The Effect of Stellar Evolution on Migrating Warm Jupiters
aa r X i v : . [ a s t r o - ph . S R ] O c t Mon. Not. R. Astron. Soc. , 1–16 (2015) Printed October 10, 2018 (MN L A TEX style file v2.2)
The Effect of Stellar Evolution on Migrating WarmJupiters
S. F. N. Frewen, ⋆ B. M. S. Hansen Division of Astronomy and Astrophysics, University of California, Los Angeles, CA 90095-1547
October 10, 2018
ABSTRACT
Warm jupiters are an unexpected population of extrasolar planets that are too nearto their host to have formed in situ, but distant enough to retain a significant eccen-tricity in the face of tidal damping. These planets are curiously absent around starslarger than two solar radii. We hypothesize that the warm jupiters are migrating dueto Kozai-Lidov oscillations, which leads to transient episodes of high eccentricity anda consequent tidal decay. As their host evolves, such planets would be rapidly draggedin or engulfed at minimum periapse, leading to a rapid depletion of the populationwith increasing stellar radius, as is observed. Using numerical simulations, we deter-mine the relationship between periapse distance and orbital migration rate for planets0.1 to 10 Jupiter masses and with orbital periods between 10 and 100 days. We findthat Kozai-Lidov oscillations effectively result in planetary removal early in the evolu-tion of the host star, possibly accounting for the observed deficit. While the observedeccentricity distribution is inconsistent with the simulated distribution for an oscil-lating and migrating warm jupiter population, observational biases may explain thediscrepancy.
Key words: warm jupiters – planet-star interactions – planets and satellites: dy-namical evolution and stability
Arguably the biggest surprise in the field of exoplanets wasthe discovery of hot jupiters (HJ): extrasolar planets withorbital periods less than 10 days but masses near that ofJupiter ( M J ) (Mayor & Queloz 1995; Butler et al. 1997).The proximity of these planets to their host precludes themfrom forming at their observed location (Bodenheimer et al.2000), indicating that they must have migrated after forma-tion. A range of migration mechanisms have been proposed,including disc migration (Lin et al. 1996) and planet-planetscattering (Rasio & Ford 1996; Weidenschilling & Marzari1996), but the existence of HJs that are inclined relative tothe spin of their host star (H´ebrard et al. 2008; Winn et al.2010; Triaud et al. 2010; Albrecht et al. 2012) indicates thatat least some migrated via a mechanism that excites theplanetary inclination to high values. One of these is theKozai-Lidov (KL) mechanism (Kozai 1962; Lidov 1962),in which an inner body oscillates between highly eccen-tric and highly inclined modes due to an inclined, externalperturber. Recent results have shown that the KL mech-anism naturally leads to misaligned and flipped planetaryorbits, indicating it may contribute significantly to the for- ⋆ E-mail: [email protected] mation of HJs (Naoz et al. 2011, 2012, 2013; Li et al. 2014;Teyssandier et al. 2013; Petrovich 2015).The efficiency of tidal circularisation is a strong functionof distance, and so it is not surprising that high-eccentricitymigration is also expected to yield a population of warmjupiters (WJs). These planets are similar to HJs but orbitat larger periods of 10 to 100 days, because their migra-tion timescales are comparable to the age of the system.Many of the systems observed in this period range have ob-served eccentricities too small for significant tidal evolution,but this can be understood as a consequence of the eccen-tricity oscillations inherent in the KL mechanism. However,the difference between a WJ population with fixed eccen-tricities and oscillating eccentricities becomes very impor-tant when we consider the WJ population around evolv-ing stars. As stars evolve off the main sequence and in-crease in size, they can tidally drag in and engulf plan-ets orbiting too closely (Rasio et al. 1996; Passy et al. 2012;Nordhaus & Spiegel 2013; Li et al. 2014). The eccentricityof a planet plays an important role in how long it survives,as eccentricity oscillations are rapid compared to stellar evo-lution timescales, and so the planets will be removed whenthe star approaches their minimum periastron. Tidal effectsare also dramatically increased for highly eccentric planets.For planets undergoing KL oscillations, the survival of their © S. Frewen & B. Hansen orbits are thus determined by their maximum rather thancurrent or observed eccentricity. An observed population (orlack thereof) of WJs around evolved stars can then give usinsight into whether the population is made up of planetswith constant eccentricities, or if most go through phasesof significantly larger eccentricity. This process may explainthe lack of HJs and WJs observed around subgiant stars(Johnson et al. 2007, 2010; Schlaufman & Winn 2013), asshown in Figure 1.In this paper we examine how a population of migrat-ing and oscillating WJs would be affected by the evolutionof their host stars compared to an observationally identicalpopulation with constant eccentricities, and determine howit compares to observations. To do so, we run numerical sim-ulations of a WJ and a perturber over the full period rangeof WJ and determine the relationship between system prop-erties, the closest approach of the planet, and how rapidlythe planets move inward. With this data we create modelpopulations that match the observed distribution, and cal-culate how such populations are winnowed by stellar evolu-tion, both in the case where eccentricities oscillate and incontrol populations with constant eccentricity. We find thatKL oscillations do cause planets to be removed much earlierin stellar evolution, in line with the observed distributionof stellar sizes for WJ hosts. The oscillations required toproduce inward migration also skew the eccentricity distri-bution to values higher than those observed, but this couldbe due to observational biases, as we shall discuss.The structure of the paper is as follows: In Section 2 weestimate the number of WJs predicted around evolved starsrelative to the number observed. In Section 3 we review therelevant dynamics taking place in systems undergoing KLoscillations. In Section 4 we setup our numerical simula-tions of migrating, oscillating WJs. In section 5 we discussour numerical results, and examine the relative effect of stel-lar expansion on oscillating and non-oscillating populationsin Section 6. In Section 7 we compare our results to obser-vations and discuss the possibility that observational biasexplains the discrepancy between the observed and simu-lated eccentricity distributions. In Section 8 we review ourconclusions.
The lack of HJs and WJs around stars larger than 2 R ⊙ isapparent from a cursory examination of Figure 1. While theshort-period ( .
10 days) orbits of HJs naturally lead to theirengulfment early on in stellar evolution, the lack of WJs atsimilar stellar sizes in spite of orbits ∼
10 times larger issurprising. However, the number of WJs around stars of allsizes, and the number of exoplanets around evolved starsat all periods, are both significantly lower than in otherregions of exoplanet period–stellar-radius parameter space.This contrast raises the question of whether the number ofWJs around larger stars is genuinely below observationalpredictions, or if they simply suffer from poor statistics.We answer this question using the observed number of WJsaround main-sequence stars ( R ∗ = 1 − R ⊙ ) and their ob-served periapse values, combined with the observed numberof lukewarm Jupiters (LJs, periods of 100 − Orbital Period (days) 10 R a d i u s o f S t a r ( R ⊙ ) Figure 1.
