The TESS–Keck Survey. XIII. An Eccentric Hot Neptune with a Similar-mass Outer Companion around TOI-1272

TOI-1272 was observed by TESS with 2 minute cadence photometry in sectors 15, 16, and 22 between UT 2019 October 10 and 2020 May 11. The time-series photometry was processed by the TESS Science Processing Operations Center pipeline (SPOC; Jenkins et al. 2016), which first detected the periodic transit signal of TOI-1272b with a wavelet-based, noise-compensating matched filter (Jenkins 2002; Jenkins et al.2010). An initial limb-darkened transit model fit was performed (Li et al. 2019) and the signature passed a suite of diagnostic tests described by Twicken et al. (2018), leading to the selection of this target as a TESS Object of Interest (TOI).

We accessed the pre-search data conditioning simple aperture photometry (PDC-SAP; Stumpe et al. 2012, 2014; Smith et al. 2012) through the Mikulski Archive for Space Telescopes (MAST), stitching together the light curves from individual TESS sectors into a single time series using Lightkurve (Lightkurve Collaboration et al. 2018). We performed outlier rejection, normalization, and detrending of the full light curve following the procedures outlined in MacDougall et al. (2021). We then searched for transits using a box least squares (BLS; Kovács et al. 2002) transit search to recover the same planetary signal detected by SPOC with signal-to-noise ratio (S/N) = 23.3. We subtracted the known transits and applied the BLS search again but identified no additional periodic transit events.

To confirm that the observed transit events were on target and not the result of a background source, we referenced additional ground-based time-series photometry taken for TOI-1272. Independent observations were collected with MuSCAT2 (Narita et al. 2019) at a pixel scale of 044 in g, r, i, and z_s filters on UT 2020 February 28 and again ∼1 year later with MuSCAT (Narita et al. 2015) at a pixel scale of 036 in g, r, and z_s filters on UT 2021 May 8. These detections confirmed that the expected transit was on target and presented no evidence of nearby eclipsing binaries. This target has one neighbor listed in Gaia Data Release 2 (DR2) within 30''. At a separation of 845 and ΔG = 5.93, the neighbor contributes <1% dilution to the light curve, which was already corrected for in the photometric data products that we used.

We characterized the planetary transit signal using a photometric light-curve model to determine whether TOI-1272b was a candidate for high eccentricity. We made this determination by comparing the planet's observed transit duration (T; mid-ingress to mid-egress) to the expected duration for a circular orbit T_circ. The ratio of these two values can be used to assess the orbital geometry of a transiting planet through the geometric relation for transit duration given by Winn (2010):

$$\begin{eqnarray}&&T=\left(\displaystyle \frac{{R}_{\ast }P}{\pi a}\sqrt{1-{b}^{2}}\right)\displaystyle \frac{\sqrt{1-{e}^{2}}}{1+e\sin \omega }.\end{eqnarray} \tag{ 1 }$$

Given the known period P = 3.316 days, fixed b = 0, and the stellar characterization from Section 4.1, TOI-1272b would have a transit duration of T_circ = 0.094 ± 0.004 days if it were on a circular orbit. The observed transit, however, had a duration that was nearly 40% shorter than this at T_obs ≈ 0.06 days. The short transit duration suggests either a high-eccentricity orbit transiting near periastron or an orbit with a high impact parameter, motivating our follow-up analysis to constrain the true eccentricity.

To characterize the transit properties of TOI-1272b more precisely, we fit the available TESS photometry with the exoplanet package (Foreman-Mackey et al. 2021). The exoplanet package uses a Hamiltonian Monte Carlo algorithm that is a generalization of the No U-Turn Sampling method (Hoffman & Gelman 2014; Betancourt 2016). We used this model to generate samples from the posterior probability density for the parameters {P, t₀, R_p /R_*, b, ρ_*, $\sqrt{e}\sin \omega$, $\sqrt{e}\cos \omega$, μ, u, v}, conditioned on the observed TESS light curve. Here, μ is the mean out-of-transit stellar flux and {u, v} are quadratic limb-darkening parameters. The model used here follows that of MacDougall et al. (2021).

We applied weakly informative priors to each of the 10 model parameters, similar to those used by Sandford & Kipping (2017). In particular, the prior used on our parameterization of eccentricity and argument of periastron {$\sqrt{e}\sin \omega$, $\sqrt{e}\cos \omega$} was uniform on both parameters, not accounting for transit probability or other astrophysically motivated considerations. Also, our prior on ρ_* was based on the stellar characterization discussed in Section 4.1. We fit the photometry of TOI-1272 with this model using 6000 tuning steps and 4000 sampling steps over four parallel chains. Figure 1 shows the final transit model sampled from the posteriors.

