Observable Predictions from Perturber-coupled High-eccentricity Tidal Migration of Warm Jupiters

Before we can construct the population of perturbers and assess their observability, we must first define a population of warm Jupiters. The properties of the warm Jupiters will affect the stability of the systems and the capability of the warm Jupiter to undergo large secular eccentricity oscillations despite general relativistic precession, both of which will dictate the requirements for the perturbers (see Section 2.4). We assume for the purpose of constructing this population that all warm Jupiters—even those observed with low eccentricities—underwent perturber-coupled high-eccentricity migration in order to reach their current orbits.

In Figure 1, we show the observed period and eccentricity distribution of Jupiter-sized planets. Hot Jupiters are plotted in red, warm Jupiters in orange, and cold Jupiters in gray. Warm Jupiters with companions are plotted as diamonds (companions discovered with RVs) or triangles (companions discovered with TTVs). These planets will be discussed further in Section 6. The black dashed line is a track of constant angular momentum and will be discussed further in Section 2.4. Note that the broad distribution of warm Jupiter eccentricities seen here is one of the motivators for researching high-eccentricity migration mechanisms.

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We base our warm Jupiter population on characteristics of the observed distribution. We begin by creating a Kepler-like sample of transiting warm Jupiters, which will be appropriate for assessing the observability of transit timing and duration variations. These planets have periods between 10 and 200 days and masses between 0.1 and 10 m_Jup. To build this population, we draw the planet's period, mass, and eccentricity from distributions that model the intrinsic properties of warm Jupiters and then discard any planets with impact parameter b < 1, which would not transit.

We draw our warm Jupiter periods from a power law with fitting coefficient P = 0.63 (Fernandes et al. 2019). Since the primary Kepler mission lasted ∼4 yr, and three transits were required to detect a planet, the period distribution is complete out to the 200 day maximum in our sample.

We draw our warm Jupiter eccentricities from a beta distribution (e.g., Kipping 2013). To find the appropriate fitting parameters for warm Jupiters, we fit a beta distribution to the sample of 102 confirmed planets on the NASA Exoplanet Archive with periods between 10 and 200 days and masses > 0.1m_Jup discovered via the RV technique (as of 2020 February 26). The functional form of this distribution is

$$\begin{eqnarray}&&P(e;\alpha ,\beta )=\displaystyle \frac{{\rm{\Gamma }}(\alpha +\beta )}{{\rm{\Gamma }}(\alpha ){\rm{\Gamma }}(\beta )}{e}^{\alpha -1}{\left(1-e\right)}^{\beta -1},\end{eqnarray} \tag{ 1 }$$

where e is the eccentricity, Γ() is the gamma function, and α and β are fitting parameters. Our best-fit parameters are α = 0.61 and β = 2.16. We discard any planets whose semimajor axis and eccentricity would result in tidal disruption: a(1 − e) < a_Roche, where a_Roche includes a scaling coefficient of f_p = 2.7 (e.g., Guillochon et al. 2011). We note that some planets with eccentricities just below this limit may not survive as warm Jupiters for long due to rapid tidal circularization, but this is strongly dependent on the particular tidal parameters of the system.

We draw our warm Jupiter inclinations from an isotropic distribution (i.e., uniform in cosi). We note, however, that after the cut for transiting planets, the inclination distribution will be strongly peaked near π/2.

Lastly, we assume a power-law mass distribution (${dN}/d\mathrm{ln}(m)\propto {m}^{\alpha }$) with coefficient α = −0.31 found by Cumming et al. (2008). These authors considered selection effects in the RV sample to estimate the mass distribution of nearby giant planets. We limit our masses to be between 0.1 and 10 m_Jup.

To finish constructing our Kepler-like warm Jupiter population, in addition to the periods (P), eccentricities (e), inclinations (i), and planet masses (m_wj) described above, we draw arguments of pericenter (ω), and mean anomalies (M) from uniform distributions before applying our impact parameter cut of b < 1. It is not necessary to cut in orbital period or eccentricity to account for detectability, because Kepler is complete for all eccentricities and periods <200 days for the mass range we are considering. For each system, we assume a solar mass and radius star. Through this method, we build a population of 10,000 warm Jupiter–star systems.

In addition to TTVs and TDVs, we want to assess the detectability of our perturbers via RVs and astrometry. Unfortunately, most Kepler systems are too distant for RV or astrometric follow-up, which can reliably detect planets around bright stars (V ≲ 12; see Perryman et al. 2014). However, TESS systems and warm Jupiters discovered by the RV method (see Section 2.3) are typically much brighter and closer, making them more amenable for follow-up. Thus, here we define a population of TESS-like warm Jupiter systems.

