A-type Stars, the Destroyers of Worlds: The lives and deaths of Jupiters in evolving stellar binaries
Hot Jupiters (HJs), gas giant planets orbiting their host stars with periods on the order of days, commonly occur in the Galaxy, including relatively massive ( M, i.e. A-type main sequence stars) and evolved stars. The majority of A-type main sequence stars have stellar binary companions, which can strongly affect the dynamical evolution of planets around either star. In this work we investigate the effects of gravitational perturbations by a far away stellar companion on the orbital evolution of gas giant planets orbiting A-type stars, the so-called Eccentric Kozai-Lidov (EKL) mechanism, including the effects of general relativity, post-main sequence stellar evolution, and tides. We find that only of all A-type stars will host HJs during their main sequence lifetime. However, we also find a new class of planets, Temporary Hot Jupiters (THJs), that form during the post-main sequence lifetime of a significant fraction (about ) of all former A-type main sequence stars. These THJs orbit on periods of tens to a hundred days and only exist for a few years before they are engulfed by their expanding host stars, but they reach the same temperatures as “classical” HJs due to the increased stellar luminosities. THJs’ spin-orbit angles will mostly be misaligned. Tidal interactions between THJs and their expanding host stars will significantly influence the stellar envelopes, while the planets’ eventual engulfment will lead to faster stellar spins and transient increased luminosity signals that should extend the possible time-frame to observe THJs far beyond the initial few years. Overall we find that approximately of all gas giant planets orbiting A-type stars will eventually be destroyed or engulfed by their star, about during the main sequence lifetime and about during post-main sequence evolution.
0000-0001-8220-0548]Alexander P. Stephan \move@AU\move@AF\@affiliationUniversity of California, Los Angeles, Department of Physics and Astronomy, CA 90095, USA \move@AU\move@AF\@affiliationMani L. Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, CA 90095, USA
University of California, Los Angeles, Department of Physics and Astronomy, CA 90095, USA \move@AU\move@AF\@affiliationMani L. Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, CA 90095, USA
The Ohio State University, Department of Astronomy, Columbus, OH 43210, USA
Exoplanets have been observed around a variety of host stars, with different masses, at all stages of stellar evolution, including main-sequence, sub giant and red giant branch (RGB) stages (e.g., Charpinet et al., 2011; Johnson et al., 2011a; Howard et al., 2012; Gettel et al., 2012; Barnes et al., 2013; Nowak et al., 2013; Reffert et al., 2015; Niedzielski et al., 2015, 2016). However, for massive or evolved stars, there appears to be a deficit in short-period or high eccentricity planets (e.g., Sato et al., 2008, 2013; Bowler et al., 2010; Johnson et al., 2007, 2008, 2010a, 2010b; Schlaufman & Winn, 2013). Furthermore, high metal abundances in so-called “polluted” white dwarf (WD) atmospheres indicate the presence of the remnants of planetary systems around these stars; the processes by which this material was brought onto the WDs is an active topic of research (e.g., Farihi et al., 2009, 2010; Klein et al., 2010, 2011; Melis et al., 2011; Zuckerman et al., 2011; Xu et al., 2013, 2017; Stephan et al., 2017).
The architectures of these planetary systems have become the focus of a rich field of research, as it was recently shown that dynamical processes play an important role in planetary system formation and evolution. These processes include resonant interactions (e.g., Lithwick & Wu, 2012; Batygin & Morbidelli, 2013; Petrovich et al., 2013; Goldreich & Schlichting, 2014), planet-planet scattering (e.g., Rasio & Ford, 1996; Nagasawa et al., 2008; Chatterjee et al., 2008; Boley et al., 2012; Beaugé & Nesvorný, 2012) and secular perturbations from a companion (either a star or a planet; e.g., Fabrycky et al., 2007; Wu et al., 2007; Naoz et al., 2011, 2012, 2013a) or from multiple planets (e.g., Wu & Lithwick, 2011; Denham et al., 2018).
