HIP 65426 is a High-Frequency Delta Scuti Pulsator in Plausible Spin-Orbit Alignment with its Directly Imaged Exoplanet

HIP 65426 hosts a young giant planet that has become the first exoplanet directly imaged with JWST. Using time-series photometry from the Transiting Exoplanet Survey Satellite (TESS), we classify HIP 65426 as a high-frequency $\delta$ Scuti pulsator with a possible large frequency separation of $\Delta \nu =$7.23$\pm$0.02 cycles day$^{-1}$. We check the TESS data for pulsation timing variations and use the nondetection to estimate a 95% dynamical mass upper limit of 12.8 Mjup for HIP 65426 b. We also identify a low-frequency region of signal that we interpret as stellar latitudinal differential rotation with two rapid periods of 7.85$\pm$0.08 hr and 6.67$\pm$0.04 hr. We use our TESS rotation periods together with published values of radius and $v \sin{i}$ to jointly measure the inclination of HIP 65426 to $i_{\star}=107_{-11}^{+12}$$^\circ$. Our stellar inclination is consistent with the orbital inclination of HIP 65426 b ($108_{-3}^{+6}$$^{\circ}$) at the $68\%$ percent level based on our orbit fit using published relative astrometry. The lack of significant evidence for spin-orbit misalignment in the HIP 65426 system supports an emerging trend consistent with preferential alignment between imaged long-period giant planets and their host stars.

