Draft version September 10, 2018Preprint typeset using LATEX style emulateapj v. 12/16/11

ROCHE-LOBE OVERFLOW IN ECCENTRIC PLANET-STAR SYSTEMS

Fani Dosopoulou1, Smadar Naoz2,3, Vassiliki Kalogera1

(1) Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy,Northwestern University;

(2) Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA(3) Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA

Draft version September 10, 2018

ABSTRACT

Many giant exoplanets are found near their Roche limit and in mildly eccentric orbits. In thisstudy we examine the fate of such planets through Roche-lobe overflow as a function of the physicalproperties of the binary components, including the eccentricity and the asynchronicity of the rotatingplanet. We use a direct three-body integrator to compute the trajectories of the lost mass in theballistic limit and investigate the possible outcomes. We find three different outcomes for the masstransferred through the Lagrangian point L1: (i) self-accretion by the planet, (ii) direct impact onthe stellar surface, (iii) disk formation around the star. We explore the parameter space of the threedifferent regimes and find that at low eccentricities, e . 0.2, mass overflow leads to disk formationfor most systems, while for higher eccentricities or retrograde orbits self-accretion is the only possibleoutcome. We conclude that the assumption often made in previous work that when a planet overflowsits Roche lobe it is quickly disrupted and accreted by the star is not always valid.Subject headings: Stars: kinematics and dynamics Binaries: dynamics

1. INTRODUCTION

Many of the shortest-period exoplanets, covering alarge mass-range, from Earth-size to Jupiter-size (hotJupiters), are found in mildly eccentric orbits and nearlyat or even interior to their Roche limit. This suggeststhat many planets are near Roche lobe overflow (RLOF)(e.g., Figure 1 in Jackson et al. 2017). For example, Li etal. (2010) suggested that WASP-12 b is in the process ofRLOF (see also Patra et al. 2017). The presence of manygas giants in similar orbits suggests that RLOF may becommon among exoplanets.

Even if a planet is not currently close enough to itshost star to overflow, tidal interactions may eventuallybecome important once planets are inside ∼ 0.1 AU andtidal orbital decay can drive the planet to the Rochelimit (e.g., Levrard et al. 2009; Jackson et al. 2009; Mc-Quillan et al. 2013; Poppenhaeger & Wolk 2014). Thisprocess tends to circularize the planet’s orbit. Further-more, high eccentricity migration predicts that a largefraction of the planets may get disrupted. Specifically,the eccentric Lidov-Kozai mechanism (e.g., Naoz 2016)can result in disrupted Jupiters (e.g., Naoz et al. 2012;Petrovich 2015). These planets can plunge in, and crossthe Roche limit, with extremely large eccentricity or withmoderate ones that may result from planet-planet inter-actions (e.g., Antonini et al. 2016; Petrovich & Tremaine2016; Hamers et al. 2017).

Most previous studies have assumed that whenever aplanet crosses the Roche limit, it is quickly disintegratedand its material is accreted by the star (e.g., Jackson etal. 2009; Metzger et al. 2012; Schlaufman & Winn 2013;Teitler & Königl 2014; Zhang & Penev 2014). However,recent studies showed that hot Jupiters might be onlypartially consumed, leaving behind lower-mass planets(Valsecchi et al. 2014, 2015; Jackson et al. 2016). These

FaniDosopoulou2012@u.northwestern.edu

studies suggest the need for further investigation of theplanetary system orbital evolution due to RLOF in ec-centric systems, e.g., using secular evolution equations(e.g., Dosopoulou & Kalogera 2016a,b). The accuracy ofpredictions for the fate of gas giants and the final orbits oftheir remnants depends on the trajectory the mass lostthrough the Lagrangian point L1 follows after its ejec-tion. In an eccentric system the latter is determined bythe eccentricity of the system as well as the masses, radiiand spins of the planet-star components. Depending onthe formation history, giant exoplanets can be tidallylocked or rotate asynchronously in their orbits aroundthe star.

In this paper we study short-period eccentric planet-star systems at the onset of RLOF. We assume RLOFtakes place at each subsequent periastron passage. Weinvestigate the possible outcomes of mass overflow for asystem with a generically asynchronous planet. We showthat for a given initial eccentricity, depending on the bi-nary mass-ratio, the planet rotation rate and the starradius, RLOF can lead to three different possible out-comes for the lost matter. These are: (i) self-accretionby the planet, (ii) direct impact on the stellar surfaceand (iii) disk formation around the star. These threedifferent regimes do not uniquely lead to the disruptionof the planet or the accretion of the lost matter by thestar as often assumed in previous studies. Here we ex-plore the parameter space of these regimes calculatingthe trajectories of the lost particle in the ballistic limit.