A particularly interesting group of discovered exoplanets are the so called “Hot Jupiters” (HJs), which are gas giants that orbit their host stars on very tight orbits with periods on the order of a few days. While this class of exoplanets seems ubiquitous in the galaxy, it is noticeably absent in our own solar system. Several models have been developed to explain the formation and existence of these HJs, including gravitational perturbations of the planets’ original orbits to high eccentricities, followed by tidal dissipation and orbit circularization and shrinking (e.g., Fabrycky & Tremaine, 2007; Naoz et al., 2011, 2012; Beaugé & Nesvorný, 2012; Petrovich, 2015; Frewen & Hansen, 2016), as well as disk migration during giant planet formation (e.g., Armitage et al., 2002; Masset & Papaloizou, 2003; Armitage, 2007). The idea that outer companions have perturbed these planets and led to high eccentricity migration is also supported by recent observational campaigns that have shown that most HJs have a far away companion, either a star or a planet (e.g., Knutson et al., 2014), though it remains unclear if most of these companions can trigger high-eccentricity migration (e.g., Ngo et al., 2016). For a three body system consisting of star-planet-star to be long-term stable the inner two bodies, the main star and the gas giant, have to be on a much tighter orbit than the third, outer, object, leading to a hierarchical configuration.
In recent years so-called “retired” A-type stars have been focused by many studies in attempts to discover exoplanets (e.g., Johnson et al., 2007, 2008; Bowler et al., 2010; Johnson et al., 2010b, 2011a, 2011b). “Retired” A-type stars are stars that would be classified as A-type during their main-sequence lifetime, but which have evolved beyond the main sequence and are sub-giant or giant stars at their currently observed life stage. While main-sequence A-type stars usually rotate rapidly and have high surface temperatures, greatly impeding exoplanet detection through radial velocity measurements, “retired” A-type stars rotate slower and are cooler and have allowed the discovery of several exoplanets. Furthermore, several HJs have also been discovered around A-type main sequence stars through exoplanet transits (e.g., Gaudi et al., 2017; Johnson et al., 2018).
The classification of “retired” A-type stars has been shown to be rather challenging, as different methods to determine stellar masses seem to have yielded different values (e.g., Lloyd, 2011; Johnson et al., 2013; North et al., 2017; Stello et al., 2017; Ghezzi et al., 2018). There seems to be uncertainty over the validity of some methods when compared to precise astroseismological measurements. In this work we avoid these classification problems by simply focusing on a particular range of stellar masses, between and (with masses above technically belonging to the low-end B-type mass range). This intermediate stellar mass range broadly coincides with classical definitions of main-sequence A-type star masses (Adelman, 2004) and we refer to this mass range when labelling a star as A-type. A further discussion concerning A-type star evolution is given in Appendix A.
Stars more massive than the sun, like A-type stars, reach the post-main sequence phase of their evolution much faster than smaller stars, and the vast majority of them has stellar companions (e.g., Raghavan et al., 2010). Indeed, several A-type stars with HJs have companion stars (e.g., Johnson et al., 2018; Siverd et al., 2018). This leads to an interesting interplay of dynamical and stellar evolution effects that needs to be considered for planets in such systems. On the one hand, a hierarchical star-planet-star configuration will lead to secular oscillations of the orbital parameters due to gravitational perturbations by the outer companion on the planet’s orbit, often leading to extreme eccentricities, the so-called Eccentric Kozai-Lidov (EKL) mechanism (Kozai, 1962; Lidov, 1962; Naoz, 2016). On the other hand, post-main sequence stellar evolution will lead to, for example, stronger tidal dissipation or engulfment of close-in planets due to stellar radial expansion, and expanded orbits due to stellar mass loss. In this work we study the combined interplay of these dynamical and stellar effects for Jupiter-sized planets in stellar binaries with A-type star primaries. We propose that observed planets around intermediate mass main and post-main sequence stars with stellar companions333Planetary companions might have different results. are consistent with our predicted results.
2 Numerical Setup
We perform large Monte-Carlo simulations that follow the dynamical evolution of hierarchical three-body systems, consisting of a relatively tight inner binary pair of a star and a Jupiter-sized planet, which are orbited by another star on a distant orbit as outer binary. We solve the hierarchical secular triple equations up to the octupole level of approximation (the so-called Eccentric Kozai-Lidov (EKL) mechanism, e.g., Naoz, 2016), including general relativity effects on both inner and outer orbit (e.g., Naoz et al., 2013b), static tides between the primary star and the planet (following Hut (1980); Eggleton et al. (1998), see Naoz (2016) for complete set of equations), where we also distinguish between tides for radiative and for convective stars (e.g., Zahn, 1977), as well as the effects of stellar evolution on masses, radii, and spins on the two stars, as derived from the stellar evolution code SSE by Hurley et al. (2000). Unlike G and F-type main sequence stars, which exhibit magnetic braking due to their convective envelopes, A-type main sequence stars are nearly completely radiative and do not experience significant magnetic braking (van Saders & Pinsonneault, 2013). Their spin rates therefore do not substantially slow down during their main sequence evolution lifetime. The interplay between EKL mechanism and stellar evolution has previously been shown to play an important role in shaping the underlying dynamics and outcome of these systems (e.g., Kratter & Perets, 2012; Michaely & Perets, 2014; Shappee & Thompson, 2013; Frewen & Hansen, 2016; Naoz et al., 2016; Stephan et al., 2016, 2017).