The giant planet HIP 65426 b orbits at a semi-major axis of 62-120 au and inclined 99-112 • to the line of sight based on orbit fitting analyses (Cheetham et al. 2019;Bowler et al. 2020;Carter et al. 2023;Do Ó et al. 2023;Blunt et al. 2023).Mass estimates of HIP 65426 b, derived from evolution models and from spectral fitting, span 6-11 M Jup (Chauvin et al. 2017;Cheetham et al. 2019;Marleau et al. 2019;Petrus et al. 2021;Carter et al. 2023).HIP 65426 was recently observed with NIRCam and MIRI as part of the JWST (Gardner et al. 2006(Gardner et al. , 2023) ) Early Release Science Program (Hinkley et al. 2022), resulting in HIP 65426 b becoming the first direct detection of an exoplanet at wavelengths beyond 5 µm (Carter et al. 2023).
Assessment of the spin-orbit alignment between a host star and its orbiting companions (i.e., obliquity) provides insight into the dynamical history of the system.While obliquity measurements have been obtained for dozens of short-period transiting exoplanets (e.g., Albrecht et al. 2022), there are relatively few constraints for long-period imaged substellar companions (e.g., Bowler et al. 2017;Bryan et al. 2020;Kraus et al. 2020).Recently, Bowler et al. (2023) analyzed the spinorbit alignments for a sample of 23 imaged substellar companions orbiting cool stars by constraining the stellar inclinations and comparing to the companion orbital inclinations.They found that misalignments are common for their sample, which mostly comprised brown dwarfs.However, the two imaged giant planet systems in their sample were consistent with alignment or nearalignment.Determining whether or not additional imaged giant planets like HIP 65426 b are aligned with their host star would contribute to our understanding of giant planet formation as a population, as well as help to clarify the dynamical history for the planet itself.
Time-series photometry from the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015) is continuing to boost the sample size of systems where obliquity assessments can be carried out.Time-series photometry can reveal the stellar rotation period if surface spot modulation is significant (e.g., Affer et al. 2012;McQuillan et al. 2014).While this has traditionally been carried out for cool stars, Balona (2011Balona ( , 2013Balona ( , 2017) ) used Kepler time-series photometry to show that as many as ∼40% of A-type stars exhibit a detectable photometric rotation frequency.This fraction persists to ∼30% when source contamination is considered (Sikora et al. 2020).A measured rotation period places a determination on the line-of-sight inclination when combined with the stellar radius and spectroscopic projected rotational velocity (v sin i), (e.g., Masuda & Winn 2020).Furthermore, time-series photometry also probes for other phenomena, including stellar pulsations and transit events, both of which have previously been studied in directly imaged exoplanet host stars using TESS (e.g., Zieba et al. 2019;Pavlenko et al. 2022;Sepulveda et al. 2022Sepulveda et al. , 2023)).The challenge for hot A-stars like HIP 65426 then becomes a matter of detecting a rotation frequency and discerning it from other phenomena, such as pulsation modes or contaminating sources in the aperture.
We analyze 2-minute cadence TESS time-series photometry of HIP 65426 in this study.We present the detection of stellar pulsations and rotational modulation in the data, and use the latter to constrain the obliquity of the system.We conclude by summarizing the implications of our analysis.(Smith et al. 2012;Stumpe et al. 2012Stumpe et al. , 2014;;Jenkins et al. 2016).We removed outlying photometry from each TESS sector via a sigma-clipping of five standard deviations from the median flux value, which is a standard practice to remove outliers potentially caused by instrumental artifacts that were not captured through standard quality flags.This resulted in only 21 cadences (< 0.05%) being removed, resulting in 16,392 cadences for Sector 11, 18,486 cadences for Sector 38, and 18,849 cadences for Sector 64.
We assessed our TESS data for any significant sources of contamination.We began by inspecting a region within a radius of 80 ′′ from HIP 65426, which is an approximate size of the default aperture used for our light curves.We identified no Gaia sources with comparable brightness to HIP 65426, which has an apparent Gaia Gband magnitude of 7.00 (Gaia Collaboration et al. 2016, 2023).The five brightest Gaia sources after HIP 65426 in this region (spanning separations of ∼17-47 ′′ ) have respective apparent G-band magnitudes of 11. 98, 13.28, 14.12, 14.59, and 15.05.Their respective G BP − G RP colors are 1.72, 0.72, 1.41, 1.47, and 1.32.HIP 65426 has no known stellar companions (Kouwenhoven et al. 2005;Chauvin et al. 2017;Petrus et al. 2021).Furthermore, the TESS Input Catalog (TIC, Stassun et al. 2018Stassun et al. , 2019) ) gives a flux contamination ratio (i.e., ratio of the total contaminant flux within 210 ′′ to HIP 65426's flux) of only 0.03 (See also Section 3.2).Overall, it is likely that significant signals in our TESS data correspond to HIP 65426.We calculated the amplitude spectrum of each TESS sector time series (Figure 1abc) using lightkurve (Lightkurve Collaboration et al. 2018) and identified several significant frequencies spanning ∼3-130 cycles day −1 (Figure 1d).For the high frequencies, we used SigSpec (Reegen 2007) to iteratively fit sine waves to the data above 10 cycles day −1 , and up to a spectral significance threshold of 10 (following, e.g., Gruberbauer et al. 2007;Zwintz et al. 2009;Sepulveda et al. 2022).This resulted in 13 significant pulsation frequencies for Sector 11 and 14 significant pulsation frequencies for Sectors 38 and 64.The pulsation frequencies span 28-131 cycles day −1 , consistent with pressure modes of δ Scuti stars (e.g., Kurtz 2022), and they are tabulated in Table 2.
Figure 2a shows HIP 65426 in a Gaia DR3 (Gaia Collaboration et al. 2016, 2023) color-magnitude diagram (CMD) compared to the Kepler δ Scuti stars from Murphy et al. (2019) as well as the Kepler γ Doradus stars from Li et al. (2020).We created the CMD following the procedure described in Sepulveda et al. (2022), which includes corrections for extinction and reddening for the Kepler stars.We used A V = 0.038 mag (Chen et al. 2012)  We attempted an asteroseismic mode identification for HIP 65426 by constraining the large frequency separation (∆ν), which is a parameter that can also constrain the mean stellar density (Aerts et al. 2010).We used the echelle package (Hey & Ball 2020) to iterate through a range of trial ∆ν values (5-9 cycles day −1 , which generously encompasses typical ∆ν values, e.g., Bedding et al. 2020;Murphy et al. 2023) and visually inspected the échelle diagrams for regular vertical sequences following Bedding et al. (2020).We arrived at a possible value of ∆ν =7.23±0.02cycles day −1 (Figure 2b).This is consistent within the range that Bedding et al. (2020) found for their sample of high-frequency δ Scuti stars, but the resulting ridge patterns of HIP 65426 are too complex to confidently identify the majority of the pulsation modes.The rapid stellar rotation period (indicated by v sin i and our analysis in Section 3.2) greatly contributes to the complication (e.g., Reese 2022; Aerts & Tkachenko 2023, and references therein).Detailed asteroseismic modeling, which may yield an independent age estimate, is beyond the scope of this study.
We checked the TESS data for pulsation timing variations that could be caused by mutual gravitation with an orbital companion (e.g., Hey et al. 2020).Such effects would manifest as a periodic (or coherent) variation of the phase of all the δ Scuti pressure modes, and is only measurable for sufficiently massive planets on long enough periods (e.g., Murphy et al. 2016;Hey et al. 2021).However, if no variation is detected, it is possible to instead place an upper limit on the mass of the orbital companion known a priori to exist.This is also a useful tool to rule out stellar mass companions.We follow the same methodology as Hey et al. (2021) to place a dynamical mass upper limit on HIP 65426 b in the absence of coherent phase variations, using our posterior predictions of the orbit from Section 3.4.This results in a 95% upper limit of 12.8 M Jup , which serves as an independent verification that HIP 65426 b is likely in the giant planet mass regime.The 99.7% upper limit is 19.1 M Jup .

