TKS III: A Stellar Obliquity Measurement of TOI-1726 c
Fei Dai, Arpita Roy, Benjamin Fulton, Paul Robertson, Lea Hirsch, Howard Isaacson, Simon Albrecht, Andrew W. Mann, Martti H. Kristiansen, Natalie M. Batalha, Corey Beard, Aida Behmard, Ashley Chontos, Ian J. M. Crossfield, Paul A. Dalba, y Courtney Dressing, Steven Giacalone, Michelle Hill, Andrew W. Howard, Daniel Huber, Stephen R. Kane, Molly Kosiarek, Jack Lubin, Andrew Mayo, Teo Mocnik, Joseph M. Akana Murphy, Erik A. Petigura, Lee Rosenthal, Ryan A. Rubenzahl, Nicholas Scarsdale, Lauren M. Weiss, Judah Van Zandt, George R. Ricker, Roland Vanderspek, David W. Latham, Sara Seager, Joshua N. Winn, Jon M. Jenkins, Douglas A. Caldwell, David Charbonneau, Tansu Daylan, Maximilian N. Günther, Edward Morgan, Samuel N. Quinn, Mark E. Rose, Jeffrey C. Smith
DDraft version September 2, 2020
Typeset using L A TEX twocolumn style in AASTeX62
The TESS-Keck Survey III: A Stellar Obliquity Measurement of TOI-1726 c
Fei Dai, Arpita Roy, Benjamin Fulton, Paul Robertson, Lea Hirsch, Howard Isaacson, Simon Albrecht, Andrew W. Mann, Martti H. Kristiansen,
9, 10
Natalie M. Batalha, Corey Beard, Aida Behmard, Ashley Chontos, ∗ Ian J. M. Crossfield, Paul A. Dalba, † Courtney Dressing, Steven Giacalone, Michelle Hill, Andrew W. Howard, Daniel Huber, Stephen R. Kane, Molly Kosiarek, Jack Lubin, Andrew Mayo, Teo Mocnik, Joseph M. Akana Murphy, ∗ Erik A. Petigura, Lee Rosenthal, Ryan A. Rubenzahl, ∗ Nicholas Scarsdale, Lauren M. Weiss, Judah Van Zandt, George R. Ricker, Roland Vanderspek, David W. Latham, Sara Seager,
18, 20, 21
Joshua N. Winn, Jon M. Jenkins, Douglas A. Caldwell,
24, 23
David Charbonneau, Tansu Daylan,
25, 26
Maximilian N. Günther,
25, 27
Edward Morgan, Samuel N. Quinn, Mark E. Rose, andJeffrey C. Smith
23, 24 Division of Geological and Planetary Sciences, California Institute of Technology, 1200 East California Blvd, Pasadena, CA 91125, USA Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E California Blvd, Pasadena, CA 91125, USA Department of Physics & Astronomy, The University of California, Irvine, Irvine, CA 92697, USA Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA, USA
501 Campbell Hall, University of California at Berkeley, Berkeley, CA 94720, USA Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,Denmark Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA Brorfelde Observatory, Observator Gyldenkernes Vej 7, DK-4340 Tølløse, Denmark DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Lyngby, Denmark Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95060, USA Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA Department of Physics and Astronomy, University of Kansas, Lawrence, KS, USA Department of Earth and Planetary Sciences, University of California, Riverside, CA 92521, USA California Institute of Technology, Pasadena, CA 91125, USA Gemini Observatory Northern Operations, 670 N. A’ohoku Place, Hilo, HI 96720, USA Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA02139, USA Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA NASA Ames Research Center, Moffett Field, CA, 94035, USA SETI Institute, Mountain View, CA, USA Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 70 VassarStreet, Cambridge, MA 02139, USA Kavli Fellow Juan Carlos Torres Fellow (Received ?; Revised ?; Accepted ?)