The mass of the primary star, , is taken from a Salpeter distribution with (Salpeter, 1955), however the mass range is restricted between and in order to ensure that the planet host star is an A-type or, at most, a small B-type star during its main sequence lifetime (Adelman, 2004). The stellar initial radii and spins are calculated using SSE by Hurley et al. (2000). Each primary star is given one planet, whose mass (), size, and spin are set equal to Jupiter’s. The mass of the outer companion star, , is determined by the binary mass ratio distribution taken from Duquennoy & Mayor (1991). The semi-major axis, of the inner binary of A-type star and Jupiter-sized planet is chosen uniformly between and AU, while the outer binary orbit’s semi-major axis, , is again taken from the distribution in Duquennoy & Mayor (1991) for the stellar companion. The inner orbit’s eccentricity is initially set to a small value, , as are the stellar and planetary spin orbit angles, since we assume that the planet was formed in a gaseous disk, while the outer orbit’s eccentricity, is chosen uniformly between and . To ensure long-term stability we reject systems where is greater than AU, as galactic tides will tend to separate such systems relatively quickly (Kaib et al., 2013), and we only consider hierarchical systems to ensure long-term stability, which requires:
and (e.g., Naoz, 2016). The inclination, between the inner and outer orbit’s angular momenta is chosen isotropically in cosine.
In total we simulate systems using these parameter conditions, of which have a primary star mass smaller than about and therefore safely classify as A-type stars during their main sequence lifetime. We calculate the dynamical evolution of these systems for Gyrs, or until a stopping condition is fulfilled. If a planet touches the surface of a star or crosses the Roche limit, we stop the integration. We define the Roche limit of a body of mass and radius in respect to an orbiting body of mass as:
Note that this equation simplifies to and if , as in the case of a star with mass being orbited by a much smaller planet of mass . The disruption of planets is highly sensitive to the Roche limit and thus to the numerical pre-factor value, chosen here to be . Numerical simulations by Guillochon et al. (2011) and Liu et al. (2013) suggested a larger value, (e.g., ) while Faber et al. (2005) simulations found . Thus, choosing here a fiducial value of means that the engulfed planets represent a lower limit on the fraction of planets that can be engulfed during post-main sequence radial stellar expansion. On the other hand, this value might lead us to overestimate the number of planets forming HJs during the main sequence lifetime (see Petrovich, 2015).
3.1 Classification of Dynamical Evolution Outcomes
From our simulated systems about ended in the destruction of the Jupiter-like planet, while in only of cases did the planet survive to the WD phase of its host star. We identify several distinct groups of orbital evolution behaviors and outcomes for Jupiters around A-type stars in stellar binaries. There are mainly four such groups (see Table 3.1 for percentages):
Classical Hot Jupiters: These are HJs on orbits shorter than about 10 days, as one would expect from previous studies of high-eccentricity migration (e.g., Fabrycky & Tremaine, 2007; Wu et al., 2007; Naoz et al., 2011, 2012; Petrovich, 2015). These planets reach their short period orbits during the host stars’ main sequence lifetime due to an interplay of EKL-caused high orbital eccentricities and tidal effects. They are ultimately engulfed and destroyed as the stars evolve and expand. About of our systems experience this outcome. We further discuss this group in Sec. 3.2.
Roche-limit crossers: These are those planets that reach extremely large eccentricities through the EKL-mechanism and cross their host stars’ Roche limits or graze the stars’ surfaces. The actual fate of these planets might be complicated (e.g., Dosopoulou et al., 2017), and some might even survived, however, for simplicity we mark them here as “RL-cross”. About of our systems lead to Roche-limit crossing, of which occur during main sequence and occur during post-main sequence evolution. Those that cross the stellar Roche-limit during post-main sequence evolution simply do not undergo high enough eccentricity excitations or are on too wide initial orbits to have short periapsis distances during the stellar main-sequence lifetime.
