Observable Predictions from Perturber-coupled High-eccentricity Tidal Migration of Warm Jupiters
Jonathan M. Jackson, Rebekah I. Dawson, Andrew Shannon, Cristobal Petrovich
DDraft version February 9, 2021
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Observable Predictions from Perturber-coupled High-eccentricity Tidal Migration of Warm Jupiters
Jonathan M. Jackson, Rebekah I. Dawson, Andrew Shannon,
2, 1 and Cristobal Petrovich
3, 4, 5 Department of Astronomy & Astrophysics, Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, UniversityPark, PA 16802 LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon,France Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA Pontificia Universidad Católica de Chile, Facultad de Física, Instituto de Astrofísica, Av. Vicuña Mackenna 4860, 7820436 Macul,Santiago, Chile Millennium Institute for Astrophysics, Chile
ABSTRACTThe origin of warm Jupiters (gas giant planets with periods between 10 and 200 days) is an openquestion in exoplanet formation and evolution. We investigate a particular migration theory in which awarm Jupiter is coupled to a perturbing companion planet that excites secular eccentricity oscillationsin the warm Jupiter, leading to periodic close stellar passages that can tidally shrink and circularizeits orbit. If such companions exist in warm Jupiter systems, they are likely to be massive and close-in, making them potentially detectable. We generate a set of warm Jupiter-perturber populationscapable of engaging in high-eccentricity tidal migration and calculate the detectability of the perturbersthrough a variety of observational metrics. We show that a small percentage of these perturbers shouldbe detectable in the
Kepler light curves, but most should be detectable with precise radial velocitymeasurements over a 3-month baseline and
Gaia astrometry. We find these results to be robust to theassumptions made for the perturber parameter distributions. If a high-precision radial velocity searchfor companions to warm Jupiters does not find evidence of a significant number of massive companionsover a 3-month baseline, it will suggest that perturber-coupled high-eccentricity migration is not thepredominant delivery method for warm Jupiters. INTRODUCTIONBoth hot (periods < 10 days) and warm (periods be-tween 10 and 200 days) Jupiters are commonly thoughtto have formed beyond the ice-line and migrated inwardto their current semi-major axes (Bodenheimer et al.2000; Rafikov 2006; see Dawson & Johnson 2018 for areview of hot Jupiter origins theories). Disk migration(Goldreich & Tremaine 1980; Ward 1997; Baruteau et al.2014) could deliver hot Jupiters, but it is difficult to rec-oncile with the wide eccentricity distribution of warmJupiters because planet-disk interactions tend to dampeccentricities (Bitsch et al. 2013; Dunhill et al. 2013).Planet-disk interactions can sometimes excite eccentric-ities, but they typically saturate at a random velocityequal to the sound speed ( e (cid:46) . for a 100 day orbit;Duffell & Chiang 2015). The eccentricity distribution isdifficult to explain with post-disk planet-planet scatter- Corresponding author: Jonathan M. [email protected] ing because eccentricity growth via scattering is limitedby v escape /v keplerian , which corresponds to an e max ∼ . for a 0.5 m Jup r Jup planet on a 50 day orbit (Goldreichet al. 2004; Ida et al. 2013; Petrovich et al. 2014).Warm Jupiters, particularly those with intermediateto large eccentricities, may have arrived at their shortorbital periods (10–200 days) through high-eccentricitytidal migration (e.g., Wu & Murray 2003). In this sce-nario, the planets form at ∼ a few AU and their orbitsare excited to large eccentricities. Possible mechanismsof eccentricity excitation include planet-planet scatter-ing (Rasio & Ford 1996), secular chaos (Wu & Lithwick2011), and stellar flybys (e.g., Kaib et al. 2013). Thesemechanisms are successful in producing a wide distribu-tion of giant planet eccentricities from multi-planet sys-tems with initially circular orbits (e.g., Jurić & Tremaine2008; Chatterjee et al. 2008). Once the planets are ex-cited to sufficiently large eccentricities, tidal dissipationin the planet during close pericenter passages shrinksand circularizes their orbits.Tidal dissipation is strongly dependent on the distancefrom the star. Although a couple warm Jupiters such as a r X i v : . [ a s t r o - ph . E P ] F e b Jackson et al.
HD 80606 b (Naef et al. 2001) and HD 17156 b (Fis-cher et al. 2007) have pericenter distances potentiallysmall enough for tidal migration, most eccentric warmJupiters do not. One possibility is that warm Jupitersare undergoing Kozai-Lidov oscillations or other secu-lar eccentricity oscillations induced by an exterior per-turber. In this scenario, the planets spend a smallpercentage of their time at eccentricities large enoughfor tidal migration, but would typically be observed atmore moderate eccentricities (e.g., Takeda & Rasio 2005;Dong et al. 2014). Eccentricity oscillations have beenstudied in some warm Jupiter systems with companions(e.g., Kane & Raymond 2014), but they are dependenton the inclination which is often unknown. The pres-ence of subjovian companions in many of these systemssuggests the high-eccentricity migration model cannotaccount for all warm Jupiters (Huang et al. 2016); How-ever, it may still explain those with intermediate to largeeccentricities. Petrovich & Tremaine (2016) determinedthat perturber-coupled high-eccentricity tidal migrationcan account for ∼ of warm Jupiters and most warmJupiters with e ≥ . .Dong et al. (2014) showed that warm Jupiters needmassive, nearby companions to engage in perturber-coupled high-eccentricity tidal migration. They derivedequations for the perturbing strength needed to over-come general relativistic precession and reach large ec-centricities. In this paper, we construct a synthetic pop-ulation of perturbers that meet these requirements andcalculate their observational signatures (including tran-sit timing variations, transit duration variations, radialvelocities, and astrometry). In Section 2 we constructour synthetic warm Jupiter-perturber systems. In Sec-tion 3 we calculate the observational signatures of theperturbers in those systems. In Section 4 we assessthe robustness of our choices of parameter distributions.In Section 5 we perform the same calculations usingthe perturber population from Petrovich & Tremaine(2016). In Section 6 we summarize the population ofobserved warm Jupiters with massive exterior compan-ions and compare these systems to our results. Lastly,we summarize our conclusions and interpret our resultsin Section 7. POPULATION CONSTRUCTIONWe construct a population of warm Jupiter-perturbersystems capable of engaging in high-eccentricity tidalmigration. We will calculate the observational signa-tures of this population of perturbers and compare themto current detection limits (Section 3). In the followingtwo subsections, we describe how we build our popula-tion of warm Jupiters (Sections 2.1, 2.2, and 2.3) and their corresponding perturbers (Section 2.4) from initialparameter distributions that are informed by observedsystems.2.1.