Our TESS-like sample is very similar to the Kepler sample defined in Section 2.1, with the exception of its period distribution. Since TESS is an all-sky mission, individual stars will be observed for a much shorter period of time than the equivalent from the Kepler mission, making the expected sample incomplete to our 200 day maximum period for warm Jupiters. To account for this incompleteness, we take the periods of planets with radius r > 8R_Earth from a simulated TESS yield (Barclay et al. 2018), drawing a period value at random from this list for each iteration in our warm Jupiter population construction process. This simulated TESS yield approximates the distribution of expected planetary parameters in the TESS sample based on current planet occurrence rate estimates and the TESS instrumental capabilities. Barclay et al. (2018) predicts the discovery of roughly 300 warm Jupiters by TESS.

We follow the method described in Section 2.1 to draw e, i, m_wj, ω, and M , applying the same cuts for transiting planets and stable warm Jupiter orbits. As with the Kepler planets, we build a population of 10,000 TESS-like warm Jupiter systems.

Lastly, we would like to assess the detectability of perturbing companions to warm Jupiters discovered by the RV method. These companions could be detectable by RVs or astrometry because the host stars are typically bright and nearby.

Our population of RV-detected warm Jupiters is more straightforward to develop than the transiting populations described in Sections 2.1 and 2.2 because the masses and eccentricities are well characterized. Thus, we can directly use the observed planets, pulling their minimum masses (${m}_{\mathrm{wj}}\sin {i}_{\mathrm{wj}}$), P, e, ω, and stellar mass (m_*) from the sample of 66 RV-detected warm Jupiters with well-characterized orbits on the NASA Exoplanet Archive (as of 2020 February 26). We draw the M of the planets from a uniform distribution and the i from an isotropic distribution, cutting out orientations in which the derived m_wj is above 10 m_Jup. The use of an isotropic inclination distribution introduces a slight bias, because we are now dealing with observed planets instead of an intrinsic population; however, this effect is small for the mass range we are considering (Ho & Turner 2011; Lopez & Jenkins 2012). We draw from our sample of warm Jupiters 10,000 times to create a population similar to our two transit-detected warm Jupiter populations.

Now we construct the population of perturbing companion planets that will each accompany a warm Jupiter. We set two criteria for accepting a given perturber: (1) the warm Jupiter-perturber system must be long-term stable, and (2) the perturber must be capable of periodically exciting the eccentricity of the warm Jupiter high enough for it to undergo tidal migration through an ongoing cycle of secular angular momentum exchange.

To implement the first criterion, we impose an analytical stability requirement from Petrovich (2015a), which, when satisfied, should result in the system evolving secularly with no close encounters:

$$\begin{eqnarray}&&\displaystyle \frac{{a}_{\mathrm{per}}(1-{e}_{\mathrm{per}})}{{a}_{\mathrm{wj}}(1+{e}_{\mathrm{wj}})}\gt 2.4{\left[\max ({\mu }_{\mathrm{wj}},{\mu }_{\mathrm{per}})\right]}^{1/3}{\left(\displaystyle \frac{{a}_{\mathrm{per}}}{{a}_{\mathrm{wj}}}\right)}^{1/2}+1.5,\end{eqnarray} \tag{ 2 }$$

where a_per(wj) is the semimajor axis of the perturber (warm Jupiter) planet, e_per(wj) is the eccentricity of 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 empirical limit on the warm Jupiter's minimum angular momentum for tidal migration:

$$\begin{eqnarray}&&a(1-{e}^{2})\lt {a}_{{\rm{f}},\mathrm{crit}}=0.1\ \mathrm{au},\end{eqnarray} \tag{ 3 }$$

where a(1 − e²) is the final semimajor axis of a tidally circularized planet when e = 0 and a_f,crit = 0.1 au is the final semimajor axis required to circularize within the age of the universe. The value of a_f,crit comes from the observed population of hot Jupiters, which tend to exhibit small eccentricities for a < 0.1 au (Socrates et al. 2012; Dong et al. 2014). In reality, the limit is likely much stricter, since there are still some moderate eccentricities among hot Jupiters with a > 0.05 and the tidal circularization timescale scales vary strongly with separation. However, we adopt the conservative value of a_f,crit = 0.1 au to ensure that the criterion presented below is a necessary requirement for perturber-coupled high-eccentricity tidal migration. Figure 1 shows the theoretical limit from Equation (3) plotted as a black dashed line over the observed Jupiter-mass planet a versus e distribution. A planet must spend time above the line on this plot 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 perturber is capable of inducing eccentricity oscillations in the warm Jupiter that can overcome precession due to general relativity and satisfy Equation (3) for some portion of its secular cycle. This is a minimum requirement for high-eccentricity tidal migration, but does not guarantee it will occur.