“Temporary” Hot Jupiters: These are planets that did not reach high enough eccentricities during the main sequence lifetime of their host stars to experience tidal circularization and orbital shrinking, but which do so as the stars leave the main sequence and become giant stars. Virtually all of them only classify as “hot” Jupiters (in terms of temperature and eccentricity) for a short part of their total lifetime. They get engulfed as their host stars continue to expand and as tides continue to drive them to the stellar surface. The engulfment process may result in energetic disturbances on the host star and might serve as an observable (e.g., MacLeod et al., 2018). We find that about of our systems experience this outcome, however about a fifth of these ( of all systems) reach this outcome even though the EKL perturbations by their companion stars are negligible, as their initial orbits are relatively close to the star, at about AU. Further details about this group are discussed in Sec. 3.3.
Surviving Jupiters: These are planets that never significantly interact with their host stars and survive until the stars become WDs. For this population, the companion stars’ EKL perturbations were too weak to cause large eccentricities and the planets’ orbits were too wide to experience strong tidal effects during post-main sequence evolution. This population mostly represents our lack of knowledge on the initial conditions of these systems. About of Jupiters survive to this stage. A small fraction of these ( of all systems) get destroyed and accrete onto the WDs as the stellar mass loss changes the orbital parameters of the system, allowing extreme excitations of the orbital eccentricities through the EKL-mechanism, ultimately driving these Jupiters to cross their own Roche limits and to get tidally disrupted by the WDs. Examples of this potential WD pollution mechanism have been discussed in detail in Stephan et al. (2017). Recent work (van Lieshout et al., 2018) also indicates that some of the planet’s material can be recycled into new planets and escape accretion.
Table 0. \Hy@raisedlink\hyper@@anchor\@currentHrefJupiter evolution outcome percentages. Listed are Classical HJ (CHJ), RL-crossing Jupiter (RL-cross), Temporary HJ (THJs), and surviving Jupiter (Survived), up to Hubble time, outcomes as percentages of the whole population of simulated systems. The percentages are given for the whole evolution of the stars (Total), and also split between main sequence (MS), post-main sequence (Post-MS), and White Dwarf (WD) phases.
In Figure 2 we show three snapshots of the different systems’ realizations during the stars’ lifetimes, mid-main sequence (top left), at the beginning of post-main sequence (top right), and post ABG phase (bottom). We consider the eccentricity as a function of the planet’s period (top panels). For comparison we over-plot the observed systems (depicted as black stars) for the relevant snapshot. For context we also show the stellar radii for all of our systems (bottom panels). When a planet crossed the Roche-limit we stop the integration and mark its orbital parameters.
3.2 Classical Hot Jupiters and Surviving Jupiters
As we show in Figure 2 (see upper panels), by the middle of the main sequence lifetime of the host stars several classical HJs have been formed through high-eccentricity migration (shown in the upper left panels as purple dots, with periods shorter than about days and eccentricities close to ), or are still in the process of forming until the end of the main sequence. However, these more massive stars expand their radius during the main sequence by about a factor of two, which results in a higher rate of Roche-limit crossing than for less massive stars, since the Roche-limit is proportional to the radius of the star (see equation 2). In total we form such HJs, or about of our systems, which is broadly consistent with previous estimates of EKL-induced high-eccentricity migration models in stellar binaries (see Naoz et al., 2012; Petrovich, 2015; Naoz, 2016). Some of these classical HJs only survive for short times at these orbits, as the interplay of the EKL-mechanism and tides keeps driving them towards the stellar surface, however many can exist for tens to hundreds of Myrs after formation. Ultimately though, all classical HJs get engulfed and destroyed as their host stars evolve and expand in radius. By the end of the main sequence most of them have been destroyed (see upper right panels of Figure 2; red dots show destroyed planets, with several red dots forming a line in the stellar radius vs. orbital period parameter space where HJs were engulfed by their expanding host stars), with the remaining ones being destroyed as the stars evolve towards the giant phase (see the orange and yellow dots in the lower, large panels).
The black stars in Figure 2 show the parameters of several observed Jupiters around A-type or retired A-type stars (Johnson et al., 2007, 2008; Collier Cameron et al., 2010; Johnson et al., 2010a, b, 2011a, 2011b; Sato et al., 2013; Bieryla et al., 2014; Wittenmyer et al., 2014; Johnson et al., 2014; Hartman et al., 2015; Wittenmyer et al., 2015a, b; Zhou et al., 2016; Borgniet et al., 2017; Beatty et al., 2017; Gaudi et al., 2017; Lund et al., 2017; Talens et al., 2017; Johnson et al., 2018; Siverd et al., 2018), and they appear to be broadly consistent with our calculations. The observed HJs agree well with our predicted parameters at some time during the middle of the main sequence lifetime of A-type stars (upper left panels), while observed Jupiters on wider orbits (periods beyond about days) around retired A-type stars agree well with our predicted parameters for Jupiters that survive to the end of the main sequence lifetime (upper right panels). Note that both observations and our predictions do not seem to show many Jupiters on intermediate orbits between and days. HJs migrate relatively fast through this part of the parameter space, on the order of a few million years (a very small fraction of the stars’ total main sequence lifetime), and are unlikely to be randomly observable.