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An independent fit to the MuSCAT2 transit photometry of TOI-1272b was performed and used to verify the results of our transit fit to the full TESS photometry (Figure 2). The raw MuSCAT2 data were reduced by the MuSCAT2 pipeline (Parviainen et al. 2019), which performed standard image calibration and aperture photometry, and modeled the instrumental systematics present in the data while simultaneously fitting a transit model to the light curve. We also applied our own transit model to the detrended MuSCAT2 photometry, achieving consistent posterior constraints on all transit parameters. The same process was repeated for independent transit photometry from MuSCAT, producing similar results.

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The photometrically constrained eccentricity posterior distribution that we measured for TOI-1272b from {$\sqrt{e}\sin \omega$, $\sqrt{e}\cos \omega$} was consistent with our high-eccentricity hypothesis, yielding a 1σ range of e = 0.18–0.60 and an ω suggestive of a transit near periastron. The individual posterior distributions of e and ω are shown in Figure 3, along with their joint 2D posterior.

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We do note, however, that our impact parameter distribution remains loosely constrained, with a 1σ range of b = 0.19–0.73, peaking in density toward the upper end of this range (Figure 3). The similarly loose constraints on both e and ω implied that our photometric characterization of the orbital geometry was complicated by e–ω–b degeneracy, as can be seen in the 2D joint posterior distributions in Figure 3. Nevertheless, the potential for a high eccentricity combined with the expected Neptune-like size of the planet and its short orbital period made TOI-1272b a prime target for follow-up radial velocity observations.

Following the procedures outlined by MacDougall et al. (2021), we characterized the bulk properties of TOI-1272 by first inferring T_eff and [Fe/H] from our Keck/HIRES template spectrum using SpecMatch-Synth (Petigura et al. 2017). We then modeled the stellar mass, radius, surface gravity, density, and age via stellar isochrone fitting with isoclassify (Berger et al. 2020; Huber et al. 2017). We report these values and their associated uncertainties in Table 2, accounting for small corrections due to model grid uncertainties discussed by Tayar et al. (2020). We note that the properties derived with isoclassify rely on Two Micron All Sky Survey (2MASS) K-band magnitude and Gaia parallax, also reported in Table 2. The properties that we measured were consistent with those reported to ExoFOP-TESS from two spectra obtained with the TRES instrument at the Whipple Observatory, analyzed using the Stellar Parameter Classification (SPC) tool (Buchhave et al. 2012, 2014).

Table 2. TOI-1272 System Properties

Parameter	Value	Notes
Stellar
RA (deg)	199.1966	A
Dec (deg)	49.86104	A
π (mas)	7.24 ± 0.021	A
mG	9.6844 ± 0.0004	A
mK	9.70 ± 0.02	B
Teff (K)	4985 ± 121	C
[Fe/H] (dex)	0.17 ± 0.06	C
log(g)	4.55 ± 0.10	C
M* (M⊙)	0.851 ± 0.049	C
R* (R⊙)	0.788 ± 0.033	C
ρ* (g cm−3)	2.453 ± 0.343	C
age (Gyr)	${3.65}_{-0.98}^{+4.17}$	D
P*,rot (days)	28.3 ± 0.6	E
SHK	0.331	F
log$R{{\prime} }_{\mathrm{HK}}$	−4.705	F
u	0.39 ± 0.05	G
v	0.09 ± 0.05	G
μ (ppm)	−4.3 ± 36.2	H
γ (m s−1)	0.7 ± 0.7	H
σjit (m s−1)	5.6 ± 0.6	H
Planet b
P (days)	3.31599 ± 0.00002	H
Tc (BJD – 2,457,000)	1713.0253 ± 0.0006	H
b	${0.45}_{-0.21}^{+0.15}$	H
Rp (R⊕)	4.14 ± 0.21	H
Mp (M⊕)	24.6 ± 2.3	H
ρp (g cm−3)	1.9 ± 0.3	H
K (m s−1)	12.6 ± 1.1	H
a (au)	0.0412 ± 0.0008	H
e	${0.338}_{-0.062}^{+0.056}$	H
ω (deg)	123.6 ± 11.5	H
Teq (K)	961 ± 32	I
Planet c
P (days)	8.689 ± 0.008	H
Tc (BJD – 2,457,000)	1885.34 ± 0.48	H
Mp sini (M⊕)	26.7 ± 3.1	H
K (m s−1)	9.4 ± 1.0	H
a (au)	0.0783 ± 0.0014	H
e	≲0.35	J
ω (deg)	$-{80.8}_{-57.3}^{+97.4}$	H
Teq (K)	697 ± 23	I

Note. A: Gaia DR2, epoch J2015.5 (Gaia Collaboration et al. 2018); B: 2MASS (Skrutskie et al. 2006); C: derived with isoclassify; D: derived with stardate (Angus et al. 2019a); E: derived with TESS-SIP (Hedges et al. 2020); F: measured from Keck/HIRES template; G: derived with LDTK (Parviainen & Aigrain 2015); H: constrained from joint RV–photometry model with juliet (Espinoza et al. 2018; Kreidberg 2015; Fulton et al. 2018; Speagle 2020); I: calculated from other parameters assuming albedo α = 0.3; J: dynamically constrained with rebound (Rein & Liu 2012).