Submitted to AASABSTRACTWe report the measurement of a spectroscopic transit of TOI-1726 c, one of two planets transiting aG-type star with V = 6.9 in the Ursa Major Moving Group ( ∼
400 Myr). With a precise age constraint [email protected] a r X i v : . [ a s t r o - ph . E P ] A ug Dai et al. from cluster membership, TOI-1726 provides a great opportunity to test various obliquity excitationscenarios that operate on different timescales. By modeling the Rossiter-McLaughlin (RM) effect, wederived a sky-projected obliquity of − +35 − ◦ . This result rules out a polar/retrograde orbit; and isconsistent with an aligned orbit for planet c. Considering the previously reported, similarly progradeRM measurement of planet b and the transiting nature of both planets, TOI-1726 tentatively conformsto the overall picture that compact multi-transiting planetary systems tend to have coplanar, likelyaligned orbits. TOI-1726 is also a great atmospheric target for understanding differential atmosphericloss of sub-Neptune planets (planet b 2.2 R ⊕ and c 2.7 R ⊕ both likely underwent photoevaporation).The coplanar geometry points to a dynamically cold history of the system that simplifies any futuremodeling of atmospheric escape. Keywords: planets and satellites: formation; INTRODUCTIONThe stellar obliquity is the angle between the rotationaxis of the host star and the normal of the orbital planeof its planet. While the planets in the Solar Systemare well-aligned with the Sun (obliquity (cid:46) ◦ ), many ofthe known exoplanets have polar or even retrograde or-bits (e.g. Sanchis-Ojeda et al. 2013; Dalal et al. 2019).These spin-orbit misalignments are often interpreted assignposts of a dynamically hot formation or evolutionhistory. Various mechanisms have been proposed to beresponsible for tilting the orbits of planets. Many ofthese mechanisms operate on different timescales: pri-mordial disk misalignment during the disk-hosting stage( (cid:46) Myr, e.g. Lai et al. 2011; Batygin 2012); nodal pro-cession induced by an inclined companion ( ∼ . Myrfor HAT-P-11b, Yee et al. 2018); the Kozai-Lidov mech-anism operates on a wide range of timescales 10 to yr depending on the system configuration (e.g. Fabrycky& Tremaine 2007); and secular chaos between longer-period giant planets can happen in 10 to yr (e.g.Wu & Lithwick 2011). A sample of obliquity measure-ments spanning a range of precise host star ages willhelp us distinguish these orbit-tilting mechanisms.Precise stellar ages for main sequence stars are hard tocome by, particularly for later-type stars which barelyevolve over a Hubble time. Our best age constraintscome from establishing cluster membership of a planethost so that the ensemble study of kinematics, stellaractivity, Li abundance, gyrochronology and isochronalfitting of other stars in the same cluster can firmly pindown the stellar age. So far, there are about a dozenplanet hosts found in young clusters (e.g. David et al.2016; Mann et al. 2016). They are crucial for our un-derstanding of various aspects of planet formation andevolution. TOI-1726 is a G-type star in the Ursa Ma- ∗ NSF Graduate Research Fellow † NSF Astronomy and Astrophysics Postdoctoral Fellow jor Moving Group (414 ± Myr, Jones et al. 2015) thathosts two transiting sub-Neptune planets with 2.2 and2.7 R ⊕ on 7 and 20-day orbits (Mann et al. 2020). Witha V -band magnitude of 6.9 and a projected rotationalvelocity v sin i of ∼ km/s, TOI-1726 provides a rareopportunity to measure the stellar obliquity of a youngsub-Neptune planet. In this work, we discuss a newmeasurement of the stellar obliquity of planet c.This letter is structured as follows. In Section 2 wepresent the spectroscopic measurements of the TOI-1726. Section 3 describes the constraints on the stellarparameters using both spectroscopy and Gaia informa-tion. In Section 4, we present a joint analysis of the TESS light curve and the Rossiter-McLaughlin (RM)effect to measure the stellar obliquity of TOI-1726 c.Section 5 discusses the implication of our finding. SPECTROSCOPIC MEASUREMENTWe obtained 49 spectra of TOI-1726 on the night ofUTC 2020 Feb 26, spanning a transit of TOI-1726 c. Weused the Automated Planet Finder (APF, Vogt et al.2014a) at the Lick Observatory. The spectra were ob-tained with an iodine cell whose dense forest of molec-ular lines provide both the wavelength solution and ameans of determining the line spread function. Thespectral resolution was ∼ KS III: A Stellar Obliquity Measurement of TOI-1726 c R ad i a l V e l o c i t y ( m / s ) O - C ( m / s ) Figure 1.
