We propose that certain white dwarf WD planets such as WD 1856534 b may form out of material from a stellar companion that tidally disrupts from common envelope CE evolution with the WD progenitor star. The disrupted companion shreds into an accretion disc out of which a gas giant protoplanet forms due to gravitational instability. To explore this scenario we make use of detailed stellar evolution models consistent with WD 1856534. The minimum mass companion that produces a gravitationallyunstable disk after tidal disruption is sim0.15mathrmModot. Planet formation from tidal disruption is a new channel for producing secondgeneration planets around WDs.
['Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics']
Secondgeneration planet formation after tidal disruption from common envelope evolution
{'No Header': '<!-- image -->', 'Second-generation planet formation after tidal disruption from common envelope evolution': 'Luke Chamandy 1 , 2 , Jason Nordhaus 3 , 4 , Eric G. Blackman 2 and Emily Wilson 5  \n- 1 National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar 752050, Odisha, India\n- 2 Department of Physics and Astronomy, University of Rochester, Rochester NY 14627, USA\n- 3 Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY 14623, USA\n- 4 National Technical Institute for the Deaf, Rochester Institute of Technology, Rochester, NY 14623, USA\n- 5 Department of Astronomy and Physics, Lycoming College, Williamsport, PA 17701, USA', 'Abstract': 'We propose that certain white dwarf (WD) planets, such as WD 1856+534 b, may form out of material from a stellar companion that tidally disrupts from common envelope evolution with the WD progenitor star. The disrupted companion shreds into an accretion disc, out of which a gas giant protoplanet forms due to gravitational instability. To explore this scenario, we make use of detailed stellar evolution models consistent with WD 1856+534. The minimum mass companion that produces a gravitationally-unstable disc after tidal distruption is ∼ 0 . 15 M glyph[circledot] . Planet formation from tidal disruption is a new channel for producing second-generation planets around WDs.  \nKeywords: planetary systems: formation - white dwarfs - stars: AGB and post-AGB - binaries: close - accretion, accretion disks  \n(Received xx xx xxxx; revised xx xx xxxx; accepted xx xx xxxx)', '1. Introduction': "Planets in short-period orbits around white dwarfs (WD) are predicted to be rare, as many are expected to be destroyed during post-main-sequence evolution or migrate to longer-period orbits (Nordhaus et al. 2010; Nordhaus and Spiegel 2013). However, the presence and characteristics of a planet in a short-period orbit around a WD can provide constraints on formation scenarios. First generation planets may migrate from large-period orbits via KozaiLidov cycles in hierarchical triple systems or perhaps if they can survive a common envelope (CE) event. Second generation planets have been suggested to form in the expanding ejecta of the CE (e.g. Kashi and Soker 2011; Schleicher and Dreizler 2014; Ledda et al. 2023). However, formation of second generation planets around isolated WD debris discs is unlikely, as the surface densities are low.  \nObservational searches via various methods, such as eclipses, have revealed a few interesting WDs such as WD 1856+534, estimated to have mass M wd = 0 . 518 ± 0 . 055 M glyph[circledot] (Vanderburg et al. 2020), 0 . 606 ± 0 . 039 (Alonso et al. 2021) or 0 . 576 ± 0 . 040 M glyph[circledot] (Xu et al. 2021), and a detected planet, WD 1856+534 b, on a 1 . 41 d orbit with mass 1 MJ < ∼ m 2 < ∼ 12 MJ (Vanderburg et al. 2020; Xu et al. 2021). Throughout this study, we scale our results to parameter values for this system.  \nWD 1856+534 b is difficult to explain using a single CE scenario (Vanderburg et al. 2020; Lagos et al. 2021; Chamandy et al. 2021, hereafter C21; O'Connor et al. 2023). In C21, we proposed that WD 1856+534 b was instead dragged in from a wider orbit  \nAuthor for correspondence: E.G. Blackman, Email: blackman@pas.rochester.edu Cite this article: L. Chamandy, J. Nordhaus, E.G. Blackman & E. Wilson, Secondgeneration planet formation from CE evolution. Publications of the Astronomical Society of Australia 00 , 1-12. https://doi.org/10.1017/pasa.xxxx.xx  \n- c © Astronomical Society of Australia ; published by Cambridge University Press  \nby a CE event involving a companion that tidally disrupts inside the envelope. In that scenario, a second CE event - the one with the observed planet - unbinds the remainder of the envelope. A somewhat different idea is that the planet was dragged in when the WD progenitor underwent a helium flash (Merlov et al. 2021). This work proposes a different scenario to explain such planets. As in C21, a CE event takes place that leads to the tidal disruption of the companion. But in our new scenario, the observed planet forms in an accretion disc that results from the tidal disruption event. Notably, a somewhat similar scenario has been proposed to explain planets orbiting millisecond pulsars. In that scenario, a WD companion is disrupted, resulting in a disc around the neutron star out of which planets form. Accretion onto the neutron star causes it to spin up to millisecond periods (e.g. Stevens et al. 1992; van den Heuvel 1992; Margalit and Metzger 2017).  \nOther alternatives for explaining WD 1856+534 are scenarios where the planet migrated from further out due to the ZeipelLidov-Kozai effect, driven either by other planets (Maldonado et al. 2021; O'Connor et al. 2022; Maldonado et al. 2022) or the distant binary M-dwarf companion system (Mu˜noz and Petrovich 2020; O'Connor et al. 2021; Stephan et al. 2021). While these scenarios are plausible, searches for other planets in this sytem have so far come up empty (Kubiak et al. 2023).  \nOur proposed scenario begins with a CE event involving a ∼ 1 . 5-2 . 5 M glyph[circledot] red giant branch (RGB) or asymptotic giant branch (AGB) star and a ∼ 0 . 15-1 M glyph[circledot] main sequence (MS) star. In such cases, the companion tidally disrupts to form an accretion disc that orbits the core of the WD progenitor. The formation and early evolution of such discs has been studied with hydrodynamic simulations, albeit with planetary or brown dwarf rather than stellar companions (Guidarelli et al. 2019, 2022). Sufficient orbital energy can be liberated before, during, and after tidal disruption to eject the remainder of the CE, leaving a system consisting of a  \nFigure 1. Relationship between primary radius and core mass, for different values of M ZAMS . Constraints on M ZAMS for the system WD 1856+534 are highlighted in the plot. The lower limit of ≈ 1 . 4 M glyph[circledot] comes from the age upper limit of ∼ 10 Gyr (Vanderburg et al. 2020 ). Larger states most likely to undergo CE are boxed (RGB, AGB, TPAGB). For curves enclosed by the magenta (middle) rectangle, only those with M ZAMS > ∼ 2 . 0 M glyph[circledot] are likely because below this the maximum radius on the RGB is greater.  \n<!-- image -->", '0.15': 'Table 1. List of stellar models with parameters, obtained by running singlestar simulations with the 1D stellar evolution code MESA. C18 refers to Chamandy et al. ( 2018 ). Quantities are the stellar mass M , mass of its zero-age MS progenitor M ZAMS , mass of its core M c , radius R , the value of the dimensionless parameter l appearing in equation (13) for the envelope binding energy E bind , which is listed in the final column, chosen so that the binding energy includes the gravitational potential energy and thermal energy. Models were evolved using MESA release 10108 with solar metallicity (z=0.02) and with mass-loss parameters on the RGB and AGB of h R = 0 . 7 and h B = 0 . 15 , respectively, so as to match the initial-final mass relation of Cummings et al. ( 2018 ), with the exception of the RGB(C18) model, which used MESA release 8845 with h R = 1 .  \nproto-WD core and an accretion disc of perhaps a few × 0 . 1 M glyph[circledot] in mass.  \nAs the disc viscously spreads, it transitions from an advective, radiation pressure-dominated state with h ∼ r (e.g. Nordhaus et al. 2011) to a radiative cooling-dominated, gas pressure-dominated state with h glyph[lessmuch] r (e.g. Shen and Matzner 2014). The disc then becomes gravitationally unstable near its outer radius and selfgravitating clumps of mass greater than the Jeans mass (of order 1 MJ) collapse to form puffed-up protoplanets that may be massive enough to clear a gap and avoid rapid inward (type I) migration down to their own tidal disruption separations (e.g. Boss 1997, 1998; Zhu et al. 2012; Schleicher and Dreizler 2014; Lichtenberg and Schleicher 2015).', '2.1. Progenitor': "To obtain realistic examples of RGB and AGB progenitors, we employ Modules for Experiments in Stellar Astrophysics (MESA) (Paxton et al. 2011, 2013, 2015, 2019). Table 1 summarizes the progenitor models we consider. We note that these models are consistent with the observed initial-final mass relationships derived from cluster observations (Cummings et al. 2018; Hollands et al. 2023) and have been extensively used for CE studies (Wilson and Nordhaus 2019, 2020, 2022; Kastner and Wilson 2021). Figure 1 shows the stellar radius as a function of core mass for stellar models of various zero-age main sequence (ZAMS) masses. The core mass increases with time, so stars evolve from left to right. Observational mass estimates for WD 1856+534 are plotted as vertical lines for reference. That the observed WD mass coincides with the AGB phase suggests that the progenitor was on the AGB at the time of the CE event. However, as discussed below, the proto-WD core mass can increase by accreting the disrupted companion, so an RGB progenitor cannot immediately be ruled out.  \nGiven that the age of the WD 1856+534 system is < ∼ 10 Gyr and the cooling age of the WD is 5 . 85 ± 0 . 5 Gyr (Vanderburg et al. 2020), the WD progenitor spent < ∼ 4 . 6 Gyr on the MS, which implies a mass of M ZAMS > ∼ 1 . 4 M glyph[circledot] . a Furthermore, an AGB progenitor would likely have had M ZAMS > ∼ 2 . 0 M glyph[circledot] ; otherwise its RGB radius would have been larger and it would have entered CE then. Given the steepness of the initial mass function above 1 M glyph[circledot] (Salpeter 1955; Hennebelle and Grudi'c 2024), we choose examples with M ZAMS ≤ 2 . 