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To properly detrend our RV data and interpret any planetary signals, we first needed to characterize the intrinsic stellar variability for TOI-1272. We measured stellar variability from the TESS 2 minute cadence SAP photometry where TOI-1272 was observed in three sectors, one of which partially overlapped with our RV observation baseline. Upon removing data that were flagged as being of poor quality, ≥5σ outliers, or part of the TOI-1272b transit events, we measured the stellar variability period from the trimmed SAP light curve using the TESS-SIP algorithm (Hedges et al. 2020). This systematics-insensitive Lomb–Scargle periodogram (Angus et al. 2016) yielded a clear variability signal at 28.3 ± 0.6 days, likely associated with stellar rotation. The corrected TESS-SIP light curve, Lomb–Scargle periodogram, and phase-folded light curve with a sinusoidal fit are shown in Figure 4.

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The stellar variability observed in the TESS photometry of TOI-1272 also allowed us to derive the expected stellar activity-driven variability in our RV measurements using the ${{FF}}^{{\prime} }$ method (Aigrain et al. 2012). This method uses the light-curve flux (F), its derivative (${F}^{{\prime} }$), an estimate of relative spot coverage (f ∼ 0.005 in this work), and a simple spot model to simulate activity-induced RV variability. We estimated that this stellar activity would produce an RV variability signal with semiamplitude K ≈ 5.0 m s⁻¹, assuming a sinusoidal signal. Given the partial overlap of our RV baseline with that of the TESS photometry, we used this RV variability estimate as the foundation for our consideration of activity-driven RV signals in Section 5.2. By doing so, we implicitly assumed that the variability signal in the region of overlap could be extrapolated out to the entire data set. Based on the measured value of log(g) and the observed values of activity metrics S_HK and log$R{{\prime} }_{\mathrm{HK}}$, moderate stellar activity-driven RV jitter was also expected for TOI-1272 based on the classifications presented in Luhn et al. (2020), σ_jit ≳ 2.5 m s⁻¹, consistent with our photometry-only estimate.

As a final consideration of stellar variability, we searched for periodic, activity-driven signals in the S_HK data series for TOI-1272 using a Lomb–Scargle periodogram. We identified a significant 28.5 day signal, consistent with our stellar variability measurement from TESS-SIP (Figure 5). We also detected additional sub-significant S_HK variability signals that did not correspond to any known sources. We consider the impact that activity may have on our RV measurements when constructing our RV-only model in Section 5.

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Given the short orbital period of TOI-1272b and the possibility of a high-eccentricity orbit, the age constraints for this system were valuable for interpreting the tidal circularization timescale of the transiting planet. Our isochrone fit using isoclassify yielded a poorly constrained age estimate of ∼1–7 Gyr. This was consistent with a first-order analytical estimate of the age of TOI-1272, 3.1 Gyr, based on G_BP − G_RP color and stellar rotation period via gyrochronology (Angus et al. 2019b).

We took this analysis a step further by using stardate (Angus et al. 2019a) to combine stellar isochrone modeling with gyrochronology to precisely measure the stellar age. Running the stardate Markov Chain Monte Carlo (MCMC) sampler for 10⁵ draws, we measured an age of ${3.65}_{-0.98}^{+4.17}$ Gyr. While this age range remained broad and consistent with our isoclassify measurement, the increased median value and reduced uncertainty from stardate provided us with better constraints on the lower bound of the age of this system.

We searched for periodic signals in our RV data using the RVSearch pipeline (Rosenthal et al. 2021). We set Gaussian priors on the period P_b and time of conjunction T_c,b of the 3.3 day planetary signal known from photometry. We then used RVSearch to iteratively search the RV data for additional Keplerian signals across the period range from 2 to 4000 days. This search yielded an eccentric Keplerian fit with K ≈ 12.6 m s⁻¹ at the known period and an outer 8.7 day Keplerian fit with K ≈ 9.4 m s⁻¹ (Figure 6). Both signals surpassed our significance threshold, with false-alarm probabilities (FAPs) measured by RVSearch FAP ≈ 10⁻⁴ and 10⁻⁵, respectively.