This paper is organized as follows. In Section 2 wedescribe the system and the adopted methodology. InSection 3 we explore the system’s parameter space iden-tifying the regions that lead to different mass overflowoutcomes. We conclude with Section 4.

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SELF ACCRETION

q = 10−3, f1 = 0.3, e0 = 0.2

Planet

Star

particle

Fig. 1.— Different outcomes of RLOF at periastron for different initial conditions (q, fp, e0) and star radii R? = (1.0, 1.5) R� withM? = 1M�. The orbits of the planet, the star and the particle are depicted in the (x−y) plane where x and y are normalized to the initialperiastron distance rp. The location of the Lagrangian point at periastron, xL1,p , is depicted with a black star while connecting lines show

the relative distance of planet-particle (red) and star-particle (yellow) in different phases along the orbit. Left panel: No impact; Possiblya disk is formed (DISK). Middle panel: Direct impact (DI) on stellar surface due to the presence of a larger star. Right panel: The lostparticle undergoes self-accretion (SA) due to the more eccentric orbit.

2. ROCHE-LOBE OVERFLOW IN PLANET-STAR SYSTEMS

2.1. System set-up

We follow the formalism developed by Sepinsky et al.(2010) and consider a generically eccentric binary con-sisting of a planet with mass Mp and radius Rp and astar with mass M? and radius R?. We define the binarymass-ratio as q = Mp/M? and in what follows we con-sider planet-star systems with mass-ratio q in the range−6 ≤ log q ≤ −2. We assume that the planet rotates uni-formly and with constant spin ΩΩΩp parallel or anti-parallelto the orbital angular velocity ΩΩΩorb. We normalize ΩΩΩp bythe orbital angular velocity at periastron ΩΩΩorb,p, i.e.,

ΩΩΩp = fpΩΩΩorb,p (1)

where now fp defines the degree of pseudo-asynchronicityof the planet at periastron and (Eq. (2.32) in Murray &Dermott 1999)

Ωorb,p =2π

Porb

(1 + e)1/2

(1− e)3/2(2)

with Porb the binary orbital period and e the binaryeccentricity. For simplicity, from now on we refer tofp as the planet’s degree of asynchronicity. We con-sider both prograde (fp > 0) and retrograde (fp < 0)planetary orbits. When fp = 1 the planet is pseudo-synchronously rotating at periastron, while fp < 1 de-notes a sub-synchronously rotating planet.

We consider short-period systems where the planet un-dergoes RLOF and mass is lost through the Lagrangianpoint L1. Sepinsky et al. (2007) investigated the exis-tence and properties of equipotential surfaces and La-grangian points in asynchronous, eccentric binary starand planetary systems. They showed that in an eccen-tric orbit the position of the Lagrangian point L1 de-pends on the phase along the orbit and it is the smallestwhen the two bodies are at periastron (e.g., Figure 8 inSepinsky et al. 2007). This implies that mass overflow ismore likely to reoccur at each subsequent periastron pas-sage. In what follows we assume RLOF only at periastonand use the method developed by Sepinsky et al. (2007)to calculate the position of the Lagrangian point L1 at

periastron, xL1,p . Sepinsky et al. (2007) provided alsofitting formulae for the volume-equivalent Roche lobe ra-dius appropriate for asynchronous eccentric systems as afunction of the binary mass ratio and the degree of asyn-chronicity of the overflowing body. Sepinsky et al. (2007)verified that for the low mass-ratio systems consideredhere, the simpler formula given in Eggleton (1983) forthe volume-equivalent Roche lobe radius at periastron,RL1,p , is still a good approximation for eccentric asyn-chronous systems (e.g., Figure 9 in Sepinsky et al. 2007).Thus, we use

RL1,p(q) = rp0.49q2/3

0.6q2/3 + ln(1 + q1/3)= rprL(q) (3)

where rp = a(1− e) is the periastron distance and a thebinary semi-major axis. We assume the following planetmass Mp and radius Rp relation (Lissauer et al. 2011;Howard 2013)

Rp =

R⊕

(MpM⊕

)1/2.06if Mp < MJ,

RJ if Mp > MJ

(4)

where R⊕,M⊕ are the Earth radius and mass and RJ,MJare the Jupiter radius and mass. We also test two differ-

ent mass-radius relations: (i) Rp = Re (Mp/2.7Me)1/1.3

(Wolfgang et al. 2016) and (ii) Rp = (3Mp/4πρp)1/3,

where ρp is the planet density in the range ρp = 0.1 −10grcm−3. We find that our results presented in Section3 are not affected by the mass-radius relation adopted.