Kepler-like Warm Jupiter Population
Before we can construct the population of perturbersand assess their observability, we must first define a pop-ulation of warm Jupiters. The properties of the warmJupiters will affect the stability of the systems and thecapability of the warm Jupiter to undergo large seculareccentricity oscillations despite general relativistic pre-cession, both of which will dictate the requirements forthe perturbers (see Section 2.4). We assume for thepurpose of constructing this population that all warmJupiters – even those observed with low eccentricities– underwent perturber-coupled high-eccentricity migra-tion in order to reach their current orbits.In Figure 1, we show the observed period and ec-centricity distribution of Jupiter-sized planets. HotJupiters are plotted in red, warm Jupiters in orange,and cold Jupiters in gray. Warm Jupiters with com-panions are plotted as diamonds (companion discoveredwith radial velocities) or triangles (companions discov-ered with transit timing variations). These planets willbe discussed further in Section 6. The black dashed lineis a track of constant angular momentum and will bediscussed further in Section 2.4. Note that the broaddistribution of warm Jupiter eccentricities seen here isone of the motivators for researching high-eccentricitymigration mechanisms.We base our warm Jupiter population on character-istics of the observed distribution. We begin by cre-ating a
Kepler -like sample of transiting warm Jupiterswhich will be appropriate for assessing the observabilityof transit timing and duration variations. These plan-ets have periods between 10 and 200 days and massesbetween 0.1 and 10 m Jup . To build this population, wedraw the planet’s period, mass, and eccentricity fromdistributions that model the intrinsic properties of warmJupiters and then discard any planets with impact pa-rameter b < , which would not transit.We draw our warm Jupiter periods from a power lawwith fitting coefficient P = 0 . (Fernandes et al. 2019).Since the primary Kepler mission lasted ∼ years and3 transits were required to detect a planet, the perioddistribution is complete out to the 200 day maximum inour sample.We draw our warm Jupiter eccentricities from a betadistribution (e.g., Kipping 2013). To find the appropri-ate fitting parameters for warm Jupiters, we fit a betadistribution to the sample of 102 confirmed planets onthe NASA Exoplanet Archive with periods between 10 EM Perturber Observables > . m Jup discovered via theradial velocity technique (as of 2020 February 26). Thefunctional form of this distribution is P ( e ; α, β ) = Γ( α + β )Γ( α )Γ( β ) e α − (1 − e ) β − , (1)where e is the eccentricity, Γ() is the gamma function,and α and β are fitting parameters. Our best fit param-eters are α = 0 . and β = 2 . . We discard any planetswhose semi-major axis and eccentricity would result intidal disruption: a (1 − e ) < a Roche , where a Roche includesa scaling coefficient of f p = 2 . (e.g., Guillochon et al.2011). We note that some planets with eccentricitiesjust below this limit may not survive as warm Jupitersfor long due to rapid tidal circularization, but this isstrongly dependent on the particular tidal parametersof the system.We draw our warm Jupiter inclinations from anisotropic distribution (i.e., uniform in cos i ). We note,however, that after the cut for transiting planets, the in-clination distribution will be strongly peaked near π/ .Lastly, we assume a power law mass distribution( dN/d ln(m) ∝ m α ) with coefficient α = − . foundby Cumming et al. (2008). These authors considered se-lection effects in the RV sample to estimate the mass dis-tribution of nearby giant planets. We limit our massesto be between 0.1 and 10 m Jup .To finish constructing our
Kepler -like warm Jupiterpopulation, in addition to the periods ( P ), eccentricities( e ), inclinations ( i ), and planet masses ( m wj ) describedabove, we draw arguments of pericenter ( ω ), and meananomalies ( M ) from uniform distributions before apply-ing our impact parameter cut of b < . It is not neces-sary to cut in orbital period or eccentricity to accountfor detectability because the Kepler is complete for alleccentricities and periods < 200 days for the mass rangewe are considering. For each system, we assume a solarmass and radius star. Through this method, we build apopulation of , warm Jupiter-star systems.2.2. TESS-like Warm Jupiter population
In addition to transit timing variations (TTVs) andtransit duration variations (TDVs), we want to assessthe detectability of our perturbers via radial velocities(RVs) and astrometry. Unfortunately, most
Kepler sys-tems are too distant for RV or astrometric followup,which can reliably detect planets around bright stars (V (cid:46) ; see Perryman et al. 2014). However, TESS sys-tems and warm Jupiters discovered by the RV method(see Section 2.3) are typically much brighter and closer,making them more amenable for followup. Thus, here - e e Figure 1.
Here we highlight the warm Jupiter populationof exoplanets (orange) in the context of hot (red) and cold(gray) Jupiters. Each point on this plot represents an exo-planet with m > . m Jup and measured eccentricity. Amongthe warm Jupiters, open circles represent planets with noknown companions, closed diamonds represent planets withcompanions detected via radial velocities, and closed trian-gles represent planets with companions detected via transittiming variations. The black dashed line represents the tidalcircularization track of constant angular momentum that amigrating warm Jupiter would follow if it were to end itsmigration as a circular hot Jupiter on a 10-day orbit. Datataken from the NASA Exoplanet Archive (as of 2020 Febru-ary 26). we define a population of
TESS -like warm Jupiter sys-tems.Our
TESS -like sample is very similar to the
Kepler sample defined in Section 2.1, with the exception of itsperiod distribution. Since
TESS is an all-sky mission,individual stars will be observed for a much shorter pe-riod of time than the equivalent from the
Kepler mis-sion, making the expected sample incomplete to our 200day maximum period for warm Jupiters. To account forthis incompleteness, we take the periods of planets withradius r > R Earth from a simulated
TESS yield (Bar-clay et al. 2018), drawing a period value at random fromthis list for each iteration in our warm Jupiter popula-tion construction process. This simulated
TESS yieldapproximates the distribution of expected planetary pa-rameters in the
TESS sample based on current planetoccurrence rate estimates and the
TESS instrumental
Jackson et al. capabilities. Barclay et al. (2018) predicts the discoveryof roughly 300 warm Jupiters by
TESS .We follow the method described in Section 2.1 to draweccentricities ( e ), inclinations ( i ), masses ( m wj ), argu-ments of pericenter ( ω ), and mean anomalies ( M ), ap-plying the same cuts for transiting planets and stablewarm Jupiter orbits. As with the Kepler planets, webuild a population of 10,000
TESS -like warm Jupitersystems.2.3.
RV-discovered warm Jupiter population
Lastly, we would like to assess the detectability of per-turbing companions to warm Jupiters discovered by theradial velocity (RV) method. These companions couldbe detectable by RVs or astrometry because the hoststars are typically bright and nearby.Our population of RV-detected warm Jupiters is morestraightforward to develop than the transiting popu-lations described in Sections 2.1 and 2.2 because themasses and eccentricities are well-characterized. Thus,we can directly use the observed planets, pulling theirminimum masses ( m wj sin i wj ), periods ( P ), eccentrici-ties ( e ), arguments of pericenter ( ω ), and stellar mass( m ∗ ) from the sample of 66 RV-detected warm Jupiterswith well-characterized orbits on the NASA ExoplanetArchive (as of 2020 February 26). We draw the meananomalies ( M ) of the planets from a uniform distribu-tion and the inclinations ( i ) from an isotropic distribu-tion, cutting out orientations in which the derived m wj is above 10 m Jup . The use of an isotropic inclinationdistribution introduces a slight bias because we are nowdealing with observed planets instead of an intrinsic pop-ulation; however, this effect is small for the mass rangewe are considering (Ho & Turner 2011; Lopez & Jenk-ins 2012). We draw from our sample of warm Jupiters10,000 times to create a population similar to our twotransit-detected warm Jupiter populations.2.4.