Confirmed exoplanets from Exoplanet Orbit Databasewith M p sin i > . M J or R p > . R J , as of June 2015. Or-bital periods are shown as a function of host star radius. The WJpopulation is seen from 10–100 days but is conspicuously absentaround stars with radii > R ⊙ , although more distant planetsremain quite common. R ∗ is N p,obs ( R ∗ ) = N ∗ , obs ( R ∗ ) f p,0 f p,S ( R ∗ ) (1)where N ∗ , obs ( R ∗ ) is the number of stars observed at a givenstellar radius, f p,0 is the initial frequency of planets aroundthe stars, and f p,S ( R ∗ ) is the fraction of planets that havesurvived prior stellar evolution.For WJs and LJs around main-sequence stars, we get N WJ (1 − R ⊙ ) = N ∗ , obs (1 − R ⊙ ) f WJ,0 f WJ,S (1 − R ⊙ )(2) N LJ (1 − R ⊙ ) = N ∗ , obs (1 − R ⊙ ) f LJ,0 f LJ,S (1 − R ⊙ ) (3)Assuming survival rates are similar without stellar evo-lution, the relative fraction of stars with WJs and LJs is: f WJ,0 f LJ,0 = N WJ (1 − R ⊙ ) N LJ (1 − R ⊙ ) (4)Equation 1 holds for larger stellar radii as well: N WJ ( > R ⊙ ) = N ∗ , obs ( > R ⊙ ) f WJ,0 f WJ,S ( > R ⊙ ) (5) N LJ ( > R ⊙ ) = N ∗ , obs ( > R ⊙ ) f LJ,0 f LJ,S ( > R ⊙ ) (6)Dividing these two equations, we get the predicted num-ber of WJs around evolved stars based on their observednumber around main-sequence stars and observed numberof HJs: N WJ ( > R ⊙ ) = N WJ (1 − R ⊙ ) N LJ (1 − R ⊙ ) f WJ,S ( > R ⊙ ) f LJ,S ( > R ⊙ ) N LJ ( > R ⊙ )(7)Assuming all LJs survive ( f LJ,S ( > R ⊙ ) = 1) gives themost conservative estimate for the number of WJs. Obser-vations provide values for N WJ (1 − R ⊙ ), N LJ (1 − R ⊙ ),and N LJ ( R ∗ > R ⊙ ), as detailed below, so an estimate for f WJ,S ( > R ⊙ ) allowed us to calculate the predicted num-ber of WJs around evolved stars, N WJ ( R ∗ > R ⊙ ). To doso we assumed the observed periapse distribution for WJsaround unevolved stars is representative of the true peri-apse distribution. We also assumed that exoplanets are re-moved when their periapse comes within 2.5 stellar radii © , 1–16 ffect of Stellar Evolution on Migrating WJs ⊙ )01020304050607080 N u m b e r o f P l a n e t s Observed, 100-1000 daysObserved, 10-100 daysPredicted, 10-100 days
Figure 2.
The red bars show the number of observed planets withorbital periods between 100–1000 days (LJ), as a function of hoststar radius. The light blue bars show the equivalent WJ popula-tion (orbital periods 10–100 days). The dark blue bars representthe expected WJ population if we simply scale the LJ populationto match the overall occurrence rate. The lack of observed (lightblue) planets relative to expected (dark blue) planets is apparantfor stellar radii > R ⊙ . of their host (based on the smallest observed periapse-to-stellar-radius ratio, 2.7, in the case of WASP-12b as reportedby Maciejewski et al. 2011).The data for this estimation came from Exoplanet Or-bit Database (Wright et al. 2011). We limited our datasetto massive planets with listed eccentricity values. We alsoexcluded possible brown dwarfs , as such massive bodiesmay have formed via a different mechanism than exoplan-ets. Finally, we used the periapse distribution for planetsaround stars with radii 1 − R ⊙ , rather than including plan-ets around smaller or larger stars . From these values wecalculated the predicted number of observed WJs as a func-tion of stellar radius.As illustrated by Figure 2, this calculation predicted asignificant population (15) of observed WJs around evolvedstars, which is inconsistent with the observed number (2). If,however, each WJ is oscillating between some minimum andmaximum value of eccentricity, then fewer will survive stellarexpansion as they are removed at their minimum periapse(at maximum eccentricity), rather than the value currentlyobserved. The actual maximum eccentricity will depend ona particular planet–perturber configuration, but we can as-sess the direction of the effect by assuming all WJs are un-dergoing these oscillations up to a maximum eccentricity of e max = 0 .
85, from which calculated a predicted number (2)that equals the value from observations (Figure 3).We make a similar estimate of the number of missingHJs using planets on periods < Due to some anomalously low values in the
MASS keyword, weincluded planets with either
MSINI > 0.1 or R > 0.5 . To avoid excluding planets on circular orbits, we used the filter
ECC > -1 . using the limit MASS < 10 . The periapse dataset used
RSTAR >= 1.0 and RSTAR < 2.0 . ⊙ )051015202530 N u m b e r o f P l a n e t s Observed, 10-100 daysPredicted (Observed e )Predicted with Kozai ( e max =0.85 ) Figure 3.
We repeat the comparison of Figure 2, comparing theobserved (light blue) WJ population as a function of host star ra-dius to that expected from a scaled population of LJ (dark blue).We now also include a prediction (green bars) which assumes thatthe scaled LJ population all oscillate in eccentricity on timescalesshort compared to the stellar evolution timescale, with a maxi-mum eccentricity = 0.85. We see that this removes most of theexcess predicted population at stellar radii above 2 R ⊙ . such short periods. Figure 4 shows that the observed numberof planets agrees with the number predicted by the currenteccentricity distribution and does not benefit from a peri-apse distribution shifted to lower values.This brief calculation illustrates that the lack of WJs isunlikely to be a simple statistical fluctuation, and necessi-tates an explanation. We note that it does include a numberof assumptions, foremost being that the relative frequencyof WJs to LJs is independent of stellar radius outside of re-moval via tides. However, the purpose of this calculation isnot to determine the precise number of WJs removed to duestellar evolution, but rather to demonstrate that an absenceexists and can be accounted for they are oscillating to highereccentricity values, as would be required for migration viatides. A more detailed analysis follows in Section 6, withfurther discussion of assumptions in Section 7. The KL mechanism results from secular (long-term) interac-tions between an inner and outer body when their mutual in-clination exceeds some nominal value, or if both bodies havesignificant eccentricity while coplanar (Li et al. 2014). In thesimplest case, where the inner body has negligible mass andthe outer body is on a circular orbit, the z-component of theangular momentum is constant for the inner body:cos i in q − e = Const (8)where i in is the inclination of the inner body (identical tothe mutual inclination in the massless case) and e in is itseccentricity (Kozai 1962; Lidov 1962).This relationship requires that a decrease in mutual in-clination between the two bodies is accompanied by an in- © , 1–16 S. Frewen & B. Hansen ⊙ )01020304050607080 N u m b e r o f P l a n e t s Observed, 100-1000 daysObserved, 0-3 daysPredicted, 0-3 days
Figure 4.
