Analytic Estimates of the Achievable Precision on the Physical Properties of Transiting Planets Using Purely Empirical Measurements
Romy Rodriguez Martinez, Daniel J. Stevens, B. Scott Gaudi, Joseph G. Schulze, Wendy R. Panero, Jennifer A. Johnson, Ji Wang
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Analytic Estimates of the Achievable Precision on the Physical Properties of Transiting PlanetsUsing Purely Empirical Measurements
Romy Rodr´ıguez Mart´ınez, Daniel J. Stevens,
2, 3, ∗ B. Scott Gaudi, Joseph G. Schulze, Wendy R. Panero, Jennifer A. Johnson,
1, 5 and Ji Wang Department of Astronomy, The Ohio State University, 140 W. 18th Avenue, Columbus, OH 43210, USA Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA Department of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA School of Earth Sciences, The Ohio State University, 125 South Oval Mall, Columbus OH, 43210, USA Center for Cosmology and AstroParticle Phyrics , The Ohio State University, 191 W. Woodruff Ave., Columbus, OH 43210, USA
ABSTRACTWe present analytic estimates of the fractional uncertainties on the mass, radius, surface gravity, anddensity of a transiting planet, using only empirical or semi-empirical measurements. We first expressthese parameters in terms of transit photometry and radial velocity (RV) observables, as well as thestellar radius R (cid:63) , if required. In agreement with previous results, we find that, assuming a circularorbit, the surface gravity of the planet ( g p ) depends only on empirical transit and RV parameters;namely, the planet period P , the transit depth δ , the RV semi-amplitude K (cid:63) , the transit duration T , and the ingress/egress duration τ . However, the planet mass and density depend on all thesequantities, plus R (cid:63) . Thus, an inference about the planet mass, radius, and density must rely uponan external constraint such as the stellar radius. For bright stars, stellar radii can now be measurednearly empirically by using measurements of the stellar bolometric flux, the effective temperature, andthe distance to the star via its parallax, with the extinction A V being the only free parameter. Forany given system, there is a hierarchy of achievable precisions on the planetary parameters, such thatthe planetary surface gravity is more accurately measured than the density, which in turn is moreaccurately measured than the mass. We find that surface gravity provides a strong constraint on thecore mass fraction of terrestrial planets. This is useful, given that the surface gravity may be one ofthe best measured properties of a terrestrial planet. Keywords: methods: analytical — planetary systems — exoplanet composition INTRODUCTIONThe internal composition and structure of small, ter-restrial planets is generally difficult to characterize. Asis well known, mass-radius relationships alone do notconstrain the internal composition of a planet beyond ameasurement of its bulk density. The internal structureis crucial, as it determines the bulk physical propertiesof planets and provides valuable insights into their for-mation, history, and present composition. Unterbornet al. (2016) found that the core radius, the presence oflight elements in the core, and the existence of an uppermantle have the largest effects on the final mass and ra-
Corresponding author: Romy Rodr´ıguez Mart´ı[email protected] ∗ Eberly Research Fellow dius of a terrestrial exoplanet. The final mass and radiusin turn directly determine the planet’s habitability. Forexample, the core mass fraction affects the strength of aplanet’s magnetic field, which shields it against harmfulradiation from the host star.At present, we have ∼
330 small planets ( < R ⊕ ) withmasses and radii constrained to better than 50% . Suchmeasurement uncertainties are generally good enoughto determine the general structure of many exoplanets.However, for low-mass terrestrial planets with thin at-mospheres, planetary masses and radii must be mea-sured to precisions better than 20% and 10%, respec- Based on data from the NASA Exoplanet Archive,https://exoplanetarchive.ipac.caltech.edu/ a r X i v : . [ a s t r o - ph . E P ] J a n tively, in order to constraint the core mass fraction andstructure (Dorn et al. 2015; Schulze et al. 2020).However, high-precision measurements of low-massexoplanets between 1 − R ⊕ are challenging. Addi-tionally, because of the large number of individual dis-coveries, and because (to date) they have been mostlydetected around faint Kepler /K2 (Borucki et al. 2010;Howell et al. 2014) targets (with typical
Kepler andK2 magnitudes of K ∼
15 and K ∼
12, respectively,Vanderburg et al. 2016), they are difficult to follow upwith high-resolution radial velocity (RV) observationsand thus, obtain precise masses and other fundamentalphysical properties. This has already begun to changewith the Transiting Exoplanet Survey Satellite mission(TESS; Ricker et al. 2015), as its main science driver isto detect and measure masses and radii for at least 50small planets ( < R ⊕ ) around bright stars. At the timeof writing, 24 such planets have already been confirmed,and almost all have masses and radii measured to betterthan 30% .The discoveries of the TESS mission will also raisevery important questions in exoplanet science. The onethat we address here relates to the achievable precisionwith which we shall be able to constrain the fundamentalparameters of a transiting planet, such as its mass, den-sity and surface gravity. Given precise photometric andspectroscopic measurements of the host of a transitingplanet system, it is possible to measure the planet sur-face gravity with no external constraints (Southworthet al. 2007). On the other hand, measuring the massor radius of a transiting planet requires some externalconstraint (Seager & Mall´en-Ornelas 2003). Since, un-til very recently, it has only been possible to measurethe mass or radius of the closest isolated stars directly,theoretical evolutionary tracks or empirical relations be-tween stellar mass and radius and other properties of thestar have often been used (e.g., Torres et al. 2010). How-ever, these constraints typically assume that the star isrepresentative of the population of systems that wereused to calibrate these relations. In the case of theoret-ical evolutionary tracks, there may be systematic errorsdue to uncertainties in the physics of stellar structure,atmospheres, and evolution, or second-order propertiesof the star, such as its detailed abundance distribution,which can manifest as irreducible systematic uncertain-ties on the stellar parameters. For example, most evo-lutionary tracks assume a fixed solar abundance patternscaled to the iron abundance [Fe/H] of the star, andthus, the same [ α /Fe] as the Sun. If the host star has https://exoplanetarchive.ipac.caltech.edu/ a significantly different [ α /Fe] than the Sun, that willlead to incorrect inferences about the properties of theplanet. By using evolutionary tracks that assume a solar[ α /Fe], one might infer an incorrect density and mass ofthe planet, and therefore an incorrect core/mantle frac-tion.Thus, a direct, empirical or nearly empirical measure-ment of the radius or mass of the star that does notrely on assumptions that may not be valid is needed(see Stassun et al. 2017 for a lengthier discussion onthe merits and benefits of using empirical or semi-empirical measurements to infer exoplanet parameters).As has been demonstrated in numerous papers (see,e.g., Stevens et al. 2017), with Gaia (Gaia Collabora-tion et al. 2018) parallaxes, coupled with the availabil-ity of absolute broadband photometry from the near-UV to the near-IR, it is now possible to measure theradii of bright ( V (cid:46)
12 mag) stars. This allows for di-rect, nearly empirical measurements of the masses andradii of transiting planets and their host stars (and, in-deed, any eclipsing single-lined spectroscopic binary) ina nearly empirical way. See Stevens et al. (2018) foran initial exploration of the precisions with which thesemeasurements can be made.In this paper, we build upon the work of Stevens et al.(2018) by also assessing the precision with which the sur-face gravity g p of transiting planets can be measured.Given that a measurement of g p only requires measure-ments of direct observables from the transit photometryand radial velocities without the need for external con-straints, the precision on g p in principle improves withever more data, assuming no systematic floor. Thus weseek to address two questions. First, with what frac-tional precision can g p be measured, and how does thiscompare to the fractional precision with which the den-sity or mass can be measured? Second, how useful is g p as a diagnostic of a terrestrial planet’s interior structureand, potentially, habitability?Answering these questions is quite important becausethe surface gravity of a planet may be a more funda-mental parameter than the radius and mass, at leastin addressing certain questions, such as its habitability(O’Neill & Lenardic 2007; Valencia & O’Connell 2009;van Heck & Tackley 2011). For example, the surfacegravity, along with the equilibrium temperature andmean molecular weight, determines the scale height ofany extant atmosphere. If a planet’s surface gravity pro-vides more of a lever arm in determining certain aspectsof the planet’s interior or atmosphere, and if we canachieve a better precision on the planet surface gravitymeasurement than the radius, then we can use that tobetter constrain the composition of the planet and, ul-timately, its habitability. Thus, given the importance ofthe planetary surface gravity, mass and radius in con-straining the habitability of a planet, it is critical tounderstand how well we can measure these properties.Here we focus on the precision with which the surfacegravity, density, mass, and radius of a transiting planetcan be measured essentially empirically. We will employmethodologies that are similar to those used in Stevenset al. (2018), and thus this work can be considered acompanion paper to that one. ANALYSISWe begin by deriving expressions for the surface grav-ity g p , mass M p , density ρ p , and radius R p , of a tran-siting planet in terms of observables from photometricand radial velocity observations, as well as a constrainton the stellar radius R (cid:63) from a Gaia parallax combinedwith a bolometric flux from a spectral energy distribu-tion (SED) (Stassun et al. 2017; Stevens et al. 2017,2018). 2.1.
