\dot{\Omega}_{\rm s,bc}|\ll|\dot{\Omega}_{\rm
bc,s}|$). If so, the set-up in Storch et al. (2014) with a migrating planet
will not drive chaotic obliquity evolution, but just regular oscillations due
to the ZKL cycles. Instead, in our model the periodic resonance crossings for
the planetary orbits $\dot{\Omega}_{\rm bc,s}\sim\dot{\Omega}_{\rm bc,B}$ may
occur at all planetary masses (Eq. 4 and 6). In other words, our model may
lead to either chaotic tumbling of the star as in Storch et al. (2014) and/or
the planetary orbits, both leading to chaotic obliquity evolution. We have
checked this by decreasing the masses of our planets by a factor of 10 or 100.

###  4.3 Primordial disk misalignment or tertiary-driven secular chaos?

As discussed by Hjorth et al. (2021), K2-290 is the first system that provides
strong support to the primordial disk misalignment theory (Batygin, 2012; Lai,
2014; Spalding & Batygin, 2014). The companion star B lies at a desired
distance of $\sim 100$ au; not too close to suppress planet formation (Moe &
Kratter, 2019) and not too far to require unrealistically long disk lifetimes.

Unlike the primordial tilting where a resonance crossing is mediated by the
dispersing disk, our proposal to cross the resonances relies on the wider-
orbit companion star C. This is an important difference as the star C is
currently observed and large obliquities are excited by a wide range of
orbital distances ($\sim 2000-4000$ au, Figure 5f) that are consistent with
the observations. The chaotic obliquity excitation is then a natural outcome
of the observed system, relying just on few-body gravitational dynamics and
with a success rate of $\sim 40-70\%$ depending on the assumed parameters,
mainly the value of $J_{2}$. This is a difficult value to estimate as it
depends on the internal structure of the star (Batygin & Adams, 2013), but we
still observe the effect for a wide range of values. As for the disk,
involving gas dynamics and evaporation processes, the dynamics is far more
uncertain.

Finally, we remark that the primordial tilting and our proposed model can work
in tandem. The chaotic obliquity excitation described here starts taking place
long after the disk disperses and it is fairly insensitive to the initial
stellar obliquity.

###  4.4 Future testable predictions

Figure 5c shows this mechanism is more effective when the orbits of stars B
and C are almost perpendicular to each other. Although this mutual inclination
decreases when $e_{\rm B}$ grows, the system spends more time with its large-
inclination state, as can be seen in Figure 2c. Thus, a prediction of our
results would be that $i_{\rm BC}$ should be large, peaking at $\sim
80^{\circ}$ or $\sim 110^{\circ}$.

As shown by Tokovinin (2017), the orbital planes are uncorrelated for tertiary
stars at $\sim 10^{3}$ au (Figure 1 therein), corroborated for larger values
of the outer separation (up to $10^{4}$ au) in Tokovinin (2021).

Given a Keplerian speed for star C (relative to A) of $\sim$0.7 km/s, the
relative RVs can be measured with current spectrographs. Also we estimate a
proper motion of $\sim$0.6 mas/yr for motion in the sky plane which is above
Gaia’s uncertainties.

###  4.5 Applications to other planetary systems

KOI-5 (Furlan et al., 2017) is a more compact triple star system with a
confirmed inclined inner planet (and an unconfirmed outer one). We ran a few
simulations to test if the effects previously mentioned could also explain
this system and found that with a sufficiently large mutual inclination of the
external star, we can even get retrograde states for KOI-5’s planets.

It is worth noting that there is nothing special about the measured stellar
obliquity of 124 degrees in our simulations. In fact, given this mechanism, we
could expect to find any retrograde angle up to 180 degrees in other systems
with a similar architecture.

More generally, the dynamics depend on the outer bodies only through the
amplitude of their tidal fields $\propto M/a^{3}$. Therefore, we may replace
the star B in our set-up for a Jovian-mass planet $\sim 10$ times closer to
get the same behavior. Correspondingly, making sure star C is close enough to
drive ZKL oscillations of such planet (i.e., unquenched by the inner planets).