Perturber Population
Now we construct the population of perturbing com-panion planets that will each accompany a warmJupiter. We set two criteria for accepting a given per-turber: (1) the warm Jupiter-perturber system must belong-term stable and (2) the perturber must be capa-ble of periodically exciting the eccentricity of the warmJupiter high enough for it to undergo tidal migrationthrough an ongoing cycle of secular angular momentumexchange.To implement the first criterion, we impose an analyt-ical stability requirement from Petrovich (2015a) which, when satisfied, should result in the system evolving sec-ularly with no close encounters: a per (1 − e per ) a wj (1 + e wj ) > . µ wj , µ per )] / (cid:18) a per a wj (cid:19) / +1 . , (2)where a per(wj) is the semi-major axis of the per-turber(warm Jupiter) planet, e per(wj) is the eccentricityof the perturber(warm Jupiter) planet, and µ per(wj) = m per(wj) /m ∗ is the perturber(warm Jupiter) planet-to-star mass ratio.To implement the second criterion, we first set an em-pirical limit on the warm Jupiter’s minimum angularmomentum for tidal migration: a (1 − e ) < a f , crit = 0 . , (3)where a (1 − e ) is the final semi-major axis of a tidallycircularized planet when e = 0 and a f , crit = 0 . AU isthe final semi-major axis required to circularize withinthe age of the universe. The value of a f , crit comes fromthe observed population of hot Jupiters which tend toexhibit small eccentricities for a < . AU (Socrateset al. 2012; Dong et al. 2014). In reality, the limit islikely much stricter since there are still some moderateeccentricities among hot Jupiters with a > . and thetidal circularization timescale scales vary strongly withseparation. However, we adopt the conservative value of a f , crit = 0 . AU to ensure the criterion presented belowis a necessary requirement for perturber-coupled high-eccentricity tidal migration. Figure 1 shows the theoret-ical limit from equation 3 plotted as a black dashed lineover the observed Jupiter-mass planet a vs. e distribu-tion. A planet must spend time above the line on thisplot in order to tidally migrate to become a hot Jupiter.We impose an analytical requirement from Dong et al.(2014) which, when satisfied, implies that the perturberis capable of inducing eccentricity oscillations in thewarm Jupiter that can overcome precession due to gen-eral relativity and satisfy equation 3 for some portionof its secular cycle. This is a minimum requirement forhigh-eccentricity tidal migration, but does not guaranteeit will occur. b per a wj < (cid:18) Gm ∗ c a wj (cid:19) − / (cid:18) m ∗ m per (cid:19) − / × (cid:114) a wj a f , crit − (cid:113) − e − / (cid:20) − a f , crit a wj − e (cid:21) / × (cid:0) − i mut sin ω wj (cid:1) / , (4) EM Perturber Observables b per = a per (1 − e ) / is the semi-minor axisof the perturber, m ∗ is the stellar mass, m per is themass of the perturber, e wj is the eccentricity of the warmJupiter, a wj is the semi-major axis of the warm Jupiter, ω wj is the argument of pericenter of the warm Jupiterin the invariable plane, i mut is the mutual inclinationbetween the two planets, and a f , crit = 0 . .The population of planets with orbital periods beyond200 days is less constrained by the observations thanthat of the warm Jupiters, so we have to make someassumptions about the distribution of orbital and phys-ical parameters. The robustness of our choices for thesedistributions is discussed in Section 4.Our population of perturber eccentricities ( e ) is drawnfrom a Beta distribution (equation 1) with best fit pa-rameters α = 0 . and β = 1 . . determined by fittingthe beta distribution to the sample of 428 long-period,Jupiter sized ( m Jup > . ) confirmed planets discoveredvia radial velocity from the NASA Exoplanet Archive(as of 2020 February 26).The perturber sky-plane inclinations ( i ) are drawnfrom an isotropic distribution. The orbital angles ω , Ω , and M are drawn from uniform distributions in theinterval [0 , π ] . For the perturber population, we drawthe masses ( m ) from the power law distributions calcu-lated by Cumming et al. (2008) with a range of 0.1 to20 m Jup . We choose the range of masses based on pre-vious studies that show consistency with the power lawfit in that range (e.g., Fernandes et al. 2019). Above 20 m Jup , companions are more likely to be a part of thelow-mass end of the brown dwarf population, which fol-lows a different mass function (Bowler 2016). We assessour choice for the upper mass limit in Section 4.5.Our perturber periods follow a symmetric brokenpower law, dNd log P ∝ (cid:16) PP break (cid:17) P P ≤ P break (cid:16) PP break (cid:17) − P P > P break (5)where P break = 859 days is the turnover point in theperiod distribution and P = 0 . is the power law co-efficient. Fernandes et al. (2019) showed that a bro-ken power law (with these fitting coefficients) is a bet-ter fit to the observed period distribution than a singlepower law or a log normal distribution. We draw ourperiods from the range 200 to 100,000 days. The lowerlimit is set to ensure these are exterior companions towarm Jupiters and the upper limit is set by the fact thatplanet-mass objects with periods beyond 100,000 daysalways fail equation 4 for the mass range we test.For a given warm Jupiter-star system, we draw per-turber properties from the distributions listed above and check against our two acceptance constraints (equations2 and 4). If the set of properties fails either of the crite-ria, we redraw the parameters and recheck against thecriteria. We repeat this process until every warm Jupiterhas an accompanying perturber. OBSERVATIONAL SIGNATURES OFPERTURBERS TO WARM JUPITERSIn this section, we compute the observational signa-tures for our sample of synthesized systems (Section2). For each warm Jupiter discovery method (i.e.,
Ke-pler transits,
TESS transits, radial velocities), we have10,000 systems on which to run our calculations of per-turber observables. In Sections 3.1, 3.2, 3.3, and 3.4,we calculate the transit timing variation (TTV), tran-sit duration variation (TDV), radial velocity (rv), andastrometric signals, respectively. Finally, in Section 3.5,we assess the overall detectability of our sample of per-turbers. 3.1.
Transit Timing Variations
First, we calculate the transit timing variation (TTV)signal of the perturbers to the
Kepler -like warmJupiters. We do not run this calculation on the
TESS -like systems because the time baselines of most
TESS planets will be too short to detect TTVs. The type ofTTVs we expect are typical of those seen in hierarchicaltriple systems (e.g., Borkovits et al. 2003, 2004).We model the TTVs over the
Kepler mission life-time of 4 years. We calculate the TTVs numerically bysimulating the 3-body motion of the two planets andtheir host star over those 4 years and recording themid-transit time for each orbit of the warm Jupiter us-ing Dawson et al. (2014)’s N -body transit time code.We then calculate the deviations from a best-fit linearephemeris at each time, which define our TTVs.We define two metrics for distinguishing detectableTTV system from non-detectable TTV systems follow-ing Mazeh et al. (2013). The first metric is the ratioof the scatter in the TTVs, s TTV , defined as the me-dian absolute deviation (MAD) of the TTV series, totheir median error, ¯ σ TT . We randomly draw ¯ σ TT fromthe median uncertainties in mid-transit time of Holczeret al. (2016)’s light curve fits to warm Jupiter Kepler candidates. The second metric is the false alarm prob-ability (FAP) of the Lomb-Scargle (LS) periodogram ofthe TTV time series. To generate the LS periodogram,we use the rvlin package (Wright & Howard 2009), as-signing the median uncertainty to each data point inour simulated TTV time series and applying a Gaus-sian scatter. We categorize perturbers as detectable viaTTVs when the s TTV / ¯ σ TT ratio is greater than 5 or the Jackson et al.
LS FAP is below × − . Note that Mazeh et al. (2013)set the threshold for s TTV / ¯ σ TT at the more conservativevalue of 15.In Figure 2, we show four representative examples ofTTV time series from our sample. The top panel showsthe signal of an undetectable perturber, the second panelshows the signal of a perturber detectable by our firstmetric ( s TTV / ¯ σ TT > ), the third panel shows the signalof a perturber detectable by our second metric (LS FAP< × − ), and the bottom panel shows the signal of aperturber detectable by either metric. Each plotted timeseries has an uncertainty drawn from the median un-certainties in mid-transit time of Holczer et al. (2016)’slight curve fits to warm Jupiter Kepler candidates and aGaussian scatter has been applied. We present a compi-lation of these results from all simulated systems in Fig-ure 3, where purple points represent detectable systemsand red dashed lines delineate our detection criteria.Using these metrics, if each
Kepler warm Jupiter is ac-companied by a perturber that fits our criteria, a smallportion of these perturbers (7.7%) should be detectableby TTVs in the
Kepler light curves. Using the more con-servative cutoff of s TTV / ¯ σ TT > , this number dropsto (4.9%). The perturbers that are detectable are typi-cally short period planets (<5000 day orbits) with largemasses. See Section 3.5 for further discussion of thedetectability of the perturbers. Since the majority ofour synthesized perturbers produce undetectable TTVsand some TTVs have been detected for Kepler warmJupiters (see Holczer et al. 2016), these results do notconstrain the high-eccentricity migration mechanism.3.2.
Transit Duration Variations
Next, we calculate the transit duration variation(TDV) signal of the perturbers to the
Kepler -like warmJupiters. We model the TDVs simultaneously with theTTVs over the 4-year
Kepler mission lifetime by track-ing the slight changes to the impact parameter and tran-sit speed of the warm Jupiter between each orbit. Wecalculate the approximate transit durations using thefollowing equation from Winn (2010) which includes acorrection for non-circular orbits: T tot = Pπ arcsin (cid:34) R ∗ a (cid:112) (1 + k ) − b sin i (cid:35) (cid:32) √ − e e sin ω (cid:33) (6)where P is the period, R ∗ is the stellar radius, a isthe semi-major axis, k is the planet-star radius ratio, b is the impact parameter, i is the inclination, e is theeccentricity, and ω is the argument of pericenter. Ourfinal TDV amplitudes are calculated by subtracting the -40-2002040 TT V ( m i n ) -40-2002040 TT V ( m i n ) -40-2002040 TT V ( m i n ) TT V ( m i n ) Figure 2.