The observed number of LJs and the observed and pre-dicted number of very hot jupiters (Period < crease in the eccentricity of the inner body. As a result,the inner body undergoes oscillations in eccentricity andinclination under the influence of the outer companion. Inthis simple case these oscillations are characterised by theirtime-scale ( P Kozai ) and maximum eccentricity (Lidov 1962;Kiseleva et al. 1998): P Kozai = 23 π P P in M tot M (1 − e ) / (9) e calc = p − (5 /
3) cos i (10)where P in and P out are the inner and outer periods, respec-tively; M and M tot are perturber mass and total mass of allthe bodies, respectively; e out is the perturber eccentricity; i is the minimum value of i in ; and e calc is the calculated max-imum eccentricity. The maximum eccentricity has a morecomplicated, non-linear form when the inner body is mas-sive. Naoz et al. (2013) derive the equation in the case of noinitial eccentricity and a perturber on a circular orbit: (cid:18) L L (cid:19) e + L L cos i + (cid:18) L L (cid:19) ! e + L L cos i − i = 0 (11)where L and L are the scaled angular momenta of theinner and outer orbit, respectively. These are defined as: L = M M M + M p G ( M + M ) a in (12) L = M ( M + M ) M + M + M p G ( M + M + M ) a out (13)where M , M , and M are the masses of the central body,inner body, and perturber, respectively, and a in and a out arethe inner and outer semi-major axes (SMA). Including theinitial eccentricities of both orbits modifies the calculationonly slightly and gives a value that differs at most by a fewpercent.The KL mechanism has been covered extensively in the literature in a number of contexts, includ-ing asteroids (Fang & Margot 2012), exoplanet systems(Naoz et al. 2011; Petrovich 2015), the dynamics ofthe Galactic Center (L¨ockmann et al. 2008), and stel-lar triple systems (Eggleton & Kiseleva-Eggleton 2001;Fabrycky & Tremaine 2007; Thompson 2011; Prodan et al.2013; Naoz & Fabrycky 2014). Recently it has been shownthat the inclusion of higher-order terms can dramaticallyalter the oscillations induced by the KL mechanism. Theseoctupole terms, which are non-zero if the inner body is notmassless or the outer body has a non-zero eccentricity, canresult in larger eccentricities, flips of the inner orbit, andchaotic behavior (Katz et al. 2011; Lithwick & Naoz 2011;Naoz et al. 2013). The full equations have no analytical so-lution for the maximum eccentricity of the inner orbit. How-ever, Equation 11 still provides an adequate first-order esti-mate of e max , which we use in Section 4 to limit our parame-ter space to those systems that could be capable of migratinginward.In the absence of any other effects, oscillating bodiescan approach within an arbitrary distance of the surface ofthe star as long as they avoid collision. However, two effectsprevent that from happening: general relativity (GR) andtides. For short-period orbits, GR causes a precession of apsideson a time-scale that depends on the properties of the orbitand the host star: P GR = P / c (1 − e )3(2 π / )( GM ) / (14)If P Kozai is longer than this time-scale, GR precessioncan damp and eliminate KL oscillations. For this reason,orbits with shorter periods require stronger perturbers toundergo oscillations: those that are more massive, closer,and/or more eccentric. As shown in Dong et al. (2014), WJsneed perturbers within 10 au to undergo the high eccentric-ity migration discussed here. This constraint is due in partto the eccentricity dependence of the GR time-scale. With-out a strong enough perturber, oscillating bodies that reachvery high eccentricities can be stranded at their maximumeccentricity. When tides are taken into account, that canlead to rapid evolution into a HJ. While the detailed effectsof GR are significantly more complicated (Naoz et al. 2013),the damping interpretation is adequate for our purposes.
Both the KL mechanism and GR conserve orbital energy,ensuring that the SMA of the inner orbit is constant. WJscan only migrate when under the influence of a dissipativeforce, which takes the form of tidal friction. The existence ofWJs at their current periods, as well as their non-zero eccen-tricities, indicate that they must have large circularizationand tidal decay time-scales as a result of weak tidal forces. Inthis section we describe tidal forces that hold for two bodies © , 1–16 ffect of Stellar Evolution on Migrating WJs in general, but in our simulations apply specifically to theplanet and star.The effects of tidal forces on orbital evolution (first in-vestigated in the context of planets and satellites, see Darwin1880) have been investigated in detail for stars, showing thattides reduce orbital energy and lead to smaller and more cir-cular orbits (Hut 1981; Eggleton et al. 1998; Kiseleva et al.1998). For two tidally interacting bodies, whether massiveplanets or stars, the strength of tides raised on an object1 by an object 2 is characterised by the tidal friction time-scale, as described in Eggleton & Kiseleva-Eggleton (2001)and Fabrycky & Tremaine (2007): t F = t V a R M ( M + M ) M (1 + 2 k ) − (15)where a is the SMA of the orbit and R is the radius of ob-ject 1. The internal structure of object 1 is included by wayof k , the classical apsidal motion constant, which representsthe quadrupolar deformability of the star or planet, and t V ,the viscous time-scale, which is a parametrization of inter-nal dissipation in the star (Zahn 1977). The physical valuesparametrizing tidal evolution are still not fully understood,although they have been investigated by a number of au-thors. The planetary k is frequently set to k P = 0 .
25, theresult for a n = 1 polytrope representing gas giants. Recentresearch has gone into matching t V to observations, includ-ing the Jupiter-Io system, the eccentricity distribution ofhot Jupiters, and the existence of high eccentricity exoplan-ets. Hansen (2010) calibrated tidal models to observationsof massive exoplanets (0.3-3 M J ) around solar-type stars us-ing a single tidal dissipation constant for each population,and found that t V for a Jupiter-mass planet with moderateeccentricity is t V p = 150 years. However, longer orbital peri-ods may imply greater dissipation as they couple to a largerfraction of the internal turbulent viscosity. Hansen (2012)finds a roughly linear increase in the dissipation rate withorbital period. Furthermore, Socrates et al. (2012) repeateda similar calibration in the high eccentricity limit and deter-mined that planets undergoing high-eccentricity migrationrequire tides equivalent to t V p = 1 year. It is this value weuse in the numerical simulations of Section 5.Hansen (2010) also found that tides raised on the solar-type host stars by Jovian-mass planets were a factor of 50weaker than those raised on the planets by the stars, al-lowing us to ignore stellar tides for our numerical simula-tions. However, this inequality does not hold as stars evolve.The strong radius dependence of t F ∗ indicates that as a starleaves the main sequence it will increase its contribution tothe planet’s orbital evolution until stellar tides dominate orthe star engulfs the planet. Once the star dominates tidaleffects the strong radius dependence will rapidly acceleratethe inward migration of the planet. Whether this increasein migration rate occurs before direct collision with the stardepends on the migration rate due to planetary tides alone,as discussed in greater detail in Section 6.2. When stellartides are included, we use the values k ∗ = 0 . n = 3 polytrope, and t V ∗ = 50 years, from the equation pro-vided in Eggleton & Kiseleva-Eggleton (2001). Stellar tidesare likely weaker than this value, as seen by the results ofHansen (2010), but our choice of t V ∗ does not affect ourconclusion as long as it is longer than t V p . Tidal forces also exert a torque on a planet, changing itsspin and aligning it on time-scales much shorter than thoserequired to circularize the orbit or move the planet inward.Planetary systems residing in the WJ period range as a re-sult of migration should have reached an equilibrium in theirspin as a result of this effect. In the case of planets not un-dergoing oscillations in eccentricity, the equilibrium spin canbe determined by the value which results in no torque, orpseudo-synchronous (PS) spin (Hut 1981). In the case oflow eccentricity, the planetary spin period is the same asits orbital period (synchronous rotation). For large valuesof eccentricity, the planet is moving much more rapidly atperiapse, where tidal forces are strongest, and as a result theplanet rotation period can be less than 1 percent of the or-bital period. Those planets rotating faster than the PS valuewill have angular momentum transferred from its rotationto its orbit, which can result in a modest increase in SMA.