Given the initial eccentricity of the system, e0, the ini-tial semi-major axis, a0, is calculated such that the planetundergoes RLOF at periastron, i.e., for Rp = RL1,pwe find a0 = Rp/(rL(q)(1 − e0)). Notice that accord-ing to Equation (4), for Mp > MJ, the initial peri-astron distance of the planet decreases with the massratio. For low initial eccentricities, 0.01 . e0 . 0.2,the value of the initial semi-major axis lies in the range0.0047 AU . a0 . 0.012 AU.

3

2.2. Ballistic limit

We assume that a particle with negligible massMloss 0).

3. RESULTS

In this section we explore the system parameter spaceand investigate the dependence of the mass overflow out-come on the initial conditions (q, fp, e0) as well as themass M? and radius R? of the star.

In Figure 2 we explore the mass ratio and asynchronic-ity parameter space of the system (q, fp) for two differ-ent initial eccentricities e0 and three different types ofstars: (i) an ultra-cool dwarf star with M? = 0.08M�and radius R? = 0.11R� similar to the one in the re-cently discovered TRAPPIST-1 planetary system (Gillonet al. 2017) (ii) a Sun-like star with mass M? = 1M�and radius R? = 1.0R� (iii) a subgiant star with massM? = 1M� and radius R? = 1.5R�.

As shown in Figure 2 for a given eccentricity, e0, thevalues of fp and q which separate the SA and DISK/DIregimes do not depend on the mass M? and radius R?

4

Fig. 2.— RLOF at periastron outcomes for e0 = 0.05 (top panels), e0 = 0.2 (bottom panels) and a ultra cool dwarf star with M = 0.08M�and R? = 0.11R� (left column) a Sun-like star with mass M = 1M� and radius R? = 1.0R� (middle column) a subgiant star with M = 1M�and R? = 1.5R� (right column). Grey areas indicate systems where the planets comes into contact with the star (CON). DISK/DI occursmostly at low eccentricities while at higher eccentricities SA takes over. For the types of systems considered here, although an increase inthe stellar radius leads to an extension of the DI and CON regime, DISK formation dominates over DI for both eccentricities.

of the star. At low eccentricities, the most likely out-come is DISK/DI, while at higher eccentricities, e & 0.2,SA takes over. Although increasing the star radius, R?,increases the parameter space for DI, for the cases weconsider in Figure 2, the formation of a DISK appears tobe a more common outcome than DI.

For the systems we consider in Figure 2, at a given R?,there are values of q such that R? > rp − xL1,p , i.e., theplanet comes into contact with the star. The parameterspace for these type of systems is shown in Figure 2 as agrey area (CON). For planets less massive than Jupiter,the periastron distance increases with a decreasing massratio. This implies that for a small enough mass ratio theplanet comes into contact with the stellar surface. Thisis depicted as the lower gray area in the right columnin Figure 2, where for the subgiant star we estimate thislower limit to be log q ∼ −5.25. For planets more massivethan Jupiter, we adopted an upper limit for the planetradius (Rp = RJ). This means that for these planetsthe periastron distance decreases with an increasing massratio. Thus, for large mass ratio above a threshold theplanet and the star come into contact. This is depictedas the upper gray area in the right column in Figure 2where for the subgiant star this upper threshold becomeslog q ∼ −2.6.

In Figure 3 we investigate the parameter space thatseparate the SA and DI/DISK regimes. As depicted in

Figure 2 the values of fp and q at which the transitionto SA occurs do not depend on M? and R?. Thus, forsimplicity, we set in Figure 3 R? = 0 and M? = 1M�,and explore the (q, fp) parameter space as a functionof the initial eccentricity, e0. As Figure 3 indicates, forany initial eccentricity, DI/DISK can occur only for sub-synchronously rotating planets (fp < 1). Increasing theinitial eccentricity restricts the parameter space regionfor DI/DISK. We find that the DI/DISK regime dom-inates at low eccentricities, e0 . 0.2, while for highereccentricities or retrograde orbits the only possible out-come is SA.