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{b}_{\mathrm{per}}}{{a}_{\mathrm{wj}}}\lt {\left(\displaystyle \frac{8{{Gm}}_{* }}{{c}^{2}{a}_{\mathrm{wj}}}\right)}^{-1/3}{\left(\displaystyle \frac{{m}_{* }}{{m}_{\mathrm{per}}}\right)}^{-1/3}\\ \times \,{\left(\sqrt{\displaystyle \frac{{a}_{\mathrm{wj}}}{{a}_{{\rm{f}},\mathrm{crit}}}}-\displaystyle \frac{1}{\sqrt{1-{e}_{\mathrm{wj}}^{2}}}\right)}^{-1/3}{\left[1-\displaystyle \frac{{a}_{{\rm{f}},\mathrm{crit}}}{{a}_{\mathrm{wj}}}-{e}_{\mathrm{wj}}^{2}\right]}^{1/3}\\ \times \,{\left(2-5{\sin }^{2}{i}_{\mathrm{mut}}{\sin }^{2}{\omega }_{\mathrm{wj}}\right)}^{1/3},\end{array}\end{eqnarray} \tag{ 4 }$$

where ${b}_{\mathrm{per}}={a}_{\mathrm{per}}{(1-{e}_{\mathrm{per}}^{2})}^{1/2}$ is the semiminor axis of the perturber, m_* is the stellar mass, m_per is the mass of the perturber, e_wj is the eccentricity of the warm Jupiter, a_wj is the semimajor axis of the warm Jupiter, ω_wj is the argument of pericenter of the warm Jupiter in the invariable plane, i_mut is the mutual inclination between the two planets, and a_f,crit = 0.1.

The population of planets with orbital periods beyond 200 days is less constrained by the observations than that of the warm Jupiters, so we have to make some assumptions about the distribution of orbital and physical parameters. The robustness of our choices for these distributions is discussed in Section 4.

Our population of perturber eccentricities (e) is drawn from a beta distribution (Equation (1)) with best-fit parameters α = 0.74 and β = 1.61 determined by fitting the beta distribution to the sample of 428 long-period, Jupiter-sized (m_Jup > 0.1) confirmed planets discovered via RV from the NASA Exoplanet Archive (as of 2020 February 26).

The perturber sky-plane inclinations (i) are drawn from an isotropic distribution. The orbital angles ω, Ω, and M are drawn from uniform distributions in the interval [0, 2π]. For the perturber population, we draw the masses (m) from the power-law distributions calculated by Cumming et al. (2008) with a range of 0.1 to 20 m_Jup. We choose the range of masses based on previous studies that show consistency with the power-law fit in that range (e.g., Fernandes et al. 2019). Above 20 m_Jup, companions are more likely to be a part of the low-mass end of the brown dwarf population, which follows a different mass function (Bowler 2016). We assess our choice for the upper mass limit in Section 4.5.

Our perturber periods follow a symmetric broken power law,

$$\begin{eqnarray}\displaystyle \frac{{dN}}{d\mathrm{logP}}\propto \left\{\begin{array}{lc}{\left(\displaystyle \frac{P}{{P}_{\mathrm{break}}}\right)}^{{P}_{1}} & P\leqslant {P}_{\mathrm{break}}\\ {\left(\displaystyle \frac{P}{{P}_{\mathrm{break}}}\right)}^{-{P}_{1}} & P\gt {P}_{\mathrm{break}}\end{array}\right.\end{eqnarray} \tag{ 5 }$$

where P_break = 859 days is the turnover point in the period distribution, and P₁ = 0.63 is the power-law coefficient. Fernandes et al. (2019) showed that a broken power law (with these fitting coefficients) is a better fit to the observed period distribution than a single power law or a log-normal distribution. We draw our periods from the range 200 to 100,000 days. The lower limit is set to ensure these are exterior companions to warm Jupiters, and the upper limit is set by the fact that planet-mass objects with periods beyond 100,000 days always fail Equation (4) for the mass range we test.

For a given warm Jupiter–star system, we draw perturber properties from the distributions listed above and check against our two acceptance constraints (Equations (2) and (4)). If the set of properties fails either of the criteria, we redraw the parameters and recheck against the criteria. We repeat this process until every warm Jupiter has an accompanying perturber.