Planet Surface Gravity
The planet surface gravity is defined as g p = GM p R p . (1)The radial velocity semi-amplitude K (cid:63) can be ex-pressed as K (cid:63) = (cid:18) πGP (cid:19) / M p sin i ( M (cid:63) + M p ) / √ − e (cid:39) . − (cid:18) P yr (cid:19) − / M p sin iM J (cid:18) M (cid:63) M (cid:12) (cid:19) − / (1 − e ) − / , (2)where M (cid:63) is the stellar mass, P and e are the planetaryorbital period and eccentricity, M J is Jupiter’s mass,and i is the inclination angle of the orbit. In the secondequality, we have assumed that M p (cid:28) M (cid:63) .Using Newton’s version of Kepler’s third law andEquation 2, the surface gravity can then be expressedas g p = 2 πP √ − e ( R p /a ) K (cid:63) sin i . (3)For the majority of the following analysis, will assumecircular orbits ( e = 0) for simplicity and drop the ec-centricity dependence. This analysis could be repeatedfor eccentric orbits, but the algebra is tedious and doesnot lead to qualitatively new insights. The assumptionof circular orbits thus provides a qualitative expecta-tion of the uncertainties on the planetary parameters.Furthermore, in many cases it is justified because we expect many of the systems to which this analysis is ap-plicable will have very small eccentricities. We also fur-ther assume that sin i = 1, which is approximately truefor transiting exoplanets. Under these assumptions, wehave that g p = 2 πK (cid:63) P (cid:18) aR p (cid:19) . (4)The semimajor axis scaled to the planet radius canbe converted to the semimajor axis scaled to the stellarradius by using the depth of the transit δ ≡ ( R p /R (cid:63) ) ,which is a direct observable: aR p = aR (cid:63) (cid:18) R (cid:63) R p (cid:19) = aR (cid:63) δ − / . (5)We can then rewrite the scaled semimajor axis a/R (cid:63) in terms of the stellar density ρ (cid:63) (see e.g., Sandford &Kipping 2017 for a precise derivation): aR (cid:63) = (cid:18) GP π (cid:19) / (cid:0) ρ (cid:63) + k ρ p (cid:1) / , (6)where k ≡ R p /R (cid:63) . Since typically k (cid:28) aR (cid:63) = (cid:18) GP ρ (cid:63) π (cid:19) / . (7)Using Equation 5, we find aR p = (cid:18) GP ρ (cid:63) π (cid:19) / δ − / . (8)Noting that, from Equation 4, we can write g p = 2 πK (cid:63) P (cid:18) aR p (cid:19) = 2 πK (cid:63) P (cid:18) aR (cid:63) (cid:19) δ − . (9)Then a/R (cid:63) can be written in terms of observables as aR (cid:63) = Pπ δ / √ T τ . (10)where the observables are the orbital period P , transittime T (full-width at half-maximum), the ingress/egressduration τ , and the transit depth δ .Inserting this into Equation 9, we find that the planetsurface gravity is given in terms of pure observables as(Southworth et al. 2007) g p = 2 K (cid:63) PπT τ δ − / . (11)Using linear propagation of uncertainties, and as-suming no covariances between the observable parame-ters , and the aforementioned assumptions ( M P (cid:28) M (cid:63) , See Carter et al. (2008) for an exploration of the covariancesbetween photometric and RV observable parameters. sin i ∼ e = 0, and k (cid:28) (cid:18) σ g p g p (cid:19) ≈ (cid:18) σ K (cid:63) K (cid:63) (cid:19) + (cid:18) σ P P (cid:19) + (cid:18) σ T T (cid:19) + (cid:18) σ τ τ (cid:19) + 14 (cid:18) σ δ δ (cid:19) . (12)2.2. Planet Mass
We now turn to the uncertainty on the planet mass.We can approach this estimate two ways. First, we canstart from Equation 2, again making the same simplify-ing assumptions, and solve for M p in terms of observ-ables. We note that this method requires the intermedi-ate step of deriving an expression for the host star massin terms of direct observables, using the fact that M (cid:63) = 4 π ρ (cid:63) R (cid:63) (13)and using Equations 5 and 10 to write ρ (cid:63) in terms ofobservables. We find M (cid:63) = 4 PπG δ / ( T τ ) − / R (cid:63) . (14)Using this, we can then derive the planet mass in termsof observables as M p = 2 πG K (cid:63) PT τ R (cid:63) δ / . (15)A more straightforward approach is to use the factthat we have already derived the planet surface gravityin terms of observables. Starting from the definition ofsurface gravity, we can write M p = 1 G g p R p = 1 G g p R (cid:63) δ. (16)Using Equation 11, we arrive at the same expression asEquation 15.Using Equation 15, we derive the fractional uncer-tainty on the planet mass in terms of the fractionaluncertainty in the observables, again assuming no co-variances and the simplifying assumptions stated before( M P (cid:28) M (cid:63) , sin i ∼ e = 0, and k (cid:28) (cid:18) σ M p M p (cid:19) ≈ (cid:18) σ K (cid:63) K (cid:63) (cid:19) + (cid:18) σ P P (cid:19) + (cid:18) σ T T (cid:19) + (cid:18) σ τ τ (cid:19) + 14 (cid:18) σ δ δ (cid:19) + 4 (cid:18) σ R (cid:63) R (cid:63) (cid:19) . (17)2.3. Planet Density
We derive the planet density ρ p in terms of observ-ables. The planet density is given by ρ p = 3 M p πR p = 3 M p πR (cid:63) δ − / . (18)We have already derived the mass of the planet in termsof observables in Equation 15. Using this expression, wefind ρ p = 32 π G K (cid:63)
PT τ δR (cid:63) . (19)From this equation, we derive the fractional uncertaintyon the planet density in terms of the fractional uncer-tainty in the observables, again assuming no covariancesand the simplifying assumptions stated before. We find (cid:18) σ ρ p ρ p (cid:19) ≈ (cid:18) σ K (cid:63) K (cid:63) (cid:19) + (cid:18) σ P P (cid:19) + (cid:18) σ T T (cid:19) + (cid:18) σ τ τ (cid:19) + (cid:18) σ R (cid:63) R (cid:63) (cid:19) + (cid:18) σ δ δ (cid:19) . (20)2.4. Planet Radius