Rapid Differential Rotation in HIP 65426
Inspection of the < 10 cycles day −1 region of the amplitude spectrum reveals several significant signals that are well-separated from the pressure modes (Figure 3).We identify two rotation frequencies, f r,1 =3.06 cycles day −1 and f r,2 = 3.60 cycles day −1 .f r,1 is detected in all three TESS sectors whereas f r,2 is detected is only Sector 64.Gaussian functions were fit to the highest peak of each frequency and the best-fitting standard deviation is the reported uncertainty (Table 1).In the case of f r,1 , our adopted value is the average of the three separate detections from each TESS sector.This yielded f r,1 =3.06±0.03cycles day −1 (7.85±0.08 hr) and f r,2 =3.60±0.02cycles day −1 (6.67±0.04hr).We note that 2 × f r,2 is similar to our estimated value for ∆ν.This is likely a coincidence, but supports the conclusion that rapid rotation will complicate the mode identification for this star.
We associated these signals with the stellar rotation period for the following reasons.Balona (2017) compared Kepler photometric rotation frequencies for a sample of 30 hot (8300-12000 K) stars with their corresponding v sin i values and found that they correlated in a manner physically consistent with rotation frequencies.We overplotted f r,1 and f r,2 compared with the v sin i value of HIP 65426 (280 km s −1 , Section 3.3) along with the sample presented in Balona (2017) in Figure 4.Both follow the expected trend, which validates our interpretation of the signals as tracers of the stellar rotation frequency.We considered if HIP 65426 could be a hybrid pulsator showing gravity modes in addition to pressure modes (e.g., Grigahcène et al. 2010).However, low-frequency gravity modes often appear in groups of several dense peaks (e.g., Li et al. 2020), as opposed to more narrow signals like in the case of HIP 65426.Moreover, the TESS amplitude spectra show signal at 2 × f r,1 and 2 × f r,2 (Figure 3) consistent with rotation harmonics and an untypical feature for pulsations (e.g., Uytterhoeven et al. 2011).The peak at ≈6.6 cycles day −1 is associated with rotation, but we do not include it in Table 1 due to the absence of a subharmonic.This is not uncommon for rotational modulation (e.g.Kawaler 2021) and could be caused by multiple spots located at different longitudes.The same is likely true for the 2 × f r,2 frequency in Sector 38 without f r,2 .
Together, we interpret the detection of f r,1 and f r,2 as evidence for surface latitudinal differential rotation in HIP 65426.The physical consistency of both frequencies with v sin i, as well as their proximity to one another, is consistent with the two frequencies tracing different stellar latitudes rotating at different velocities.Moreover, while each TESS sector is only ∼27 days, they are separated by ∼2 years, which corroborates a picture of spot evolution occurring on those timescales.This would also explain why f r,2 is only detected in one sector and why f r,1 shows amplitude variations between sectors.This is consistent with Balona & Abedigamba (2016), who analyzed the photometric rotation frequency spread for a sample of 522 hot (7400-10000 K) A and F stars and found that ≈ 60% of this sample was consistent with exhibiting differential rotation.Their analysis used Kepler light curves with a continuous time baseline of ∼4 yr, while only three sectors of TESS data are available for HIP 65426, limiting more detailed investigation.
We noted in Section 2 an absence of nearby comparatively bright sources to HIP 65426, as well as a lack of stellar companions.We further assessed the likelihood that our reported rotation frequencies correspond to HIP 65426 and not a faint contaminating background source in the TESS aperture.To this end, we used the TESS localize package (Higgins & Bell 2023), which is an algorithm designed to identify the most likely Gaia source in the TESS aperture responsible for producing a set of input frequencies.In very brief summary, the algorithm models the expected relative flux distribution of the input frequencies (i.e., the frequency amplitude) across TESS pixels to estimate the location of the variability source, assuming that only one source varies at the input frequencies.As an initial baseline, we began by running the routine to evaluate the three most significant pulsation frequencies of HIP 65426 (Table 2), separately for Sectors 11, 38, and 64, and down to a Gaia magnitude limit of 18.The corresponding per-sector relative probabilities that the pulsation frequencies cor-   respond to HIP 65426 are respectively 97.9%, >99.9%, and >99.9%.We next ran the routine in the same manner but now to evaluate the rotation frequencies in each of the three TESS sectors (Table 1).The resulting relative probabilities that these rotation frequencies correspond to HIP 65426 are respectively 88.1%, >99.9%, and >99.9%.This evaluation contributes additional confidence that our TESS rotation frequencies (and pulsation frequencies) intrinsically correspond to HIP 65426.