First, we calculate the TTV signal of the perturbers to the Kepler-like warm Jupiters. 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 of TTVs we expect are typical of those seen in hierarchical triple systems (e.g., Borkovits et al. 2003, 2004).

We model the TTVs over the Kepler mission lifetime of 4 yr. We calculate the TTVs numerically by simulating the three-body motion of the two planets and their host star over those 4 yr and recording the mid-transit time for each orbit of the warm Jupiter using Dawson et al.'s (2014) N-body transit time code. We then calculate the deviations from a best-fit linear ephemeris at each time, which define our TTVs.

We define two metrics for distinguishing detectable TTV system from non-detectable TTV systems following Mazeh et al. (2013). The first metric is the ratio of the scatter in the TTVs, s_TTV, defined as the median absolute deviation of the TTV series, to their median error, ${\bar{\sigma }}_{\mathrm{TT}}$. We randomly draw ${\bar{\sigma }}_{\mathrm{TT}}$ from the median uncertainties in mid-transit time of Holczer et al.'s (2016) light-curve fits to warm Jupiter Kepler candidates. The second metric is the false-alarm probability (FAP) of the Lomb-Scargle (LS) periodogram of the TTV time series. To generate the LS periodogram, we use the rvlin package (Wright & Howard 2009), assigning the median uncertainty to each data point in our simulated TTV time series and applying a Gaussian scatter. We categorize perturbers as detectable via TTVs when the ${s}_{\mathrm{TTV}}/{\bar{\sigma }}_{\mathrm{TT}}$ ratio is greater than 5 or the LS FAP is below 3 × 10⁻⁴. Note that Mazeh et al. (2013) set the threshold for ${s}_{\mathrm{TTV}}/{\bar{\sigma }}_{\mathrm{TT}}$ at the more conservative value of 15.

In Figure 2, we show four representative examples of TTV time series from our sample. The top panel shows the signal of an undetectable perturber, the second panel shows the signal of a perturber detectable by our first metric (${s}_{\mathrm{TTV}}/{\bar{\sigma }}_{\mathrm{TT}}\gt 5$), the third panel shows the signal of a perturber detectable by our second metric (LS FAP <3 × 10⁻⁴), and the bottom panel shows the signal of a perturber detectable by either metric. Each plotted time series has an uncertainty drawn from the median uncertainties in the mid-transit time of Holczer et al.'s (2016) light-curve fits to warm Jupiter Kepler candidates, and a Gaussian scatter has been applied. We present a compilation of these results from all simulated systems in Figure 3, where purple points represent detectable systems, and red dashed lines delineate our detection criteria.

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Using these metrics, if each Kepler warm Jupiter is accompanied by a perturber that fits our criteria, a small portion of these perturbers (7.7%) should be detectable by TTVs in the Kepler light curves. Using the more conservative cutoff of ${s}_{\mathrm{TTV}}/{\bar{\sigma }}_{\mathrm{TT}}\gt 15$, this number drops to (4.9%). The perturbers that are detectable are typically short-period planets (<5000 day orbits) with large masses. See Section 3.5 for further discussion of the detectability of the perturbers. Since the majority of our synthesized perturbers produce undetectable TTVs, and some TTVs have been detected for Kepler warm Jupiters (see Holczer et al. 2016), these results do not constrain the high-eccentricity migration mechanism.

Next, we calculate the TDV signal of the perturbers to the Kepler-like warm Jupiters. We model the TDVs simultaneously with the TTVs over the 4 yr Kepler mission lifetime by tracking the slight changes to the impact parameter and transit speed of the warm Jupiter between each orbit. We calculate the approximate transit durations using the following equation from Winn (2010), which includes a correction for noncircular orbits:

$$\begin{eqnarray}&&{T}_{{tot}}=\displaystyle \frac{P}{\pi }\arcsin \left[\displaystyle \frac{{R}_{* }}{a}\displaystyle \frac{\sqrt{{\left(1+k\right)}^{2}-{b}^{2}}}{\sin i}\right]\left(\displaystyle \frac{\sqrt{1-{e}^{2}}}{1+e\sin \omega }\right)\end{eqnarray} \tag{ 6 }$$

where P is the period, R_* is the stellar radius, a is the semimajor axis, k is the planet–star radius ratio, b is the impact parameter, i is the inclination, e is the eccentricity, and ω is the argument of pericenter. Our final TDV amplitudes are calculated by subtracting the minimum from the maximum duration among all of the transits.