Finally, the planet radius uncertainty can be triviallyderived from the definition of the transit depth δ , as-suming no limb darkening: δ = (cid:18) R p R (cid:63) (cid:19) . (21)Then, R p = √ δR (cid:63) , (22)and the fractional uncertainty on the planet radius issimply (cid:18) σ R p R p (cid:19) ≈ (cid:18) σ R (cid:63) R (cid:63) (cid:19) + 14 (cid:18) σ δ δ (cid:19) . (23)We note that, by assuming that δ is a direct observ-able, we are fundamentally assuming no limb darkeningof the star. Of course, in reality the presence of limbdarkening means that the observed fractional depth ofthe transit is not equal to δ , and thus the uncertainty in δ is larger than one would naively estimate assuming nolimb darkening. However, assuming that the limb dark-ening is small (as it is for observations in the near-IR), orthat it can be estimated a priori based on the propertiesof the star, or that the photometry is sufficiently precisethat both the limb darkening and δ can be simultane-ously constrained, the naive estimate of the uncertaintyon δ assuming no limb darkening will not be significantlylarger than that in the presence of limb darkening. COMPARING THE ESTIMATEDUNCERTAINTIES ON THE PLANET MASS,DENSITY, AND SURFACE GRAVITYComparing the expressions for the fractional uncer-tainty on g p , M p , and ρ p (Equations 12, 17, and 20,respectively), we can make some broad observations onthe precision with which it is possible to measure thesethree planetary parameters.First, comparing the uncertainties on g p and M p , wenote that the only difference is that σ M p /M p requiresthe additional term 4( σ R (cid:63) /R (cid:63) ) . Stevens et al. (2018)estimates that it should be possible to infer the stel-lar radii of bright hosts ( G (cid:46)
12 mag) to an accuracyof order 1% using the final
Gaia data release paral-laxes, currently-available absolute broadband photome-try, and spectrophotometry from
Gaia and the Spectro-Photometer for the History of the Universe, Epoch ofReionization, and Ices Explorer (SPHEREx; Dor´e et al.2018). The exact level of accuracy will depend on thestellar spectral type and the final parallax precision. Itis likely that R (cid:63) may dominate the error budget rela-tive to the other terms, with the possible exception ofthe uncertainty in τ . We note that TESS is able tomeasure τ more precisely than either Kepler or K2 wereable to for systems with similar physical parameters andnoise properties, primarily because the TESS bandpassis redder than that of
Kepler , and thus the stellar limbdarkening is smaller and less degenerate with τ . Over-all, we generically expect the planetary surface gravityto be measured to smaller fractional precision than theplanet mass.We now turn to the uncertainty on planetary den-sity. When comparing the expressions for the uncer-tainty in M p to ρ p , we note that the uncertainty due tothe depth enters as (1 / σ δ /δ ) for M p , whereas it en-ters as simply σ δ /δ for ρ p . For large planets, the depthshould be measurable to a precision of ∼
1% or better,particularly in the TESS bandpass, similar to the bestexpected precision on R (cid:63) . Thus, we expect σ δ to becomparable to σ R (cid:63) , and thus both should contribute atthe ∼
1% level to σ ρ p . On the other hand, we expect σ R (cid:63) to dominate over the transit depth for M p . Thus,for any given system, we generally expect the followinghierarchy: σ M p /M p > σ ρ p /ρ p > σ g p /σ g p > σ R p /R p .Similarly, there is a hierarchy in the precision withwhich the observed parameters T , P , K (cid:63) , δ , τ , and R (cid:63) are measured. For the relatively small sample of plan-ets confirmed from TESS so far, we find that in general,the most precise observable parameter is the orbital pe-riod, followed by the stellar radius, the transit depth,the RV semi-amplitude, and the transit duration, suchthat: σ T /T > σ K /K > σ δ /δ > σ R (cid:63) /R (cid:63) > σ P /P . The ingress/egress time τ is not always reported in discov-ery papers, so we do not include it in this comparison.However, we generally expect that it will be measuredto a precision that is worse than T (Carter et al. 2008;Yee & Gaudi 2008).This hierarchy is in agreement with the findings ofCarter et al. (2008) and Yee & Gaudi (2008), who de-rived the following approximate relations for the uncer-tainties in the parameters of a photometric transit (as-suming no limb darkening): σ δ δ (cid:39) Q − (24) σ T T (cid:39) Q − (cid:114) τT (25) σ τ τ (cid:39) Q − (cid:114) Tτ , (26)where Q is the signal-to-noise ratio of the combinedtransits, defined as Q ≡ ( N tr Γ phot T ) / δ, (27)where N tr is the effective number of transits that wereobserved, and Γ phot is the photon collection rate . Wenote that Equation 27 implicitly assumes uncorrelatedphotometric uncertainties. Since, in general, σ τ > σ T ,we have that σ δ /δ < σ T /T < σ τ /τ .In the above equations, we have ignored the uncer-tainty in the transit midpoint t c as it does not enterinto the expressions for the uncertainties in R p , M p , ρ p ,or g p . We also assumed that the uncertainty in the base-line (out of transit) flux is negligible, which is generally agood assumption, particularly for space-based missionssuch as Kepler , K2, and TESS, where the majority ofthe measurements are taken outside of transit.We note that, particularly for small planets, when thelimb darkening is significant, and when only a handful oftransits have been observed, τ may be poorly measured(i.e., precisions of (cid:38) VALIDATION OF OUR ANALYTIC ESTIMATES Alternatively, assuming all measurements have a fractional pho-tometric uncertainty σ phot , and there are N measurements intransit, the total signal-to-noise ratio can be defined as Q ≡√ N ( δ/σ phot ). Model-Independent Uncertainty L i t e r a t u r e U n c e r t a i n t y Fiducial HJ KELT-26bHD 21749bK2-106bKepler-93b KELT-26b M p / M pg p / g p p / pR p / R p Figure 1.