HIP 65426 Stellar Inclination
We used our stellar rotation period together with published measurements of v sin i and stellar radius to con-strain the stellar inclination.These parameters are related by a geometric relation (e.g., Doyle et al. 1984;Campbell & Garrison 1985): Masuda & Winn (2020) noted that v sin i is not statistically independent of the rotational velocity, v (computed using P rot and R ⋆ ), and thus it is not appropriate to apply Monte Carlo propagation of uncertainties.They provided a Bayesian framework that properly computes cos i given measurements of P rot , R ⋆ , and v sin i. Bowler et al. (2023) expanded on this by deriving analytic expressions that approximate the framework of Masuda & Winn (2020) as long as P rot is measured to ≲ 20%.We used the Bowler et al. (2023) expressions in this work to derive the inclination posterior of HIP 65426, P (i ⋆ | P rot , R ⋆ , v sin i).Their expressions (Eqn.9 and 10, Bowler et al. 2023) assume uniform priors on the measured inputs, an isotropically uniform prior for cos i, and Gaussian-distributed uncertainties on the inputs.Chauvin et al. (2017) reported an isochrone-derived stellar radius of R ⋆ = 1.77 ± 0.05 R ⊙ , which is consistent with the TIC value of R ⋆ = 1.74 ± 0.05 R ⊙ (Stassun et al. 2018(Stassun et al. , 2019) ) derived using the Stefan-Boltzmann law.As an input radius for HIP 65426, we used the mean of these two literature values.We note however that rapidly rotating intermediate mass stars are better described as oblate spheroids than spheres (e.g., van Belle et al. 2001;Domiciano de Souza et al. 2003;McAlister et al. 2005;Monnier et al. 2007).Therefore, HIP 65426 is likely not well described by a single radius but rather by a radius gradient that increases from polar to equatorial latitudes.HIP 65426 is too distant (i.e., its angular size is too small) to resolve with optical/infrared interferometry, a technique that can directly constrain the polar (R pol ) and equatorial (R eq ) radii for nearby stars with large projected angular sizes (e.g., van Belle 2012).To account for systematic uncertainty in our input radius due to unresolved oblateness, we consider R pol and R eq of Altair, a nearby δ Scuti star whose surface has been interferometrically resolved and who is of similar mass and rotational velocity as HIP 65426 (e.g., Buzasi et al. 2005;Monnier et al. 2007;Bouchaud et al. 2020).Using R pol = 1.661R ⊙ and R eq = 2.022R ⊙ (Monnier et al. 2007) we calculate a fractional uncerainty of 10%, which we adopt for our input value for HIP 65426 (i.e., R ⋆ = 1.76 ± 0.18 R ⊙ .) For the input v sin i, we used the mean of two reported values measured using HARPS spectroscopy: 299±9 km s −1 (Chauvin et al. 2017) and 261±2 km s −1 (Petrus et al. 2021).For the uncertainty, we used the standard error on the mean of the HARPS values, yielding 280±13 km s −1 .We note that in the presence of latitudinal differential rotation, the spectroscopic v sin i (derived without accounting for differential rotation) will represent a weighted average of the set of latitudinal velocities and not soley represent the equatorial velocity.This is supported by interferometric imaging of the nearby A-star Altair, which has a v sin i of 241 km s −1 from detailed modeling (Monnier et al. 2007) and a v sin i from methods that ignore differential rotation of 216 km s −1 (mean Altair v sin i from: Bernacca & Perinotto 1970;Abt & Morrell 1995;Royer et al. 2002;Schröder et al. 2009).Detailed modeling of the differential rotation (and its effect on line broadening) in HIP 65426 is beyond the scope of this paper.
For the input rotation period, we took the mean of our two TESS rotation periods from Section 3.2.By taking the mean we appropriately match the weighted average velocity implied by the HARPS v sin i measurements.We opted to keep the larger of the two individual period uncertainties for our input mean period , i.e., 7.26±0.08hr.
The resulting posterior for P (i ⋆ | P rot , R ⋆ , v sin i) is shown in Figure 5.Here we have assumed a prograde orbit of HIP 65426 b relative to its host star, whereby the planet orbits clockwise on the sky (i.e., is inclined at 90 • < i < 180 • , Section 3.4), and thus the inclination angle parameter space spans [90,180] • .The posterior median and 68% credible interval is i ⋆ = 107 +12 −11 • , with a maximum a posteriori of 107 • .This indicates that HIP 65426 is most likely oriented somewhat edge-on, consistent with a priori expectation from the literature v sin i measurements.The 95% and 99.7% credible intervals span 91-127 • and 90-133 • , respectively.