As with the TTVs, we define two metrics for distinguishing detectable TDV systems from non-detectable TDV systems. Unlike TTVs, where any linear trend is subtracted out, the slope of a TDV signal is meaningful. Therefore, our first metric is the strength of the linear trend in the TDV data, or more specifically, the ratio of the estimated slope over the estimated error in the slope (∣slope∣/e_slope) following Kane et al. (2019). We randomly draw the median uncertainty in transit duration from Holczer et al.'s (2016) light-curve fits to warm Jupiter Kepler candidates. The second metric is the FAP of the LS periodogram of the TDV time series. As with the TTVs, we generate the LS periodogram using the rvlin package (Wright & Howard 2009), assigning the median error to each data point and applying a Gaussian scatter. We categorize perturbers as detectable via TDVs when the ∣slope∣/e_slope is greater than 3.5 or the LS FAP is below 3 × 10⁻⁴ (Kane et al. 2019).

In Figure 4, we show four representative examples of TDV time series from our sample. The top panel shows the signal of an undetectable perturber, the second panel shows the signal of a perturber detectable by our first metric (∣slope∣/e_slope > 3.5), the third panel shows the signal of a perturber detectable by our second metric (LS FAP <3 × 10⁻⁴), and the bottom panel shows the signal of a perturber detectable by either metric. Each plotted time series has an uncertainty drawn from the median uncertainties in transit duration from Holczer et al.'s (2016) light-curve fits to warm Jupiter Kepler candidates and a Gaussian scatter has been applied. We present a compilation of these results from all simulated systems in Figure 5, where yellow points represent detectable systems, and red dashed lines delineate our detection criteria.

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If each Kepler warm Jupiter is accompanied by a perturber that fits our criteria from Section 2.4, a small percentage of these perturbers (13.5%) should be detectable by TDVs in the Kepler light curves. Like with the TTVs, short-period perturbers with large masses are the easiest to detect. See Section 3.5 for further discussion of the TDV detectability of the perturbers. As with the TTVs (Section 3.1), these results alone cannot significantly constrain the high-eccentricity migration mechanism.

Next, we calculate the RV signal of the synthesized perturbers to our TESS-like warm Jupiters and our RV-detected warm Jupiters. Since RV follow-up of known planetary systems does not come from a single instrument or survey, we model the signal with two representative time baselines (3 months and 3 yr). The periods of our perturbers span the range of 200–100,000 days, so some signals may appear weak over a short time baseline, but strong once a larger percentage of their orbit is observed.

To calculate the observed ΔRV over a given time baseline, we first model the radial motion of the star due to both planets. For the 3 month time baseline, we assume 20 data points, and for the 3 yr baseline, we assume 100 data points. For each point, we assume a 1 m s⁻¹ uncertainty, which could arise from instrument error and/or stellar variability. Once we have our RV signal, we fit a one-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 two-planet signal with the best-fit one-planet solution subtracted out. This approach is more conservative than simply calculating the outer planet's signal because, for a real system, we do not precisely know all of the inner planet parameters a priori, and they might be conflated in the two-planet fit.

In order to determine which signals are detectable, we calculate the ratio of the slope of the RV signal to the error in the slope (∣slope∣/e_slope). As with the TDV signals, we categorize perturbers for which this metric is greater than 3.5 as detectable since those systems would have RV trends that are statistically distinguishable from zero. We also calculate an LS periodogram of the RV time series, but it is ineffective at discovering the long-period signals of our perturbers, so we omit it here.

The results of both the 3 month calculation and the 3 yr calculation are plotted in Figure 6. We find that 77.2(91.2)% of our companions to TESS-like warm Jupiters and 31.4(83.0)% of our companions to RV-detected warm Jupiters would produce detectable signals over 3 months(3 yr) of RV data. The bump near 0.0001 in the slope ratio for the 3 month RV-detected warm Jupiter companions is caused by degeneracies between the signals of the inner and outer planets that are broken with a longer time baseline. Note that e_slope is strongly dependent on the stellar activity of the host star, and our assumed RV uncertainty of 1 m s⁻¹ is representative of state-of-the-art RV instruments observing quiet stars. For noisier stars or less precise instruments, the detectability of our perturbers decreases significantly (e.g., our TESS perturbers are 64.4% detectable over 3 months of observations at 3 m s⁻¹ uncertainty or 32.9% detectable at 10 m s⁻¹ uncertainty). The perturbers that are detectable by their RV slope cover the full mass range tested, with larger masses allowing for longer period detections. See Section 3.5 for further discussion of the detectability of the perturbers and Section 6 for a discussion of the known companions to RV-detected warm Jupiters.