Reported fractional uncertainties in M p (dia-monds), g p (squares), ρ p (circles), and R p (triangles) versusour model-independent analytic estimates for a variety oftransiting planets. These include a fiducial hot Jupiter (darkblue), Kepler-93b (pink), KELT-26b (sky blue), KELT-26bwithout external constraints (green, open symbols), K2-106b(brown), and HD 21749b (gold). For K2-106b and HD21749b, arrows point from the fractional uncertainties re-ported in the discovery papers to our ‘forensic’ estimatesof the uncertainties that could be achieved had the authorsadopted only empirical constraints. The open symbols aresystems for which no external constraints were used. Adashed, gray one-to-one line is plotted for reference. We test the analytic expressions derived in Section 2using four confirmed exoplanets and one fiducial hotJupiter simulated by Stevens et al. (2018). The con-firmed exoplanets are KELT-26b (Rodr´ıguez Mart´ınezet al. 2020), HD 21749b (Dragomir et al. 2019), K2-106b (Adams et al. 2017; Guenther et al. 2017), andKepler-93b (Ballard et al. 2014; Dressing et al. 2015).These systems have masses and radii between ∼ M ⊕ and ∼ R ⊕ .We estimated the expected analytic uncertainties onthe planet parameters by inserting the values of T , P , K (cid:63) , δ , τ , R (cid:63) , and their respective uncertainties fromthe discovery papers into Equations 12, 17, 20, and 23.Then, we compared the analytic uncertainties to thosereported in the discovery papers, which were derivedfrom MCMC analyses and not using the analytic approx-imations presented here. For parameters with asymmet-ric uncertainties, we took the average of the upper andlower bounds and adopted that as the uncertainty. We note that the discovery papers of all of the ex-amples we present here (except for KELT-26b) do notprovide the transit duration T (or the full width at halfmaximum of the transit), but rather T , which is de-fined as the difference between the fourth and first con-tact (see, e.g., Carter et al. 2008). Since we are generallyinterested in T , we calculate it from the given observ-ables using T = T − τ (28)and we estimate its uncertainty with the relationshipfrom Carter et al. (2008) and Yee & Gaudi (2008): σ T T = (cid:32)(cid:114) τT (cid:33) σ τ τ . (29)We use Equations 28 and 29 to calculate the transitduration values and uncertainties for all the systems forwhich only T is given.There are other ways of estimating the uncertainty in T , such as by propagating the uncertainty on T fromEquation 28, or by assuming that the uncertainty in T is approximately equal to that of T . However, theseapproaches overestimate the uncertainty on T as com-pared to Equation 29 because they do not account forthe covariance between the measurements of T and τ .Therefore, we adopt the uncertainty in T from Equa-tion 29 for all the exoplanets referenced here.Finally, for systems where the transit depth δ is notprovided, but rather the planet-star radius ratio R p /R (cid:63) ,we use linear propagation of error to estimate σ δ , finding (cid:18) σ δ δ (cid:19) = 4 (cid:18) σ R p /R (cid:63) R p /R (cid:63) (cid:19) . (30)And we adopt the fractional uncertainties on R (cid:63) as re-ported in the papers. In some cases these were derivedusing external constraints, such as stellar models, andthus may be underestimates or overestimates of the em-pirical uncertainty in R (cid:63) derived from the stellar SEDand parallax.The fractional uncertainties calculated using our ana-lytic approximations for the five planets in our sampleare listed in Table 1 and shown in Figure 1. As is clearfrom Figure 1, our estimates are broadly in agreementwith the fractional uncertainties quoted in the discoverypapers. However, we note that the fractional uncertain-ties we predict for certain quantities are systematicallylarger or smaller than those reported in the papers. Af-ter a careful ‘forensic’ analysis, we have tracked downthe reason for these discrepancies. In the two most dis-crepant cases, it is because the authors used externalconstraints on the properties of the host star (such as T eff , [Fe/H], and log g (cid:63) ) combined with stellar evolution-ary tracks, to place priors on the stellar parameters R (cid:63) and M (cid:63) . In one case (HD 21749), the resulting con-straint on ρ (cid:63) is tighter than results from empirical con-straint on the stellar density ρ (cid:63) from the light curve. Inthe other case, the adopted constraint on ρ (cid:63) is weaker than results from empirical constraint on the stellar den-sity ρ (cid:63) from the light curve, but nevertheless the external(weaker) constraints on M (cid:63) and R (cid:63) were adopted, ratherthan the (tighter) empirical constraints. In the remain-ing cases, either external constraints were not assumed,and as a result their parameter uncertainties agree wellwith our analytic estimates, or the external constraintswere negligible compared to the empirical constraintsand thus the empirical constraints dominated, againleading to agreement with our analytic estimates. Weultimately conclude that our analytic estimates are reli-able; however, we describe in detail our forensic analysisof the systems for pedagogical purposes. In the followingsubsections, we discuss each system in further detail.Before doing so, however, we stress that the advantageof empirical, model-independent approximations like theones presented here is that they do not assume that thephysical properties of any particular system is repre-sentative of the systems used to calibrate the empiricalmodels, or that the properties of the systems necessar-ily agree with the theoretical predictions. For example,theoretical models that make assumptions about the el-emental abundances of the host star may not apply tothe particular system under consideration. Therefore,although our empirical approach may lead to weakerconstraints on the parameters of the planets, we believeit leads to more robust constraints on these parameters.4.1. A Fiducial Hot Jupiter
Stevens et al. (2018) simulated the photometric timeseries and RV measurements for a typical hot Jupiter( M p = M J and R p = R J ) on a 3 day orbit transiting a G-type star using VARTOOLS (Hartman & Bakos 2016).They injected a Mandel-Agol transit model (Mandel &Agol 2002) into an (out-of-transit flux) normalized lightcurve, and simulated measurement offsets by drawingfrom a Gaussian distribution with 1 millimagnitude dis-persion. They furthermore assumed a cadence of 100seconds. They note that these noise properties are typi-cal of a single ground-based observation of a hot Jupiterfrom a small, ∼ − precision (which they assumed was equal to thescatter, or ‘jitter’). They then performed a joint pho-tometric and RV fit to the simulated data using EXO-FASTv2 (Eastman 2017; Eastman et al. 2019) to model and estimate the star and planet’s properties. They sim-ulated three different cases: a circular ( e = 0) orbit andequatorial transit, or an impact parameter of b = 0, aneccentric orbit with e = 0 . b = 0, and a circularorbit and b = 0 .
75. We consider the parameters anduncertainties for the case of a circular orbit and equato-rial transit, for which our equations are most applicable,and use the best-fit values and uncertainties from Table1 in Stevens et al. (2018).The fractional uncertainties in the planet mass, sur-face gravity, and planetary bulk density quoted inStevens et al. (2018) are all roughly 5%, whereas thefractional uncertainty in the planet radius is 1.7%.These uncertainties are in very good agreement with ouranalytic estimates, as Figure 1 and Table 1 show.4.2.
A Real Hot Jupiter
KELT-26b is an inflated ultra hot Jupiter on a 3.34day, polar orbit around a early Am star characterizedby Rodr´ıguez Mart´ınez et al. (2020). It has a mass andradius of M p = 1 . +0 . − . M J and R p = 1 . +0 . − . R J ,respectively. The photometry (which included TESSdata) and radial velocity data were jointly fit using EX-OFASTv2, and included an essentially empirical con-straint on the radius of the star from the spectral energydistribution and the Gaia
Data Release 2 (DR2) paral-lax, as well as theoretical constraints from the MESAIsochrones and Stellar Tracks (MIST) stellar evolutionmodels (Dotter 2016; Choi et al. 2016; Paxton et al.2011, 2013, 2015). Therefore, unlike the fiducial hotJupiter discussed above, this system was modeled usingboth external empirical constraints and external theo-retical constraints. The uncertainties in the planet pa-rameters reported by Rodr´ıguez Mart´ınez et al. (2020)are ∼
34% for the mass, ∼
33% for the surface gravity, ∼
33% for the bulk density, and 3.8% for the planet’sradius. These are very close to our estimates of the frac-tional uncertainties of these parameters, implying thatthe constraints from the MIST evolutionary tracks havelittle effect on the inferred parameters of the system.To test this hypothesis, we reanalyzed this systemwith EXOFASTv2 without using the external theoreti-cal constraints from the MIST isochrones, that is, onlyusing the spectral energy distribution of the star, its par-allax from
Gaia
DR2, and the light curves and radialvelocities. The uncertainties from this analysis are 35%for the planetary mass, surface gravity, and the density,and 3.3% for the radius. These are consistent with theuncertainties derived from the analysis using the MISTevolutionary tracks as constraints. The fractional uncer-tainties from the original paper and the analysis with-out constraints are shown in sky blue (with constraints)
Table 1.