Finally, for these three-body systems to exist they must bestable. While KL oscillations require a strong perturber toavoid damping by GR, a perturber that is too near to theinner orbit will destabilize the system. The limit for stabilityin mutually inclined systems with an eccentric perturber wascalculated by Mardling & Aarseth (2001): a out a in > . q ) / (1 + e out ) / (1 − e out ) / (cid:18) − . i tot ◦ (cid:19) (16)where q = M / ( M ∗ + M p ). This criterion has been used inprior investigations of KL oscillations in exoplanet systems,including Teyssandier et al. (2013) and Rice (2015).With these effects in mind, the planetary systems wewant to investigate are those that are undergoing KL oscil-lations, requiring P Kozai < P GR over the full range of eccen-tricities that the planet reaches. The maximum eccentricitydue to oscillations should be large enough to induce tidaldecay, but over a time-scale large enough that a populationof WJs would be detectable. The orbital evolution of an oscillating planet depends onboth the maximum eccentricity and the distribution of ec-centricity values over time. These properties of the systemdo not have an analytical form. The maximum eccentricitydeviates from e calc in Equation 11 due to octupole terms,while the eccentricity distribution has no analytic form evenwithout octupole terms. In order to understand the orbitalevolution of migrating WJs, we need to use numerical sim-ulations spanning the parameter space of interesting sys-tems. These simulations allow us to determine the orbitaldecay as a function of initial period, mass, and eccentric-ity, as well as determine the relationship between calcu-lated ( e calc ) and true maximum eccentricity. In our simu-lations we use the code of S. Naoz, which integrates thethree-body secular equations up to the octupole level ofapproximation as described in Naoz et al. (2013), includ-ing GR effects for the inner and outer orbits and tidal © , 1–16 S. Frewen & B. Hansen effects following Eggleton & Kiseleva-Eggleton (2001) andFabrycky & Tremaine (2007). This code has been used ex-tensively in numerous calculations e.g. Naoz et al. (2011,2012); Naoz & Fabrycky (2014); Li et al. (2014, 2015).
Each system is composed of a central star, an inner body(hereafter referred to as “planet”), and an outer body (here-after referred to as “perturber”). For our star, we selected amass of 1.2 M ⊙ and ignored the contribution of tides raisedon the star to the evolution of our planetary orbit (see Sec-tion 3.3.1). The other properties of our systems were chosento produce all three of the following properties in the planet: • Warm ( P = 10 −
100 days) Jupiters ( M p = 0 . − M J ) • Undergoing KL oscillations • Experiencing tidal migration on a plausible timescaleEach of these requirements introduces constraints onto thepopulation. The first constrains the mass and period of theplanets, the second constrains the perturber such that GRtime-scale is longer than the KL time-scale, and the thirdconstrains the planet to reach high eccentricities during os-cillations. The consequences for perturber and planet prop-erties are discussed below.
We limited the parameter space of our primary simulationsby keeping the perturber constant across them. We selectedits properties such that it caused KL oscillations in the sys-tems with semi-major axis ∼ . M J body at 2 auwith an eccentricity of 0.13, similar to the planets that aresometimes detected as companions to HJs (Knutson et al.2014). Plugging these numbers into Equation 16, we findthat the limit for stability is a out /a in > .
5. Our systemshave a out /a in = 4 . −
20, clearly in the stable regime. Ad-ditionally, we ran two other sets of simulations: one withsimply a larger eccentricity (0.35), and one with a larger(30 M J ), more eccentric ( e = 0 .
64) perturber at a larger dis-tance (10 au), both discussed in Section 5.3.2. These simu-lations showed that while the perturber plays a pivotal rolein the planetary oscillations and migration, the perturberproperties did not impact our general results.
We defined our population of WJs to have masses 0 . − M J and orbital periods from 10 −
100 days, or semi-major axes0 . − .
45 au. For simplicity, we set the size of all planets to 1Jupiter radius, with k p = 0 .
25 and t V = 1 year as describedin Section 3.3.1. While planets at the low-mass end of ourpopulation are unlikely to be this large, there is not a firmmass-radius relationship for extrasolar planets at this point.We discuss the impact of this assumption in Section 5.2.2.To generate the properties of our planet, we first randomlysampled the mass range, initial eccentricity, and mutual in-clination. We did so logarithmically in mass and uniformly C a l c u l a t e d M a x i m u m E cc e n t r i c i t y Figure 5.
A randomly generated distribution of planets basedon our limits (black points), along with the bins (blue lines), anda set of selected systems (red dots). This approach ensured wesimulated a wide range of systems rather than be dominated bythe portions of parameter space with the majority of points. in initial eccentricity and inclination, limiting the latter to0 − . ◦ − ◦ . We chose these bound-aries based on the properties of the Kozai-Lidov oscillationswhich are needed to match our requirement that the mini-mum eccentricities be close to circular – to match observa-tions – while still allowing planets to reach sufficiently largeeccentricities to evolve tidally on a short enough timescale.From these properties and those of the perturber, wecalculated e calc for all samples using Equation 11. We thenselected a uniform distribution in initial eccentricity and e calc by dividing the parameter space up into a grid andselecting systems from each grid box, as shown in Figure5. This approach allowed us to probe the wide range of be-havior caused by different minimum periapse values whilestill limiting computation time. We constrained the initialeccentricity to between 0 and 0.1 and e calc between 0.75and 1.0 to produce oscillating systems that had eccentric-ity values enabling migration. While e calc is only accuratefor systems without any contribution from octupole terms,it gave us a first approximation and allowed us to excludesystems that are unlikely to migrate. We also set both ar-guments of periapse to zero in order to limit our parameterspace, as other groups have done (Teyssandier et al. 2013).Generally speaking, this assumption is equivalent to max-imizing the effect of the companion, leading to the largestpeak eccentricity. As described in Section 3.3.2, planetary rotation can havea significant effect on orbital evolution via tides. We testedthis by running a set of simulations with a range of planetaryrotation rates and found that as expected planets spinningfaster than the equilibrium (PS) rate migrated more slowly,or in some cases migrated outwards. Because these systemsare assumed to be migrating WJs, which originated beyondperiods of 100 days, they should have already reached PSrotation. For our main simulations, we used e calc to esti-mate the PS rotation period. However, the planets nearest © , 1–16 ffect of Stellar Evolution on Migrating WJs to the star (10 and 20 days) reached maximum eccentricityvalues significantly different from e calc , due to GR and tidaleffects, resulting in planets spinning too rapidly. To correctfor this, we performed a linear fit between the calculated andsimulated maximum eccentricity, and used the derived ec-centricity value to set the correct rotation rate. We also seta lower limit on the spin period by capping the eccentricityused in its calculation to 1 − R ⊙ /a , the value that bringsthe planetary periapse to 2 R ⊙ . This limit avoided spin ratesthat were unreasonably fast, exceeding the maximum phys-ical rotation rate of a Jupiter-mass planet. We ran a total of 1,320 simulations across 6 period values:10, 20, 30, 50, 70 and 100 days. We also ran another 384 withdifferent perturbers and 192 with reduced viscous time-scale,both across the same period range. The goal is not to simu-late the transition from WJ to HJ, because we are interestedin those which have not yet completed such a transition onan astrophysical timescale yet are evolving fast enough tohave moved to their current locations. All simulations wererun for 10 years or until the orbit of the planet decayedby 10 percent, whichever occurred first. While this is shortcompared to the full migration time it is long enough toencapsulate many eccentricity oscillations and thus charac-terise the rate of migration at the observed stage.For comparison to systems not undergoing oscillations,we also ran an additional set of 192 simulations over thesame period bins without the effects of a perturber. Thesesimulations spanned both the same mass range (3 bins: 0.1,1, 10 M J ) and the full eccentricity range (19 bins from 0 to0.95), and began with the theoretical value for PS rotationat their eccentricity. As a case study, we selected a system with M p = 0 . M J at50 days (0.28 au), with an initial eccentricity of e = 0 . i = 72 . ◦ . Using Equa-tion 11 we determined e calc = 0 .