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Finally, we calculate the astrometric signal of the synthesized perturbers to our TESS-like warm Jupiters and our RV-detected warm Jupiters as they would be observed by Gaia (Lindegren & Perryman 1996; Perryman et al. 2001). As with the RV calculation, we leave out Kepler warm Jupiters here because their host stars are typically too far away for Gaia to successfully detect an astrometric planetary signal. Specifically, Gaia is expected to detect planets out to 500 pc (Perryman et al. 2014), while nearly all Kepler warm Jupiter hosts are outside of that radius. We model the transverse motion of the stellar host on the sky due to the gravitational pull of the perturber, then subtract off a linear fit, which would be removed as proper motion. The following is our model for the star's motion in R.A. and decl.:

$$\begin{eqnarray}&&\xi (t)={\alpha }_{0}^{* }+\left(t-{t}_{0}\right){\mu }_{{\alpha }^{* }}+{BX}(t)+{GY}(t)\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\eta (t)={\delta }_{0}+\left(t-{t}_{0}\right){\mu }_{\delta }+{AX}(t)+{FY}(t),\end{eqnarray} \tag{ 8 }$$

where α* = α cos δ, $({\alpha }_{0}^{* },{\delta }_{0})$ is the reference position of the star, $({\mu }_{{\alpha }^{* }},{\mu }_{\delta })$ 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 its orbit as a function of the eccentric anomaly, E. We calculate E iteratively from the mean anomaly, M, and the eccentricity, e, following Danby (1988). For a detailed description of the astrometric model, see Quirrenbach (2010).

Gaia is expected to observe every star in its catalog ∼70 times over the course of its 5 yr primary mission lifetime (Jordan 2008). For representative distances that are consistent with typical TESS warm Jupiters (200 parsecs) and typical RV warm Jupiters (50 parsecs), all of which will appear in the Gaia astrometric catalog, we calculate the maximum angular distance, Δθ, between data points in the model for each of our simulated systems after subtracting off the linear proper motion fit. We plot the results of this calculation for TESS-like warm Jupiter systems and RV-detected warm Jupiter systems in Figure 7. Also plotted in Figure 7 is a line indicating the conservative Gaia detection limit for stars brighter than G = 12 mag of ∼100 μas (Perryman et al. 2014). 16.7(40.3)% of the perturbers in TESS-like warm Jupiter systems and 9.8(29.6)% of the perturbers in RV-discovered warm Jupiter systems would be detectable with Gaia astrometry at distances of 200(50) parsecs. Only perturbers with large masses (>1 M_Jup) are detectable with this method, and intermediate periods are preferred. See Section 3.5 for further discussion of the detectability of the perturbers.

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As shown in Sections 3.1 and 3.2, a small portion of our perturbers should be detectable in the current Kepler light curves via TTVs or TDVs (19.0%, combined). However, because most of the Kepler stars are too dim and too far away for RV or astrometric planet detection, most of the perturbers in those systems will remain hidden, even if they exist. On the other hand, many of the perturbers in TESS and RV-discovered warm Jupiter systems (77.2% and 31.4%, respectively) are observable with high-precision RV follow-up with even a short time baseline (see Section 3.3). Gaia will also be able to detect many perturbers associated with TESS and RV-discovered warm Jupiters (16.7% and 29.6%, respectively; see Section 3.4). Furthermore, when combining the RV and astrometric detection methods together, nearly all of the parameter space for perturbers to TESS warm Jupiter and most of the parameter space for RV-discovered warm Jupiters that satisfy the two constraints in Section 2.4 are detectable with current or near-future instruments (78.3% and 45.8%, respectively).