Analytic and reported fractional uncertainties
Planet Analytic Literature Reference M p g p ρ p R p M p g p ρ p R p Fiducial HJ 0.06 0.04 0.06 0.01 0.05 0.05 0.05 0.01 Stevens et al. (2018)KELT-26b 0.34 0.34 0.34 0.03 0.33 0.37 0.35 0.03 Rodr´ıguez Mart´ınez et al. (2020)KELT-26b (cid:63) † † Notes.
The first four columns are the analytic uncertainties in M p (Eqn 17), g p (Eqn 12), ρ p (Eqn 20),and R p (Eqn 23), while the next four are the uncertainties in those parameters reported in the literature.Planets with a † were analyzed using external constraints from stellar evolutionary models. KELT-26b (cid:63) is KELT-26 analyzed without external constraints. The quantities in parentheses below HD 21749band K2-106b are the values we recover if we assume external constraints, as explained in Sections 4.4and 4.5. and green (without) in Figure 1. We conclude that theinferred parameters of the system derived using purelyempirical constraints are as precise (and likely more ac-curate) than those inferred using theoretical evolution-ary tracks. Therefore, at least for systems similar toKELT-26, we see no need to invoke theoretical priors.4.3. Kepler-93b
Kepler-93b is a terrestrial exoplanet on a 4.7 dayperiod discovered by Ballard et al. (2014). It has amass and radius of M p = 4 . ± . M ⊕ and R p =1 . ± . R ⊕ . With a radius uncertainty of only1.2%, it is one of the most precisely characterized exo-planets to date. Ballard et al. (2014) used asteroseismol-ogy to precisely constrain the stellar density, and thenused it as a prior in their MCMC analysis, leading to theremarkably precise planet radius. Their analysis did notuse external constraints from stellar evolutionary mod-els, however. Dressing et al. (2015) revisited Kepler-93and collected HARPS-N (Mayor et al. 2003) spectra,which they combined with archival Keck/HIRES spec-tra to improve upon the planet’s mass estimate. Theythus reduced the uncertainty in the mass of Kepler-93bfrom ∼
40% (Ballard et al. 2014) to ∼ T , τ , and δ ) from Ballardet al. (2014) and the semi-amplitude K (cid:63) from Dressinget al. (2015) to test our analytic estimates. We com-pared our results to the reported uncertainties in M p , g p and ρ p from Dressing et al. (2015), since they pro-vide slightly more precise properties. The uncertaintiesin the properties of Kepler-93b are all ∼ ρ (cid:63) does not significantly improvethe overall constraints on the system.4.4. HD 21749b
HD 21749b is a warm sub-Neptune on a 36 day orbittransiting a K4.5 dwarf discovered by Dragomir et al.(2019). The planet has a radius of 2 . +0 . − . R ⊕ deter-mined from TESS data, and a mass of 22 . +2 . − . M ⊕ con-strained from high-precision, radial velocity data fromthe HARPS spectrograph at the La Silla Observatoryin Chile. Dragomir et al. (2019) performed an SED fitcombined with a parallax from Gaia
DR2 to constrainthe host star’s radius to R (cid:63) = 0 . ± . R (cid:12) . Theythen used the Torres et al. (2010) relations to derive astellar mass of M (cid:63) = 0 . ± . M (cid:12) , although they donot specify what values of T eff , [Fe/H], and log g (cid:63) theyadopt as input into those equations, or from where theyderive these values. We assume they were determinedfrom high-resolution stellar spectra. Finally, they per-formed a joint fit of their data and constrained the plan-etary parameters with the EXOFASTv2 modeling suite,using their inferred values of M (cid:63) and R (cid:63) as priors.When comparing our analytic approximations of thefractional uncertainties in M p , g p , and ρ p to the uncer-tainties in the paper, we find that our estimates are sys-tematically larger than those of Dragomir et al. (2019)by 34% ( M p ), 60% ( g p ), and 80% ( ρ p ).Understanding the nature of such discrepancies re-quires a closer examination of the methods employedby Dragomir et al. (2019) as compared to ours. Thefundamental difference is that their uncertainties in theplanetary properties are dominated by their more pre-cise a priori uncertainties on M (cid:63) and R (cid:63) (and thus ρ (cid:63) ),rather than the empirically constrained value of ρ (cid:63) fromthe light curve and radial velocity measurements. Onthe other hand, we estimate the uncertainty on ρ (cid:63) di-rectly from observables (e.g., the light curve and the RVdata).Because their prior on ρ (cid:63) is more constraining thanthe value of ρ (cid:63) one would obtain from the light curve,and because the inferred planetary parameters criticallyhinge upon ρ (cid:63) , this ultimately leads to smaller uncer-tainties in the planetary parameters than we obtainpurely from the light curve observables.To show why this is true, we begin by comparing theirprior in ρ (cid:63) (the value they derive from their estimate of M (cid:63) and R (cid:63) , which we will denote ρ (cid:63), prior ) to the uncer-tainty in ρ (cid:63) from observables (denoted ρ (cid:63), obs ).Their prior on ρ (cid:63) can be trivially calculated from ρ (cid:63) =3 M (cid:63) / πR (cid:63) , and its uncertainty, through propagation oferror, is therefore simply (cid:18) σ ρ (cid:63), prior ρ (cid:63), prior (cid:19) ≈ (cid:18) σ M (cid:63) M (cid:63) (cid:19) + 9 (cid:18) σ R (cid:63) R (cid:63) (cid:19) . (31)Inserting the appropriate values from Dragomir et al.(2019) yields ρ (cid:63), prior = 3 . ± .
49 g cm − . This repre-sents a fractional uncertainty of σ ρ (cid:63), prior /ρ (cid:63), prior = 0 . ρ (cid:63), obs and its uncertainty in terms of transit observablesas ρ (cid:63), obs = (cid:18) PGπ (cid:19) δ / ( T τ ) − / . (32)Therefore, (cid:18) σ ρ (cid:63), obs ρ (cid:63), obs (cid:19) ≈ (cid:18) σ P P (cid:19) + 916 (cid:18) σ δ δ (cid:19) + 94 (cid:18) σ T T (cid:19) +94 (cid:18) σ τ τ (cid:19) . (33)Inserting the fractional uncertainties on P , R p , T , and τ from the discovery paper into Equation 33, we find σ ρ (cid:63), obs /ρ (cid:63), obs = 0 .
37. This is larger and less constrain-ing than the fractional uncertainty in the prior on ρ (cid:63), obs from Dragomir et al. (2019) by a factor of 2.3. Thus,we expect the prior on ρ (cid:63), obs to dominate over the con-straint from the light curve. However, despite being con-siderably less constraining than the prior, the empirical We note that the actual value reported in Section 3.1 of Dragomiret al. (2019) is ρ (cid:63) = 3 . ± .
23 g cm − , but after careful analysis,we believe that this value is probably a typographical error, as itdiffers from the value we derive and from the posterior value inTable 1 of the paper. constraint on ρ (cid:63), obs can still influence the posterior valueif the central value is significantly different than the priorvalue. Inserting the values of P , δ , T , and τ in Equa-tion 32, we find a central value of ρ (cid:63), obs = 5 . ± .
06 gcm − . This value is (5 . − . / .