##  5 Conclusions

In this work we show that the striking retrograde stellar obliquity of planets
K2-290b and c (124$\pm$6 degrees) is a natural outcome of the long-term spin-
orbit dynamics driven by the distant stars B and C. Here, the star C drives
eccentricity and/or inclination oscillations in the AB orbit, triggering
widespread chaos in the evolution of the planets inclinations and stellar
obliquities.

Since our model works for a wide range of initial conditions, independent of
the host star’s spin history, we suggest that the previously proposed
explanation relying on the primordial tilt of the birth protoplanetary disk
may not be required.

Finally, we remark that, for our estimate of J2 for the central star, we can
only achieve the observed obliquity if star C is present, and so, the effect
of this perturber, even though it is very distant to the planetary system,
should not be ignored.

We thank the anonymous reviewer for their feedback, which improved this
manuscript. The authors would also like to thank Julio Chanamé, Gijs Mulders,
Diego Muñoz, Smadar Naoz, Barbara Rojas Ayala and J. J. Zanazzi for their
helpful comments. C.P. acknowledges support from ANIDMillennium Science
Initiative-ICN12_009, CATA-Basal AFB-170002, ANID BASAL project FB210003,
FONDECYT Regular grant 1210425, and CONICYT+PAI (Convocatoria Nacional
subvencion a la instalacion en la Academia convocatoria 2020, PAI77200076).
S.B. acknowledges additional support from the National Agency for Research and
Development (ANID) / Scholarship Program / Doctorado Nacional grant 2021 -
21211921.

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##  Appendix A Equations of motion and definitions

To describe the orbits of each body we use the vectors of eccentricity ${\bf
e}=e{\bf\hat{e}}$ and specific angular momentum ${\bf
j}=(1-e^{2})^{1/2}{\bf\hat{\textbf{\j}}}$ with the convention of ${\bf e}$
pointing in the direction of the periapsis and ${\bf j}$ pointing in the
direction of the angular momentum. Then, given the potential in terms of these
vectors (from Tremaine & Yavetz 2014), we can get their secular evolution:

$\displaystyle\phi=-\frac{\phi_{\rm bc,A}}{2}\left[({\bf\hat{s}}\cdot{\bf
j}_{\rm bc})^{2}-\tfrac{1}{3}\right]-\frac{\phi_{\rm bc,B}}{2}\left[({\bf
j}_{\rm bc}\cdot\hat{\textbf{\j}}_{\rm
B})^{2}-\tfrac{1}{3}\right]-\frac{\phi_{\rm B,C}}{2}\left[-5({\bf e}_{\rm
B}\cdot\hat{\textbf{\j}}_{\rm C})^{2}+({\bf j}_{\rm
B}\cdot\hat{\textbf{\j}}_{\rm C})^{2}+2e_{\rm B}^{2}-\tfrac{1}{3}\right]$ (A1)

where the amplitudes are given by:

$\displaystyle\phi_{\rm bc,A}$ $\displaystyle=$
$\displaystyle\frac{3J_{2}GM_{\rm A}R_{\rm A}^{2}}{2}\left(\frac{M_{\rm
b}}{a_{\rm b}^{3}}+\frac{M_{\rm c}}{a_{\rm c}^{3}}\right),$ (A2)
$\displaystyle\phi_{\rm bc,B}$ $\displaystyle=$ $\displaystyle\frac{3GM_{\rm
B}M_{\rm c}a_{\rm c}^{2}}{4b^{3}_{\rm B}},$ (A3) $\displaystyle\phi_{\rm B,C}$
$\displaystyle=$ $\displaystyle\frac{3GM_{\rm A}M_{\rm B}M_{\rm C}a_{\rm
B}^{2}}{4(M_{\rm A}+M_{\rm B})b^{3}_{\rm C}},$ (A4)

and $b=a(1-e^{2})^{1/2}$ is the semi-minor axis.