3.3 Temporary Hot Jupiters
During post main-sequence evolution stars begin to expand and those planets that have short pericenter distances are engulfed by their host stars. Furthermore, our calculations show that a large number of moderately eccentric giant planets will undergo significant tidal interactions with the expanding red giant stars, leading to orbital shrinking and circularization before the eventual crossing of the Roche lobe or engulfment by the star, as shown by the example system evolution in Figure 3.1.The strength of tidal effects on the star increases significantly due to the stellar radial expansion and the increased size of the convective envelope during the red giant phase. During their main-sequence lifetime A-type stars are mostly radiative, which severally reduces the effectiveness of their tides (Zahn, 1977). We consequently observe a switch in the role of tides between the main sequence and post-main sequence evolution; while during the main sequence the tidal effects on the planet were more significant, during post-main sequence evolution the tides on the star dominate. Tidal interactions and planet engulfment can be expected to have significant effects on the red giant stars’ envelope and mass loss evolution, as well as changing the stellar rotation rate; such giant planet interactions with evolved stars have been used to explain observed irregularities in the shape of the horizontal branch in the Hertzsprung-Russel diagram (e.g., Soker, 1998; Soker & Harpaz, 2000; Livio & Soker, 2002). Our “Temporary” HJs would most likely lead to such interactions, though we stop our calculations at the entering of the stellar Roche lobe. Further investigation would require a full hydro-dynamical treatment of the star-planet interactions. We assume that the further evolution of the stellar envelope and post-AGB remnant will be altered due to these interactions. THJ engulfment could also lead to lithium enrichment in the giant stars’ atmospheres (e.g., Aguilera-Gómez et al., 2016).
The lower large panels in Figure 2 show the final orbital parameters of our systems in eccentricity vs. period (upper frame) and host-star radius vs. period space (lower frame). Those dots colored magenta, yellow, green, cyan, and blue show the final parameters before planets enter the stellar Roche lobe during the different post-main sequence and pre-WD phases. The vast majority of these, those that form a distinct line on the left in the stellar radius vs. period frame, experience at least some degree of high-eccentricity migration, and reach equilibrium temperatures comparable to classical HJs (see Figure 3.3). These planets begin to experience significant tidal interactions with their host stars as those stars expand in radius, facilitated by increasing orbital eccentricities due to the EKL-mechanism. However, due to the continued stellar evolution of the host stars and the significantly enhanced tides, most of these THJs enter their host stars’ Roche lobe even before they can be fully circularized, making their status as HJs very short-lived. We therefore name this class of HJs “Temporary” Hot Jupiters (THJs). The THJ shown in Figure 3.1 only lives for about years once tidal orbital decay becomes efficient before it reaches the stellar Roche limit. This process should, as mentioned above, have significant effects on the further evolution of the giant star’s outer envelope.
While most () of the THJs we discuss here are products of high eccentricity migration caused by the EKL mechanism and tidal dissipation and therefore depend on the presence of a companion star, some THJs can also be formed without a companion. If a gas giant’s initial orbit is sufficiently close, in general smaller than about AU, post-main sequence expansion of the host star will eventually lead to tidal interactions and engulfment of the planet even without increased orbital eccentricity, consistent with previous works investigating the evolution of gas giants around giant stars (e.g., Villaver & Livio, 2009; Spiegel & Madhusudhan, 2012; Lopez & Fortney, 2016). However, one can distinguish between these two cases considering the spin-orbit alignment of the planet and the main star. If a THJ is formed without companion excitations, the orbital plane will not change from its initial orientation, which we assume to be aligned with the stellar spin. If the THJ was formed through the EKL-mechanism, however, the spin-orbit angle will be from a nearly uniformly random distribution between and . We can see this in Figure 3.3. The two histograms on the right side of the figure show the distribution of spin-orbit angles among THJs. Both histograms show the same population, only that the far right one labeled “THJs (all)” includes THJs formed both with and without EKL, while the one labeled “THJs (EKL)” only shows those formed through EKL. Those THJs formed without EKL effects, for which the stellar companion was irrelevant, are all aligned and form the large peak at , while those formed with EKL are nearly evenly spread across the whole range of angles between and . While the peak is large, it is extremely narrow and only includes of our total number of THJs.