In Figure 8 we show the detectability of all of our synthetic perturbers with current and near-future instruments in terms of their masses and periods. We classify a perturber to a Kepler warm Jupiter as detectable via its TTV signal if ${s}_{\mathrm{TTV}}/{\bar{\sigma }}_{\mathrm{TT}}\gt 5$ or LS FAP < 3 × 10⁻⁴ (Purple circles). We classify a perturber to a Kepler warm Jupiter as detectable via its TDV signal if ∣slope∣/e_slope > 3.5 or LS FAP < 3 × 10⁻⁴ (yellow crosses). We classify a perturber to a TESS or RV-discovered warm Jupiter as detectable via its observable RV trend if ∣slope∣/e_slope > 3.5 over 3 months of observations (blue circles). Lastly, we classify a perturber to a TESS or RV-discovered warm Jupiter as detectable via its astrometric signal (Δθ) if Δθ > 100 μas assuming a distance of 200 parsecs for TESS systems and 50 parsecs for RV systems (green crosses). Red points represent undetectable companions in each sample. The background shading demonstrates the percentage of perturbers that are detectable in a given grid space, with whiter areas being more detectable. The lower-right corner in each panel is an area of parameter space that did not produce any perturbers capable of passing the criteria from Section 2.4. The three panels in Figure 8 show the detectability of perturbers in our Kepler-like (top), TESS-like (middle), and RV-discovered (bottom) samples. As stated above, 19.0% of our perturbers to Kepler warm Jupiters should currently be detectable in the Kepler light curves, and 78.3(45.8)% of our perturbers to TESS(RV-discovered) warm Jupiters will be detectable in the near future with RV and astrometric follow-up. Those that are undetectable tend to have low masses and intermediate-to-long periods, as these areas of parameter space are the most difficult to detect with RVs and astrometry and tend to produce weaker TTV and TDV signals.

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If, after sufficient RV follow-up of TESS warm Jupiters and the release of the planets detected by Gaia, we do not find a large number of strong perturbers accompanying warm Jupiters, we can infer that perturber-coupled high-eccentricity migration may not be a common mechanism for delivering warm Jupiters.

Thus far in this study, we have considered perturbers to warm Jupiters from the full range of periods (10–200 days). Here, we focus only on short-period warm Jupiters (P < 50 days) and rerun our calculations in order to determine if a more focused future search might produce more robust results. We find that this period cut slightly shrinks the allowed area of the parameter space for perturbers, which, in turn, increases the detectability of the perturbers. This effect is very small for the TTV, TDV, and astrometric detection techniques 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 warm Jupiters is approximately double that of our prior results (62.7% over a 3 month baseline).

Until this point, we have also assumed that all warm Jupiters may have been delivered by the high-eccentricity tidal migration mechanism, even those with low observed eccentricities. However, it may be likely that most low-eccentricity warm Jupiters arrived via a different formation or migration method (e.g., Dawson & Murray-Clay 2013; Petrovich & Tremaine 2016). To test the effects of our choice on the results of this paper, we rerun our calculations considering only perturbers to warm Jupiters with large eccentricities (e > 0.4). We find that the detectability of these perturbers is not significantly larger than our prior result for any of the considered detection techniques.

We now examine the effects of the inclination distribution of our perturbers. Because the true inclination distribution of giant planet companions is not well-constrained, we used an isotropic distribution for our fiducial population. Here we compare this to the other extreme: coplanar companions. In the coplanar case, we assign the i and longitudes of ascending node (Ω) of the perturbers to be identical to those of the warm Jupiter. We find the observed RV trends for both populations of warm Jupiters to be indistinguishable between the coplanar case and fiducial isotropic case. The astrometric signal of the coplanar perturbers is shifted toward smaller Δθ in the transiting warm Jupiter case. This can be attributed to the fact that, in edge-on systems, one dimension of astrometric movement is lost. In the RV-discovered warm Jupiter case, this discrepancy is not present because the sky-plane inclinations of these systems are widely distributed. In general, our assumption of isotropic perturbers has a small effect on our results, but may predict more astrometric detections for TESS companions than we would expect from a coplanar inclination distribution.

We now examine the eccentricity distribution of our perturbers. Our fiducial perturber population drew eccentricities from a beta distribution with α = 0.74 and β = 1.61 (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 observationally motivated, it is consistent with assumptions used by previous theoretical studies on this topic (Dong et al. 2014; Petrovich & Tremaine 2016). Additionally, we test the two extreme cases of zero eccentricity and uniform eccentricities between e = 0–1, the latter of which is more representative of the observed binary stellar companion sample (Duchêne & Kraus 2013).

We find that, even including the extreme cases, all of our eccentricity distributions produce similar distributions of observables for each case we test. As shown in Figure 9, each eccentricity test closely follows the fiducial population histogram.

Next, we consider how the underlying perturber orbital period distribution we draw from affects our calculated observables. Our fiducial periods are drawn from a broken power-law distribution. We note that our perturber periods, masses, and eccentricities are drawn before applying the two analytical cuts described in Section 2.4. Since any perturbers that do not satisfy the stability or perturber strength criteria are discarded and redrawn, the final distributions are different from the underlying distributions we draw from. The filtering by these criteria is particularly important for orbital periods, where short and long periods are disfavored by the two cuts, and intermediate periods are selectively chosen.