06 = 1 . σ discrepantfrom the prior value. Thus, there is a weak tension be-tween the empirical and prior values of ρ (cid:63), obs that shouldbe explored.If we include the eccentricity in the expression for ρ (cid:63) , we find much closer agreement between ρ (cid:63), obs and ρ (cid:63), prior .From Winn (2010), we can express the scaled semi-major axis as a function of eccentricity as aR (cid:63) = Pπ δ / √ T τ (cid:18) √ − e e sin ω (cid:19) . (34)We can then combine this equation with Equation 7to find the ratio between the inferred ρ (cid:63) assuming acircular orbit ( ρ (cid:63), obs , c ) and that for an eccentric orbit( ρ (cid:63), obs , e ): ρ (cid:63), obs , e = ρ (cid:63), obs , c (cid:18) √ − e e sin ω (cid:19) . (35)Inserting the values from the paper ( e = 0 .
188 and ω = 98 ◦ ) yields ρ (cid:63), obs = 5 .
56 g cm − × .
568 = 3 .
16 gcm − , and assuming the same fractional uncertainty as ρ (cid:63), obs , c of 0 .
37 (which we discuss below), we get a valueof ρ (cid:63), obs , e = 3 . ± .
17 g cm − , which is ∼ . σ greaterthan the prior, and in much better agreement than ourestimate without including eccentricity. The reason whythe eccentricity significantly affects ρ (cid:63), obs in this case,despite the fact that it is relatively small ( e = 0 . ω (cid:39) ◦ , which implies that the transit occurs near pe-riastron, and thus the transit is shorter than if the planetwere on a circular orbit by a factor of T e T c (cid:39) √ − e e = 0 . . (36)Thus τ is shorter by the same factor. Since ρ (cid:63) ∝ ( T τ ) − / , by assuming e = 0, one overestimates thedensity by factor of (cid:18) √ − e e (cid:19) − = 0 . − , (37)approximately recovering the factor above.The eccentricity also affects the uncertainty in ρ (cid:63) inthe following way: ρ (cid:63), obs , e ρ (cid:63), obs , c ∝ (cid:18) √ − e e sin ω (cid:19) (cid:39) (1 − / e )(1 − e sin ω ) (cid:39) − e sin ω, (38)0where we have assumed that e (cid:28)
1. Propagating the un-certainty leads to a final value of ρ (cid:63), obs , e = 3 . ± .
04 gcm − , which is only ∼ σ greater than ρ (cid:63), prior . Thus,the eccentricity plays a significant role in the parame-ter uncertainties for this system. The uncertainty in theprior constraint on ρ (cid:63) is a factor of ∼ ρ (cid:63) from the data( ρ (cid:63), obs ). This also explains why their final value anduncertainty in ρ (cid:63) (3 . +0 . − . g cm − ) is so close to theirprior ( ρ (cid:63), prior = 3 . ± .
49 g cm − ).Assuming that the uncertainty on their priors for M (cid:63) and R (cid:63) indeed dominates the fractional uncertainty inthe resulting planet parameters, we can reproduce theiruncertainties in M p , g p , and ρ p using their prior to re-cover their reported fractional uncertainties as follows.For the surface gravity g p , we have g p = GM p R p , (39)and M p = (cid:18) P πG (cid:19) / M / (cid:63) (1 − e ) / K (cid:63) (40)while the planet radius can be expressed as R p = δ / R (cid:63) (41)Therefore, g p = (cid:18) P πG (cid:19) / M / (cid:63) K (cid:63) δ − R − (cid:63) (1 − e ) / . (42)Instead of simplifying g p in terms of observables (aswe have done in Equation 11), we express it in terms of M (cid:63) and R (cid:63) . Using propagation of error, the uncertaintyis (cid:18) σ g p g p (cid:19) ≈ (cid:18) σ P P (cid:19) + 49 (cid:18) σ M (cid:63) M (cid:63) (cid:19) + (cid:18) σ K (cid:63) K (cid:63) (cid:19) + (cid:18) σ δ δ (cid:19) + 4 (cid:18) σ R (cid:63) R (cid:63) (cid:19) . (43)where we have assumed (1 − e ) / ≈ σ g p /g p = 0 .
14, which is 12.5% different from the valuereported in Dragomir et al. (2019), and thus agrees muchbetter with their results than our initial estimate. For the planet’s mass, we start from Equation 40 andpropagate its uncertainty as (cid:18) σ M p M p (cid:19) ≈ (cid:18) σ P P (cid:19) + 49 (cid:18) σ M (cid:63) M (cid:63) (cid:19) + (cid:18) σ K (cid:63) K (cid:63) (cid:19) , (44)implying a fractional uncertainty in the mass of σ M p /M p = 0 . ∼ ρ p = 3 M p πR p = 3 M p πR (cid:63) δ − / . (45)Therefore, ρ p = 34 π (cid:18) P πG (cid:19) / M / (cid:63) K (cid:63) δ − / R − (cid:63) (1 − e ) / . (46)And the uncertainty in ρ p is thus (cid:18) σ ρ p ρ p (cid:19) ≈ (cid:18) σ P P (cid:19) + 49 (cid:18) σ M (cid:63) M (cid:63) (cid:19) + (cid:18) σ K (cid:63) K (cid:63) (cid:19) +94 (cid:18) σ δ δ (cid:19) + 9 (cid:18) σ R (cid:63) R (cid:63) (cid:19) , (47)which leads to σ ρ p /ρ p = 0 .
18, while the paper reports σ ρ p /ρ p = 0 .
21, which is a ∼
14% difference.In summary, we can roughly reproduce the uncertain-ties in Dragomir et al. (2019) to better than 15% if weassume that such uncertainties are dominated by the pri-ors on the stellar mass and radius. In Figure 1, we plotboth our initial fractional uncertainties and the recov-ered uncertainties as pairs connected by golden arrowsthat point in the direction of the ‘recovered’ uncertain-ties based on our forensic analysis.4.5.
K2-106b
K2-106b is the inner planet in a system of two tran-siting exoplanets discovered by Adams et al. (2017) andlater characterized by Guenther et al. (2017). It is onan ultra short, 0.57 day orbit around a G5V star. It hasa mass and radius of 8 . +0 . − . M ⊕ and 1 . ± . R ⊕ ,leading to a high bulk density of ρ p = 13 . +5 . − . g cm − .Guenther et al. (2017) used data from the K2 mis-sion combined with multiple radial velocity observationsfrom the High Dispersion Spectrograph (HDS; Noguchiet al. 2002), the Carnegie Planet Finder Spectrograph(PFS; Crane et al. 2006), and the FIber-Fed EchelleSpectrograph (FIES; Frandsen & Lindberg 1999; Teltinget al. 2014) to confirm and analyze this system. Theyperformed a multi-planet joint analysis of the data us-ing the code pyaneti (Barrag´an et al. 2017) and derived1the host star’s mass and radius using the PARSEC modelisochrones and the interface for Bayesian estimation ofstellar parameters from da Silva et al. (2006).As with HD 21749b, we found large discrepancies( ∼ ρ (cid:63), obs to the densityfrom the prior ρ (cid:63), prior .First, we have that the uncertainty in ρ (cid:63) based purelyon the prior fractional uncertainties on M (cid:63) and R (cid:63) isgiven by (cid:18) σ ρ (cid:63), prior ρ (cid:63), prior (cid:19) ≈ (cid:18) σ M (cid:63) M (cid:63) (cid:19) + 9 (cid:18) σ R (cid:63) R (cid:63) (cid:19) . (48)Inserting the values of σ M (cid:63) /M (cid:63) and σ R (cid:63) /R (cid:63) from thepaper, we derive a fractional uncertainty on the densityof the star from the prior of σ ρ (cid:63),prior /ρ (cid:63),prior = 0 .