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We corroborated the significance of the 8.7 day signal by performing an independent search of the RV data set using an l₁ periodogram (Hara et al. 2017), which minimizes the aliasing seen in a general Lomb–Scargle periodogram by evaluating all frequencies simultaneously rather than iteratively. We implemented our l₁ periodogram with jitter σ = 5.0 m s⁻¹, correlation time τ = 0, and maximum frequency 1.5 cycles/day across the period range from 1.1 to 1000 days. Both the 3.3 and 8.7 day signals were clearly detected by this l₁ periodogram search, with consistent FAP values of ∼10⁻⁴ and ∼10⁻⁵, respectively. Given the significance of the 8.7 day period and the lack of a corresponding signal in either the S_HK activity data or photometric time series (see Section 4.2), we concluded that this Keplerian signal was of planetary origin. A close inspection of the phase-folded and detrended TESS photometry at the RV-constrained period and time of conjunction for the outer RV signal showed no evidence for a corresponding transit event.

While we did not identify any additional signals in our RV data that met our significance criteria, we did detect a sub-significant Keplerian signal at a 14.1 day period using both RVSearch and an l₁ periodogram search. This signal persisted throughout our entire observing baseline and was detectable in the residuals to a preliminary two-planet RV fit with Keplerian modeling code RadVel (Fulton et al. 2018). We concluded that this signal was the first harmonic (P_rot/2) of the 28.3 day stellar rotation as measured from the S_HK time series and TESS photometry. The P_rot/2 harmonic of a star's rotation period is known to induce strong periodic activity signatures such as this in RV time-series data (Boisse et al. 2011).

Preliminary RV modeling revealed no other RV signals and insignificant detections of a trend and curvature in our RV time series, providing no evidence of further companions. We also found an estimated RV jitter of σ ≈ 5.5 m s⁻¹. This jitter measurement was consistent with both our ${{FF}}^{{\prime} }$ estimate of RV variability and the RV semiamplitude of the marginal 14.1 day signal (K = 4 ± 1 m s⁻¹). Given the low significance of this additional signal and its sub-jitter amplitude, we chose to consider only the two planetary signals in our final RV models. We therefore interpreted the spectroscopic data for TOI-1272 to reveal two planetary signals (3.3 days and 8.7 days), with a sub-significant activity signal driven by stellar rotation (P_rot/2 ≈ 14.1 days).

We performed a two-planet fit to the RV time series for TOI-1272 using RadVel, a Python package used to characterize planets from Keplerian RV signals by applying maximum a posteriori model fitting and parameter estimation via MCMC (Fulton et al. 2018). Our model consisted of two planetary Keplerian signals with periods of 3.3 and 8.7 days. We modeled the data by fitting the following free parameters for both planets: P, T_c , K, $\sqrt{e}\cos \omega$, and $\sqrt{e}\sin \omega$. Our model also included RV offset γ and RV jitter term σ to account for astrophysical white noise and instrumental uncertainty. The best-fit RV-only RadVel model confirmed the existence of two eccentric sub-Jovian mass planets orbiting TOI-1272, and we used these results to inform the priors for a joint RV–photometry model.

We obtained the most precise planet parameters for the TOI-1272 system by performing global RV–photometry modeling using juliet (Espinoza et al. 2018), a robust tool for modeling both transiting and non-transiting exoplanets. We used juliet to jointly fit the radial velocities through RadVel and the transit photometry through batman (Kreidberg 2015), with proper handling of limb-darkening coefficients (Kipping 2013). Estimation and comparison of Bayesian evidences and posteriors was performed directly by the dynamic nested sampling package dynesty (Speagle 2020), one of several such tools offered through the juliet interface. Unlike the Monte Carlo algorithm used in our initial transit-only analysis, nested sampling algorithms break up complex posterior distributions into simpler nested slices, sampling from each slice individually then recombining the weighted results to reconstruct the complete posterior. This method becomes more efficient in the higher-dimensional posterior spaces of joint models.

We directly fit for each transit and Keplerian property with priors informed from our previous photometry-only and RV-only models. For the final global model, we fit for photometry-only properties {R_p/R_*, b, ρ_*, μ, u, v}, joint properties {P_b, t_0,b, $\sqrt{{e}_{{\rm{b}}}}\sin {\omega }_{{\rm{b}}}$, $\sqrt{{e}_{{\rm{b}}}}\cos {\omega }_{{\rm{b}}}$}, and RV-only properties {P_c , t_0,c, $\sqrt{{e}_{{\rm{c}}}}\sin {\omega }_{{\rm{c}}}$, $\sqrt{{e}_{{\rm{c}}}}\cos {\omega }_{{\rm{c}}}$, K_b, K_c, γ, σ}. Our final measurements are included in Table 2 and the corresponding maximum a posteriori RV model is shown in Figure 7.