The measured radial velocities during the transit of TOI-1726 c. The red line is the best-fit model; the blue shadedregion represent the 68% confidence region from the posterior distribution. The data suggest a stellar obliquity of − +35 − ◦ thatfavors a prograde, and likely aligned orbit for TOI-1726 c . Visually, there are also hints of a red noise component towards theend of observation. We investigated the source of this red noise component with line profile analysis and its effect on obliquitymeasurement with a Prayer’s Beads analysis in Section 4. et al. 2010), therefore the high SNR, iodine-free HIRESspectrum should serve adequately as the template spec-trum for reducing the APF dataset. More details ofour forward-modeling Doppler pipeline are described inHoward et al. (2010). The radial velocities and uncer-tainties are plotted in Fig. 1 and reported in Table 1. STELLAR PARAMETERSWe constrained the spectroscopic parameters ( T eff , log g and [Fe/H]) of TOI-1726 using the iodine-free spectra from Keck/HIRES and the SpecMatch pipeline (Pe-tigura et al. 2017). In short, SpecMatch models ob-served optical spectra with interpolated model spectrafrom the precomputed grid (Coelho et al. 2005) of dis-crete T eff , [Fe/H], log g and v sin i values. Line broad-ening effects from both rotation and macroturbulenceare included by convolving the model spectra with thekernel described by Hirano et al. (2011). Instrumen- https://github.com/petigura/specmatch-syn Dai et al. T i m e f r o m M i d - t r an s i t ( H ou r s ) O-C (m/s)
20 10 0 10 20Velocity (km/s)10123456 T i m e S i n c e M i d - T r a n s i t ( H o u r s ) L i n e P r o f il e R e s i d u a l s
20 10 0 10 20Velocity (km/s)10123456 T i m e S i n c e M i d - T r a n s i t ( H o u r s ) L i n e P r o f il e R e s i d u a l s Figure 2. Top Left : The residuals of the RM time series same as Figure 1.
Top Right : The measured line profile residualsas a function of time and velocity. The vertical gray lines indicate the vsin i of the host star. The horizontal gray line indicatethe end of the transit t IV . Some localized patterns can be seen which are likely due to a combination of stellar activity andinstrumental drifts. Bottom : The simulated planetary shadow of TOI-1726c on a well-aligned orbit. The signal is about oneorder of magnitude lower than the uncertainties seen in the measurements (note the different color coding in these two panels);and remains undetected with the current measurement. tal broadening is modeled as a Gaussian function with aFWHM of 3.8 pixels, a value that provides a good matchto the widths of telluric lines. We calculate the weightedaverage of spectroscopic parameters of five ∼ ÃĚspectral segments. The final output spectroscopic pa-rameters are corrected for known systematic effects fromprevious comparison with standard stars. Particularly,SpecMatch systematically yields higher ( ∼ g for earlier-type stars when comparedwith asteroseismic results of standard stars (Huber et al.2013a). This effect is empirically corrected for with a scaling relation log g ( T eff ,[Fe/H]). See Petigura (2015a)for detail.To derive the stellar parameters, we further make useof Gaia parallax information (Gaia Collaboration et al.2018). We followed the procedure described in detail byFulton & Petigura (2018). To summarize, we link thestellar effective temperature, the parallax measurementfrom Gaia and the K -band magnitude (which is lessaffected by extinction) together with StefanâĂŞBoltz-mann Law for an independent constraint on the radiusof the star. In practice, we put in the priors on spectro-scopic parameters and the parallaxes into the Isoclas-
KS III: A Stellar Obliquity Measurement of TOI-1726 c p /R050100150200 ( deg ) HD 3167 c Kepler-56 b Kepler-56 c
Other Methods Single-TransitingOther Methods Multi-TransitingRM/DT Single-TransitingRM/DT Multi-TransitingTOI-1726 c Age (Gyr)050100150200 ( deg ) DS Tuc A b Kepler-63 b
Figure 3.