We solve the motion using the Milankovitch set of equations (e.g., Tremaine &
Yavetz 2014) as

$\displaystyle\frac{d{\bf s}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{S}\left(\nabla_{{\bf s}}\phi\times{\bf s}\right)$ (A5)
$\displaystyle\frac{d{\bf j}_{\rm k}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{L_{\rm k}}\left(\nabla_{{\bf j}_{\rm k}}\phi\times{\bf
j}_{\rm k}+\nabla_{{\bf e}_{\rm k}}\phi\times{\bf e}_{\rm k}\right),$ (A6)
$\displaystyle\frac{d{\bf e}_{\rm k}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{L_{\rm k}}\left(\nabla_{{\bf e}_{\rm k}}\phi\times{\bf
j}_{\rm k}+\nabla_{{\bf j}_{\rm k}}\phi\times{\bf e}_{\rm k}\right),$ (A7)

where the sub-index $\mbox{k}=\left\\{{\rm bc,B,C}\right\\}$ 666Except for the
eccentricity of the planets. As the potential does not depend on this
parameter (which starts at zero), the planetary orbits remain circular, as was
required by the fact that both planets are coupled. and $L_{\rm bc}=M_{\rm
b}\sqrt{GM_{\rm A}a_{\rm b}}+M_{\rm c}\sqrt{GM_{\rm A}a_{\rm c}}$, $L_{\rm
B}=\frac{M_{\rm B}M_{\rm A}}{M_{\rm B}+M_{\rm A}}\sqrt{GM_{\rm A}a_{\rm B}}$
and $L_{\rm C}=\frac{M_{\rm C}(M_{\rm A}+M_{\rm B})}{M_{\rm C}+M_{\rm
B}+M_{\rm A}}\sqrt{G(M_{\rm A}+M_{\rm B})a_{\rm C}}$ are the orbital angular
momenta for circular orbits and $S=k_{q}M_{\rm A}R_{\rm A}^{2}\Omega_{\star}$
is the magnitude of the angular momentum spin of the central star.


# PID performance of the ALICE-TOF detector in Run 2

University of Bologna and INFN
E-mail For the ALICE Collaboration

###### Abstract:

In these proceedings we report on the status of the ALICE Time-Of-Flight (TOF)
detector. The running performance of the Run 1 (2009-2013) and Run 2
(2015-present) data taking campaigns are compared. The Particle IDentification
(PID) capabilities of the detector are presented and discussed in the light of
the improved detector calibration that allowed to reach a timing resolution of
56 ps.

##  1 Introduction

The ALICE experiment [1, 2] is one of the four main experiments recording
collisions at the CERN Large Hadron Collider. One main point of its physics
program is the study of the dense and hot matter created in ultra-relativistic
heavy-ion collisions i.e. the so called Quark-Gluon-Plasma. The ALICE
experiment is not limited to the study of heavy-ion collisions but it extends
to smaller systems such as pp and p–Pb collisions, these cover interesting
physics cases on themselves but are also be used as a useful reference for the
heavy-ion case. Identifying the particles produced in the collisions is
fundamental in many analyses. This is the reason why the ALICE detector is
capable of providing Particle IDentification (PID) over a continuous interval
of momentum [3]. The ALICE detector was designed to have good tracking
capabilities down to very low $p_{\rm T}$ ($\sim 100$ MeV/c). In order to
achieve this performance it uses a moderate magnetic field in the mid-rapidity
region (0.2 or 0.5 T) coupled with a very low material budget. The high
density of charged particles created in a single heavy-ion collision requires
high granularity detectors. The tracking is performed in the region close to
the collision vertex with a Silicon detector (Inner Tracking System or ITS)
and then continued in a Time Projection Chamber (TPC) towards the outer layers
of the detector. Both detectors provide an analogic measurement of the
particle energy loss and this information allows to differentiate between
different particle species for what concerns low momenta (below $\sim 800$
MeV/c). The Time-Of-Flight (TOF) [4, 5] is used to complement these detectors
and extend the momentum reach of the PID for charged particles.