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In summary, we measured mass constraints for TOI-1272 b and c at significance levels ∼11σ and ∼9σ, respectively, reflecting the strengths of the two periodogram signals discussed in Section 5.1. We also measured a high eccentricity of e_b = 0.34 ± 0.06 for TOI-1272b, consistent within 1σ with our photometry-only eccentricity constraint from Section 2.3. The eccentricity of the outer planet was loosely constrained to ${e}_{c}={0.12}_{-0.08}^{+0.1}$. We note, however, that a model fit with e_c = 0 performed nearly identically to the eccentric model, suggesting that the eccentricity of TOI-1272c is only marginally significant. We discuss these constraints on eccentricity further in Section 6.1. Our global model also served to minimize degeneracies between e, ω, and b and allowed us to obtain more precise b and R_p,b values than with our photometry-only model. Our loose posterior constraint on impact parameter from Figure 3 was improved to $b={0.45}_{-0.21}^{+0.15}$, subsequently yielding our final radius measurement of R_p,b = 4.14 ± 0.21 R_⊕.

Despite the compact architecture of the TOI-1272 system, both planets had moderate RV-constrained eccentricities that were inconsistent with zero to ∼5σ and ∼1σ significance, respectively. Such excited dynamics put TOI-1272 b and c at risk of dynamical instability if orbit crossing were to occur:

$$\begin{eqnarray}&&\displaystyle \frac{{a}_{c}\left(1-{e}_{c}\right)}{{a}_{b}\left(1+{e}_{b}\right)}\gt 1.\end{eqnarray} \tag{ 2 }$$

Given our RV-constrained measurements of eccentricity and orbital separation in this system, we found the left-hand side of the above equation to be 1.22 ± 0.15, or <2σ from the orbit-crossing threshold. We note, however, that our confidence in long-term orbital stability based on this value is highly sensitive to our eccentricity uncertainties. Assuming a fixed true value of e_c = 0.05, this system would be firmly out of reach of geometric orbit crossing given its current configuration.

With the ambiguity in our orbit-crossing stability result, we also calculated the dynamical stability of the TOI-1272 system according to the stricter criterion from Petrovich (2015):

$$\begin{eqnarray}&&\displaystyle \frac{{a}_{c}\left(1-{e}_{c}\right)}{{a}_{b}\left(1+{e}_{b}\right)}-2.4{\left(\max \left({\mu }_{b},{\mu }_{c}\right)\right)}^{1/3}{\left(\displaystyle \frac{{a}_{c}}{{a}_{b}}\right)}^{1/2}\gt 1.15,\end{eqnarray} \tag{ 3 }$$

where μ is M_p /M_*, drawn from our joint model results. This threshold marks an estimated empirical boundary in two-planet system stability, determined by applying a support vector machine algorithm to a large number of numerical integrations. Planet–planet interactions resulting in the ejection of a planet into the star or out of the system were considered by Petrovich (2015) in developing this criterion.

Systems that satisfy the condition in Equation (3) are expected to maintain dynamical stability for integrations out to at least 10⁸ orbits of the inner planet. When computed for this system, we measured the left-hand side of Equation (3) to be 1.1 ± 0.15, or <1σ below the stated stability threshold of 1.15. Similar to the orbit-crossing criterion, TOI-1272 straddles the stability boundary for the Petrovich (2015) empirical threshold. We again note that a fixed outer planet eccentricity of e_c = 0.05 would promote the long-term stability of the TOI-1272 system according this stability criterion.

We followed up these inconclusive analytical predictions of long-term stability with a full N-body treatment of the stability of the TOI-1272 system. Drawing initial conditions from our RV–photometry model posteriors, we ran 10⁴ N-body simulations with rebound (Rein & Liu 2012) for ∼10⁶ orbits of the inner planet. We restricted the initialized eccentricities of our simulations to avoid starting on crossing orbits, and we considered a simulation to be "unstable" after an orbit-crossing event or close dynamical encounter. Overall, ∼81% of simulations remained "stable" for the entirety of our integration time, suggesting that the eccentricities and masses measured from our RV model were largely consistent with a stable architecture on moderate timescales (Figure 8). Our rebound simulations also showed that stable configurations of this system exhibit Laplace–Lagrange oscillations in eccentricity with a secular timescale on the order of ∼10² yr.