31. Onthe other hand, using Equation 33, the fractional uncer-tainty in the stellar density from pure observables ρ (cid:63), obs is σ ρ (cid:63), obs /ρ (cid:63), obs = 0 .
15, a factor of ∼ ρ (cid:63) estimated fromthe prior.We can compute the uncertainty in the planetary massassuming the fractional uncertainty on M (cid:63) from theprior and the fractional uncertainty on the measuredsemi-amplitude K (cid:63) using Equation 44: (cid:18) σ M p M p (cid:19) ≈ (cid:18) σ M (cid:63) M (cid:63) (cid:19) + (cid:18) σ K (cid:63) K (cid:63) (cid:19) , (49)where we have assumed that σ P /P (cid:28)
1. We find σ M p /M p = 0 .
11, which is only 1% different from theuncertainty reported in Guenther et al. (2017).Further, we infer that Guenther et al. (2017) estimatedthe density of the planet by combining their estimate ofthe mass of the planet by adopting the prior value of M (cid:63) , along with the observed values of K (cid:63) and P , withthe radius of the planet derived by adopting the priorvalue of R (cid:63) and the observed value of transit depth (andthus R p /R (cid:63) ). Thus we infer that Guenther et al. (2017)estimated the uncertainty in the planet density via, (cid:18) σ ρ p ρ p (cid:19) ≈ (cid:18) σ M p M p (cid:19) + 9 (cid:18) σ R (cid:63) R (cid:63) (cid:19) + 94 (cid:18) σ δ δ (cid:19) , (50)again assuming that σ P /P (cid:28)
1. Substituting the val-ues quoted in Guenther et al. (2017) into the expression above, we find σ ρ p /ρ p = 0 .
32, whereas they quote a frac-tional uncertainty of σ ρ p /ρ p = 0 .
34, a ∼
6% difference.On the other hand, if we analytically estimate the frac-tional uncertainty on the density of K2-106b using pureobservables (Equation 20), but assume their reportedvalue and uncertainty on R (cid:63) , we find σ ρ p /ρ p ∼ . ∼ ∼ M (cid:63) ∝∼ R (cid:63) . This constraint on ρ (cid:63) causes the prior estimates of the stellar mass, radius,and their uncertainties to cancel out in the expressionfor the planet density: g p ∝ P / K (cid:63) M / (cid:63) δ − R − (cid:63) . (51)Assuming M (cid:63) ∝ ρ (cid:63) R (cid:63) , and ρ (cid:63) ∼ constant, we find g p ∝ P / K (cid:63) R (cid:63) δ − R − (cid:63) = P / K (cid:63) δ − . (52)The reason why Equation 52 and Equation 11 do notagree is because Equation 52 does not include the fullcontributions of the uncertainties in the light curve ob-servables P , δ , T , and τ .Figure 1 shows the fractional uncertainties in M p , g p ,and ρ p for K2-106b and brown arrows pointing from theoriginal values we estimate to the ‘recovered’ values.We reanalyzed K2-106 with EXOFASTv2 to derivestellar and planetary properties without using either theMIST stellar tracks, the Yonsei Yale stellar evolutionarymodels (YY; Yi et al. 2001), or the Torres et al. 2010relationships that are built into EXOFASTv2. We firstconstrained the stellar radius by fitting the star’s SEDto stellar atmosphere models to infer the extinction A V and bolometric flux, which when combined with its dis-tance from Gaia
EDR3 (Gaia Collaboration et al. 2020)provides a (nearly empirical) constraint on R (cid:63) . We finda fractional uncertainty in R (cid:63) of 2.4%, while Guentheret al. (2017) derive a fractional uncertainty of 10% us-ing the measured values of T eff , log g (cid:63) , and [Fe/H] fromtheir HARPS and HARPS-N spectra, combined withconstraints from the PARSEC model isochrones (da Silvaet al. 2006).We used our estimate of the stellar radius to recalcu-late the fractional uncertainties in M p , g p , ρ p , and R p us-ing our analytic expressions, and the constraints on theempirical parameters P, K (cid:63) , T, τ , and δ from Guentheret al. (2017). Our derived fractional uncertainty in theplanetary radius is 3.0%, whereas Guenther et al. (2017)2 Figure 2. σ and 2 σ mass-radius ellipses for K2-229b (Daiet al. 2019). The red ellipses assume M p and R p are uncor-related, random variables. The black ellipses are the resultof correlating M p and R p via the added constraint of sur-face gravity. Planets whose masses and radii lie along theblue solid solid line would have a constant core mass frac-tion of 0.565, whereas those that lie along the blue dottedline would have a constant core mass fraction of 0.29. Plan-ets forming with iron abundances as expected from K2-229Fe/Mg abundances will follow the blue dotted solid line. find 10% . Our derived fractional uncertainty on thedensity of the planet is a factor of 2.3 times smaller thanreported by (Guenther et al. 2017). This is because theradius of the star enters into their estimate of ρ p as R − (cid:63) (Eqn. 45), whereas our estimate of ρ p only depends lin-early on R (cid:63) (Eqn. 19). We estimate an uncertainty inthe planet mass of 15%, a bit larger than that reportedby Guenther et al. (2017), as it scales as the square of theradius of the star (Eqn. 15). Finally, the planetary sur-face gravity uncertainty that we estimate is 14%, almostthe same as that estimated by Guenther et al. (2017),as it does not depend directly on σ R (cid:63) /R (cid:63) .We conclude that a careful reanalysis of the K2-106system using purely empirical constraints may well re- We note that when σ R p /R (cid:63) (cid:28) σ R (cid:63) /R (cid:63) , the fractional uncer-tainty on the planetary radius is equal to fractional uncertaintyon the radius of the star (Eqn. 22). While this is approximatelythe case given fractional uncertainty in R (cid:63) estimated by Guen-ther et al. (2017), for our estimate the uncertainty in R p /R (cid:63) of1 .