##  2 The ALICE Time-Of-Flight detector and its PID capabilities

The TOF detector covers a large area ($\sim 141{\rm m}^{2}$) with cylindrical
geometry with internal radius $\sim 3.7$ m from the beam axis. With respect to
the interaction point the TOF system uniformly covers a pseudorapidity
($\eta$) region of 0.9 with axial symmetry, for a total coverage of 1.8 units
of $\eta$ in the full azimuthal angle. The sensitive unit of the TOF detector
is based on the Multigap Resistive Plate Chamber (MRPC) technology, built from
two stacks of five single gas gaps. The whole detector is segmented into 1593
MRPC strips, each one covering $120\times 7.4\ {\rm cm}^{2}$ and mounted
inside 87 separate modules which are grouped into 18 sectors covering the
whole $2\pi$ angle. To achieve the high granularity needed to guarantee a low
occupancy even at the high charged particle densities of Pb–Pb collisions, the
single MRPC strip is segmented into two rows of 48 pickup pads of $3.5\times
2.5\ {\rm cm}^{2}$, accounting for 96 readout pads per strip and 152928 total
readout channels. In addition to its PID capabilities, the TOF system provides
a dedicated trigger for cosmic rays. In all its years of operation the TOF
detector proved itself to be very reliable: an example can be seen in Fig.
1(a) where the trigger rate for cosmic events is shown as a function of time
and it is found to be stable, indicating no sign of detector degradation so
far. The same conclusion can be drawn from the measurement of the current
passing trough all 1593 MRPCs. In Fig. 1(c) the current without any beam
circulating in the LHC is plotted as a function of time and it is found to be
very stable with the exception of few points measured just after the
finalization of the detector installation and due to the initial MRPCs HV
conditioning. The stability of the current and its low value indicate that the
detector has low noise and no signs of ageing is observed from 2009 till 2017.
The current is also shown as a function of the instantaneous luminosity in
Fig. 1(b), the stability and the linearity of the response without any sign of
saturation make the TOF detector a good candidate for a luminometer.

(a)

(b)

(c)

Figure 1: Summary of the running conditions of TOF during Run 1 and Run 2. In
figure (1(a)) the rate of the cosmic events triggered by the TOF detector is
shown as a function of time. In figure (1(b)) and figure (1(c)) the total
current passing trough the TOF MRPCs is shown as a function of the
instantaneous luminosity (with beam) and as a function of time (without beam)
respectively.

The ALICE TOF MRPC was designed to ensure large particle detection
efficiencies ($\sim 99\%$), nonetheless this efficiency has to be coupled with
the detector geometrical acceptance and the global tracking algorithm of the
ALICE experiment, resulting in a lower effective efficiency shown in Fig. 2
for p–Pb collisions as measured from Run 1 and Run 2 data. The discrepancies
between the two periods can be understood as follows: during the Long Shutdown
1 phase ALICE finished the installation of all the modules of the TRD detector
which is located between TPC and TOF, thus adding more material for particles
to cross. These changes are the cause for the difference in the overall track
matching efficiencies, indicating again no sign of ageing of the MRPCs.

Figure 2:  Overall track matching efficiency as measured in p–Pb at
$\sqrt{s_{NN}}=5.02$ TeV collisions during Run 1 and Run 2. The difference of
the two efficiencies is due to the installation of all TRD modules.

The data collected in the first months after the start of Run 2 was used to
extract an improved calibration. Among the several technical changes used for
the calibration it is important to mention the single channel granularity (in
the past 8-channel groups were used) and the use of a finer point by point
parameterization of the correction as a function of the measured Time-Over-
Threshold (in the past limited to a $5^{th}$ order polynomial). Both standard
and improved calibration curves are shown for one channel in Fig. 3(a). The
effect of the calibration on a single channels is shown in Fig. 3(b) and
during the whole 2017 the new procedure proved to be very stable as 88% of all
TOF channels showed RMS lower than 20 ps. The
$Time\operatorname{-}Of\operatorname{-}Flight$ measurement can simply be
obtained by subtracting the event collision time ($t_{event}$) from the time
measured with the TOF detector ($t_{TOF}$), as reported in Eq. 1:

$\displaystyle\begin{split}Time\operatorname{-}Of\operatorname{-}Flight\equiv
TOF=t_{TOF}-t_{event}\end{split}$ (1)
$\displaystyle\begin{split}\sigma_{TOF}&=\sigma_{t_{TOF}}^{2}+\sigma_{tracking}^{2}+\sigma_{t_{event}}^{2}\\\
\sigma^{2}_{t_{TOF}}&=\sigma^{2}_{MRPC}+\sigma^{2}_{TDC}+\sigma^{2}_{FEE}+\sigma^{2}_{Calibration}\end{split}$
(2)
$\displaystyle\begin{split}\sigma_{t_{event}}&\sim\frac{\sigma_{t_{TOF}}}{\sqrt{n_{\rm
tracks}}}\end{split}$ (3)