4 Observational Signatures
4.1 Equilibrium Temperature
The equilibrium temperature of planets is a potentially observable signature (e.g., Gaudi et al., 2017). We calculate the time-averaged equilibrium temperature for elliptical orbits for the planets in our calculations, following equations by Méndez & Rivera-Valentín (2017), namely:
where K, Earth’s equilibrium temperature, is the planetary albedo (assumed to be zero), is the host star’s luminosity (in ), and are the planet orbit’s eccentricity and semi-major axis (in AU), respectively, and and are coefficients for the planet’s heat distribution and emissivity, respectively (both assumed to be for simplicity).
The CHJs in our calculations have typical temperatures of about to K, consistent with estimated temperatures of observed HJs, as can be seen in the top frame of Figure 3.3, which shows our CHJs as magenta dots and observed HJs as black stars in the “Hot Planets Zone”. CHJs remain at these temperatures for potentially millions of years, until they either evaporate or are engulfed by the host star as it leaves the main sequence (see the red dots in the middle frame of Figure 3.3. We find that THJs reach temperatures between to K just before entering the stellar Roche lobe, as shown in the bottom frame of Figure 3.3. The THJs only exist at such high temperatures for a short time, on the order of a few years, as seen in the top right frame of Figure 3.1, as the increase in temperature follows from the planets’ rapid orbital decay and the rapid increase in stellar luminosity due to post-main sequence stellar evolution. Considering that the stellar luminosity increases tremendously during post-main sequence stellar evolution, even THJs will produce a rather small contribution to the overall stellar spectra. At a wavelength of around micron the contrast between THJ and stellar emissions will be around . Additionally, the expanding stars’ large size will make planet transit detection signals smaller as well, however the larger size will also increase the chance that a transit can occur. At least for THJs around “small” giant stars (RR), such as for stars during the helium burning phase, models suggest that transits are indeed observable (e.g., Assef et al., 2009), and some giant planets transiting giant stars of such sizes have indeed been observed (e.g., Lillo-Box et al., 2014), though it remains unclear if such transits would currently be observable for stars with sizes of hundreds of solar radii.
4.2 Stellar Obliquity
Detecting planets through the transit method makes it possible to observe the (projected) angle between the stellar spin axes and the planets’ orbital planes through the Rossiter-McLaughlin effect (Gaudi & Winn, 2007). Our calculations show that THJs, like CHJs, should preferentially be misaligned to the stellar spin axis, with a nearly uniform distribution of spin-orbit angles, showing only a small preference against fully retrograde orbits, as can be seen from the THJ spin-orbit angle histograms in Figure 3.3. Using the spin-orbit angle it should also be possible to distinguish THJs born through EKL effects from THJs without that effect, as the later will remain aligned to their original spin-orbit angle, which can be assumed to be small, close to in our case. As can be also seen from the left frame of Figure 3.3, observed projected spin-orbit angles for HJs also broadly confirm a broad range of misalignment angles, potentially caused by EKL effects.
4.3 Orbital Eccentricity
The eccentricities of planetary orbits is an important factor for understanding the architectures of planetary systems, as larger eccentricities can be indicators of significant dynamical interactions, such as planet-planet scattering events or secular perturbations, while small eccentricity values are expected from disk models of planetary formation and a subsequently quiescent dynamical history. Massive evolved stars show a deficiency of short period eccentric planets (Sato et al., 2008; Bowler et al., 2010; Johnson et al., 2007, 2008, 2010a, 2010a, 2013; Schlaufman & Winn, 2013). Our calculations show that this feature is in agreement with the dynamical evolution of giant planets in stellar binaries. As shown in Figure 2, top panels, both the MS-phase as well as the post-MS phase are in agreement with the observed giant planet eccentricities (the latter depicted as black stars). We predict that there is a large population of very highly eccentric planets around retired A-type stars, as shown in Figure 3.3 by the blue histogram, however, most of these high eccentricity planets have orbits longer than days (see Figure 2, top right panel), making them difficult to observe. Interestingly, the highest eccentricity giant planet observed, which orbits the retired A-type star HD 120084, is very consistent with these predictions, having an eccentricity of and a period of days (Sato et al., 2013).