9% contributes somewhat to the our estimated fractional un-certainty in R p . sult in a significantly more precise constraint on the den-sity of K2-106b, which is already a strong candidate foran exceptionally dense super-Earth. DISCUSSIONHere we discuss the importance of achieving high-precision measurements of planetary masses, surfacegravities, densities and radii, and their overall role ina planet’s habitability.The mass and radius of a planet are arguably its mostfundamental quantities. The mass is a measure of howmuch matter a planet accreted during its formation andis also tightly connected to its density and surface grav-ity, which we discuss below. The mass also determineswhether it can acquire and retain a substantial primor-dial atmosphere. Atmospheres are essential for a planetto maintain weather and thus life (see, e.g., Dohm &Maruyama 2013). In addition, the planetary core massand radius (themselves a function of the total mass)are related to the strength of a planet’s global magneticfield, although the strength of the field does depend onother factors, such as the rotation rate of the planet andother aspects of its interior. The presence of a substan-tial planetary magnetic field is vital in shielding againstharmful electromagnetic radiation from the host star.This is especially true for exoplanets orbiting M dwarfs,which are much more active than Sun-like stars. With-out a magnetic field to shield against magnetic phenom-ena such as flares and Coronal Mass Ejections (CMEs),planets around such stars may undergo mass loss andatmospheric erosion on relatively short timescales (see,e.g., Kielkopf et al. 2019). The initial mass may also de-termine whether planets will have moons, a factor whichhas been hypothesized to play a role in the habitabilityof a planet, as it does for the Earth. Some authorshave even proposed that Mars- and Earth-sized moonsaround giant planets may themselves be habitable (see,e.g., Heller et al. 2014; Hill et al. 2018).The mean density of a planet is also important as it isa first-order approximation of its composition. Basedon their density, we can classify planets as predomi-nantly rocky (typically Earth-sized and super Earths)or gaseous (Neptune-sized and hot Jupiters). A reliabledetermination of the density and structure of a planethelps to constrain its habitability.Next, we briefly discuss a few aspects of the impor-tance of knowledge of a planet’s surface gravity. First,the surface gravity dictates the escape velocity of theplanet, as well as the planet’s atmospheric scale height, h , defined as h = k b T eq µg p , (53)3where k b is the Boltzmann constant, T eq is the planetequilibrium temperature, and µ is the atmospheric meanmolecular weight. The surface gravity is connected tomass loss events and the ability of a terrestrial planetto retain a secondary atmosphere. Perhaps most impor-tantly, gravity may be a main driver of plate tectonics ona terrestrial planet. One of the most fundamental ques-tions about terrestrial or Earth-like planets is whetherthey can have and sustain active plate tectonics or ifthey are in the stagnant lid regime, like Mars or Venus(van Heck & Tackley 2011). On Earth, plate tectonicsare deeply linked to habitability for several crucial rea-sons. Plate tectonics regulate surface carbon abundanceby transporting some CO out of the atmosphere andinto the interior, which helps maintain a stable climateover long timescales (Sleep & Zahnle 2001; Unterbornet al. 2016). An excess of carbon dioxide can result in arunaway greenhouse effect, as in the case of Venus. Platetectonics also drive the formation of surface features likemountains and volcanoes, and play an important role insculpting the topography of a rocky planet. Weathercan then bring nutrients from mountains to the oceans,contributing to the biodiversity of the oceans. Some au-thors have argued that plate tectonics, dry land (such ascontinents), and continents maximize the opportunitiesfor intelligent life to evolve (Dohm & Maruyama 2013).However, the origin and mechanisms of plate tectonicsare poorly understood on Earth, and are even more sofor exoplanets. The refereed literature on this topic in-cludes inconsistent conclusions regarding the conditionsrequired for plate tectonics, and in particular how thelikelihood of plate tectonics depends on the mass of theplanet. For example, there is an ongoing debate aboutwhether plate tectonics are inevitable or unlikely on su-per Earths. Valencia & O’Connell (2009) used a convec-tion model and found that the probability and ability ofa planet to harbor plate tectonics increases with planetsize. On the other hand, O’Neill & Lenardic (2007) cameto the opposite conclusion, finding that plate tectonicsare less likely on larger planets, based on numerical sim-ulations. The resolution to this debate will have impor-tant consequences for our assessment of the likelihoodof life on other planets.5.1. Surface gravity as a proxy for the core massfraction
The surface gravity of a planet may also play an im-portant role in constraining other planetary parameters,like the core mass fraction. Here, we considered K2-229b( R p = 1 . +0 . − . R ⊕ and M p = 2 . +0 . − . M ⊕ , Daiet al. 2019), a potential super Mercury first discoveredby Santerne et al. (2018). This planet has well measured properties and the prospects for improving the precisionof the planet parameters are good given the brightnessof the host star.We calculate the core mass fraction of K2-229b as ex-pected from the planet’s mass and radius, CMF ρ , whichis the mass of the iron core divided by the total mass ofthe planet: CMF ρ = M Fe / M p . We compare this to theCMF as expected from the refractory elemental abun-dances of the host star, CMF star . This definition as-sumes that a rocky planet’s mass is dominated by Feand oxides of Si and Mg. Therefore, the stellar Fe/Mgand Si/Mg fractions are reflected in the planet’s coremass fraction. The mass and radius of K2-229b areconsistent with a rocky planet with a 0.57 core massfraction (CMF), while the relative abundances of Mg,Si, and Fe of the host star K2-229 (as reported in San-terne et al. 2018) predict a core mass fraction of 0.29(Schulze et al. 2020). Figure 2 shows mass-radius (M-R) ellipses for K2-229b when the mass and radius areassumed to be uncorrelated (red) and correlated via theadded constraint of surface gravity (black). While ap-parently enriched in iron, the enrichment is only signif-icant at the 2 σ level. The surface gravity, however, iscorrelated to the mass and gravity, reducing the uncer-tainty in CMF ρ (black): the M-R ellipse that includesthe surface gravity constraint reduces the uncertainty inthe differences of CMF measures. This arises becausethe planet’s density and surface gravity only differ byone factor of R p . Because the black contours closely fol-low the line of constant 0.57 CMF, we assert that surfacegravity and planet radius may be a better proxy for coremass fraction than mass and radius. Indeed, at the cur-rent uncertainties, we calculate that the additional con-straint of surface gravity reduces the uncertainty in theCMF ρ of K2-229b from 0.182 to 0.165. This is impor-tant given that we have demonstrated that the surfacegravity of a planet is likely to be one of the most pre-cisely measured properties of the planet. Furthermore,the fractional precision of the surface gravity measure-ment can be arbitrarily improved with additional data,at least to the point where systematic errors begin todominate. CONCLUSIONSOne of the leading motivations of this paper was theanswer to the question: “given photometric and RV ob-servations of a given exoplanet system, can we measurea planet’s surface gravity better than its mass?” At firstglance, the surface gravity depends on the mass itself,so it seems that the gravity should always be less con-strained. However, upon expressing the mass, gravityand density as a function of photometric and RV pa-4rameters, we see that the mass and density have an ex-tra dependence on the stellar radius, which makes thesurface gravity generically easier to constrain to a givenfractional precision than the mass or density. When ex-pressed in terms of pure observables, a hierarchy in theprecisions on the planet properties emerges, such thatthe surface gravity is better constrained than the den-sity, and the latter is in turn better constrained thanthe mass. The surface gravity is a crucial planetaryproperty, as it dictates the scale height of a planet’s at-mosphere. It is also a potential driver of plate tectonics,and as we show in this paper, can be an excellent proxyto constrain a planet’s core mass fraction to better facil-itate the discrimination of planet composition as differ-ent from its host star. With current missions like TESS,we expect to achieve high precisions in the photometricparameters. State-of-the-art RV measurements can nowreach precisions in the semi-amplitude of < τ and the host star radius R (cid:63) may be the limiting fac-tors in constraining the properties of low-mass terrestrialplanets. ACKNOWLEDGMENTSWe would like to thank Andrew Collier Cameronfor his suggestion that the surface gravity of a tran-siting planet may be more well constrained than itsmass, radius, or density. R.R.M. and B.S.G. were sup-ported by the Thomas Jefferson Chair for Space Ex-ploration endowment from the Ohio State University.D.J.S. acknowledges funding support from the EberlyResearch Fellowship from The Pennsylvania State Uni-versity Eberly College of Science. The Center for Exo-planets and Habitable Worlds is supported by the Penn-sylvania State University, the Eberly College of Science,and the Pennsylvania Space Grant Consortium. Theresults reported herein benefited from collaborationsand/or information exchange within NASA’s Nexus forExoplanet System Science (NExSS) research coordina-tion network sponsored by NASA’s Science Mission Di-rectorate. J.G.S. acknowledges the support of The OhioState School of Earth Sciences through the Friends ofOrton Hall research grant. W.R.P. was supported fromthe National Science Foundation under Grant No. EAR-1724693. Software:
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