The components that contribute to the uncertainty on $t_{TOF}$ and $t_{event}$
are listed in Eq. 2 and Eq. 3. The resolution on the simple time measurement
($\sigma_{t_{TOF}}$) depends mostly on fixed elements which are intrinsic of
the detector such as the MRPC resolution ($\sigma_{MRPC}$) and the one given
by electronic readout chain ( $\sigma_{TDC}$ and $\sigma_{FEE}$). Thanks to
the improvements described above the contribution due to the calibration
($\sigma_{Calibration}$) was significantly reduced and allowed to achieve
resolution of $\sim 56ps$, as reported in Fig. 4(a). The overall effect on the
separation power is depicted in Fig. 4(b).

TOF Ch. # (a)

TOF Ch. # (b)

Figure 3:  Summary of the status for the improved calibration procedure. In
figure (3(a)) the old (dashed line) and new (solid line) parameterization for
the time slewing correction are drawn. In figure (3(b)) the time signal
distribution for a particular channel is shown before and after the time
slewing correction is applied, it is clear that the correction improves the
signal significantly.

(a)

(b)

Figure 4:  figure (4(a)) Signal distribution with resolution as measured by
the full ALICE TOF detector during Run 2. figure (4(b)) the impact of the
improved resolution on the PID is depicted together with lines at 2 and 3
sigma.

The determination of the event collision time needed to compute the
$Time\operatorname{-}Of\operatorname{-}Flight$ can be performed autonomously
by the TOF detector itself with the iterative procedure described in [6].
Thanks to this method the measurement of the
$Time\operatorname{-}Of\operatorname{-}Flight$ can benefit further from the
improved precision on $t_{TOF}$ as reported in Eq. 3. The $\sigma_{t_{event}}$
depends on the number of tracks available for the $t_{event}$ evaluation
($n_{\rm tracks}$). The improvement between Run 1 and Run 2 of the event
collision time precision is shown in Fig. 5(a). With the improved calibration
the resolution is found to be better than 5 ps for events with more than $\sim
600$ tracks matched to TOF (Fig. 5(b)). All the developments and procedures
described above are summarized in Fig. 6 where the $\beta$ of each particle is
shown as a function of its momentum as measured in Pb–Pb collision at
$\sqrt{s_{NN}}=$ 5.02 TeV. The momentum slices of the $\beta$ distribution are
shown in Fig. 7, where it is possible to appreciate the separation of the
signals of each particle species. Starting at 400 MeV/c in Fig. 7(a) it is
possible to identify the signals of pions, muons and electrons. At larger
momenta (Fig. 7(b)) the peaks of protons and kaons are visible, while light
nuclei can also be identified starting from $\sim 3$ GeV/c (Fig. 7(c)). As
expected from Fig. 4(b) the proton and kaon peaks are still separated at 5
GeV/c (Fig. 7(d)).

(a)

(b)

Figure 5:  Collision event collision time resolution as measured with the TOF
detector in both Run 1 and Run 2 as a function of the number of reconstructed
tracks with a time measurement. Two ranges are proposed: in figure (5(a)) the
event collision time resolution is shown up to multiplicities of 70 and
compared across multiple collision systems, in figure (5(b)) the multiplicity
range is enlarged so as to fit the values reached in heavy-ion collisions.
Figure 6: Particle velocity ($\beta$) distribution as a function of momentum
for tracks reconstructed with the ALICE experiment in Pb–Pb collisions at
$\sqrt{s_{NN}}=$ 5.02 TeV.

##  3 Conclusions

The working performance of the ALICE Time-Of-Flight detector was discussed and
it was shown that the detector does not appear to suffer from ageing in any
way. This information is crucial as the TOF detector will continue the data
taking also after Run 2 with limited modifications to the readout chain.
During Run 2 an improvement in the TOF calibration made it possible to reach
timing resolutions of $\sim 56$ ps, thus improving significantly the Particle
IDentification capabilities and extending its reach. In the light of these new
developments the PID performance now allows an improved separation of
electrons, muons, pions, kaons, protons and light nuclei over a wide range of
momenta.

(a)

(b)

(c)

(d)

Figure 7: Distribution of the particle velocity ($\beta$) as measured with the
TOF detector in different momentum intervals. The signals from each particle
species are clearly visible up to high momenta.

## References

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