91, which is large enoughto drive inward migration as long as the planet is not spin-ning extremely rapidly. This eccentricity corresponds to aperiastron of 5 R ⊙ at closest approach, well outside of the2 R ⊙ limit we set for PS rotation calculations in Section4.1.3. We set the rotation period to the calculated valueof P rot /P orb = 42, or P rot = 1 . . × yrs, froma direct numerical simulation.Figure 7 shows the relative fraction of the time spent ateach eccentricity when the simulation is run for 10 years.We found that the eccentricity peaked at a value of 0.93,slightly larger than our calculation (Figure 7). Additionally,the difference between the full 10 year distribution and thedistribution in the last 10 years (thick line) showed that theminimum eccentricity during oscillations increased slightlyover time. Given the distribution of eccentricity as a functionof time, a system with these properties would most likely be Figure 6.
The upper panel shows the eccentricity evolution andthe lower panel shows the inclination evolution. This shows thecharacteristic sweep from the initial low eccentricity to very highvalues, as the inclination falls to its minimum.
Period: 50 days, e =0.08, i =72.2 ◦ Full simulationLast 10 percent
Figure 7.
Eccentricity distribution of our 50-day case studyplanet, migrating inwards with ∆ a/ ∆ t = − year simulation (thin line) and the final 10 years (thickline). The vertical lines show the 75th, 90th and 98th percentileof the distribution of observed WJ eccentricities (dashed, fromthe Exoplanet Orbit Database) and our simulated eccentricitydistribution (dotted). We see that, for this case, the simulatedeccentricity distribution is biased high relative to the observedone. observed with e < .
4, but would be detected 25 percent ofthe time with e > .
7. Migrating inward at 3 au per Gyr,such a planet would survive for significantly less than themigration time-scale a/ ˙ a = 0 .
094 Gyr, due the increase inthe strength of tides as the SMA gets smaller. A migrationtime-scale of this duration is short compared to the agesof WJ host stars, indicating that this type of system couldhave migrated to its current location from farther out.
Repeating this process on all 1,320 systems produces theresults shown in Figure 8: an instantaneous migration rate © , 1–16 S. Frewen & B. Hansen −11 −10 −9 −8 −7 −6 −5 −4 −3log(∆a/∆t) (AU/yr)12345678910 L a r g e t p e r i a p e ( R ⊙ ) Vi cou time- cale: 1.00 year
10 day 20 day 30 day 50 day 70 day 100 day
Figure 9.
The largest minimum periapse that resulted in theplanet migrating at a given rate, for each orbital period. Thesystems used in Section 6 fall between the dotted and dashedlines. (∆ a/ ∆ t ) as a function of minimum periapse/maximum ec-centricity and planetary mass. These plots illustrate the ex-tremely strong dependence of migration rate on maximumeccentricity, as expected. In all simulations with planet pe-riods longer than 10 days, orbital migration only took placewhen the maximum eccentricity exceeded 0.8. This result issignificant, as all observed WJs have eccentricities below thisvalue (see Figure 18), which will be discussed in Section 7.Notably, the same periapse distance results in similar migra-tion rates regardless of period (Figure 9). This result playsan important role when determining how the population ofWJs is affected by stellar evolution in Section 6. The partialexception to this phenomenon are those planets at 10 dayperiods, which are near enough to their host to experiencetidal effects with even moderate eccentricity.Additionally, some systems at larger periods migratedoutward rather than inward. In these cases our estimate forthe rotation rate was too high, possibly due to the effectof octupole terms in the KL oscillations or tidal effects,and they experienced outward migration due to their spindown. The relative symmetry between the outward and in-ward moving planets is due to the magnitude of migrationbeing set primarily by the product of the tidal friction time-scale (Equation 15) and a function of eccentricity dominatedby a (1 − e ) − / coefficient. We also note that none of oursystems spent any time on retrograde orbits. This result isin line with the findings of Teyssandier et al. (2013), whichshowed that highly inclined systems are significantly poorerat causing flips in the planet. In our simulations only highlyinclined systems produce large eccentricities, and as a resultall stayed prograde. The eccentricity distributions of individual planets, alongwith the setup of the system, determined the magnitude ofmigration. Systems rapidly migrating ( da/dt > au/Gyr)tended to peak more strongly at the high-eccentricity value,as seen in Figure 10. Systems migrating on smaller time- Period: 30 days, e =0.07, i =72.8 ◦ Full simulationLast 10 percent
Figure 10.
Eccentricity distribution of a rapidly migrating planetat 30 days ( da/dt = − × au/Gyr), illustrating that the ma-jority of time is spent at high eccentricities during the last 10percent of simulation time. This planet migrates too fast to be aplausible WJ candidate as it should rapidly circularise into a HJorbit. Period: 10 days, e =0.04, i =84.8 ◦ Full simulationLast 10 percent
Figure 11.
Eccentricity distribution of a planet at 10 dayswith damped eccentricity oscillations, where the KL time-scaleis roughly equal to the GR time-scale. scales (10 − − au/Gyr) generally peaked near e = 0 − . The relationship between mass and migration rate, withmore massive planets migrating slower and less massiveplanets migrating rapidly, showed up in all periods (Figure8) as a result of Equation 15. The strength of tides dependson planetary mass and radius, with more massive planetshaving stronger surface gravity and correspondingly weakertides. Massive planets do produce larger tides in their hoststar, but our simulations ignored stellar tides due to theirrelative weakness, even with large planetary mass. As a re-sult of keeping a constant perturber and planetary radius,small planets migrated the fastest and larger planets theslowest. © , 1–16 ffect of Stellar Evolution on Migrating WJs J ) Minimum Periapse (R ⊙ )-10 -10 -10 -10 -1 -10 -2 -2 -1 ∆ a / ∆ t ( A U / G y r ) Min: dadt =0.1 AU/GyrMax: dadt =100 AU/GyrMin: dadt =0.1 AU/GyrMax: dadt =100 AU/Gyr0.0.50.70.80.90.95 Eccentricity
Period: 10 days Minimum Periapse (R ⊙ )-10 -10 -10 -10 -1 -10 -2 -2 -1 ∆ a / ∆ t ( A U / G y r ) Min: dadt =0.1 AU/GyrMax: dadt =100 AU/GyrMin: dadt =0.1 AU/GyrMax: dadt =100 AU/Gyr0.0.50.70.80.90.95 Eccentricity
Period: 20 days Minimum Periapse (R ⊙ )-10 -10 -10 -10 -1 -10 -2 -2 -1 ∆ a / ∆ t ( A U / G y r ) Min: dadt =0.1 AU/GyrMax: dadt =100 AU/GyrMin: dadt =0.1 AU/GyrMax: dadt =100 AU/Gyr0.0.50.70.80.90.95 Eccentricity
Period: 30 days Minimum Periapse (R ⊙ )-10 -10 -10 -10 -1 -10 -2 -2 -1 ∆ a / ∆ t ( A U / G y r ) Min: dadt =0.1 AU/GyrMax: dadt =100 AU/GyrMin: dadt =0.1 AU/GyrMax: dadt =100 AU/Gyr0.0.50.70.80.90.95 Eccentricity
Period: 50 days Minimum Periapse (R ⊙ )-10 -10 -10 -10 -1 -10 -2 -2 -1 ∆ a / ∆ t ( A U / G y r ) Min: dadt =0.1 AU/GyrMax: dadt =100 AU/GyrMin: dadt =0.1 AU/GyrMax: dadt =100 AU/Gyr0.0.50.70.80.90.95 Eccentricity
Period: 70 days Minimum Periapse (R ⊙ )-10 -10 -10 -10 -1 -10 -2 -2 -1 ∆ a / ∆ t ( A U / G y r ) Min: dadt =0.1 AU/GyrMax: dadt =100 AU/GyrMin: dadt =0.1 AU/GyrMax: dadt =100 AU/Gyr0.0.50.70.80.90.95 Eccentricity
Period: 100 days
Figure 8.