2 M glyph[circledot] (Table 1), but more massive progenitors are possible.", '2.2. Disc formation and survival': "A fraction of the disrupted companion forms an accretion disc around the AGB or RGB core, inside the remaining common envelope (Reyes-Ruiz and L'opez 1999; Blackman et al. 2001; Nordhaus and Blackman 2006; Nordhaus et al. 2011; Nordhaus and Spiegel 2013; Guidarelli et al. 2019, 2022). The hydrodynamic simulations performed by Guidarelli et al. (2022) involved 1030 MJ companions undergoing tidal disruption inside the envelope of an AGB star. About 60% of the mass of the disrupted companion formed a disc and the other 40% constituted an outwardly moving but gravitationally bound tidal tail that would eventually fall back. This is reminiscent of work by Shen and Matzner (2014) on tidal disruption event discs around supermassive black holes. These authors find that a large fraction of the material from the disruption falls back and collides with itself, settling at a radius somewhat larger than the disruption radius. Guidarelli et al. (2022) find that the disc becomes quasi-Keplerian within a few dynamical times, with an aspect ratio 0 . 05 < ∼ h / r < ∼ 0 . 2. Metzger et al. (2021) models the tidal disruption of a star in a cataclysmic variable system and the disc that subsequently forms around the WD; the model explored in this work is in some ways similar to their model.  \nGuidarelli et al. (2019) simulated a disc surrounded by a hot AGB envelope, and estimated that the disc is likely to be stable for at least 10 times the duration of the simulation, or 100 orbits at the outer radius of the disc. For a disc of outer radius equal  \nto the minimum disruption separation ∼ 0 . 3 R glyph[circledot] (Figure 2a and Section 2.5), this corresponds to ∼ 3 d, whereas taking the disc outer radius to be the present orbital separation of WD 1856+534b of 4 . 4 R glyph[circledot] gives ∼ 0 . 4 yr. The actual disc survival time might be much longer, especially considering that the envelope would have expanded and spun up due to orbital energy and angular momentum transfer from the companion, reducing the destabilizing effect of shear on the disc.  \nIf the envelope were to truncate the evolution of the disc and outlive it, this could provide challenges for the formation, survival, and emergence of the planet in its current orbit. However, we find that the envelope is likely removed before, during, or shortly after disc formation. In Appendix E, we discuss what processes may power envelope ejection, and estimate how long it takes. One possibility is that tidal disruption of the companion is preceded by a Roche lobe overflow (RLOF) phase that leads to unstable mass transfer and orbital tightening, and the transfer of orbital energy ejects the envelope prior to disruption. In this case, the disc of disrupted companion material would not be surrounded by a hot envelope, so would easily survive. Another possibility is that the envelope is removed by the release of energy during an early, possibly brief phase of advection-dominated accretion. We spectulate that envelope unbinding might also be assisted by shocking during disc formation and nuclear reactions due to accretion onto the core.  \nIf the envelope is not rapidly unbound by these processes, then it could be unbound by subsequent accretion at close to the Eddington rate, but this takes ∼ 10 2 years. Whether the disc can survive for this long is not clear. In cases where early rapid removal of the envelope does not occur, survivability of the disc may be a bottleneck in our planet formation scenario that may help to explain why such planets are rare. In any case, after the envelope is ejected, ionizing radiation from the hot central core photo-evaporates the disc in ∼ 10 Myr (Appendix A).", '2.3. Minimum required companion and protoplanet masses': "From the Toomre criterion (Toomre 1964), the disc is gravitationally unstable if  \nQ = h W 2 p G S < ∼ 1 , (1)  \nwhere h is the disc scale height,  \nW = ( GM c r 3 ) 1 / 2 (2)  \nis the angular rotation speed with M c the mass of the core of the giant star, and S is the gas surface density. We approximate the surface density as (e.g. Armitage and Rice 2005; Raymond and Morbidelli 2022)  \nS = S out ( r r out ) -b , (3)  \nwith 1 < ∼ b < ∼ 3 / 2. A fraction f of the disrupted companion forms the disc, while the rest either accretes prior to disc formation or mixes with envelope material.  \nIn Appendix B, we show that condition (1) leads to the following constraint on the disc mass:  \nf m 1 > ∼ 0 . 12 M glyph[circledot] 1 2 -b ( q 0 . 1 )( M c 0 . 576 M glyph[circledot] )( r r out ) -( 2 -b ) . (4)  \nSince f ≤ 1, gravitational instability requires m 1 > ∼ 0 . 12 M glyph[circledot] , with m 1 the mass of the disrupted companion. (Throughout, we use subscript '1' to refer to the disrupted companion and subscript '2' to refer to the extant companion, i.e. the planet.) For equation (2), we neglected the disc self-gravity but it can be important if f m 1 is comparable to M c.  \nThe mass of the collapsing object must also exceed the Jeans mass (e.g. Binney and Tremaine 2008),  \nM Jeans = p 5 / 2 c 3 s 6 G 3 / 2 r 1 / 2 , (5)  \nwhere c s is the sound speed. Substituting c s = h W , r = S / 2 h , h = q r , W from equation (2), and S from equations (3) and (22), we obtain  \nM Jeans ≈ 2 . 7 MJ 1 ( 2 -b ) 1 / 2 ( M c 0 . 576 M glyph[circledot] ) 3 / 2 ( f m 1 0 . 3 M glyph[circledot] ) -1 / 2 × ( q 0 . 1 ) 7 / 2 ( r r out ) -1 + b / 2 . (6)  \nIf WD 1856+534 b formed this way, all of the material in the collapsing protoplanet was incorporated into the planet, and the planet retained the same mass up to the present time, then this would imply that m 2 > ∼ M Jeans, but some protoplanetary mass could be lost due to inefficiencies or ablation in the formation process. Regardless, this result is consistent with the observational estimate, 0 . 84 < ∼ m 2 / MJ < ∼ 11 . 7 (Vanderburg et al. 2020; Xu et al. 2021), as well as with the lower limit of 2 . 4 MJ found by Alonso et al. (2021).", '2.4. Timescale for planet formation': 'The protoplanet contracts on the free-fall timescale, t ff < ∼ 1 d (see Appendix C). This can be compared with the viscous dissipation timescale  \nt visc = r 2 n ≈ r 2 a SS c s h ≈ r 2 a SS h 2 W , (7)  \nwhere a SS is the viscosity parameter (Shakura and Sunyaev 1973). Making use of equation (2) we obtain  \nt visc ≈ 6 yr ( r 4 . 4 R glyph[circledot] ) 3 / 2 ( q 0 . 1 ) -2 ( M wd 0 . 576 M glyph[circledot] ) -1 / 2 ( a SS 0 . 01 ) -1 , (8)  \nwhich easily exceeds t ff, facilitating giant gaseous protoplanet (GGPP) by direct collapse (Boss 1998). Transformation of the GGPP into a full-fledged planet takes much longer, and is mostly independent of the disc evolution other than possible migration.', '2.5. Companion mass upper limit and the WD progenitor': 'The disrupted companion must have been massive enough for the disc to satisfy condition (4); but not so massive as to have avoided disruption by unbinding the envelope. The orbital separation at which tidal disruption occurs is given by (e.g. Nordhaus  \nFigure 2. Panel (a): Orbital separation where tidal disruption occurs a d1 (solid lines), and where RLOF is initiated (if the companion has not already disrupted) (dotted lines) for the stellar models listed in Table 1, as a function of the companion mass. The companion radius is shown as a dashed line (equation 11) and the difference between the observed WD mass according to Xu et al. ( 2021 ) and the core mass in the stellar model is shown in the legend. Panel (b): Maximum allowed value of the common envelope efficiency parameter a CE as a function of the companion mass; above this value the companion unbinds the envelope before it can be disrupted (solid lines) or before RLOF is initiated (dotted lines). It is possible that the envelope could be ejected before disruption if RLOF leads to unstable mass transfer and inspiral down to a d1 . Thus, for a given progenitor, the parameter space above the dotted line is excluded, and that below the dotted line but above the solid line is viable only if RLOF is initiated and ultimately leads to disruption. Panel (c): As the top panel, but now zoomed in to show the relevant parameter space for the planet , and with mass shown in units of MJ ≈ 10 -3 M glyph[circledot] . The planet cannot be formed in the hatched region, which corresponds to separations less than its tidal disruption separation a d2 .  \n<!-- image -->  \nand Blackman 2006)  \na d1 ≈ ( 2 M c m 1 ) 1 / 3 r 1 , (9)  \nwhere M c is the mass of the core of the giant and r 1 is the radius of the companion. Figure 2a shows a d1 as a function of m 1 (solid lines). The colours represent stellar models listed in Table 1. Dotted lines show the orbital separation at which RLOF occurs (Eggleton 1983),  \na RLOF = 0 . 6 q 2 / 3 + ln ( 1 + q 1 / 3 ) 0 . 49 q 2 / 3 r 1 , (10)  \nwhere q = M c / m 1. If a RLOF > a d1, then the companion begins losing mass at around a RLOF. Assuming this mass transfer phase is unstable, the separation continues to decrease and the companion tidally disrupts at around a d1. If a RLOF < a d1, then tidal disruption happens directly, without any RLOF phase. To estimate the radius of the disrupted companion and that of the planet, we use the following approximation to the results of Chabrier et al. (2009):  \nr R glyph[circledot] =                0 . 1 if m / M glyph[circledot] ≤ 0 . 077; 0 . 1 ( m / M glyph[circledot] 0 . 077 ) 0 . 8 if 0 . 077 < m / M glyph[circledot] ≤ 0 . 4; 0 . 1 ( 0 . 4 0 . 077 ) 0 . 8 ( m / M glyph[circledot] 0 . 4 ) 1 . 075 if m / M glyph[circledot] > 0 . 4 .  \n(11)  \nThe value of r 1 is shown as a black dashed line in Figure 2a. It can be seen that r 1 > a d1 in some cases for large values of m 1, which means the model is not fully self-consistent for that small region of parameter space. This limitation of the model stems from its simplicity. b However, as such lack of precision in the modelling does not affect our final conclusions, we choose to keep the model as simple as possible.  \nThe envelope remains bound if the liberated orbital energy D E orb multiplied by the efficiency parameter a CE is smaller than the envelope binding energy E B, i.e.,  \na CE D E orb < E B , (12)  \nwhere  \nE B = GM ( M -M c ) l R , (13)  \nM and R are the mass and radius of the WD progenitor, and l is a parameter of order unity that depends on the stellar density and temperature profiles. To satisfy constraint (12) at the tidal disruption separation, we substitute D E orb ≈ GM c m 1 / 2 a d1, where we have neglected the initial orbital energy. To satisfy the same constraint at the separation at which RLOF begins, we instead substitute D E orb ≈ GM c m 1 / 2 a RLOF.  \nFigure 2b shows the upper limit to a CE below which the envelope is not completely unbound before the tidal disruption separation is reached (solid lines). The value of a CE is not well constrained and likely varies from one system to another, but estimates for low-mass systems are typically between 0 . 05 and 0 . 5 (e.g. Iaconi and De Marco 2019; Wilson and Nordhaus 2019; Scherbak and Fuller 2023). The same lines can be used to find the  \nb For example, assuming synchronous rotation of the companion and taking into account the centrifugal force (but still ignoring tidal deformation) increases a d1 by the factor ( 3 / 2 ) 1 / 3 ≈ 1 . 145, which is sufficient to rectify this inconsistency for the values of m 1 plotted.  \nmaximum value of m 1 below which the envelope remains bound, for a given value of a CE. However, RLOF inside the envelope may lead to runaway mass loss, orbital tightening, and tidal disruption, even if the envelope were ejected shortly after the onset of RLOF. The dotted lines again show upper limits on a CE, but now requiring that the envelope remains bound until the onset of RLOF, which occurs at the separation a RLOF. For m 1 > ∼ 0 . 1 M glyph[circledot] , we see from Figure 2a that a RLOF > a d1, so RLOF could occur for m 1 > ∼ 0 . 1 M glyph[circledot] .  \nFigure 2b shows that the more distended models (RGB tip, shown in orange, and high/low mass thermally pulsing AGB (TPAGB), shown in violet/red) require m 1 < ∼ 0 . 08 M glyph[circledot] if a CE = 0 . 2, using the more conservative limit (solid lines). If a CE = 0 . 1, then the maximum value of m 1 lies within the range ( 0 . 13 , 0 . 28 ) for these three models. Models with a CE ≥ 0 . 2 are ruled out because condition (4) is not satisfied: the orange, violet and red lines all intersect a CE at m 1 < 0 . 1 M glyph[circledot] . If 0 . 1 < a CE < 0 . 2, the allowed range of m 1 is quite narrow. By constrast, the less distended RGB and AGB models accommodate a larger range of companion mass because the magnitude of the envelope binding energy is larger. Focussing, conservatively, on the solid rather than the dotted lines, the AGB model (green) admits companion masses up to ≈ 1 M glyph[circledot] if a CE = 0 . 2, up to ≈ 0 . 3 M glyph[circledot] if a CE = 0 . 3, and up to ≈ 0 . 18 M glyph[circledot] if a CE = 0 . 4. The RGB(C18) model (brown) allows for slightly larger upper limits to the companion mass.  \nCE evolution culminating in tidal disruption of a companion massive enough to form a planet is thus more likely for less distended primary stars. Using the less conservative upper limit on a CE corresponding to RLOF (dotted lines), the allowed ranges of a CE and m 1 are larger, but less distended progenitors are still favoured.', '2.6. Implications of accretion for the WD mass': 'In the legend of Figure 2, we show the approximate difference between M wd ≈ 0 . 576 M glyph[circledot] (Xu et al. 2021) and the core mass of the primary (Table 1). This difference could be explained by accretion of material from the disrupted companion. For the RGB models, ∼ 0 . 12-0 . 21 M glyph[circledot] must be accreted, whereas for the AGB/TPAGB models only ∼ 0 . 01-0 . 06 M glyph[circledot] must be accreted. The AGB model (green) and RGB(C18) model (brown) can accommodate a larger range of disrupted companion masses than the other models (Section 2.5), but the AGB model requires less accretion of disrupted companion material onto the proto-WD core and may thus be more likely. In Appendix D we estimate that ∼ 10 -3 -10 -2 M glyph[circledot] is accreted during a phase of advectiondominated accretion, but that another ∼ 0 . 1 M glyph[circledot] may accrete in the next 10 3 -10 4 yr during an Eddington-limited phase.', '2.7. Location of planet formation': 'The planet must form outside of its own tidal disruption separation a d2, plotted in Figure 2c as a function of the planet mass m 2. For 1 < m 2 / MJ < 12, the range is 0 . 5 < ∼ a d2 < ∼ 1 . 0 R glyph[circledot] , but a d1 < 0 . 5 R glyph[circledot] for m 1 < ∼ 0 . 4 M glyph[circledot] , and a d1 < 1 R glyph[circledot] for m 1 < ∼ 1 . 0 M glyph[circledot] , as shown in Figure 2a, ignoring m 1 < 0 . 01 M glyph[circledot] , which is excluded by the lower limit (4). Thus, if the planet formed at a d1, this would imply a separate lower limit for m 1 in the range 0 . 5-1 . 0 M glyph[circledot] due to the minimum in the a d1 vs. mass relation plotted in Figure 2a. But could the planet instead form at a larger separation, or, for WD1856+534 b, at its present location a 2 ≈ 4 . 4 R glyph[circledot] ?  \nFor q = 0 . 