Four representative examples of TTV time seriesin our
Kepler -like population. The top panel shows the sig-nal of an undetectable perturber ( s TTV / ¯ σ TT = 0 . ; LS FAP= 0.31), the second panel shows the signal of a system de-tectable by the ratio of scatter to median error in the TTVs( s TTV / ¯ σ TT = 8 . ; LS FAP = 0.50), the third panel showsthe signal of a system detectable by its periodogram signal( s TTV / ¯ σ TT = 1 . ; LS FAP = . × − ), and the bottompanel shows the signal of a perturber detectable by eithermetric ( s TTV / ¯ σ TT = 7 . ; LS FAP = . × − ). EM Perturber Observables TTV / s TT -12 -10 -8 -6 -4 -2 L S F AP Detectable Not detectable
Figure 3.
The majority of synthesized perturbers (92.3%)in our
Kepler -like warm Jupiter sample induce transit tim-ing variations (TTVs) in the warm Jupiters that are belowthe thresholds necessary for detection (gray points). Systemswith s TTV / ¯ σ TT above 5 or LS FAP below × − are cat-egorized as detectable (purple circles). The plot shows thevalue of these metrics for a random sample of 1000 systemsin our Kepler -like population. minimum from the maximum duration among all of thetransits.As with the TTVs, we define two metrics for distin-guishing detectable TDV systems from non-detectableTDV systems. Unlike TTVs where any linear trend issubtracted out, the slope of a TDV signal is meaningful.Therefore, our first metric is the strength of the lineartrend in the TDV data, or more specifically, the ratioof the estimated slope over the estimated error in theslope ( | slope | / e slope ) following Kane et al. (2019). Werandomly draw the median uncertainty in transit dura-tion from Holczer et al. (2016)’s light curve fits to warmJupiter Kepler candidates. The second metric is thefalse alarm probability (FAP) of the Lomb-Scargle (LS)periodogram of the TDV time series. As with the TTVs,we generate the LS periodogram using the rvlin pack-age (Wright & Howard 2009), assigning the median errorto each data point and applying a Gaussian scatter. Wecategorize perturbers as detectable via TDVs when the | slope | / e slope is greater than 3.5 or the LS FAP is below × − (Kane et al. 2019).In Figure 4, we show four representative examples ofTDV time series from our sample. The top panel shows -20020 T D V ( m i n ) -20020 T D V ( m i n ) -20020 T D V ( m i n ) T D V ( m i n ) Figure 4.
Four representative examples of TDV time seriesin our
Kepler -like population. The top panel shows the sig-nal of an undetectable perturber ( | slope | / e slope = 1 . ; LSFAP = 0.32), the second panel shows the signal of a sys-tem detectable by the ratio of its TDV slope to the error inslope ( | slope | / e slope = 15 . ; LS FAP = 0.06), the third panelshows the signal of a system detectable by its periodogramsignal ( | slope | / e slope = 0 . ; LS FAP = . × − ), and thebottom panel shows the signal of a perturber detectable byeither metric ( | slope | / e slope = 39 . ; LS FAP = . × − ). Jackson et al. slope -12 -10 -8 -6 -4 -2 L S F AP Detectable Not detectable
Figure 5.
Same as Figure 3 but for our transit durationvariation (TDV) calculation. The majority of synthesizedperturbers (86.5%) in our
Kepler -like warm Jupiter sam-ple induce TDVs in the warm Jupiters that are below thethresholds necessary for detection (gray points). Systemswith | slope | / e slope above 3.5 or LS FAP below × − arecategorized as detectable (yellow circles). The plot shows thevalue of these metrics for a random sample of 1000 systemsin our Kepler -like population. the signal of an undetectable perturber, the second panelshows the signal of a perturber detectable by our firstmetric ( | slope | / e slope > . ), the third panel shows thesignal of a perturber detectable by our second metric(LS FAP < × − ), and the bottom panel shows thesignal of a perturber detectable by either metric. Eachplotted time series has an uncertainty drawn from themedian uncertainties in transit duration from Holczeret al. (2016)’s light curve fits to warm Jupiter Kepler candidates and a Gaussian scatter has been applied. Wepresent a compilation of these results from all simulatedsystems in Figure 5, where yellow points represent de-tectable systems and red dashed lines delineate our de-tection criteria.If each
Kepler warm Jupiter is accompanied by a per-turber that fits our criteria from Section 2.4, a smallpercentage of these perturbers (13.5%) should be de-tectable by TDVs in the
Kepler light curves. Like withthe TTVs, short-period perturbers with large masses arethe easiest to detect. See Section 3.5 for further discus-sion of the TDV detectability of the perturbers. As withthe TTVs (Section 3.1), these results alone cannot signif- icantly constrain the high-eccentricity migration mech-anism. 3.3.
Radial Velocity Trends
Next, we calculate the radial velocity (RV) signalof the synthesized perturbers to our
TESS -like warmJupiters and our RV-detected warm Jupiters. SinceRV followup of known planetary systems does not comefrom a single instrument or survey, we model the signalwith two representative time baselines (3 months and 3years). The periods of our perturbers span the range of200 to 100,000 days, so some signals may appear weakover a short time baseline, but strong once a larger per-centage of their orbit is observed.To calculate the observed ∆ RV over a given time base-line, we first model the radial motion of the star due toboth planets. For the 3-month time baseline, we assume20 data points and for the 3-year baseline, we assume 100data points. For each point, we assume a 1 m/s uncer-tainty, which could arise from instrument error and/orstellar variability. Once we have our RV signal, we fita 1-planet model to it using mpfit (Markwardt 2009),fixing the period and mean longitude λ for the TESS -like population. Our final calculated RV is the 2-planetsignal with the best-fit 1-planet solution subtracted out.This approach is more conservative than simply calcu-lating the outer planet’s signal because, for a real sys-tem, we do not precisely know all of the inner planetparameters a priori and they might be conflated in thetwo-planet fit.In order to determine which signals are detectable, wecalculate the ratio of the slope of the RV signal to theerror in the slope ( | slope | / e slope ). As with the TDVsignals, we categorize perturbers for which this met-ric is greater than 3.5 as detectable since those systemswould have RV trends that are statistically distinguish-able from zero. We also calculate a Lomb-Scargle (LS)periodogram of the RV time series, but it is ineffectiveat discovering the long-period signals of our perturbers,so we omit it here.The results of both the 3-month calculation and the3-year calculation are plotted in Figure 6. We findthat 77.2(91.2)% of our companions to TESS -like warmJupiters and 31.4(83.0)% of our companions to RV-detected warm Jupiters would produce detectable sig-nals over 3 months(3 years) of RV data. The bump near0.0001 in the slope ratio for the 3-month RV-detectedwarm Jupiter companions is caused by degeneracies be-tween the signals of the inner and outer planets thatare broken with a longer time baseline. Note that e slope is strongly dependent on the stellar activity of the hoststar and our assumed RV uncertainty of 1 m/s is rep- EM Perturber Observables
TESS perturbers are 64.4% detectable over 3months of observations at 3 m/s uncertainty or 32.9%detectable at 10 m/s uncertainty). The perturbers thatare detectable by their RV slope cover the full massrange tested, with larger masses allowing for longer pe-riod detections. See Section 3.5 for further discussionof the detectability of the perturbers and Section 6 fora discussion of the known companions to RV-detectedwarm Jupiters. 3.4.