In our simulations we assumed that all giant planets start with very small eccentricities and treated them as singular planets around their host stars. This ignores the potential for planet-planet scattering during the further evolution. In the normalized histograms in Figure 3.3 we see that this artifact of our initial conditions produces an extremely high peak at small eccentricities for our simulated systems during the post-main sequence phase (see blue histogram). This peak simply represents those systems that lie outside the parameter space that can undergo EKL oscillations. We propose that the actual initial eccentricity distribution should be more broad, probably caused by planet-planet interactions after dissipation of the planet-forming disk.
4.4 Effects of Post-MS Roche Lobe Crossing and Engulfment of THJs
A giant planet entering an expanding post-main sequence star’s Roche lobe, and subsequently the stellar envelope, can have a multitude of effects. If the giant planet is massive and dense enough, it might resist dissipation and begin to accrete stellar material, potentially becoming a “stellar” companion undergoing complex common-envelope evolution with the main star (e.g., Soker et al., 1984). Planet engulfment should also lead to the deposition of angular momentum into the star, changing the spin rate (see Carlberg et al., 2009), which has similar consequences during the main-sequence phase (Carlberg et al., 2009; Qureshi et al., 2018), as well as the deposition of energy into the stellar envelope due to drag forces and orbital decay, increasing the luminosity of the star or producing bright UV and X-ray transients (e.g., Metzger et al., 2012; MacLeod et al., 2018). The strength of drag forces depends especially on the density of the stellar envelope and the orbital speed of the engulfed planet, which would lead to different strengths of this effect between main sequence and post-main sequence stars or between eccentric and circular planetary orbits. Furthermore, giant planets can have a range of masses beyond the simple Jupiter-analogs we have considered here, and can vary in density, especially if the planets are inflated due to increased temperatures (Lopez & Fortney, 2016). This would lead to further variations in the possible merger outcomes (e.g., Metzger et al., 2012; Siverd et al., 2012), while also produce lithium enrichment of the giant planets of varying strengths (e.g., Aguilera-Gómez et al., 2016). Overall, our predicted THJs should have a, potentially significant, influence on the evolution of their host stars post-engulfment, which could present another observable signature of THJs through increased spin rates, larger stellar luminosities, and lithium enrichment. The effects on the further stellar evolution should also be more long-lived than the predicted few years of existence as THJs, making indirect detection of THJs more promising than direct detection.
4.5 Occurrence Rate of HJs and THJs
To gain a better understanding of the importance of HJs and THJs as part of the overall planet population we can estimate the fraction of systems that will produce HJs or THJs in the following way:
where is the fraction of stars in binary systems, close to for A-type stars (e.g., Raghavan et al., 2010), is the fraction of Jupiter-mass planets formed at distances of a few AU from their stars, which is highly uncertain, and which we extrapolate here to be from values for G-type stars (e.g., Wright et al., 2012; Bowler, 2016). Lastly, is the fraction of simulated systems that has undergone one of the possible events specified in Table 3.1. For example, the percentage of systems that form HJs (during the main sequence) of all A-type stars is ( of stars have a Jupiter, and of Jupiters become HJs), while the percentage for THJs is . Interestingly, of all A-type star systems will consume a Jupiter during their main-sequence lifetime and about during their post-main sequence evolution.
We estimate the number of stars in the galaxy as , of which about are in the mass range we consider here (e.g., Salpeter, 1955). Thus, we can also estimate the rate at which a post-main sequence Roche-limit crossing will take place in the galaxy (and might result in a luminosity or spin rate signature). To first order, considering the average lifetime of an A-type star to be on the order of Gyr and assuming roughly uniform formation and death rates, the post-MS Roche-limit crossing rate is approximately per year. We predict that most of these events will be caused by THJ formation and engulfment and that, given that THJs go through their orbital decay phase on the order of a few years, there will be a few to tens of thousands of THJs in the galaxy at any given moment. The length and strength of the increased luminosity signal in red giant stars caused by THJ engulfment and orbital energy deposition is difficult to estimate, but if it is comparable to the Kelvin-Helmholtz timescale, which is on the order of a few ten thousand years for red giants, there should be thousands of such stars with enhanced luminosities in the galaxy at any given moment. These rates indicate that THJs and their effects on post-main sequence stars should be observable and have a strong effect on the luminosity function of intermediate mass red giant stars.
In this work we have explored the dynamical evolution of single giant planets around A-type stars in hierarchical binaries. Considering initially circular planetary orbits between and AU, we identify four principal evolution outcomes:
Classical Hot Jupiters (CHJs): Giant planets that undergo high-eccentricity migration to short period orbits ( days) during the main sequence lifetime of the main star, caused by an interplay of the EKL mechanism and tidal effects. These planets can typically reach temperatures of K and are eventually engulfed by the star as it expands during post-main sequence stellar evolution. About of our giant planets in binaries lead to this result.