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While our e_b posterior remained mostly unchanged by this N-body model, our stability constraints on e_c allowed us to determine an upper bound of e_c ≲ 0.35. Upon this redefinition of e_c , we inferred that the true eccentricity of TOI-1272c was likely in the lower tail of the acceptable range. Given that a RadVel model with e_c = 0 performed nearly equivalently to the nonzero eccentricity model (Section 5.4), this interpretation is consistent with our RV-only analysis.

Along with our N-body integration, we also used rebound to model the transit-timing variations (TTVs) expected to be observed in this system over a similar baseline to our TESS photometry (∼215 days). We estimated a TTV O–C rms of 0.3 ± 0.3 minutes, below our threshold of sensitivity for individual transits. We verified this empirically by modeling the TESS photometry with exoplanet, similar to Section 2.3 but this time including TTVs as an additional model parameter. From this fit, we measured a TTV O–C rms of 2 ± 2 minutes, consistent with the estimate from rebound.

With the additional photometric observations from MuSCAT, we extended our TTV search to a total photometric baseline of ∼600 days. A fit to the single transit measured by MuSCAT yielded a transit mid-point of BJD – 2,457,000 =2313.223, only ∼3 minutes off from the predicted mid-point of 2313.221 ± 0.003 and within the ∼5 minute uncertainty of this prediction. The extended photometric baseline demonstrated again that the TTVs in this system are negligible, within ∼1σ of showing no evidence for TTVs.

The lack of TTVs was also consistent with the nonresonant orbital period ratio between TOI-1272 b and c: P_c /P_b ≈ 2.62. This period ratio is outside of the resonant width of any strong resonances, lying most closely to the 3:1 second-order mean motion resonance, with a ∼14% difference in ratio. However, we cannot rule out the possibility of past planet migration leading to resonance-crossing, which could have played a role in the planet–planet excitation of e_b discussed briefly in Section 7.3.

The age measurement of TOI-1272 from Section 4.3 is valuable when considering the potential tidal eccentricity decay of TOI-1272b. According to Millholland et al. (2020), which draws from Leconte et al. (2010), the timescale of orbital circularization due to tidal eccentricity damping for an eccentric orbit is given by

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{e} & = & \displaystyle \frac{4}{99}\left(\displaystyle \frac{Q^{\prime} }{n}\right)\left(\displaystyle \frac{{M}_{P}}{{M}_{* }}\right){\left(\displaystyle \frac{a}{{R}_{P}}\right)}^{5}\\ & & \times \,{\left({{\rm{\Omega }}}_{e}\left(e\right)\cos \epsilon \left(\displaystyle \frac{{\omega }_{\mathrm{eq}}}{n}\right)-\displaystyle \frac{18}{11}{N}_{e}\left(e\right)\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 4 }$$

Here, the mean motion is given by $n=\sqrt{{{GM}}_{* }/{a}^{3}}$ and the reduced tidal quality factor $Q^{\prime}$ can be rewritten as $Q^{\prime} =3Q/2{k}_{2}$, with specific dissipation function Q and tidal Love number k₂ (Murray & Dermott 1999; Mardling & Lin 2004). We defined ω_eq as the spin rotation frequency of TOI-1272b at equilibrium, which we found to be 3 ± 0.5 day⁻¹ following the procedure outlined in Millholland et al. (2020). We assumed the obliquity to be 0°. We have also introduced functions of eccentricity Ω_e (e) and N_e (e) given by

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{e}\left(e\right)=\displaystyle \frac{1+\tfrac{3}{2}{e}^{2}+\tfrac{1}{8}{e}^{4}}{{\left(1-{e}^{2}\right)}^{5}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{N}_{e}\left(e\right)=\displaystyle \frac{1+\tfrac{15}{4}{e}^{2}+\tfrac{15}{8}{e}^{4}+\tfrac{5}{64}{e}^{6}}{{\left(1-{e}^{2}\right)}^{13/2}}.\end{eqnarray} \tag{ 6 }$$

A typical Neptune-like planet is generally assumed to have a tidal quality factor of $Q^{\prime} \approx {10}^{5}$, but the true value is highly uncertain. Assuming this fixed value for $Q^{\prime}$ and drawing the other parameters in Equation (4) from our previous analysis, we estimated a circularization timescale of τ_e ≈ 0.21 ±0.09 Gyr. This nominal value of τ_e is >3σ below our age measurement of ${3.65}_{-0.98}^{+4.17}$ Gyr, suggesting that TOI-1272b has experienced significant eccentricity decay due to tides. This is not reflected in the anomalously high eccentricity that we measured, suggesting that another mechanism must be driving the excited state of this system. We note, however, that $Q^{\prime}$ is highly uncertain and ${\tau }_{e}\propto Q^{\prime}$, so a tidal quality factor of 2 × 10⁶ would make τ_e consistent with the age of the system.