Astrometry
Finally, we calculate the astrometric signal of the syn-thesized perturbers to our
TESS -like warm Jupiters andour RV-detected warm Jupiters as they would be ob-served by
Gaia (Lindegren & Perryman 1996; Perrymanet al. 2001). As with the RV calculation, we leave out
Kepler warm Jupiters here because their host stars aretypically too far away for
Gaia to successfully detect anastrometric planetary signal. Specifically,
Gaia is ex-pected to detect planets out to 500 pc (Perryman et al.2014), while nearly all Kepler warm Jupiter hosts areoutside of that radius. We model the transverse motionof the stellar host on the sky due to the gravitationalpull of the perturber, then subtract off a linear fit, whichwould be removed as proper motion. The following isour model for the star’s motion in right ascension anddeclination ξ ( t ) = α ∗ + ( t − t ) µ α ∗ + BX ( t ) + GY ( t ) (7) η ( t ) = δ + ( t − t ) µ δ + AX ( t ) + F Y ( t ) , (8)where α ∗ = α cos δ , ( α ∗ , δ ) is the reference position ofthe star, ( µ α ∗ , µ δ ) is the proper motion vector, A , B , F ,and G are the Thiele-Innes constants, and (X(t),Y(t))is the vector describing the motion of the star in itsorbit as a function of the eccentric anomaly, E. We cal-culate E iteratively from the mean anomaly, M, and theeccentricity, e, following Danby (1988). For a detaileddescription of the astrometric model, see Quirrenbach(2010). Gaia is expected to observe every star in its catalogue ∼
70 times over the course of its 5 year primary missionlifetime (Jordan 2008). For representative distances thatare consistent with typical
TESS warm Jupiters (200parsecs) and typical RV Warm Jupiters (50 parsecs),all of which will appear in the
Gaia astrometric cata-logue, we calculate the maximum angular distance, ∆ θ ,between data points in the model for each of our sim-ulated systems after subtracting off the linear proper R e l a t i v e N u m be r slope R e l a t i v e N u m be r TESS WJsRV WJs
Figure 6.
A large fraction of synthesized perturbers in our
TESS -like (top) and RV-detected (bottom) warm Jupitersystems are detectable via their RV signal. Here we showhistograms of the | slope | / e slope metric for two representativetime baselines: 3 months (black) and 3 years (blue). For eachsystem, we calculate the 2-planet RV signal and subtract offa 1-planet warm Jupiter fit to derive at an isolated compan-ion signal. The red dashed line is placed at 3.5, above whichwe describe the RV trend as detectable. Jackson et al. motion fit. We plot the results of this calculation for
TESS -like warm Jupiter systems and RV-detected warmJupiter systems in Figure 7. Also plotted in Figure 7 is aline indicating the conservative
Gaia detection limit forstars brighter than G = 12 mag of ∼ µ as (Perrymanet al. 2014). 16.7(40.3)% of the perturbers in TESS -likewarm Jupiter systems and 9.8(29.6)% of the perturbersin RV-discovered warm Jupiter systems would be de-tectable with Gaia astrometry at distances of 200(50)parsecs. Only perturbers with large masses (>1 M Jup )are detectable with this method and intermediate peri-ods are preferred. See Section 3.5 for further discussionof the detectability of the perturbers.3.5.
Detectability
As shown in sections 3.1 and 3.2, a small portion ofour perturbers should be detectable in the current
Ke-pler light curves via TTVs or TDVs (19.0%, combined).However, because most of the
Kepler stars are too dimand too far away for RV or astrometric planet detec-tion, most of the perturbers in those systems will remainhidden, even if they exist. On the other hand, manyof the perturbers in
TESS and RV-discovered warmJupiter systems (77.2% and 31.4%, respectively) are ob-servable with high precision RV followup with even ashort time baseline (see Section 3.3).
Gaia will also beable to detect many perturbers associated with
TESS and RV-discovered warm Jupiters (16.7% and 29.6%,respectively; see Section 3.4). Furthermore, when com-bining the RV and astrometric detection methods to-gether, nearly all of the parameter space for perturbersto
TESS warm Jupiter and most of the parameter spacefor RV-discovered warm Jupiters that satisfy the twoconstraints in Section 2.4 are detectable with currentor near future instruments (78.3% and 45.8%, respec-tively).In Figure 8 we show the detectability of all of oursynthetic perturbers with current and near-future in-struments in terms of their masses and periods. Weclassify a perturber to a
Kepler warm Jupiter as de-tectable via its transit timing variation (TTV) signal if s TTV / ¯ σ TT > or LS FAP < × − (Purple circles).We classify a perturber to a Kepler warm Jupiter asdetectable via its transit duration variation (TDV) sig-nal if | slope | / e slope > . or LS FAP < × − (yellowcrosses). We classify a perturber to a TESS or RV-discovered warm Jupiter as detectable via its observableradial velocity (RV) trend if | slope | / e slope > . over 3months of observations (blue circles). Lastly, we classifya perturber to a TESS or RV-discovered warm Jupiteras detectable via its astrometric signal ( ∆ θ ) if ∆ θ > µ as assuming a distance of 200 parsecs for TESS sys- R e l a t i v e N u m be r D q ( m as)0.000.050.100.15 R e l a t i v e N u m be r
200 pc 50 pc
TESS WJsRV WJs
Figure 7.
Same as Figure 3 but for the astrometric signalof our perturbers to
TESS warm Jupiters (top) and RV-discovered warm Jupiters (bottom). The black histogramrepresents the astrometric signal assuming a 200 pc distance(typical of
TESS warm Jupiter systems) and the green his-togram represents the same calculation with a 50 pc distance(typical of RV warm Jupiter systems). For each system, wecalculate the isolated astrometric signal due to the perturberand subtract off a linear proper motion fit. The red dashedline represents the expected minimum
Gaia sensitivity of 100 µ as. EM Perturber Observables
Ke-pler -like (top),
TESS -like (middle), and RV-discovered(bottom) samples. As stated above, 19.0% of our per-turbers to
Kepler warm Jupiters should currently bedetectable in the
Kepler light curves and 78.3(45.8)% ofour perturbers to
TESS (RV-discovered) warm Jupiterswill be detectable in the near future with RV and as-trometric followup. Those that are undetectable tendto have low masses and intermediate-to-long periods, asthese areas of parameter space are the most difficult todetect with RVs and astrometry and tend to produceweaker TTV and TDV signals.If, after sufficient RV followup of
TESS warm Jupitersand the release of the planets detected by
Gaia , we donot find a large number of strong perturbers accompany-ing warm Jupiters, we can infer that perturber-coupledhigh-eccentricity migration may not be a common mech-anism for delivering warm Jupiters. TESTING OUR ASSUMED PARAMETERDISTRIBUTIONSHere we consider alternative choices for warm Jupiterand perturber population parameters that are not well-constrained observationally. Our fiducial warm Jupiterpopulations are described in Sections 2.1, 2.2, and 2.3and consist of gas giant planets with periods between10 and 200 days and a wide range of eccentricities. Weexamine these parameter ranges in Section 4.1. Ourfiducial perturber population is described in Section 2.4and includes masses between 0.1 and 20 m Jup drawnfrom the Cumming et al. (2008) power law fit, periodsdrawn from the Fernandes et al. (2019) broken powerlaw fit, eccentricities drawn from a beta distribution fit,sky plane inclinations drawn from an isotropic distribu-tion, and all other orbital angles drawn from uniformdistributions between 0 and π . We test these parame-ter distributions in Sections 4.2, 4.3, 4.4, and 4.5. Be-cause the fraction of Kepler -like perturbers detectableby their transit timing and duration variations is small,we focus on only the
TESS -like and RV-discovered sys-tems in these sections. We note, however, that when weapply these same tests to the
Kepler -like population, thefraction of detectable perturbers remains near ∼ inall cases. Figure 9 shows the calculated RV and astrometry sig-nals for all of the perturber parameter distributions wetest in Sections 4.2, 4.3, and 4.4. We assume a 3 monthtime baseline for the RV calculations. For the astro-metric calculations, we assume a distance of 200 pc forthe TESS -like warm Jupiters and 50 pc for the RV-discovered systems. The details of each alternative pa-rameter distribution are discussed in the following sub-sections, but Figure 9 generally shows that our resultsare insensitive to the particular perturber populationused for the calculations. We test the sensitivity of ourresults to changes in the inclination (Section 4.2), eccen-tricity (Section 4.3), and period (Section 4.4) distribu-tions. For all of these tests, we maintain the power lawmass distribution fit from Cumming et al. (2008). Al-though the mass and period distributions are dependenton one another, Bryan et al. (2016) and Fernandes et al.(2019) both find consistency with the Cumming et al.(2008) mass distribution, but inconsistency with theirperiod distribution. We test the impact of the range ofmasses we explored in Section 4.5.4.1.