The projected stellar obliquity λ plotted against the planetary radius (Upper) and stellar age (Lower). Themajority of stellar obliquity measurements are performed for single-transiting planets which are believed to have a dynamicallyhot history. We highlighted measurements of relatively unexplored multi-transiting systems with filled symbols. TOI-1726 isa unique opportunity for obliquity measurement for multi-transiting sub-Neptune planetary systems with a well-determinedyoung age. The green shading in the lower panel qualitatively captures the magnitude of the high-energy radiation from thehost star that is responsible for driving photoevaporation. These high-energy radiation dwindles with the first few hundredMyr: a timescale future observations of TOI-1726 are poised to probe. Dai et al. sify package of Huber et al. (2017) which then comparesthese parameters with the MESA Isochrones & StellarTracks (MIST, Choi et al. 2016) to determine the pos-terior distribution of various stellar parameters. Theresults are summarized in Table 2. JOINT LIGHT CURVE AND RM ANALYSISTOI-1726 was observed by
TESS (Ricker et al. 2014)in Sector 20 from UT 2019 Dec 24 to 2020 Jan 20. Wedownloaded the reduced light curve from the MikulskiArchive for Space Telescopes website . We only keptdata points with a Quality Flag of 0, i.e., those with noknown problems.We started from the transit ephemerides reported bythe TESS team. We first removed the data spanningthe transits of both planet b and c from the light curve.This enabled us to measure the stellar rotation period ofTOI-1726 by applying the Lomb-Scargle periodogram.We detected a strong rotational modulation at a pe-riod of . +0 . − . days where the uncertainties are de-rived from the full width half maximum of the peak. Asa consistency check, we calculated v = 2 πR (cid:63) /P rot , therotation period of . +0 . − . days and the stellar radiusof . ± − . R (cid:12) together give a rotational velocity v of . +0 . − . km/s which is consistent with the v sin i of . ± . km/s determined from the spectroscopic anal-ysis alone. Using the procedure described in Masuda &Winn (2020), the orbital inclination of the host star is>45 ◦ at 95% confidence level. This agreement of v sin i and v is supporting evidence for a prograde and per-haps well-aligned orbit of TOI-1726 c, in addition to theanalysis of the RM effect described later in the paper.We then analyzed the in-transit light curve simulta-neously with the Rossiter-McLaughlin effect. We iso-lated data taken within one transit duration of the tran-sit midpoint. We used the Batman package (Kreidberg2015) to model the transit light curves. We adopted aquadratic limb-darkening law, imposing Gaussian pri-ors on the coefficients with medians taken from precom-puted limb darkening coefficients from
EXOFAST (East-man et al. 2013) and with widths of 0.3. We put a prioron the mean stellar density based on the analysis in Sec-tion 3. We sampled P orb , R p /R (cid:63) and a/R (cid:63) uniformly inlogarithmic space. We put a uniform prior on the im-pact parameter b [-1,1] and on the midtransit time ( T c ).We assumed that both planets are on circular orbits.The current RV dataset (Hirsch et al. in prep) only pro-vide weak constraints on the orbital eccentricities andare consistent with being circular for both planets. https://archive.stsci.edu astroutils.astronomy.ohio-state.edu/exofast/limbdark.shtml. To model the RM effect, we followed the prescriptionof Hirano et al. (2011). The additional parameters arethe sky-projected obliquity λ , the projected rotationalvelocity v sin i the radial velocity offset γ and the localgradient of the offset ˙ γ . We also included a jitter pa-rameter to account for any additional astrophysical orinstrumental noise. The likelihood function of the RMmodel was combined with the likelihood function of thetransit model.We sampled the posterior distribution using theMarkov Chain Monte Carlo technique implemented inthe e mcee code (Foreman-Mackey et al. 2013). We used128 walkers and ran until the Gelman-Rubin conver-gence statistics dropped below 1.03. We first includeda prior on the rotational modulation v sin i of . ± . km/s from spectroscopic analysis in Section 3. Thesky-projected obliquity has a posterior distribution of − +28 − ◦ i.e. favor a prograde and possibly aligned or-bit for planet c. The posterior distribution also favorsa slightly higher v sin i of . ± . km/s. When weremoved the prior on v sin i altogether, the data are con-sistent with a broader