1, the disc can viscously spread out to to r out ∼ a 2 on the viscous timescale, ∼ 6 yr (equation 8). But if the flow is advection-dominated then q ∼ 1, lowering the diffusion time by a factor of ∼ 100. At some point the disc transitions from the advection-dominated geometrically thick regime to the gaspressure-dominated geometrically thin regime (e.g. Shen and Matzner 2014). Protoplanet formation can occur once Q drops below unity and Q is proportional to q and to a negative power of r / r out (equation 25). Thus, when the disc transitions, q and Q drop by an order of magnitude so this transition might trigger gravitational instability in the outer disc, leading to protoplanet formation at separations > a d1. Thus, WD 1856+534 b might have formed at or close to its present separation.  \nBy the end of the simulations of Guidarelli et al. (2022), the expanding tidal tail formed during the disruption extends an order of magnitude larger than the tidal disruption separation a d1. This material may fall back and extend the disc (e.g. Shen and Matzner 2014). Mass transfer and disc formation may also begin in the RLOF phase, which begins at separations > a d1 (Figure 2a). Moreover, the companion likely inflates by accreting a quasihydrostatic atmosphere (Chamandy et al. 2018), which would, in principle, cause it to overflow its Roche lobe at still larger separation. RLOF might also lead to mass transfer through the L2 point and circumbinary disc formation (e.g. MacLeod et al. 2018). These processes could increase the outer radius of the disc, facilitating planet formation at separations > a d1.', '2.8. Minimum mass to prevent type I migration': 'Once formed, the protoplanet can avoid destruction due to type I inward migration by opening a gap if (e.g. Papaloizou 2021)  \nm 2 > ∼ 2 MJ ( M wd 0 . 576 M glyph[circledot] )( q 0 . 1 ) 3 (14)  \nand  \nm 2 > ∼ 2 MJ ( M wd 0 . 576 M glyph[circledot] )( q 0 . 1 ) 2 ( a SS 0 . 01 ) , (15)  \nwhich happen to be lower limits of similar magnitude to each other and to M Jeans (equation 6). Therefore, two separate lines of reasoning favour a planet mass > ∼ 2 MJ; this lower limit is thus a fairly robust prediction of our model.', '3. Summary and Conclusions': 'We have presented a second generation planet formation scenario that may explain the occurrence of giant planets in low-period orbits around WD. In this scenario, planets form from gravitational instability of an accretion disc formed out of material from a low-mass star that was tidally disrupted due to CE interaction with the WD progenitor core. We showed that WD 1856+534 b may be explained by this scenario if: (i) the WD progenitor was a relatively compact AGB star with M > ∼ 2 M glyph[circledot] , (ii) the mass of the disrupted companion was in the range 0 . 15 < ∼ m 1 / M glyph[circledot] < ∼ 1 M glyph[circledot] , and (iii) the mass of the observed planet m 2 > ∼ 2 MJ. The latter can be considered as a prediction for WD 1856+534 b, and is similar to the lower limits obtained using transmission spectroscopy of 2 . 4 MJ (Alonso et al. 2021) and 0 . 84 MJ (Xu et al. 2021), while remaining comfortably below the upper limit of ∼ 12 MJ, which is based on the non-detection of thermal emission (Vanderburg et al. 2020).  \nWD 1856+534 b might have formed at the orbital separation inferred from observations or might have migrated to it. Detailed models (e.g. Masset and Papaloizou 2003) show that the sense and timescale of type II migration can vary in time and can be sensitive to various parameters. The interplay between migration and evaporation by the hot white dwarf core of the primary may account for the rarity of systems like WD1856+534 b, depending on where the planet forms before migration and how long it spends close enough to the WD core to be evaporated (Gallo et al. 2024). Future work is needed to address these issues.  \nAcknowledgements. We are grateful to the first referee for comments that helped to improve the presentation of this work. L.C. thanks R.I.T. for supporting his visit from 4-10 June, 2023.  \nFunding Statement. Financial support for this project was provided by the U. S. Department of Energy grants DE-SC0001063, DE-SC0020432 and DE-SC0020103, the U. S. National Science Foundation grants AST-1515648, AST-1813298, PHY-2020249, AST-2009713, and AST-2319326, and the Space Telescope Science Institute grant HST-AR-12832.01-A.  \nCompeting Interests. None  \nData Availability Statement. 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Photo-evaporation timescale': "The ionizing luminosity is sensitive to the effective temperature. The hottest white dwarfs are observed to have T eff ∼ 10 5 K (B'edard et al. 2017). Theoretical models which track WD properties as a function of time predict that they are born with T eff ≈ 0 . 9-1 . 1 × 10 5 K and radius R wd ≈ 0 . 0220 . 027 R glyph[circledot] for M wd ∼ 0 . 55-0 . 60 M glyph[circledot] (B'edard et al. 2020). c Or perhaps the exposed core would resemble a hot subdwarf B (sdB) star. Such stars are remnants of evolved stars often found in post-CE binary systems, and have T eff ≈ 2-7 × 10 4 K (Heber 2016; Ge et al. 2024).  \nFor T eff = 10 5 K, the Planck spectrum peaks at 29 nm, in the extreme ultraviolet. Let us assume, conservatively, that all photons are ionizing. The radiative flux impinging on the disc can then be equated with the wind power, with an efficiency factor e ,  \n4 p R 2 c s T 4 eff q p / 2 e ≈ 1 2 ˙ M w 2 GM c r , (16)  \nwhere q = h / r is the disc aspect ratio (assumed to be small), and we have used the escape speed from the central core for the wind speed. This gives a wind mass-loss rate of  \n˙ M w ≈ 4 × 10 -8 M glyph[circledot] yr -1 ( e 0 . 1 ) ( q 0 . 1 )( R c 0 . 025 R glyph[circledot] ) 2 × ( T eff 10 5 K ) 4 ( r 4 . 4 R glyph[circledot] )( M c 0 . 576 M glyph[circledot] ) -1 , (17)  \nwhere we have scaled r to the present orbital separation of WD 1856+534 b, assuming a circular orbit. If we instead adopt typical values for sdB stars, with R c about 5 times higher and T eff about 2 times lower than the above values, we obtain approximately the same numerical estimate for ˙ M w.  \nAlternatively, we can try to apply detailed models from the literature which were designed for classical protoplanetary discs. We try the model of Alexander et al. (2006) (see also Alexander et al. 2014 and Kunitomo et al. 2020), which takes as input the number of ionizing photons emanating from the star per unit time F . We estimate  \nF ≈ 4 p R 2 c s T 4 eff h n max , (18)  \nwith n max given by Wien's displacement law. Thus, we obtain  \nF ≈ 6 × 10 45 s -1 ( R c 0 . 025 R glyph[circledot] ) 2 ( T eff 10 5 K ) 3 . (19)  \nThen, using the Alexander et al. (2006) model with CD = 1, a = 6 and m = 1, and taking the outer disc radius to be much larger than the inner disc radius, we find  \n˙ M w ≈ 6 × 10 -9 M glyph[circledot] yr -1 ( q 0 . 1 ) -1 / 2 ( F 6 × 10 45 s -1 ) 1 / 2 × ( r in 4 . 4 R glyph[circledot] ) 1 / 2 , (20)  \nwhere r in is the inner radius of the disc. Thus, this estimate gives a mass-loss rate that is of the same order of magnitude as that obtained in equation (17). A disc of mass 0 . 3 M glyph[circledot] would take ∼ 10 Myr to evaporate, which is long compared to the timescales of other key processes, as discussed below.", 'Appendix B. Gravitational instability of the disc': 'The disc mass can be obtained by integrating equation (3), which gives  \nf m 1 = 2 p S out r b out ∫ r out r in r 1 -b dr ≈ 2 p S out r 2 out 2 -b , (21)  \nc See https://www.astro.umontreal.ca/ ∼ bergeron/CoolingModels/.  \nwhere f is the fraction of the disrupted companion mass incorporated in the disc and we have assumed b < 2 and ( r in / r out ) 2 -b glyph[lessmuch] 1. Rearranging, we obtain  \nS out ≈ ( 2 -b ) f m 1 2 p r 2 out . (22)  \nThe volume density of the disc is given by  \nr = S 2 h = r out ( r r out ) -( b + 1 ) , (23)  \nwhere r out = S out / ( 2 q r out ) with q ≡ h / r the disc aspect ratio. Substituting expression (22) for S out into this expression for r out and then substituting into equation (23) gives  \nr ≈ ( 2 -b ) f m 1 4 p r 3 out q ( r r out ) -( b + 1 ) . (24)  \nAdisc with q = 0 . 1, f m 1 = 0 . 08 M glyph[circledot] and r out = a 2 ≈ 4 . 4 R glyph[circledot] has r ( r out ) ≈ 4 × 10 -3 ( 2 -b ) g cm -3 . If r out = 100 R glyph[circledot] , then r ( 4 . 4 R glyph[circledot] ) ≈ 2 × 10 -4 g cm -3 if b = 1 or ≈ 5 × 10 -4 g cm -3 if b = 3 / 2. On the other hand, the density of the envelope of a ZAMS 2 M glyph[circledot] AGBstar is r e ≈ 10 -4 g cm -3 at r = 4 . 4 R glyph[circledot] . Thus, the disc is expected to have slightly higher density than the original envelope at the present orbital separation of WD 1856+534 b. However, at this stage, the envelope would have already experienced expansion and at least partial ejection, so r e would be significantly smaller than the above estimate and thus much smaller than the disc density. d  \nCombining equations (1), (2), (3) and (22), we obtain  \nQ ≈ 2 q 2 -b M c f m 1 ( r r out ) -( 2 -b ) . (25)  \nThus, Q decreases with r , and reaches a minimum at the disc outer radius r out. Now we can impose the condition for instability Q < ∼ 1 (constraint 1), which leads to constraint (4) on the mass of the disrupted companion.', 'Appendix C. Free-fall timescale for planet formation': 'The free-fall timescale is given by  \nt ff = ( 3 p 32 G r ) 1 / 2 . (26)  \nMaking use of equation (24) we obtain  \nt ff ≈ 0 . 2 d ( 2 -b ) 1 / 2 ( q 0 . 1 ) 1 / 2 ( r out 4 . 4 R glyph[circledot] ) 3 / 2 ( f m 1 0 . 3 M glyph[circledot] ) -1 / 2 × ( r r out ) ( b + 1 ) / 2 , (27)  \nwhich is somewhat smaller than the observed orbital period of 1 . 41 d for WD1856+534 b (Vanderburg et al. 2020).', 'Appendix D. Accretion onto the proto-WD': 'At early times, accretion of disc material onto the core of the primary may occur on the viscous timescale. Using equation (8) with r = a d1 (equation 9), we find  \n˙ M visc ≈ f m 1 t visc ≈ f m 3 / 2 1 G 1 / 2 a SS q 2 2 1 / 2 r 3 / 2 1 ≈ 91 M glyph[circledot] yr -1 f q 2 ( a SS 0 . 01 ) ( m 1 0 . 3 M glyph[circledot] ) 3 / 2 ( r 1 0 . 4 R glyph[circledot] ) -3 / 2 , (28)  \nwhere q ∼ 1 since the disc initially cannot cool efficiently. This accretion rate is several orders of magnitude higher than the Eddington rate of (C21)  \n˙ M Edd ≈ 2 . 