Continuing with the assumption of $Q^{\prime} \approx {10}^{5}$, we estimated the initial eccentricity that would have been needed for TOI-1272b to reach its currently observed e_b after 3.65 Gyr of tidal eccentricity decay. Assuming constant $Q^{\prime}$ and τ_e , we followed the procedures of Correia et al. (2020) to derive the required post-formation eccentricity of e_b ≈ 0.8. Without a significant restructuring of the TOI-1272 system architecture, however, such a high eccentricity would not have allowed for a stable companion at the orbital separation of TOI-1272c. Ruling out this "hot-start" scenario, we are left to consider whether the anomalously high eccentricity of the inner planet is due to an underestimated $Q^{\prime}$ or excitation by some other dynamical mechanism. We discuss such formation and evolution scenarios further in Section 7.3.

TOI-1272b is a Neptune-like planet for which we measured a mass of 24.6 ± 2.3 M_⊕ and a radius of 4.1 ± 0.2 R_⊕, yielding a density of 1.9 ± 0.3 g cm⁻³. This system also contains a similar-mass outer companion (M_p sini = 26.7 ± 3.1 M_⊕) that is not transiting. Planets in this size and mass range have been reported frequently in the literature; only a few of them also fall within the Hot Neptune Desert (Mazeh et al. 2016; Owen & Lai 2018). At moderate planet sizes (∼2–6 R_⊕; M_p ≲ 100 M_⊕) and low orbital separations (P ≲ 5 days), a relative paucity of planets has been observed.

The triangular regions of R_p –P and M_p –P parameter space shown in Figure 9 highlight this phenomenon, as defined by Mazeh et al. (2016). TOI-1272b can be seen here among the small subset of Neptunes that fall within this otherwise sparse parameter space. Some notable inhabitants of the Hot Neptune Desert include GJ 436b (Lanotte et al. 2014) and HAT-P-11b (Yee et al. 2018) along with more recent finds from TESS photometry including LP 714-47b (Dreizler et al. 2020) and TOI-132b (Díaz et al. 2020). While the the exact mechanism responsible for clearing out this R_p –P and M_p –P region remains unknown, some models support a combination of photoevaporation and tidal disruption following high-eccentricity migration (Mazeh et al. 2016; Lundkvist et al. 2016; Owen & Lai 2018).

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Planets within the Hot Neptune Desert range from dense, atmosphere-poor mini-Neptunes to atmosphere-rich, "puffy" super-Neptunes. TOI-1272b lies in the middle of this spectrum, with an elevated density relative to the upper end of the Weiss & Marcy (2014) relation. We used a two-component composition model to determine the relative abundances of solid core and gaseous envelope for this dense Neptune, following the procedure of MacDougall et al. (2021). We interpolated over a 4D grid of stellar and planetary properties to derive the expected envelope mass fraction for TOI-1272b using the Lopez & Fortney (2014) planet structure models. Assuming an Earth-like core composition and a solar-composition H/He envelope, we estimated f_env = 10.9% ± 2.0% and a core mass of 21.9 ± 2.0 M_⊕. Given the strong stellar irradiance experienced by this planet, with T_eq ≈ 960 K (assuming albedo α = 0.3), TOI-1272b could have begun as a more atmosphere-rich Neptune similar to GJ 3470b (Kosiarek et al. 2019) and experienced subsequent atmosphere loss. TOI-1272b may then serve as a strong candidate for follow-up atmospheric observations, following the treatment of similar targets like those discussed by Crossfield & Kreidberg (2017).

The outer companion in this system, TOI-1272c, likely falls into the same size category as TOI-1272b, with M_p,c sin i = 27.4 ± 3.2 M_⊕. However, since no transit was detected in TESS photometry, we were unable to make any claims regarding its density or composition. One might suppose that a sufficiently low-radius planet on an 8.7 day orbit could produce a transit signal below the detection threshold of S/N ≈ 7.1. Assuming a transit duration of T₁₄ ≈ 0.15 days and the same noise properties as the TOI-1272b transit, this would require R_p,c ≲ 2.3 R_⊕ and ρ_p,c ≳ 12.0 g cm⁻³. While this density is not entirely unreasonable (see, e.g., Kepler-411b; Sun et al. 2019), it is unlikely given the known sample of similar planets.