The change in semi-major axis over the duration of each simulation, grouped by orbital period. The x-axis is minimumperiapse/maximum eccentricity while the color of the points gives the mass of planet. The dotted (maximum) and dashed (minimum)black lines correspond to the approximate limits on migration rate necessary to produce the WJ population. If they migrate too fast,they will be observed as HJ, and if they migrate too slowly, they will remain in the LJ population. The blue dotted line indicates thetidal disruption radius. © , 1–16 S. Frewen & B. Hansen −11 −10 −9 −8 −7 −6 −5 −4 −3log(∆a/∆t) (AU/yr)246810121416 L a r g e t p e r i a p e ( R ⊙ ) Vi cou time- cale: 0.01 year
10 day 20 day 30 day 50 day 70 day 100 day
Figure 12.
Same as Figure 9, but in the case of a much smaller(0.01) viscous time-scale. Note the larger scale on the y-axis dueto smaller maximum eccentricity at similar migration rates.
A viscous time-scale of 0.01 year resulted in planets withhigher migration rates and smaller maximum eccentricitiesthan our primary ( t V = 1 year) simulations (Figure 12).Planets on 10-day periods did require lower eccentricity tomigrate inward, and reached much smaller maximum eccen-tricities than those with longer periods due to their rapidcircularization. This shorter viscous time-scale led to fewerplanets reaching very high eccentricities, which is more sim-ilar to what is seen in observations (Figure 18). However,the smaller maximum eccentricities result in larger periapsedistances, which results in a larger population surviving tolarger stellar radii. Comparison to observations will be dis-cussed more thoroughly in Section 7. As described in Section 4.1.1, our simulations also includedtwo smaller samples with altered perturbers for comparison.In the first of these, we increased the perturber eccentricityto near the limit of stability, 0.35, while leaving the otherproperties (period and mass) the same. The increase in ec-centricity resulted in a small shift to larger maximum eccen-tricities and a resulting slight increase in the overall inwardmigration rate. The effect was extremely minor, as seen inFigure 13.The second sample had a dramatically different per-turber, one on a significantly larger orbit (10 au = 10 dayperiod), with a larger mass (30 M J ), and greater eccentric-ity ( e = 0 . da/dt = 0 than with the close in perturber,due to larger maximum eccentricities, but fewer reached veryhigh migration rates. Period: 30 days, e =0.05, i =70.5 ◦ e =0.133e =0.35 Figure 13.
Eccentricity frequency distribution of a 30-day periodplanet (in arbitrary units), for both the default perturber eccen-tricity (0.133) and the larger value (0.35). The more eccentricperturber produces marginally higher eccentricities on average.
Regardless of perturber or planetary properties, signifi-cant da/dt required the planet to reach very large eccentric-ity values in the vast majority of those with periods longerthan 10 days. Additionally, of those undergoing migrationwithout reaching large maximum eccentricity, the majoritydid so due to a minimum eccentricity above 0.2, a valuehigher than most observed WJs.
The essence of our model is that currently observed plan-etary eccentricities substantially underrepresent the rate oftidal migration experienced by the system because of tran-sient episodes in which the eccentricity oscillates to muchlarger values. Thus, as a comparison set we can examine apopulation of planets whose eccentricities do not oscillate.These unperturbed planets behaved as expected, migratingby much larger amounts as compared to oscillating systemsof equal maximum eccentricity. As shown by the lines in Fig-ure 14, the migration magnitude is well fit by an analyticalformula of the form dadt = − × − f e ( e ) (cid:16) a p . (cid:17) − (cid:18) M p M J (cid:19) − . au/Gyr(17)where f e ( e ) is a function of eccentricity derived from thetidal equations in the case of PS rotation (see Equation A7).Direct calculation of the migration rate leads to a differentdependence on SMA and planetary mass (see Equation A9).The discrepancy is likely due to the planet rotating slightlyfaster in our simulations. Similar to the oscillating systems,the larger planets migrated less due to experiencing weakertides from the star, and the larger periods required corre-spondingly larger eccentricities. To test if KL oscillations can account for the missing WJsaround evolved stars, we must determine the stellar size re-quired to remove each of our simulated planets in the case ofboth oscillating and constant eccentricity. Here we focus onthe planetary systems that could be observed as WJ aroundother stars. For this reason, we limit the migration rate to10 − − au/Gyr as indicated by the lines in Figures 9, © , 1–16 ffect of Stellar Evolution on Migrating WJs Minimum Periapse (R ⊙ )-10 -10 -10 -10 -1 -10 -2 -2 -1 ∆ a / ∆ t ( A U / G y r ) Min: dadt =0.1
AU/GyrMax: dadt =100
AU/Gyr10 M J J J Period: 100 days
Figure 14.
Migration rate as a function of periapse dis-tance/eccentricity for 100-day planets without a perturber, andthus non-oscillating eccentricity. The dependence of migrationrate on planetary mass and maximum eccentricity is apparent.
8, and 14. This rate is rapid enough that WJs can haveentered the 10 −
100 day-period regime in the lifetime oftheir star, but long enough that a significant number areobserved there. Furthermore, we limit the planet mass to0 . − M J . This limit is to avoid being influenced both byboth low-mass planets, whose large migration rates may beinaccurate due to our choice of uniform planetary radius,and massive planets, which are more likely to be affected bytides raised on the star that we did not include.65 systems across the six orbital period bins meet thesecriteria. We create a population of planets for comparison bydrawing from the period bins according to the observed WJdistribution (Figure 15). With each draw from a given pe-riod, we randomly select one of the systems and a value fromits eccentricity distribution. We repeat this process until weobtain a final set of 848 planets, each with an eccentricity,period, and planet mass, which match the observed perioddistribution. These systems represent what the oscillatingsystems, or an analogous population with constant eccen-tricity, would look like in observations. With our populationof oscillating- and constant-eccentricity planets, we then de-termine the criteria for removal. It is only only appropriate to assume the planet is removedat its maximum eccentricity if the evolution time-scale ofthe star ( R ∗ / ˙ R ∗ ) is significantly longer than the KL time-scale. In the case of very brief evolution time-scales, the WJeccentricity would remain relatively constant and the plan-ets would be removed at whatever eccentricity they hap-pened to have at that point in stellar evolution. Using MESA(Paxton et al. 2011, 2013) models, we calculated the expan-sion time-scale of the host star to be > years through R ∗ = 40 R ⊙ , or roughly half the size of our largest plane-tary orbits. The KL time-scale for our simulations rangedfrom 7 × − × years, depending on the period of theplanet, which are orders of magnitude shorter than the stel-lar evolution time. As a result, we safely assume the maxi- N u m b e r o f P l a n e t s Period distribution of observed WJs
Figure 15.
The period distribution of observed WJs, binned ac-cording to our simulation periods. This distribution informed ushow many eccentricity samples to draw from each period. mum eccentricity determines when the planet is removed viacontact. In addition, this portion of stellar evolution occurswithout any measurable change in mass, so we can safelyignore the effect of mass loss on the planetary orbits.