Temporary Hot Jupiters (THJs): THJs form during post-main sequence evolution, as the stars expand. These giant planets can either form like CHJs through high eccentricity migration caused by EKL and tides, or their initial orbits were close enough to their stars to be eventually heated up and engulfed by their stars even with low eccentricities ( AU). These planets only exist as HJs for a few years before Roche-lobe crossing and engulfment, but can have significant effects on the stellar envelope and can reach temperatures of K before entering the stellar Roche lobe of the expanding star. About of our systems lead to this outcome.
Roche-limit crossers: These are giant planets that undergo very strong EKL effects that are too strong to be counteracted by tidal forces, thus crossing the Roche limit or grazing the stellar surface at high eccentricities and velocities. About of our giant planets experience this result during the stellar main sequence, while a further do so in the post-main sequence. During the AGB-phase the stars lose a significant part of their mass, changing the orbital parameters of some systems enough to increase EKL strength significantly. This leads to about of our giant planets to accrete and pollute the White Dwarf remnants through high eccentricity Roche-limit crossing.
Surviving Jupiters: Gas giant planets that originally had large orbital periods and did never undergo strong enough EKL effects to lead to significant interactions with their host stars. This is the case for about of our systems.
Overall, only of the planets will survive to the White Dwarf phase without stellar interactions, while will be engulfed at some point in their evolution. The engulfed planets can have significant effects on the stellar rotation rates and luminosities. The EKL mechanism greatly enhances the fraction of planets that end up being engulfed; about of engulfed planets have undergone significant EKL effects. Overall we predict a THJ engulfment rate of per year, which, depending on the length and strength of the engulfment effects onto the red giant stars’ envelopes, could translate to thousands or tens of thousands of red giants with THJ engulfment effects at any given moment in our galaxy.
From our calculation results we also predict that there is a large population of high eccentricity giant planets around A-type stars with orbital periods days, which is difficult to observe, but is consistent with the known high eccentricity giant planets. Our results are also consistent with the observed large, nearly isotropic spread of spin-orbit misalignment angles, further suggesting that stellar binary dynamics are crucial for the understanding of giant planet orbits around A-type star.
SN acknowledges the partial support from the Sloan foundation. SN and APS also acknowledge the partial support from the NSF through grant No. AST-1739160. We thank John Johnson for many inspiring conversations between him and SN during the time she was a postdoc at the ITC. We also thank Ahmed Qureshi for assisting in the continuing improvement of modeling the stellar spins. Calculations for this project were performed on the UCLA cluster Hoffman2.
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Appendix A A-type Stars: Definitions and Evolution
Here we give a short overview of A-type stars’ evolutionary phases. A-type main sequence stars are usually defined to have masses between and M with surface temperatures between about and K (Adelman, 2004). Figure A shows the evolution of temperature vs. luminosity and radius for three example star masses (, , and M), calculated using SSE (Hurley et al., 2000).
As A-type stars evolve along the main sequence, shown in red in Figure A, they slowly expand in radius by about a factor of two, cool down by K, and increase in luminosity by about a factor of two as well. Note that this evolution is slightly different from Sun-like G-type stars, which initially heat up during their main sequence evolution before finally cooling down. The main sequence phase lasts for about Gyrs for M A-type stars and Gyrs for M. After they have expended their core hydrogen fuel, the stars then evolve through the Hertzsprung gap, shown in orange, rapidly expanding by another factor of two, cooling down to about K, and halving their luminosity over the course of Myrs (high to low mass).
Afterwards, the First Giant Branch phase begins, shown in yellow, lasting for Myrs. This phase progresses very differently for low vs. high mass A-type stars. A M star grows from to R, increases in luminosity from to L and cools down from to K, while a M star grows only from to R, increases in luminosity from to L and cools down from to K. At the end of the First Giant Branch phase, the stars contract to their previous radius, luminosity, and temperature, as they begin to burn helium in their cores, shown in green. The helium burning phase lasts for about Myrs for M stars, Myrs for M stars, and Myrs for M stars. After expending their helium fuel, the stars rapidly evolve to become AGB giants, shown as in cyan (first ascent) and dark blue (second ascent), expanding to sizes of about R, cooling down to K, and increasing their luminosities to about L over the course of Myrs. The stars then expel their outer layers and lose mass, becoming white dwarfs.
In total, M stars need Gyrs, M stars need Gyrs, and M stars need Gyrs to reach the white dwarf phase.