range of v sin i of . +4 . − . km/s;while the posterior distribution of stellar obliquity alsowidened λ − +33 − ◦ . Table 2 reports the summary of theposterior distribution for the key parameters. DOPPLER TOMOGRAPHY AND RED NOISEMITIGATIONWe tried to look for the Doppler shadow of planet cin the subtle variation of the line profiles using the non-Iodine part of the spectra ( − ÃĚ). Our analysisis similar to that of Albrecht et al. (2013). In short, wecleaned the spectrum from outliers with 5-sigma clip-ping. We removed the continuum and blaze functionwith a polynomial fit to the 95 % percentile flux levelin each Echelle order. We cross-correlated the individ-ual spectrum with the bestfit SpecMatch spectrum be-fore rotational/instrumental broadening is applied. Wethen subtracted the globally averaged line profile fromthe individual line profiles to extract the subtle varia-tions that may be caused by the shadow of the transit-ing planet (Figure 2). However, given the small transitdepth of the planet ( ∼ KS III: A Stellar Obliquity Measurement of TOI-1726 c
Python package lmfit . We recordedthe corresponding residuals and cyclically permuted theresiduals before adding them back to the best-fit model.This generated a series of mock datasets that containsthe same correlated noise component as the originaldataset. We found the maximum likelihood model foreach mock dataset. Focusing on the stellar obliquity,the resultant distribution of obliquity is λ = − +35 − ◦ .This is a broader distribution compared to that fromthe white-noise-only model in Section 4; but qualita-tively these two models both favor a prograde, possiblyaligned orbit for TOI-1726c. DISCUSSION6.1.
Obliquity of Multi-Transiting Systems
It has been noted in several previous works thatthe underlying orbital architectures of
Kepler single-transiting (here we refer to the observed multiplicity, tobe distinguished from planets that only transited hoststars once during the time span of observation) andmulti-transiting systems may be different. Specifically,single-transiting systems seem to have a broader distri-bution of orbital eccentricities whereas multi-transitingsystems mostly favor circular orbits (Van Eylen & Al-brecht 2015; Xie et al. 2016; Mills et al. 2019). In addi-tion, Fang & Margot (2012) and Zhu et al. (2018) sug-gested that the mutual inclination dispersion is largerwhen the observed multiplicity of a planetary system issmaller. A plausible explanation of this architecturaldifference is the dynamical interaction between the sub-Neptune planets or that with a more distant giant plan-ets. Zhu & Wu (2018) and Bryan et al. (2019) indepen-dently arrived at the conclusion that
Kepler -like sub-Neptune planets are much more likely to have a coldJupiter companion (>1AU) than randomly chosen stars(Cumming et al. 2008; Clanton & Gaudi 2014). Ma-suda et al. (2020) further showed that when the inner planetary system only has one transiting planet, its coldJupiter is likely inclined by tens of degrees relative to theinner planetary system. The interpretation is that thedynamical interaction of an inclined cold Jupiter can stirup the initially co-planar planetary systems while excit-ing larger mutual inclinations and eccentricities. Thesingle-transiting systems represent the dynamically hotsub-sample while the multi-transiting systems are dy-namically colder.It will be interesting to see if the same architecturaldifference carries over to the stellar obliquity distribu-tion. So far, there are about 150 obliquity measurementsin the literature. Traditional RM effect is more easilydetected for planets with larger radii and more frequenttransits. As a result, the vast majority of existing mea-surements were performed for hot Jupiters or hot Nep-tunes. Intriguingly, it is often the case that these hotJupiters and hot Neptunes are single-transiting planetswith spin-orbit misalignments both of which hint at adynamically hot past (Dong et al. 2018). On the otherhand, multi transiting systems tend to display low obliq-uities (Albrecht et al. 2013). Unfortunately there areonly ∼ obliquity measurements obtained for multi-transiting systems to date (see Fig. 3). We note the mostcomplete census of spin-orbit angle of multi-transitingsystems was done by Winn et al. (2017). They com-pared the projected rotational velocity v sin i and the ro-tational velocity v = 2 πR (cid:63) /P rot . If a system is grosslymisaligned, v sin i would be much smaller than v . Winnet al. (2017) found that the majority of Kepler -like