A WJ can be removed in one of two ways.The first is if the planet comes into direct contact withthe stellar atmosphere. Hydrodynamic drag during a singleperiastron passage will remove binding energy of the orderof ∆ E ∼ × ergs ρ (cid:18) R p R J (cid:19) M ∗ M ⊙ (18)where we have assumed a periastron distance ∼ stellar ra-dius, R p is the planet radius and ρ is the density of thestar at the radius of interaction. This is large comparedto the orbital binding energy ∼ × ergs of a Jupiter-mass planet with an orbital period of 100 days, as long as ρ > − g.cm − . This condition is satisfied very close to thesurface of moderately evolved stars of a few solar radii ex-tent, and so the hydrodynamic drag-down of a planet occurson a few orbital timescales once the planet starts to impactthe stellar atmosphere.A second removal mechanism occurs if the planet cantidally migrate interior to 10 days, becoming a HJ untilit undergoes direct contact. This migration occurs morequickly as the star evolves than during the main sequencebecause the star expands to the point where stellar tidesdominate the tidal decay (Villaver et al. 2014). Once thisoccurs, inward migration increases dramatically with con-tinued stellar expansion due to the R dependence in thetidal friction time-scale (Equation 15). A planet will mi-grate out of the WJ period space on a time-scale of roughly P T = − a/ ( da/dt ).The value of da/dt caused by tides in the planet, pairedwith some assumptions about stellar and planetary tides,allow us to quantify the increase in migration rate due tostellar expansion. We take the contribution from the star to © , 1–16 S. Frewen & B. Hansen be P T = P T p R ∗ /R ∗ ,eq ) (19)where P T p is the migration time-scale before stellar evolution(due only to planetary tides) and R ∗ ,eq is the size of thestar at which stellar tides match planetary tides. The valueof P T p for oscillating systems comes from our simulationresults, while P T p for constant eccentricity systems comesfrom Equation 17. Importantly, f (0) = 10 − in Equation 17,so that even planets on circular orbits are migrating slowlyinward.To determine the value of R ∗ ,eq , we set the tidal time-scales of the star equal to that of the planet times a coef-ficient, which accounts for the different spins between thetwo: t F ∗ = f s t F p . We assume viscous time-scales of t V ∗ = 50years based the planet-to-star strength from Hansen (2010),and f s = 0 . f ( e , Ω), our function f ( e ) with a non-PS spin value. R ∗ ,eq = R p (cid:18) f s t V ∗ t V p (cid:19) / (cid:18) M ∗ M p (cid:19) / (20)We note that this equation assumes the viscous time-scalefor the star stays constant over stellar evolution, which isnot strictly true. However, the very weak dependence on t V ∗ means it should not have a significant effect. Using thecalculation from Zahn (1977), t V ∗ ∝ ( L/R ) − / ∝ T − / .For our stellar model, the surface temperature drops from6300K to 3200K as the star grows to 50 R ⊙ , correspondingto an increase in viscous time-scale by a factor of 2.5, or a12% increase in R ∗ ,eq at its largest.We consider a planet with a migration time-scale P T
93 53 − . Table 1.
Migration values and stellar size at planetary engulf-ment ( R rem ) for a simulated oscillating planet and 10 constant-eccentricity realisations. The da/dt values for the latter groupwere calculated using Equation 17. Stellar radius (R ⊙ )0.00.20.40.60.81.0 C u m u l a t i v e f r a c t i o n s u r v i v i n g Warm jupiter removal by stellar expansion
Oscillating EccentricityConstant Eccentricity
Figure 16.
The fraction of oscillating and constant-eccentricityWJs that survive as a function of stellar radius. While the frac-tion of oscillating planets drops off dramatically above 3 R ⊙ , thefraction with constant eccentricity is significant even as the stel-lar radius exceeds 20 R ⊙ , which indicates that the lack of WJsaround evolved stars can be effectively explained by eccentricityoscillations. with the star. The constant-eccentricity population dropsoff much more slowly, with some planets surviving until thestar is over 50 R ⊙ . A small fraction of these planets are onvery eccentric orbits to match the high end of the distribu-tion of oscillating eccentricities. As a result, those planetsare removed by collision with the star. In general, however,most had low or moderate eccentricity (as seen in Figure 17)and are removed when the star dominates their migrationrate. We note that varying t V ∗ does have an effect for thosesystems, but only serves to shift the constant-eccentricitypopulation to stellar sizes larger by a factor of 2 − da/dt at a given period: all low-eccentricity planets of a givenperiod have similar da/dt values and are removed at similarstellar radii. When the minimum migration rate is removed,the bumps are smoothed out. The 100-day population be- © , 1–16 ffect of Stellar Evolution on Migrating WJs tween 50 and 60 R ⊙ is negligible, due to the small numberof such planets in the observed WJ period distribution. The model described here is motivated by the claimed deficitof WJs around moderately evolved stars (Johnson et al.2007, 2011), as seen in Figure 1 and Section 2. We pos-tulate that the reason for this deficit is that the observedeccentricity distribution of WJs around main sequence starsis really a snapshot of a population whose eccentricities areoscillating via the KL mechanism while they migrate in-wards due to tidal friction. The fact that the oscillation time-scale is short compared to the characteristic time-scale forthe stellar evolution means that planets are removed fromthe observed sample when their periapsides oscillate to theminimum value and interact with the host star. Figure 15shows the result of such a model and demonstrates that,under these conditions, a pre-existing WJ population willbe largely removed by the time the stars evolve to 4 R ⊙ ,in contrast to the case where the eccentricities of the ob-served population do not oscillate. The exact location of WJremoval depends on the details of tidal forces and the per-turber, but the general behavior is well described by Figure16. However, our results do not match all observations. Fig-ures 17 and 18 show the distribution of eccentricities for oursimulated systems and observed WJs (from the ExoplanetOrbit Database), respectively, with our simulated popula-tion drawn from the same period distribution. In both casesthe systems are restricted to the Jupiter mass range (0.3-3) and the period range of 10 −
100 days. Comparison ofthese two populations using the Kolmogorov–Smirnov (KS)test gives a p -value of 1 × − , indicating they are un-likely to be drawn from the same underlying population.The discrepancy is primarily due to the significant fraction(15%) of simulated WJs with high eccentricity ( e > . The discrepancy at high eccentricities is a direct conse-quence of our underlying model. In order for host stars toremove their orbiting WJs early on in stellar evolution, asobservations imply, the minimum periapsides must be quitesmall. As a result planets must undergo KL oscillations to P r e d i c t e d F r e q u e n c y D e t e c t i o n E ff i c i e n c y Figure 17.
The distribution of eccentricity values drawn from oursimulated systems, with the same period distribution as observed(grey). The detection efficiency of 100-day planets with signal-to-noise of 10 (dashed line), obtained from Cumming (2004), dropsoff dramatically at high eccentricities and produces the eccentric-ity distribution predicted in observations (dark grey). O b s e r v e d D i s t r i b u t i o n Figure 18.
The eccentricity distribution for observed WJs, takenfrom the Exoplanet Orbit Database. The small number of highlyeccentric planets differs significantly from the oscillating distribu-tion, but that may be a result of low detection efficiency.. large eccentricities, leading to a small but significant fractionof WJs inhabiting that portion of the eccentricity distribu-tion at any given time. Even oscillating systems peakingstrongly near e = 0 have a significant tail at high eccentric-ities, in conflict with observations.However, if eccentric planets are more difficult to detectthan low-eccentricity or circular planets, then the dearth ofhigh-eccentricity systems could be an observational effect,not a physical one. Studies of exoplanet detectability in ra-dial velocity surveys (Cumming 2004; O’Toole et al. 2009)have shown that that appears to be the case above eccen-tricities ∼ .