7 × 10 -5 M glyph[circledot] yr -1 ( R c 0 . 013 R glyph[circledot] ) . (29)  \nd To get an idea of how fast the envelope density near the companion can decrease relative to the initial value at that radius in the envelope, see, e.g., Ricker and Taam (2012), Iaconi et al. (2017), and Chamandy et al. (2019).  \nAt this stage, radiation is trapped and advected with the flow (e.g. Narayan and Yi 1995; Nordhaus et al. 2011; Shen and Matzner 2014). Most of the mass may be directed into winds/jets (Blandford and Begelman 1999; Armitage and Livio 2000; Hawley and Balbus 2002; Ohsuga et al. 2005), while a fraction accretes onto the central core.  \nThis phase may be sustained by outflows or it may transition into an Eddington-limited phase if the accretion is quenched due to the buildup of gas pressure, which would happen on roughly a dynamical timescale,  \nt dyn ∼ 1 W ∣ ∣ ∣ ∣ a d1 ≈ ( 2 r 3 1 Gm 1 ) 1 / 2 ≈ 17 min ( m 1 0 . 3 M glyph[circledot] ) -1 / 2 ( r 1 0 . 4 R glyph[circledot] ) 3 / 2 , (30)  \nwhere we made use of equations (2) and (9). The mass accreted during this time is  \nM adv ≈ ˙ M visc t dyn ≈ f a SS q 2 m 1 ≈ 3 × 10 -3 M glyph[circledot] f q 2 ( a SS 0 . 01 ) ( m 1 0 . 3 M glyph[circledot] ) , (31)  \nThus, during the advection-dominated phase, a few MJ of material would be deposited into an envelope around the WD (c.f. Nordhaus et al. 2011). Subsequently, the WD may accrete a large fraction of the remaining disc material on the timescale  \nt acc ∼ m acc ˙ M Edd ∼ 4 × 10 3 yr ( m acc 0 . 1 M glyph[circledot] )( R c 0 . 0126 R glyph[circledot] ) -1 , (32)  \nwhere m acc is the mass of accreted material. Remaining disc material would gradually disperse on a timescale of perhaps ∼ 10 Myr, as estimated in Appendix A.', 'Appendix E. Powering envelope ejection': 'The rate of energy release during this advection-dominated phase is roughly given by  \nL adv ≈ GM c ˙ M visc 2 R c ≈ 2 . 5 × 10 44 erg s -1 f q 2 ( a SS 0 . 01 ) ( m 1 0 . 3 M glyph[circledot] ) 3 / 2 × ( r 1 0 . 4 R glyph[circledot] ) -3 / 2 ( M c 0 . 576 M glyph[circledot] )( R c 0 . 0126 R glyph[circledot] ) -1 , (33)  \nwhere we have scaled R c to the measured WD radius R wd = 0 . 01263 ± 0 . 0050 R glyph[circledot] (Xu et al. 2021). The energy liberated is estimated as  \nE adv ≈ L adv t dyn ≈ 3 × 10 47 erg f q 2 ( a SS 0 . 01 ) ( m 1 0 . 3 M glyph[circledot] ) × ( M c 0 . 576 M glyph[circledot] )( R c 0 . 0126 R glyph[circledot] ) -1 , (34)  \nwhich is approximately the same as the initial binding energy of the envelope E bind (Table 1). If about 10% of the original binding energy remains at the time of tidal disruption, then the energy liberated is about an order of magnitude larger than the envelope binding energy. Thus, what remains of the envelope can be rapidly unbound if the energy transfer efficiency of the unbinding process is > ∼ 0 . 1. Nuclear burning of accreted material may also assist envelope unbinding (Siess and Livio 1999 a ,b; Nordhaus et al. 2011; C21). The disc, on the other hand, is not as susceptible to unbinding both because its binding energy is larger E d ∼ GM c f m 1 / a d1 ∼ 10 48 erg - and because the released energy may be transported perpendicular to the disc.  \nIf accretion proceeds at the Eddington rate thereafter (equation 32), then energy would be released at the rate  \nL Edd = 7 . 2 × 10 37 erg s -1 ( M c 0 . 576 M glyph[circledot] ) . (35)  \nIf, as argued above, ∼ 3 × 10 47 erg must be released to unbind the envelope (which already factors in the efficiency), we find that the envelope can be ejected in ∼ 10 2 yr by Eddington-limited accretion alone.  \nAlternatively, the envelope might be removed before tidal disruption but after the onset of RLOF. In this case, once the envelope is ejected, further orbital tightening would need to be driven by a mechanism other than CE drag. Torques may arise at a ≈ a RLOF that lead to orbital decay on timescales of a few hundred orbital periods (c.f. MacLeod et al. 2018). Mass transfer is expected to be unstable for low-mass MS stars (e.g. Stevens et al. 1992; Jones 2020). Thus, even if the envelope is ejected at a separation a d1 < a < a RLOF, orbital decay down to a d1 is still likely to occur. To determine whether this scenario is plausible, we estimate the orbital energy released between a RLOF and a d1,  \n-D E orb = GM c m 1 2 ( 1 a d1 -1 a RLOF ) . (36)  \nFrom Figure 2a for m 1 = 0 . 3 M glyph[circledot] and the AGB model (green) with M c = 0 . 52 M glyph[circledot] , we find a RLOF ≈ 0 . 72 R glyph[circledot] and a d1 ≈ 0 . 44 R glyph[circledot] , which gives -D E orb ≈ 3 × 10 47 erg for m 1 = 0 . 3 M glyph[circledot] , which is the same energy estimated to be released by accretion during the advection-dominated accretion phase (equation 34). As we have already argued, this amount of energy is probably sufficient to unbind the remaining envelope. The value of D E orb is not sensitive to which of the models in Table 1 is adopted for the primary star.  \nEnvelope removal may also be powered by shocking during accretion disc formation (e.g. Rees 1988; Piran et al. 2015; Bonnerot et al. 2021; Ryu et al. 2023; Steinberg and Stone 2024).'}