sys-tems (systems with several sub-Neptune planets within1AU) are well-aligned with their host star. The six high-obliquity suspects Winn et al. (2017) identified weredominated by hot Jupiters. This result revealed a pic-ture that planets in multi-planet systems are generallywell-aligned as one would expect from a cold dynamicalhistory. Coming back to the multi-transiting systemsthat have their stellar obliquities explicitly measured,most of these measurements were often obtained withalternative methods, rather than the RM effect, suchas asteroseismology (e.g. Huber et al. 2013b) or spot-crossing anomalies (e.g. Sanchis-Ojeda et al. 2012). Theresults mostly yield well-aligned orbits. We note thatthe only exceptions are the polar orbit of HD 3167 c(Dalal et al. 2019) and 50 ◦ inclined orbit of Kepler-56b and c (Huber et al. 2013b). What kind of forma-tion channel gave rise to misaligned multi-planet sys-tems have been a topic of interests for the theorists (e.g.Li et al. 2014; Spalding & Batygin 2015). It will beinteresting to see if these two systems are indeed rareoccurrences. Our result on TOI-1726 c is one crucialstep towards enlarging that sample of multi-transiting Dai et al. planetary system. Although the obliquity constraintson planet b and planet c individually are weak: +41 − ◦ (Mann et al. 2020) versus − +35 − ◦ , the fact that bothplanets transit and posterior distribution of obliquityboth center at 0 seems to favor a coplanar, likely aligned,dynamically quiet architecture for TOI-1726.6.2. Obliquity in Time
As we mentioned briefly in the introduction, many dif-ferent theories have been offered to explain the observeddiversity of stellar obliquities (e.g. Fabrycky & Tremaine2007; Wu & Lithwick 2011; Lai et al. 2011; Batygin2012; Yee et al. 2018). Since these theories operate onvery different timescales, a potential way to test some ofthem is to obtain obliquity measurements for a sampleof planets with well-determined ages. For example, ifyoung planetary systems rarely display spin-orbit mis-alignment, it is reasonable to say that the orbit-tiltingmechanisms that only operate during the disk-hostingstage (e.g. Lai et al. 2011; Batygin 2012) cannot be thedominant channel to generate spin-orbit misalignment.The cluster membership of TOI-1726 (Mann et al. 2020)provides a firm and precise age estimate for the host star.In Fig. 3, we plotted all obliquity measurements for sys-tems with better than 20% age estimates. TOI-1726c is the third youngest planet with obliquity measure-ment. Moreover TOI-1726 c is a sub-Neptune whichis the predominant product of planet formation in theGalaxy (Petigura et al. 2013), whereas a group of plan-ets for which obliquity measurements have been lacking(Fig. 3).6.3.
A great system for studying atmospheric losses
The bimodal radius distribution and the presence ofthe so-called "Hot Neptune Desert" both suggest thatatmospheric loss from sub-Neptune planets is a com-mon if not ubiquitous phenomena (Fulton et al. 2017).TOI-1726 is a great system for a study of atmosphericloss. The star is 400 Myr old which is comparable tothe timescale where high-energy radiation from the hoststar begins to diminish (Ribas et al. 2005) and the pho-toevaporation starts to come to a conclusion (see Fig.3). Moreover, the system contains two sub-Neptuneplanets whose low surface gravity make them the plan-ets most amenable to photoevaporation (Wang & Dai2018). The two planets are suited to comparative studysince they orbit around the same host star. In otherwords, the planets are bathed in the same high-energyradiation environment except for a difference in orbitaldistance. Any difference in the outcome of atmosphericloss has to come from the different planetary parame-ters e.g. orbital period and planetary mass etc. The prograde and coplanar orbits of both planet b (Mannet al. 2020) and planet c together disfavor a violent eventsuch as high-eccentricity migration or giant impact colli-sion that would have disrupted the planets’ coplanarityand complicated the evolution of the atmospheres. Wealso note that there is no compelling evidence for a coldJupiter that may generate dynamical instability of theinner planetary system ( ∼ Facilities:
Automated Planet Finder (Levy),
TESS
Software:
Batman (Kreidberg 2015), Emcee (Foreman-Mackey et al. 2013), EXOFAST (Eastman et al. 2013),
KS III: A Stellar Obliquity Measurement of TOI-1726 c
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KS III: A Stellar Obliquity Measurement of TOI-1726 c Table 1.