HIP 65426 b Orbital Inclination
We conducted an orbit fit utilizing all the relative astrometry of HIP 65426 b available in the literature (Chauvin et al. 2017;Cheetham et al. 2019;Stolker et al. 2020;Carter et al. 2023), which includes three recently reported high-precision measurements with VLTI/GRAVITY (Blunt et al. 2023).Where multiple astrometric measurements from the same instrument and same night of observing are reported, we took their mean and treated them as a single epoch as follows.From Cheetham et al. (2019) Cheetham et al. (2019).We chose to use the values reported in Stolker et al. (2020), specifically using their final reported measurements after they applied their astrometric bias corrections.This in total amounts to 13 astrometric epochs spanning a ∼7 year baseline (2016.41-2023.35).
We considered inflating the astrometric uncertainties to account for potential systematics following the approach of some recent orbit fitting studies with relative astrometry (Bowler et al. 2020;Sepulveda & Bowler 2022).In short, a linear model is assumed for ρ and θ as a function of time, and a reduced χ 2 (χ 2 ν ) is calculated.For any astrometric measurements only reported in [∆RA,∆Dec] form, we converted to [ρ, θ] in a Monte Carlo fashion.If χ 2 ν was greater than unity, we iteratively increased a "jitter" uncertainty term added in quadrature to the base uncertainty until χ 2 ν reaches unity.However, for the HIP 65426 b dataset used in this study, χ 2 ν was already less than unity for both ρ and θ (respectively 0.4 and 0.3, with ν=11), which suggests that the uncertainties are likely reasonable and we thus incorporated no additional uncertainty.A summary figure is displayed as Figure 8.
We fitted Keplerian orbits to our astrometric dataset using orbitize!(Blunt et al. 2020).We used ptemcee mode (Foreman-Mackey et al. 2013;Vousden et al. 2016), which utilizes the parallel-tempered affineinvariant ensemble sampler of Markov chain Monte Carlo (Goodman & Weare 2010).The varied parameters are standard for a Keplerian orbit with relative astrometry: a (semi-major axis), e (eccentricity), i (inclination), ω (argument of periastron of the planet's orbit), Ω (position angle of the ascending node), τ (time of last periapsis, as defined in Blunt et al. 2020 with t ref =MJD 58849), ϖ (parallax), and M tot (total mass of the system).We chose a Gaussian M tot prior of 2.0±0.1 M ⊙ , consistent with previous mass estimates of HIP 65426 (Tetzlaff et al. 2011;Chauvin et al. 2017;Bochanski et al. 2018).The ϖ prior was a Gaussian of 9.303±0.035mas from Gaia DR3 (Gaia Collaboration et al. 2016Collaboration et al. , 2023)).The prior for a was log-uniform ranging from 1-500 au.For i we used an isotropic (P (i) ∝ sin i) prior ranging from 0-π rad.We use linearly uniform priors for the remaining parameters with ranges: 0-1 for e; 0-2π rad for ω and Ω; and 0-1 for τ .To assist convergence, we chose to initialize the walkers at positions near the best-fit values as guided from preliminary orbit fits that we conducted as well as from prior orbit fits in the literature (e.g., Carter et al. 2023;Blunt et al. 2023).We ran ptemcee with 1500 walkers,  18 temperatures, and for 1.6 × 10 5 steps per walker per temperature.A burn-in size of the first 50% of steps are removed, and we also apply a thinning factor of 20 to mitigate the effect of correlation.Keeping only the samples of the lowest temperature yields a final total of 6 × 10 6 posterior orbit samples.A sample of skyprojected posterior orbits are displayed as Figure 6, and a selection of posterior orbital elements are displayed in Appendix B.
The most important result for the purposes of our study is the orbital inclination posterior (i HIP 65426 b ), which is shown in Figure 5.The median and 68% credible interval is 108 +6 −3 • ; the 95% credible interval spans 103-139 • .Our inclination posterior is consistent within the 68% uncertainties to those of recent studies (e.g., Carter et al. 2023;Do Ó et al. 2023;Blunt et al. 2023), despite minor differences in the choices of input astrometry and priors.