As shown in Figure 3, DI/DISK can occur only for e0 <0.2. Within this restricted regime we identify in Figure4 two regions, depending on the star radius R? and themass-ratio q. As we mentioned before, for a given q, thereexists an upper limit to the radius of the star, R?,max,above which the planet and the star are in contact. Herewe compute this upper limit setting R?,max = rp−RL1,p(notice that for low mass ratios xL1,p ' RL1,p). Thisupper limit R?,max is plotted in Figure 4 as a function ofthe mass ratio q. If the star radius R? falls within theregion labeled as “DISK”, mass overflow leads always tothe formation of a disk regardless of the values chosenfor fp and e0. This is because within the “DISK” regimethe star radius R? is always smaller than the minimumparticle distance to the star. In the region labeled as “DI

5

Fig. 3.— Color-shaded regions depict the parameter space (q, fp)for which either DI or DISK occurs. Outside these areas SA takesover. Darker tone of red refers to larger initial eccentricity whichrestricts the DI/DISK regime. DI/DISK dominates at low eccen-tricities while for higher eccentricities, e0 & 0.2, the only possibleoutcome is SA. Black region refers to the highest eccentricity thatallows for DISK/DI.

Fig. 4.— Here we consider low-eccentricity systems, e0 < 0.2,and set M? = 1M�. Color-shaded regions separate the “DISK”regime (always a DISK forms regardless of (fp, e0)) from the “DIor DISK” regime (DISK or DI depending on (fp, e0)). The blackline refers to R?,max(q), i.e., in the white area the planet and thestar are in contact. At low eccentricities RLOF at periastron leadsto DISK formation for most systems.

or DISK” the mass overflow outcome can be either DISKor DI depending on the specific values of fp and e0. Asdepicted in Figure 4, as a star begins to expand withoutlosing mass (e.g., as the star leaves the Main Sequence), aplanet undergoing RLOF in a low-eccentricity orbit maylead to DI, before the planet gets in contact with the

stellar surface. However, as shown in Figure 4, the regionfor which a DISK forms overall dominates the parameterspace. This implies that at low eccentricities, RLOF atperiastron leads to DISK formation for most systems.

4. CONCLUSIONS

We investigated the possible outcomes of Roche lobeoverflow at periastron for eccentric planet-star systemswith a planet in an asynchronous orbit. We explore thesystem parameter space identifying the regimes thatlead to different outcomes of the planet’s mass loss.

The main results of this paper are summarized below:

1) Roche lobe overflow at periastron leads to one ofthe three possible outcomes for the mass trans-ferred though the Lagrangian point L1: (i) self-accretion by the planet (ii) direct impact on thestellar surface (iii) disk formation around the star(see Figure 1).

2) Direct impact or disk formation can occur only atlow eccentricities and only for systems with sub-synchronously rotating planets in prograde orbits.For higher eccentricities, e & 0.2, mass overflowleads always to self-accretion (see Figure 3). In thecase of retrograde orbits the only possible outcomeis SA independent of the system eccentricity.

3) For the low eccentric planet-star systems, e . 0.2,and within the direct impact/disk formation pa-rameter space regime, the region where a disk isformed dominates the region that leads to directimpact (see Figure 2). Although increasing thestar radius for a given stellar mass (e.g., as thestar evolves beyond the Main Sequence) may leadto direct impact, at low eccentricities Roche lobeoverflow leads to disk formation for most systems(see Figure 4).

We have considered as a proof of concept the ballistic ap-proach to study Roche-lobe overflow in eccentric planet-star systems. We have shown that at low eccentricitiesRoche lobe overflow leads to disk formation for most sys-tems. For eccentric systems we speculate that the dy-namics may lead to the survival of giant planets nearthe Roche limit, as observed (Jackson et al. 2017). Us-ing the formalism described in Dosopoulou & Kalogera(2016a,b) the secular evolution of these systems can beinvestigated to test the speculation mentioned above andto compare to observations.

ACKNOWLEDGEMENTS

We want to thank our colleague Jeremy Sepinsky whoshared with us the three-body code and Fred Rasio foruseful discussions. F.D. and V.K. acknowledge supportfrom grant NSF AST-1517753 and Northwestern Uni-versity through the Reach for the Stars program. S.N.acknowledges partial support from a Sloan FoundationFellowship.

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http://arxiv.org/abs/1703.06582

ABSTRACT1 Introduction2 Roche-lobe overflow in planet-star systems2.1 System set-up2.2 Ballistic limit2.3 Possible outcomes of Roche-lobe overflow

3 Results4 Conclusions