Warm Jupiter Parameter Ranges
Thus far in this study, we have considered perturbersto warm Jupiters from the full range of periods (10-200 days). Here, we focus only on short-period warmJupiters (
P < days) and rerun our calculations in or-der to determine if a more focused future search mightproduce more robust results. We find that this periodcut slightly shrinks the allowed area of the parameterspace for perturbers, which, in turn, increases the de-tectability of the perturbers. This effect is very smallfor the TTV, TDV, and astrometric detection tech-niques as well as for the RV-detected perturbers to TESS warm Jupiters which tend to have short periods anyway.However, The RV-detectability of RV-discovered warmJupiters is approximately double that of our prior results(62.7% over a 3-month baseline).Until this point, we have also assumed all warmJupiters may have been delivered by the high-eccentricity tidal migration mechanism, even those withlow observed eccentricities. However, it may be likelythat most low eccentricity warm Jupiters arrived via adifferent formation or migration method (e.g., Dawson& Murray-Clay 2013; Petrovich & Tremaine 2016). Totest the effects of our choice on the results of this paper,we rerun our calculations considering only perturbers towarm Jupiters with large eccentricities ( e > . ). We findthat the detectability of these perturbers is not signifi-cantly larger than our prior result for any of the consid-ered detection techniques.4.2. Perturber Mutual Inclination Jackson et al. M a ss ( M J up )
500 5000 50000Period (days)0.1110 Detectable via TTVs Detectable via TDVs Detectable via RVs Detectable via astrometry Not detectable
Kepler WJsTESS WJsRV WJs
Figure 8.
The majority of our simulated perturbers to
TESS (middle) and RV-discovered (bottom) warm Jupiters are detectableby their radial velocity and astrometric signal, while only a small percentage of our simulated perturbers to
Kepler (top) warmJupiters are detectable by their transit timing and duration variations. Here we show the mass vs. period of these perturbers,as these both have strong effects on the signal of all four observables. In each panel, we plot a random sample of only 1000systems for clarity. The black shading is a contour map showing the ratio of detectable perturbers to undetectable perturberssuch that whiter areas of parameter space imply more detections. The lower right area of parameter space (red shaded area) islargely undetectable, but there are no perturbers in those regions that satisfy our perturber strength criteria from Section 2.4.
EM Perturber Observables i ) and longitudes of ascendingnode ( Ω ) of the perturbers to be identical to those of thewarm Jupiter. We find the observed RV trends for bothpopulations of warm Jupiters to be indistinguishable be-tween the coplanar case and fiducial isotropic case. Theastrometric signal of the coplanar perturbers is shiftedtowards smaller ∆ θ in the transiting warm Jupiter case.This can be attributed to the fact that, in edge-on sys-tems, one dimension of astrometric movement is lost. Inthe RV-discovered warm Jupiter case, this discrepancyis not present because the sky-plane inclinations of thesesystems are widely distributed. In general, our assump-tion of isotropic perturbers has a small effect on ourresults, but may predict more astrometric detections for TESS companions than we would expect from a copla-nar inclination distribution.4.3.
Perturber Eccentricity
We now examine the eccentricity distribution of ourperturbers. Our fiducial perturber population drew ec-centricities from a Beta distribution with α = 0 . and β = 1 . (see Section 2.4). In comparison to this,we first test a uniform eccentricity distribution between e =0-0.5. While a maximum e of 0.5 is not strongly ob-servationally motivated, it is consistent with assump-tions used by previous theoretical studies on this topic(Dong et al. 2014; Petrovich & Tremaine 2016). Addi-tionally, we test the two extreme cases of zero eccentric-ity and uniform eccentricities between e =0-1, the latterof which which is more representative of the observed bi-nary stellar companion sample (Duchêne & Kraus 2013).We find that, even including the extreme cases, all ofour eccentricity distributions produce similar distribu-tions of observables for each case we test. As shown inFigure 9, each eccentricity test closely follows the fidu-cial population histogram.4.4. Perturber Orbital Period
Next, we consider how the underlying perturber or-bital period distribution we draw from affects our calcu-lated observables. Our fiducial periods are drawn froma broken power law distribution. We note that our per-turber periods, masses, and eccentricities are drawn be-fore applying the two analytical cuts described in Sec-tion 2.4. Since any perturbers that do not satisfy thestability or perturber strength criteria are discarded and redrawn, the final distributions are different from theunderlying distributions we draw from. The filtering bythese criteria is particularly important for orbital peri-ods, where short and long periods are disfavored by thetwo cuts and intermediate periods are selectively chosen.In Figure 10, we show the four period distributionsthat we test, before and after applying our two cuts: alog-uniform distribution, power law fits from Cumminget al. (2008) and Bryan et al. (2016), and a broken powerlaw fit from Fernandes et al. (2019). Cumming et al.(2008) found evidence supporting a rising power law pe-riod distribution for giant planets, but had relatively fewdata points and were limited to periods under 2000 days.Using new RV data, Bryan et al. (2016) confirmed therising power law trend in orbital period for short periodgiant planets. However, when they assessed the perioddistribution for companions to short period giant plan-ets discovered by long term radial-velocity monitoringand adaptive optics (AO) imaging, they found evidencefor a decreasing power law coefficient for giant planetsat larger periods and speculated that a turnover in theperiod distribution must occur between 3 and 10 au.Here we use their single power law fit for planets withmasses between 0.5 and 20 M Jup and semi-major axesbetween 5 and 50 au, but note that the results of thisfit were very sensitive to the particular mass and semi-major axis ranges chosen. With an even larger set ofRV data, Fernandes et al. (2019) modeled the turnoverin the period distribution as a broken power law, whichwe use as our fiducial period distribution. We use thepower law fits to the RV data from each study in ourcomparison here, whose fitting variables are recorded inTable 1.The four different functions used to create our perioddistributions are significantly different in their shapesand the locations of their peaks. However, as shown inFigure 10, after accounting for perturber strength andstability, all four functions peak at intermediate periodsand drop off toward longer or shorter periods. Becausethe final distributions after the criteria are applied inall cases are similarly shaped, the RV signal is insensi-tive to the particular underlying period distribution cho-sen. The astrometric signal is slightly affected by thischoice due to its strong dependence on planet’s semi-major axis. Thus, the Bryan et al. (2016) fit, which hadthe longest peak period, produces a slightly stronger as-trometric signal. However, these effects are small andthe Fernandes et al. (2019) distribution, which we usein the analysis of our results, is the most conservative ofthe four in terms of its astrometric signal.If all giant planets form beyond the ice line and thosethat we observe interior to that point are the result of4
Jackson et al. R e l a t i v e N u m be r slope R e l a t i v e N u m be r TESS WJsRV WJs R e l a t i v e N u m be r D q ( m as)0.000.050.100.15 R e l a t i v e N u m be r coplanar zero e uniform e [0,0.5] uniform e [0,1] log uniform periodsCumming+(2008) periods Bryan+(2016) periods fiducial TESS WJsRV WJs
Figure 9.
The results of our calculated observables are largely insensitive to changes in the underlying perturber parameterdistributions. Each histogram on this plot shows the recalculated RV and astrometric signals for our perturbers assumingdifferent underlying distributions for their period, eccentricity, and inclination, as described in Section 4. migration, we might expect the companions to planetsmigrating via high-eccentricity migration to remain atlarge separations for their entire lifetime. Throughoutthis work, we have allowed our coupled perturbers toexist anywhere outside of a 200 day orbit. We test howthis assumption affects our results by removing any per-turbers interior to the ice line and rerunning our calcu-lations. The exact location of the ice line is under inves- tigation, but here we use 2.7 au as an illustrative valueLecar et al. (2006). We find that the RV detectability ofthese planets is not significantly affected by this change,while the TTV/TDV detectability is decreased, and theastrometric detectability is increased. This is consistentwith our previous findings that, if these perturbers ex-ist, we do not expect to find many in the
Kepler data,but should find many by following up
TESS and RV-
EM Perturber Observables Table 1.
Fitting parameters for each tested period distribution.Distribution Parameter 1 Parameter 2 Referencelog uniform() - - -power law( α ) 0.26 - Cumming et al. (2008)power law( α ) 0.38 - Bryan et al. (2016)broken power law( P , P break ) 0.63 859 days Fernandes et al. (2019)
200 2000 20000Perturber Period (days)0.000.010.020.030.040.05 R e l a t i v e nu m be r Log UniformCumming+(2008)Bryan+(2016)Fernandes+(2019)
Figure 10.