A Lack of Evidence for Misalignment
The inclination posteriors for HIP 65426 b and its host star are compared in Figure 5.They are consistent within their 68% uncertainties, which we interpret as a lack of significant evidence for misalignment.In the absence of knowledge of the stellar rotation axis orientation, the best available proxy for the true star-planet obliquity is the inclination difference of HIP 65426 b and its host star, |∆i| = |i ⋆ − i HIP 65426 b |, which represents a minimum inclination difference with respect to the true obliquity.We place 68% and 95% upper limits of |∆i| < 13 • and |∆i| < 25 • , respectively, for the HIP 65426 system.

Formation of HIP 65426 b
The wide orbit of HIP 65426 b is particularly interesting with respect to its formation history because core accretion is not expected to operate efficiently at separations of several tens of au (e.g., Pollack et al. 1996;Dodson-Robinson et al. 2009;Rafikov 2011).Coupled with the large eccentricity implied by early orbit fitting studies, planet-planet scattering has been proposed as a possible contributing mechanism for HIP 65426 b's presently-observed orbit.Marleau et al. (2019) explored planet-planet scattering scenarios in detail using the system properties together with N -body modeling.If HIP 65426 b formed via core accretion at a location interior to its present-day orbit, Marleau et al. (2019) suggest the presence of additional yet-undetected interior planets that would have participated in gravitationally scattering HIP 65426 b.Our lack of significant evidence for misalignment stands to disfavor a core ac-cretion followed-by planet-planet scattering scenario because the latter is generally expected to result in high mutual inclinations (Chatterjee et al. 2008;Ford & Rasio 2008).Although with incomplete geometric information, it is still possible that the true obliquity of HIP 65426 could be larger than |∆i|.
It is important to note that the orbital eccentricity of HIP 65426 b, which has played a key role in preliminary interpretations of its history, is not yet concretely determined.The most recent orbit fit by Blunt et al. (2023) using high-precision astrometry from VLTI/GRAVITY (as well as this work, which incorporated their GRAV-ITY data) disfavor the largest orbital eccentricities that were previously plausible based on eccentricity posteriors from earlier studies.Blunt et al. (2023) also showed that even with three GRAVITY epochs, the choice of eccentricity prior still influences the posterior.Do Ó et al. (2023) previously demonstrated this as well, where their adoption of an observable-based eccentricity prior (O'Neil et al. 2019) yielded an eccentricity posterior for HIP 65426 b more consistent with lower eccentricities compared to that derived with a uniform eccentricity prior.Continued astrometric monitoring to determine an accurate orbital eccentricity that is prior-independent will be needed to better understand the formation of HIP 65426 b.