5, where the largest difference between observedand simulated populations exists. To test how much this ef-fect can improve the fit between our results and observa-tions, we apply the detection efficiency (DE) of Figure 4 inCumming (2004) to our eccentricity distribution. We use the © , 1–16 S. Frewen & B. Hansen
DE found for fitting to a Lomb-Scargle periodogram with N = 39 observations short-period (100 day) planets, with asignal-to-noise ratio of 10, shown in Figure 17. After appli-cation, the KS test gives a p -value of 4 × − , somewhatimproved compared to the distribution without correctingfor DE. The remaining mismatch is now a consequence ofthe excess of low eccentricities. This can be demonstratedby assuming that the observations contain 10 percent of thepopulation in circular planets, leading to a p -value of 0.03.Thus, with the correction for DE, a population primarilyoscillating is consistent with observations. A population ofcircular WJs this small would not have a high probability ofbeing detected around evolved stars even if they existed, andcould have originated via an alternative migration mecha-nism, such as disc migration.Given the specificity of this detection efficiency functionand the inclusion of a separate, distinct population, we can-not claim that this calculation proves that our populationmatches that of observations. An in-depth examination ofthe detection efficiency of WJs around evolved stars is out-side the scope of our work. However, this calculation doesshow the conditions required to satisfy observations in such amanner that the KL migration offers a plausible physical ex-planation for the rapid removal of WJ by stellar evolution.As more eccentricities are determined in systems detectedvia the transit method, these biases may be reduced, allow-ing us a better view of the underlying eccentricity distribu-tion. We finish by noting that our DE-corrected distributionpredicts that ∼ Our simulations ignored the effect of stellar tides, which wereonly included in the evolved star calculations. Larger stel-lar tides would increase the tidal decay for a planet witha smaller maximum eccentricity and strengthen that de-cay as the star evolved. However, limits can be placed onthe strength of tides in stars from the population of WJs(Hansen 2012). Stellar tides must be weak enough that plan-ets can exist on orbits shorter than one day for an observa-tionally significant amount of time. For that reason, it is un-likely that stellar tides can dominate the evolution of mostplanets except for the most massive ones. As a test, we sim-ulated 32 systems at 50 days with identical properties to ourprimary simulations, but with the stellar tidal time-scale setto 50 years. The simulations were qualitatively identical tothose without stellar tides, indicating they would need tobe significantly stronger than the current limits in order toaccount for the lack of observed eccentric WJs.Our simulations also assumed the equilibrium model fortides, which is an approximation. Tidal effects may differ sig-nificantly, both in the star and in the planet, when they areforced on an eccentric orbit. The existence of HJs would notconstrain such effects due to their uniformly near-circularorbits. Additionally, we ignored the size difference between 1 M J planets and 0.1 M J planets. Correcting for this wouldlikely reduce the migration rate for low-mass planets, leadingto a larger population in our defined migrating region. How-ever, many WJs are Jupiter-mass and above, and the issueof a large periapse preventing prompt removal remains. A number of planets have been found around evolved stars,but there appears to be a lack of massive planets in-terior to 0.6 au (Johnson et al. 2007; Bowler et al. 2010;Johnson et al. 2011). Two possibilities exist: either the un-derlying population of planets differs around the unevolvedprogenitors of these generally more massive ( > . M ⊙ )stars, or stellar evolution has led to their removal. The re-sults of Lloyd (2011, 2013) have called into question whetherthe evolved stars truly originate from a more massive popu-lation, supporting the latter reason for the absence of WJs.Additionally, Schlaufman & Winn (2013) showed that someevolved stars have a different population of planets thantheir unevolved progenitors of the same mass, indicatingthat stellar evolution is a cause in at least some cases.Most recently, Johnson et al. (2014) showed that at leastone evolved star in the disputed population has a mass trulygreater than 1.5 M ⊙ , as they claimed in prior works. Takentogether, these results leave considerable ambiguity for theexplanation of missing WJs.Here we have simulated planets undergoing KL oscil-lations as part of their migration inward and examinedhow the population decays with stellar evolution. By us-ing a model population of WJs and their perturbing com-panions, we have shown that KL oscillating WJs explainthe observed absence around evolved stars better than aconstant-eccentricity population. A population of migrat-ing, KL oscillating WJs is almost entirely removed aroundan evolving star by the time it reaches 5 R ⊙ , while an ob-servationally identical population with constant eccentricitysurvives stellar expansion beyond 40 R ⊙ . Finally, althoughwe have adopted a stellar mass of 1.2 M ⊙ in our simulations,it should be noted that the rapid removal of WJs migratingvia KL oscillations is applicable regardless of stellar mass.Therefore the absence observed by Johnson et al. (2007) andrelated works need not indicate that WJs are absent aroundmore massive stars in general. ACKNOWLEDGMENTS
This research is supported by the Dissertation Year Fellow-ship at UCLA. We thank Smadar Naoz for the use of hercode and helpful comments, as well Jean-Luc Margot for hiscomments.
APPENDIX A: PLANETARY MIGRATIONDURING PSEUDO-SYNCHRONOUSROTATION
The orbital evolution of a planet due to tides is given by da/dta = − (cid:20) W p + W ∗ + e − e ( V p + V ∗ ) (cid:21) (A1) © , 1–16 ffect of Stellar Evolution on Migrating WJs where the subscripts p and ∗ correspond to the planetand host star, respectively. V and W are given inEggleton & Kiseleva-Eggleton (2001): V = 9 t F (cid:26) / e + (15 / e + (5 / e (1 − e ) / (A2) − n / e + (1 / e (1 − e ) (cid:27) W = 1 t F (cid:26) / e + (45 / e + (5 / e (1 − e ) / (A3) − Ω n e + (3 / e (1 − e ) (cid:27) where n is the mean motion of the orbit and Ω is the rotationrate of the body. A migrating WJ will have already reachedpseudo-synchronous rotation, which occurs when W p = 0(Hut 1981). The rotation rate in that case is given byΩ ps n = 1 + (15 / e + (45 / e + (5 / e (1 + 3 e + (3 / e )(1 − e ) / (A4)Plugging in Ω ps , we get the strength of tides for a pseudo-synchronous planet: V p (Ω ps ) = 9 t F p (cid:26) e + 14336 e (A5)+ 5480 e + 1020 e + 25 e (cid:27) − e ) / Assuming planetary tides dominate during the main se-quence ( V p >> V ∗ ) and that the planet is in PS rotation( W p = 0), we can simplify the tidal decay equation: da/dta = − (cid:20) W ∗ + e − e ( V p ) (cid:21) = − W ∗ − e − e t F p (cid:26) e + 14336 e + 5480 e + 1020 e + 25 e (cid:27) − e ) / = − f e ( e ) t F p (A6)where f e ( e ) = t F p W ∗ + (cid:26) e + 5760 e + 14336 e + 5480 e + 1020 e + 25 e (cid:27) − e ) / (A7)We have retained W ∗ because it tends to t − F ∗ as e → f e ( e ) are therefore t F p /t F ∗ near e = 0 and 3 . − e ) / for e ∼
1. From thedefinition of t F , Equation 15: t F p t F ∗ = t V p t V ∗ (cid:18) R p R ∗ (cid:19) − (cid:18) M p M ∗ (cid:19) (cid:18) k p k ∗ (cid:19) − (A8)In Section 3.3.1 we assume the following values for the plan-etary systems: t V p = 1 year, t V ∗ = 50 years, M ∗ = 1 . M ⊙ , M p = 0 . − M J , k p = 0 .
25, and k ∗ = 0 . − −
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