Perturber period distributions before (dashedline) and after (solid line) perturber strength and stabilitycuts (see Section 2.4) are applied. These cuts reduce thenumber of short and long period perturbers and favor inter-mediate periods near ∼ discovered warm Jupiters. In fact, if the perturbers areall beyond the ice line, Gaia will be much more effectiveat finding them.4.5.
Perturber Mass Range
Lastly, we consider the effects of our choice for theupper limit on the mass of our perturbers. All of the perturbers described above are of planetary mass ( < m Jup ). Stellar perturbers could also produce the de-sired eccentricity excitation in the warm Jupiter. How-ever, since binary systems harboring planets are prefer-entially widely separated (Eggenberger et al. 2011; Moe& Kratter 2019) and Equation 4 favors smaller semi-major axes as well as larger masses, the allowed windowfor stellar perturbers in our parameter space is small(masses between ∼ m (cid:12) and semi-major axes be-tween ∼
100 and 150 AU). Moreover, widely separatedstellar-mass perturbers generally lead to fast migrationand, therefore, produce very few warm Jupiters (Petro-vich 2015b; Petrovich & Tremaine 2016; Anderson et al.2016). Thus, we focus on planetary companions in thispaper.We chose an upper limit on our perturber mass rangeof m Jup for consistency with Fernandes et al. (2019).Other occurrence rate studies, however, use more conser-vative mass cut-offs of m Jup (Cumming et al. 2008) or m Jup (Bryan et al. 2016). Here we test how excludingthe most massive perturbers in our study affects the de-tectability of the planets. When we limit the mass of theperturber to m Jup , we find that the overall detectabil-ity of the perturbers is ∼ TESS and RV-discoveredwarm Jupiters should be detectable within the next fewyears with RV and astrometric instruments. OBSERVABLES OF WARM JUPITERPERTURBERS FROM A POPULATIONSYNTHESIS STUDYNext we compute observability metrics for a popula-tion of warm Jupiter perturbers from a population syn-thesis study. Petrovich & Tremaine (2016) investigated6
Jackson et al. the properties of hot and warm Jupiters produced byplanet-planet Kozai-Lidov high-eccentricity tidal migra-tion. This study assumed Sun-like host stars; a limitedrange of inner and outer planet semi-major axes, eccen-tricities, and masses; and uniformly distributed mutualinclinations. The narrow semi-major axis range allowedfor migration from ∼ au orbits, while preserving thedynamical stability of the system (see, e.g., Antoniniet al. 2016). They evolved these systems forward intime until one of the following outcomes occurred: (1)the inner planet evolved into a hot Jupiter, (2) the innerplanet was tidally disrupted, or (3) the inner planet sur-vived for the length of the simulation (1 Gyr). 2.6% ofinner planets ended their simulations as warm Jupiters.We take the set of warm Jupiter-perturber end-statesfrom Petrovich & Tremaine (2016) and calculate theobservables of the perturbers following the procedurefrom Sections 3.1, 3.2, 3.3, and 3.4. The warm Jupiter-perturber systems from Petrovich & Tremaine (2016)are different from our earlier systems because (1) theyare drawn from narrower parameter distributions and(2) they migrate in the simulations while our systemsonly pass a minimum analytical requirement for migra-tion (equation 4).We present the results of our TTV, TDV, RV, andastrometric calculations on this additional data set inFigure 11. We find that 33% of these perturbers are de-tectable via their transit light curves (purple and yellowcircles), assuming Kepler sensitivity and time baselines.This general result is consistent with our results fromSections 3.1 and 3.2. However, in this case, no planetsare detectable by their periodogram signal. This resultcan be attributed to the fact that the end state orbitalperiods of the simulated warm Jupiters are consistentlylonger than those drawn from the power law distributionin Section 2. Since these planets have fewer transits and,therefore, fewer points in the TTV and TDV times se-ries, the periodogram power is weaker and no perturbersare detectable.For both the RV and astrometric signals, the resultsfor these perturbers occupy a narrow range, primarilydue to the fact that the planets occupy narrow rangesin their orbital periods and masses. The only exceptionto this is that the distribution of short-baseline RV sig-nals has an extended tail due to degeneracies with sub-tracting out the warm Jupiter, as we saw in Section 3.3.We find that nearly all of the perturbers are detectablewith 3 years of RV data (blue histogram) or by
Gaia astrometry at 50 pc (green histogram), but nearly allare undetectable with 3 months of RV data or by
Gaia astrometry at 200 pc (black histograms). If these sim-ulated systems are more representative of the true set of systems that have engaged in perturber-coupled high-eccentricity migration than our synthesized systems, theperturbers will be more difficult to detect. OBSERVED WARM JUPITER COMPANIONSIf perturber-coupled high-eccentricity migration is acommon pathway for warm Jupiter formation, the ob-served sample may already contain some strongly per-turbing companions in warm Jupiter systems, particu-larly via radial velocity measurements in RV-discoveredwarm Jupiter systems or Kepler warm-Jupiter systemsbright enough for RV followup. Since there has beenno uniform survey searching for long-period planets inwarm Jupiter systems, we cannot directly compare ourresults to the observations. However, considering thecurrently known companions to warm Jupiters can giveus a sense of what such a survey might find. Here wecompile and discuss the known massive companions towarm Jupiters. We note that Bryan et al. (2016) sur-veyed for companions to short-period planets, but didnot distinguish warm Jupiters from other short-periodplanets they followed up.Of the 102 RV-discovered warm Jupiters (10-200 dayperiods, .1-10 m Jup masses), 26 have known massivecompanions exterior to 200 days (NASA ExoplanetArchive, 2020 August 18). Three of these systems (HIP14810, Wright et al. 2007; HIP 57274, Fischer et al. 2012;55 Cnc, Butler et al. 1997) also contain interior com-panions, making the high-eccentricity migration modelunlikely. Since the inclinations of the other 23 com-panion planets are unknown, we cannot directly sim-ulate the eccentricity evolution of the warm Jupiters,which then prevents us from assessing the viability ofperturber-coupled high-eccentricity migration in thesesystems. However, if this mechanism is common, a uni-form RV study targeting warm Jupiter systems over along time baseline should reveal more massive compan-ions in these systems. Note that all 26 of the warmJupiter systems with massive exterior companions passthe minimum perturber strength requirement (equation4) outlined in Section 2.4.Bryan et al. (2016) did conduct a uniform RV surveyof 123 exoplanet systems to search for long-period com-panions through which they found massive companionswithin 20 au in 8 of their systems as well as long-term RVtrends indicative of the presence of long-period compan-ions in an additional 20 systems. However, most of thesesystems were not warm Jupiter systems. A full analy-sis of the companions to only the warm Jupiter systemsfrom this study is beyond the scope of this paper, buttheir overall results still provide useful context.
EM Perturber Observables TTV / s TT -12 -10 -8 -6 -4 -2 L S F AP Detectable Not detectable 0.01 0.1 1.0 10 100 1000|slope|/e slope -12 -10 -8 -6 -4 -2 L S F AP Detectable Not detectable 0.0001 0.01 1 100|RV slope|/e slope R e l a t i v e N u m be r D q ( m as)0.000.050.100.150.200.250.30 R e l a t i v e N u m be r
200 pc 50 pc
Figure 11.
Here we show the detectability of the simulated perturbers from Petrovich & Tremaine (2016) based on theirTTV signal ( s TTV / ¯ σ TT , periodogram false alarm probability; top left), TDV signal ( | slope | / e slope , periodogram false alarmprobability; top right), RV signal ( | slope | / e slope ; bottom left), and astrometric signal ( ∆ θ ; bottom right). For the RV calculation,time baselines of 3 months (black) and 3 years (blue) are shown. For the astrometric calculation, distances of 50 pc (green) and200 pc (black) are shown. Detection limits are shown as red dashed lines. Jackson et al.