Trends for Directly-Imaged Giant Planets
The lack of evidence for misalignment for HIP 65426 b furthers an emerging trend where imaged long-period giant planets appear preferentially aligned with their host stars (Bowler et al. 2023).We illustrate this in Figure 7 where orbital inclinations and host star inclinations are compared for 6 directly imaged exoplanet systems comprising 11 total companions: HR 8799 bcde (Marois et al. 2008(Marois et al. , 2010)), 51 Eri b (Macintosh et al. 2015;De Rosa et al. 2015), PDS 70 bc (Keppler et al. 2018;Haffert et al. 2019;Mesa et al. 2019), β Pic bc (Lagrange et al. 2009(Lagrange et al. , 2010(Lagrange et al. , 2019;;Nowak et al. 2020), AF Lep b (Franson et al. 2023;Mesa et al. 2023;De Rosa et al. 2023), and HIP 65426 b (Chauvin et al. 2017).The stellar and orbital inclination references are as follows: HR 8799 (Sepulveda & Bowler 2022;Sepulveda et al. 2023), 51 Eri (Dupuy et al. 2022;Bowler et al. 2023), PDS 70 (Wang et al. 2021;Bowler et al. 2023), β Pic (Zwintz et al. 2019;Brandt et al. 2021), AF Lep (Zhang et al. 2023), and HIP 65426 (this work).We use median values (or alternatively maximum a posteriori values if no median is reported) and 68% uncertainties to repre- Dusty debris disks, which represent extrasolar Kuiper Belt analogs (e.g., Hughes et al. 2018;Marino 2022, and references therein), were also shown to be preferentially aligned with their host stars based on early studies with small (N=8-10) sample sizes (Watson et al. 2011;Greaves et al. 2014).Recently, Hurt & MacGregor (2023) evaluated |∆i| for a larger sample size of 31 debris disk systems.While they found some examples of significantly misaligned systems, the lack of significant evidence for misalignment persisted in ≳80% of their sample, which still suggests that most debris disks are preferentially aligned with their host stars.Additional observations will show whether the link in the formation and dynamical evolution of imaged giant planets and debris disks continues to hold for obliquities.If this trend does hold true for imaged giant planets, it would contrast with the emerging obliquity trends found for imaged brown dwarfs, which imply that misalignments are common (Bowler et al. 2023).Together with distinctions in other fundamental properties (e.g., underlying orbital eccentricity distributions, Bowler et al. 2020;Nagpal et al. 2023;Do Ó et al. 2023) this could be further illustrating brown dwarfs and giant planets as distinct classes of objects with differing formation pathways.Future direct imaging discoveries that can include a TESS analysis of the host star will help to create a larger and more diverse sample size to determine whether this population-level obliquity trend for imaged giant planets holds true.

CONCLUSION
We analyzed TESS time-series photometry of the directly-imaged exoplanet host star HIP 65426.We detected several pulsation modes consistent with classification as a high-frequency δ Scuti pulsator.We find a preliminary estimate of ∆ν =7.23±0.02cycles day −1 , consistent with other young δ Scuti stars (Bedding et al. 2020).We also used the nondetection of pulsation timing variations to estimate a 95% dynamical mass upper limit of 12.8 M Jup for HIP 65426 b, independently supporting its planetary mass.
From the TESS data we also detected rapid photometric rotation frequencies of 7.85±0.08hr and 6.67±0.04hr that we interpreted as evidence of latitudinal differential rotation in HIP 65426.Using published radius and v sin i constraints together with our TESS rotation frequencies, we jointly measured the inclination of HIP 65426 to i ⋆ = 107 +12 −11 • .This near edge-on inclination of the host star is consistent with HIP 65426 b's near edge-on inclination of 108 +6 −3 • that we determined from our orbit fit of the available relative astrometry.We thus found no significant evidence for spin-orbit misalignment in the HIP 65426 system.This finding further supports an emerging trend of preferential alignment between imaged long-period giant planets and their host stars (Figure 7), which is similar to the emerging trend for debris disks and dissimilar to that for imaged brown dwarfs.
This work adds to the synergy of using space-based photometric data and direct imaging data to probe the outer architectures of extrasolar systems at an epoch after planet formation has occurred.Evaluating the timeseries photometry of additional host stars of imaged exo-planet, brown dwarf, and debris disk systems will be key to investigating the emerging obliquity trends.Likewise, dedicated missions and surveys to discover new systems and robustly characterize the inclinations of their constituents will also be a complementary necessity.
Acknowledgments.AGS thanks Brendan P.Here we tabulate the HIP 65426 pulsation frequencies that we extracted from each sector of TESS photometry (Table 2).We report the cumulative spectral significance and Fourier-domain phase angles as defined in Reegen (2011).Uncertainties were calculated following Kallinger et al. (2008).
B. HIP 65426 B ORBIT FIT Here we display the relative astrometry used in our orbit fit along with the best-fitting linear models used to evaluate the astrometric uncertainties (Figure 8).The 1D marginalized posterior distribution for a selection of orbital parameters are displayed in Figure 9.The orbital period (P ) posterior is generated using our a and M tot posterior samples and applying Kepler's 3rd Law in a monte carlo fashion.We note that our posterior for M tot , which is not displayed in Figure 9, is unconstrained by the astrometric data and merely recovers our prior probability choice.

Figure 2 .
Figure 2. (a) Gaia color-magnitude diagram for a sample of Kepler stars generated following Sepulveda et al. (2022).HIP 65426 is overplotted as a large cyan star.The orange diamonds are the δ Scuti stars, blue squares are the γ Dor stars, and the small, grey, semi-transparent dots are the general population of Kepler stars.(b): échelle diagram for HIP 65426 generated using our suggested large frequency separation of ∆ν =7.23 cycles day −1 .This plot uses an amplitude spectrum calculated from the concatenated TESS time series.
High-Frequency δ Scuti Pulsations in HIP 65426 for HIP 65426.The G BP − G RP color and abso-lute G magnitude of HIP 65426 are consistent with the population of δ Scuti stars from the Kepler Mission (Figure 2a) as well as with the population of high-frequency δ Scuti stars recently detected in the Pleiades open cluster with TESS (Bedding et al. 2023).