Lick/APF Radial Velocities
Time (BJD) RV (m/s) RV Unc. (m/s) S index S Unc.2458905.618603 6.54 3.66 0.374 0.0022458905.626056 15.82 3.52 0.381 0.0022458905.633579 13.99 3.47 0.373 0.0022458905.640951 10.81 3.41 0.372 0.0022458905.648485 7.82 3.41 0.375 0.0022458905.655846 2.39 3.35 0.377 0.0022458905.663311 13.29 3.37 0.382 0.0022458905.670752 10.23 3.17 0.385 0.0022458905.678159 0.61 3.31 0.375 0.0022458905.685705 10.49 3.33 0.384 0.0022458905.693031 12.14 3.57 0.386 0.0022458905.700588 1.24 3.41 0.389 0.0022458905.708180 -1.23 3.61 0.381 0.0022458905.715564 -3.11 3.65 0.382 0.0022458905.722925 7.94 3.36 0.380 0.0022458905.730262 4.18 3.45 0.383 0.0022458905.737843 5.43 3.42 0.381 0.0022458905.745470 -1.55 3.22 0.379 0.0022458905.752749 6.28 3.35 0.385 0.0022458905.760260 6.17 3.41 0.381 0.0022458905.767575 5.20 3.28 0.381 0.0022458905.775190 4.44 3.20 0.391 0.0022458905.782655 11.99 3.08 0.380 0.0022458905.790061 2.87 3.21 0.384 0.0022458905.797492 6.12 3.40 0.384 0.0022458905.804945 8.19 3.35 0.381 0.0022458905.812386 2.52 3.11 0.379 0.0022458905.819793 0.77 3.44 0.367 0.0022458905.827362 6.47 3.46 0.391 0.0022458905.834758 2.63 3.59 0.382 0.0022458905.842257 4.15 3.46 0.378 0.0022458905.849699 10.48 3.43 0.379 0.0022458905.857059 9.96 3.69 0.381 0.0022458905.864721 2.41 3.66 0.375 0.0022458905.872093 8.32 3.48 0.380 0.0022458905.879396 4.79 3.59 0.374 0.0022458905.886930 11.96 3.41 0.380 0.0022458905.894476 12.24 3.67 0.376 0.0022458905.901883 18.33 3.53 0.384 0.0022458905.909255 14.24 3.88 0.385 0.0022458905.916893 12.73 3.85 0.382 0.0022458905.924254 14.29 3.69 0.383 0.0022458905.931707 26.95 3.83 0.387 0.0022458905.939126 11.95 3.92 0.386 0.0022458905.946590 4.19 3.66 0.394 0.0022458905.954113 16.51 4.02 0.383 0.0022458905.961497 16.13 4.24 0.388 0.0022458905.968869 8.97 4.26 0.383 0.0022458905.976681 9.64 4.76 0.391 0.002 Dai et al.
Table 2.
Stellar and Transit Parameters of planet c
Parameter Symbol Posterior DistributionSky-projected Obliquity (deg) λ − +35 − Projected Stellar Rotation (km/s) v sin i . ± . Radial Velocity Offset (m/s) γ . +1 . − . Radial Velocity Trend (m/s/day) ˙ γ . +8 . − . Planet/Star Radius Ratio R p /R (cid:63) . +0 . − . Planetary Radius ( R ⊕ ) R p . ± . Time of Conjunction (BJD-2457000) t . ± . Impact Parameter b . ± . Scaled Semi-major Axis a/R (cid:63) . +1 . − . Orbital Period (days) P orb . +0 . − . Jitter (m/s) σ . +0 . − . Effective Temperature ( T eff ) K ± Surface Gravity (dex) log g . ± . Metallicity (dex) [Fe/H] . ± . Projected Stellar Rotation from Spectroscopy (km/s) v sin i . ± . Stellar Mass ( M (cid:12) ) M (cid:63) . ± . Stellar Radius ( R (cid:12) ) R (cid:63) . ± . Stellar Density (g/cm ) ρ (cid:63) . ± . Rotation Period (days) P rot . +0 . − ..