There are 7 transiting warm Jupiters with massivecompanions exterior to 200 days discovered via TTVsand RVs. Of these, 2 systems (Kepler-56, Boruckiet al. 2011; Kepler-88, Nesvorný et al. 2013) containinterior companions making them unsuitable for high-eccentricity migration. Two others, however, have theirperturber inclinations constrained well enough to di-rectly simulate the eccentricity evolution of the warmJupiters (Kepler-419, Dawson et al. 2012; Kepler-693,Masuda 2017). Of these, Kepler-419b is unlikely to haveengaged in perturber-coupled high-eccentricity migra-tion unless another undetected object is also perturbingit (Dawson et al. 2014; Jackson et al. 2019), while thismigration mechanism is plausible for Kepler-693b.As of writing, no massive companions exterior to 200days have been found around
TESS warm Jupiters, butRV follow up efforts will have more targets availablethan for
Kepler due to the brightness of the host stars.In fact, RV followup of the newly discovered eccentricwarm Jupiter, TOI-677 b (Jordán et al. 2020) shows evi-dence of a linear trend that may suggest the presence ofa long-period companion. If perturber-coupled migra-tion is common, we should expect
TESS
RV followup todiscover many such companions (see Section 3.3).We note that many of the most eccentric warmJupiters do have known companions (see Figure 1). Thistrend is discussed by Socrates et al. (2012) and Bryanet al. (2016) and may suggest that many of the low-eccentricity warm Jupiters are delivered through othermethods. DISCUSSION AND CONCLUSIONSIn this work, we synthesized three populations ofwarm Jupiter systems: a
Kepler -like population, a
TESS -like population, and an RV-discovered population(Sections 2.1, 2.2, and 2.3). In each warm Jupiter sys-tem, we added a stable companion planet with enoughperturbing strength to excite large eccentricities in thewarm Jupiter in certain orbital configurations (Section2.4). These large eccentricities could have resultedin high-eccentricity tidal migration that delivered thewarm Jupiters to their current orbital separations. Wethen calculated a set of observables (consisting of tran-sit timing variations, transit duration variations, radialvelocities, and astrometric signals) that would be in-duced by the perturbing companion in these systemsand compared them with current and near-future ob-servational limits to determine their detectability (Sec-tion 3). We then assessed the sensitivity of these re-sults to our choices for the perturber parameter distri-butions and reran our calculations with a populationof warm Jupiters and perturbers from simulations (Sec- tions 4 and 5). Lastly, we discussed the current listof known massive companions to warm Jupiters in thecontext of the predictions from our calculations (Section6). We find that many perturbing companions shouldbe detectable in
TESS and RV warm Jupiter systemsin the near future if perturber-coupled high-eccentricitymigration is common.Our specific findings are as follows. If we assume thatall warm Jupiters have a perturbing companion thatcan induce significant eccentricity oscillations leading tohigh-eccentricity migration, ∼ of these companionsin Kepler warm Jupiters systems should be detectablevia TTVs and TDVs in the
Kepler light curves (Sections3.1 and 3.2). With the same assumptions, companionsto
TESS warm Jupiters should be detectable with just3 months of high-precision RV data in ∼ of sys-tems around quiet stars and by their Gaia astrometricsignal in ∼ of systems within 200 pc (Sections 3.3and 3.4). With the same assumptions, companions toRV-discovered warm Jupiters should be detectable with3 months of high-precision RV data in ∼ of systemsaround quiet stars and by their by their Gaia astromet-ric signal in ∼ of systems within 50 pc (Section3.3 and 3.4). Between RV followup and Gaia astrom-etry, a large fraction of the allowed parameter spacefor strongly perturbing companions in
TESS and RV-discovered warm Jupiter systems is detectable (Section3.5). These results are largely robust to the particularparameter distribution from which the perturbers aresynthesized (Section 4) and the detectability is improvedfor companions to RV-discovered planets if we consideronly short-period warm Jupiters (Section 4.1). Lastly,if we run the same calculations on warm Jupiters andperturbers generated by simulations (see Petrovich &Tremaine 2016), we find that most of these perturbers (1 m Jup planets at 5 au) are not detectable with 3 monthsof RV data, Gaia astrometry at 200 pc, or
Kepler -likelight curves, but nearly all are detectable with 3 years ofRV measurements or
Gaia astrometry at 50 pc (Section5).Our results suggest that if the perturber-coupled high-eccentricity migration mechanism is a common deliverypathway for warm Jupiters, many massive companionsto these planets should soon be detectable with RV fol-lowup of
TESS and RV-discovered warm Jupiters as wellas
Gaia astrometry. Bryan et al. (2016) surveyed forcompanions to RV-discovered short-period planets, butdid not distinguish between warm Jupiters and otherplanets. A reanalysis of their sample considering onlythe warm Jupiter systems may begin to reveal trends forwarm Jupiter companions, but the sample size wouldbe small. The
TESS mission, however, will soon pro-
EM Perturber Observables
Gaia as-trometry will also add a significant number of systems tothat sample with nearby stellar hosts. If the perturber-coupled high-eccentricity migration model is commonin the universe, a search for massive companions inthese systems should be fruitful. If not, such a searchwould set a hard upper-limit on the number of warmJupiter systems that can be explained by this mecha-nism. Petrovich & Tremaine (2016) argue that, basedon their simulations, only ∼ of warm Jupiters canbe explained by this process and Dawson et al. (2015)found a dearth of the super-eccentric warm Jupiters pre-dicted by high-eccentricity tidal migration. If only thesubset of warm Jupiters that resemble those producedby Petrovich & Tremaine (2016)’s simulations arrivedthrough perturber-coupled high-eccentricity migration,a longer RV time baseline would be needed to find theirperturbers.Alternatives to high-eccentricity tidal migration arethat warm and hot Jupiters may have arrived by diskmigration (Goldreich & Tremaine 1980; Ward 1997;Baruteau et al. 2014) or formed in situ (e.g., Batyginet al. 2016; Boley et al. 2016. Recent results by Beckeret al. (2017) show that all 6 of the transiting hot Jupiterswith known exterior companions in their sample musthave low mutual inclinations, thus disfavoring Kozai-Lidov migration for hot Jupiters. Lai et al. (2018), how-ever, argue that the spin-orbit coupling parameter isvery sensitive to semi-major axis and that the Beckeret al. (2017) result can not be generalized for all hotJupiters with companions. Huang et al. (2016) disfa-vor significant migration of any kind for warm Jupitersbecause they tend to have more sub-jovian companionsthan their hot Jupiter counterparts. This is problematicbecause the large observed eccentricities of some warmJupiters seem to be incompatible with in situ formationsince eccentricity excitation from planet-planet scatter- ing is limited by ν escape /ν keplarian (Goldreich et al. 2004;Ida et al. 2013; Perryman et al. 2014). Recent stud-ies have shown, however, that the challenge of reachinglarge eccentricities on short period orbits may be eas-ier to overcome in systems with 3 or more giant planets(Frelikh et al. 2019; Anderson et al. 2020). These al-ternative explanations may be necessary to explain thepresence of intermediate eccentricity warm Jupiters ifRV and astrometric followup to TESS warm Jupitersdoes not yield significant numbers of massive compan-ions, as we have shown that such companions are de-tectable if they exist and are producing intermediateeccentricity warm Jupiters by perturber-coupled high-eccentricity migration.ACKNOWLEDGMENTSWe thank the anonymous referee for their helpful com-ments on this paper. We gratefully acknowledge supportfrom grant NNX16AB50G awarded by the NASA Exo-planets Research Program and the Alfred P. Sloan Foun-dation’s Sloan Research Fellowship. Computations forthis research were performed on the Pennsylvania StateUniversity’s Institute for Computational and Data Sci-ences’ Roar supercomputer. The Center for Exoplanetsand Habitable Worlds is supported by the Pennsylva-nia State University, the Eberly College of Science, andthe Pennsylvania Space Grant Consortium. AS receivedfunding from the European Research Council under theEuropean Community’s H2020 2014-2020 ERC GrantAgreement No. 669416 “ LuckyStar ” . CP acknowledgessupport from the Bart J. Bok fellowship at Steward Ob-servatory and from ANID – Millennium Science Initia-tive – ICN12_009. This research has made use of theNASA Exoplanet Archive, which is operated by the Cal-ifornia Institute of Technology, under contract with theNational Aeronautics and Space Administration underthe Exoplanet Exploration Program. Software: rvlin (Wright & Howard 2009), mpfit (Markwardt 2009)REFERENCES
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