Figure 3 .
Figure3.TESS amplitude spectra of HIP 65426, zoomed in on a region of low frequencies.Vertical dashed lines denote the locations of fr,1, fr,2 as described in Section 3.2, as well as the locations of 2 × fr,1 and 2 × fr,2.

Figure 5 .
Figure 5. Normalized inclination posteriors for the HIP 65426 system, where P (i⋆ | Prot, R⋆, v sin i) represents the host star rotational inclination and i HIP 65426 b represents the orbital inclination of HIP 65426 b.Median and 68% credible intervals are written in the figure.

Figure 6 .
Figure 6.The left panel displays a sample of 100 sky-projected posterior orbits resulting from our fit of HIP 65426 b.The cyan star represents the position of HIP 65426 and the orange dots represent the relative astrometry of HIP 65426 b (with no error bars on this panel).The right panels compare the model orbit separation and position angles from the same 100 posterior orbits to the input astrometric data.

Figure 7 .
Figure 7. Orbital inclination compared to stellar inclination for a sample of 6 directly imaged exoplanet systems (comprising 11 total planets) assuming prograde orbits with respect to the host star.The solid black line denotes the 1:1 relation, and the light cyan region bordered by dashed black lines encapsulates an inclination difference ±10 • of the black line.References for the inclination angles are given in Section 4.3.

1β
Pictoris is a rare exception among directly-imaged exoplanet systems where the full obliquity has been constrained.The proximity of β Pictoris enabledKraus et al. (2020) to interferometrically determine the sky-projected position angle of its equator.Together with the asteroseismic stellar inclination fromZwintz et al. (2019),Kraus et al. (2020) determined a true mutual inclination angle of ≤ 3 ± 5 • for β Pic b and its host star.

Figure 8 .
Figure8.Relative astrometry of HIP 65426 b used in our orbit fit (Section 3.4) compared to our best-fitting linear models used to evaluate the astrometric uncertainties.Even with the inclusion of high-precision measurements from GRAVITY, the uncertainties overall appear reasonable.

Figure 9 .
Figure 9. 1D posterior probability densities from our orbit fit of HIP 65426 b.

Table 1 .
TESS rotation frequencies of HIP 65426.Equivalent rotation periods are included.
, we averaged the two SPHERE 2018.36 epochs into [ρ, θ] = [824.65 ± 2.2 mas, 149.87 ± 0.16 • ].From Carter et al. (2023), we averaged the two reported astrometric measurements from MIRI (2022.54)as well as five from NIRCam (2022.57)into two epochs, respectively [ρ, θ] = [829.5 ± 13 mas, 149 ± 1 • ] and [ρ, θ] = [819 ± 6.2 mas, 149.84 ± 0.42 • ].Stolker et al. (2020) reprocessed two NaCo epochs originally reported in Ochsenbein et al. (2000)or collaborative discussions that helped expedite this work, as well as Mitchell T. Dennis for assistance with operating the HPC clusters Mana and Koa.This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1842402 and 2236415.D.H. acknowledges support from the Alfred P. Sloan Foundation, the National Aeronautics and Space Administration (80NSSC21K0784), and the Australian Research Council (FT200100871).This work has benefitted from The UltracoolSheet (Best et al. 2020), maintained by Will Best, Trent Dupuy, Michael Liu, Rob Siverd, and Zhoujian Zhang, and developed from compilations by Dupuy & Liu (2012), Dupuy & Kraus (2013), Liu et al. (2016), Best et al. (2018), and Best et al. (2021).The technical support and advanced computing resources from University of Hawaii Information Technology Services -Cyberinfrastructure, funded in part by the National Science Foundation MRI award # 1920304, are gratefully acknowledged.This paper includes data collected by the TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST).Funding for the TESS mission is provided by the NASA's Science Mission Directorate.This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI : 10.26093/cds/vizier).The original description of the VizieR service was published inOchsenbein et al. (2000).This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.This research has made use of NASA's Astrophysics Data System Bibliographic Services.

Table 2 .
Significant pulsation frequencies of HIP 65426 from TESS photometry.