In Figure 10, we show the four period distributions that we test, before and after applying our two cuts: a log-uniform distribution, power-law fits from Cumming et al. (2008) and Bryan et al. (2016), and a broken power-law fit from Fernandes et al. (2019). Cumming et al. (2008) found evidence supporting a rising power-law period distribution for giant planets, but had relatively few data points and were limited to periods under 2000 days. Using new RV data, Bryan et al. (2016) confirmed the rising power-law trend in orbital period for short-period giant planets. However, when they assessed the period distribution for companions to short-period giant planets discovered by long-term RV monitoring and adaptive optics imaging, they found evidence for a decreasing power-law coefficient for giant planets at larger periods and speculated that a turnover in the period distribution must occur between 3 and 10 au. Here we use their single power-law fit for planets with masses between 0.5 and 20 M_Jup and semimajor axes between 5 and 50 au, but note that the results of this fit were very sensitive to the particular mass and semimajor axis ranges chosen. With an even larger set of RV data, Fernandes et al. (2019) modeled the turnover in the period distribution as a broken power law, which we use as our fiducial period distribution. We use the power-law fits to the RV data from each study in our comparison here, whose fitting variables are recorded in Table 1.

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The four different functions used to create our period distributions are significantly different in their shapes and the locations of their peaks. However, as shown in Figure 10, after accounting for perturber strength and stability, all four functions peak at intermediate periods and drop off toward longer or shorter periods. Because the final distributions after the criteria are applied in all cases are similarly shaped, the RV signal is insensitive to the particular underlying period distribution chosen. The astrometric signal is slightly affected by this choice due to its strong dependence on planet's semimajor axis. Thus, the Bryan et al. (2016) fit, which had the longest peak period, produces a slightly stronger astrometric signal. However, these effects are small, and the Fernandes et al. (2019) distribution, which we use in the analysis of our results, is the most conservative of the four in terms of its astrometric signal.

If all giant planets form beyond the ice line and those that we observe interior to that point are the result of migration, we might expect the companions to planets migrating via high-eccentricity migration to remain at large separations for their entire lifetime. Throughout this work, we have allowed our coupled perturbers to exist anywhere outside of a 200 day orbit. We test how this assumption affects our results by removing any perturbers interior to the ice line and rerunning our calculations. The exact location of the ice line is under investigation, but here we use 2.7 au as an illustrative value Lecar et al. (2006). We find that the RV detectability of these planets is not significantly affected by this change, while the TTV/TDV detectability is decreased, and the astrometric detectability is increased. This is consistent with our previous findings that, if these perturbers exist, we do not expect to find many in the Kepler data, but should find many by following up TESS and RV-discovered warm Jupiters. In fact, if the perturbers are all beyond the ice line, Gaia will be much more effective at finding them.

Lastly, we consider the effects of our choice for the upper limit on the mass of our perturbers. All of the perturbers described above are of planetary mass ( < 20m_Jup). Stellar perturbers could also produce the desired eccentricity excitation in the warm Jupiter. However, since binary systems harboring planets are preferentially widely separated (Eggenberger et al. 2011; Moe & Kratter 2019), and Equation (4) favors smaller semimajor axes as well as larger masses, the allowed window for stellar perturbers in our parameter space is small (masses between ∼1 and 2 m_☉ and semimajor axes between ∼100 and 150 au). Moreover, widely separated stellar-mass perturbers generally lead to fast migration and, therefore, produce very few warm Jupiters (Petrovich 2015b; Anderson et al. 2016; Petrovich & Tremaine 2016). Thus, we focus on planetary companions in this paper.

We chose an upper limit on our perturber mass range of 20m_Jup for consistency with Fernandes et al. (2019). Other occurrence rate studies, however, use more conservative mass cut-offs of 10m_Jup (Cumming et al. 2008) or 13m_Jup (Bryan et al. 2016). Here we test how excluding the most massive perturbers in our study affects the detectability of the planets. When we limit the mass of the perturber to 10m_Jup, we find that the overall detectability of the perturbers is ∼2%–4% lower. Alternatively, if we were to include more brown dwarfs, the detectability would slightly increase.

We find that our results are largely insensitive to changes in the underlying perturber parameter distributions. For a variety of assumptions about the eccentricity, inclination, period distribution, and mass range, most perturbers that meet the criteria for strength and stability necessary for perturber-coupled high-eccentricity tidal migration of TESS and RV-discovered warm Jupiters should be detectable within the next few years with RV and astrometric instruments.