The eccentricity distribution of hot Neptunes was discussed in depth by Correia et al. (2020), who noted that such planets exhibit elevated eccentricities despite being on compact orbits. We reconsidered this claim using a more recent set of confirmed planet data from the NASA Exoplanet Archive (Akeson et al. 2013; NASA Exoplanet Archive 2022), including TOI-1272b in our sample. We considered Neptunes to have radii ∼2–6 R_⊕ and "hot" planets to have P < 5 days. Constraining our sample to only planets with eccentricity uncertainties less than 0.1 (Figure 10), we found that hot Neptunes (N = 17) displayed a broader eccentricity distribution than their longer-period counterparts (N = 75). We verified the distinction between the two distributions using a Kolmogorov–Smirnov (KS) test, finding p ≈ 0.008. TOI-1272b contributed to this significant trend among hot Neptunes.

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On the other hand, planets with radii >6 R_⊕ showed the opposite trend, as can also be seen in Figure 10, verified by a KS test with p ≪ 0.01. In this radius range, 66 planets had P < 5 days and 88 had P ≥ 5 days. We did not consider planets with radii <2 R_⊕ in this analysis due to the low sample size of such planets with eccentricity uncertainties <0.1. However, a cursory examination of this small subset suggested similar eccentricity distributions between shorter- and longer-period planets in this size range.

While these findings are statistically significant based on KS tests given the current data, several confounding factors led us to determine that these eccentricity trends are suggestive rather than definitive at this time. These factors include the small sample size of hot Neptunes, the reliability of the data reported by the NASA Exoplanet Archive, and potential observational biases. Nonetheless, the possible disagreement between the eccentricity trend for hot Neptunes and that for hot Jupiters is an active area of research. The low eccentricities of hot Jupiters were consistent with the rapid tidal circularization timescales expected at lower semimajor axes. Conversely, hot Neptunes seem more likely to violate the rule.

A nominal empirically derived periastron distance of r_peri ≈ 0.03 au is often used to approximate the boundary of rapid tidal eccentricity decay, shown in Figure 11. Here, we see that only a small subset of well-characterized planets inhabit the high-eccentricity area of parameter space beyond this boundary, including TOI-1272b and a few other eccentric hot Neptunes. The well-studied planet GJ 436b is among these Neptunes with orbits that disagree with tidal circularization, making it a near-twin to TOI-1272b based on mass, radius, eccentricity, and period.

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The sparsity of the Hot Neptune Desert along with the counterintuitive trend in hot Neptune eccentricities suggests a unique evolutionary pathway for hot planets within the Neptune size regime. Several studies have sought to explain the dearth of planets within the "desert" region of M_p –P and R_p –P parameter space. The leading hypothesis suggests a combination of photoevaporation (Owen & Wu 2013; Owen & Lai 2018) and high-eccentricity migration (Mazeh et al. 2016; Matsakos & Königl 2016). Interestingly, photoevaporation is also cited as a possible mechanism for maintaining nonzero eccentricities among hot Neptunes (Correia et al. 2020), along with planet–planet excitation (see, e.g., Jurić & Tremaine 2008; Chiang & Laughlin 2013) or an eccentric Kozai–Lidov (EKL) effect from a distant giant companion (Naoz 2016). The persisting eccentricities of some hot Neptunes could also simply be a result of $Q^{\prime}$ values underestimated by an order of magnitude or more, which would make them inconsistent with the $Q^{\prime}$ values measured for Neptune and Uranus through interior modeling.

An underestimated $Q^{\prime}$ could certainly be the case for TOI-1272b, contributing to a longer τ_e and slower rate of eccentricity decay. TOI-1272b may also be undergoing significant photoevaporation given its f_env and close-in r_peri, contributing to both its location in the middle of the Hot Neptune Desert and its high eccentricity. However, TOI-1272b differs from the plausible formation and evolution pathways of other hot Neptunes due to the presence of a stable, nearby outer companion. Both high-eccentricity migration and perturbations from a distant companion through EKL effects are complicated by the presence of the mildly eccentric companion on an 8.7 day orbit. Such excitation mechanisms would have likely caused an orbit-crossing event and subsequent ejection of one or both planets.

Instead, we propose that, along with photoevaporation, TOI-1272b has experienced minor planet–planet excitation events with TOI-1272c, possibly involving close approaches or resonance-crossing events during migration (Ford & Rasio 2008). These events could have contributed to both the high eccentricity and possible inward migration of TOI-1272b into the Hot Neptune Desert region of parameter space, similar to the proposed evolution of mini-Neptune HIP-97166b (MacDougall et al. 2021). The elevated eccentricity of the inner planet may then persist in spite of strong tidal forces through Laplace–Lagrange oscillations that continue to force the eccentricity, as seen in our dynamical simulations mentioned in Section 6.1. However, additional considerations such as the relative inclination of the two planets may be necessary for a more detailed description of the dynamical evolution of this system.