Measuring Model-independent Masses and Radii of Single-lined Eclipsing Binaries: Analytic Precision Estimates

The light curve of a single-lined EB can be described by the total flux of the system, eclipse duration, ingress/egress duration, period, transit depth, and limb darkening. To make the problem analytically tractable, we will first assume no limb darkening and approximate the transit as a trapezoidal shape; we explore limb-darkening effects in Section 3.3. We also assume that no transmission of the primary's light through the companion's atmosphere occurs when the companion passes in front of the primary. Finally, we assume that each flux measurement is normalized to the median out-of-eclipse value, i.e., that f₀ = 1. In this case, the observables are the period P, the FWHM T of the transit, the ingress/egress duration τ, the transit depth δ, and the time of transit center T_c (Carter et al. 2008).

If we define the diameter crossing time ${\tau }_{0}\equiv {R}_{1}P/(2\pi a)$, then the aforementioned observables are related to the physical parameters ${\tau }_{0}$, b, and k by

$$\begin{eqnarray}&&T\equiv 2{\tau }_{0}\sqrt{1-{b}^{2}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\tau \equiv 2{\tau }_{0}\displaystyle \frac{k}{\sqrt{1-{b}^{2}}},\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&\delta \equiv {k}^{2},\end{eqnarray} \tag{ 3 }$$

and the inverse mappings are given by

$$\begin{eqnarray}&&b=\sqrt{1-k\displaystyle \frac{T}{\tau }},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\tau }_{0}=\sqrt{\displaystyle \frac{T\tau }{4k}},\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&k=\sqrt{\delta }.\end{eqnarray} \tag{ 6 }$$

The diameter crossing time as defined above assumes that, during transit, the transiting body's orbital velocity vector lies entirely within the plane of the sky—i.e., that it has no radial component. In reality, as the companion moves across the face of the primary star, it travels in an arc, such that the sky-projected velocity is less than the orbital velocity. For simplicity, we approximate that arc as a chord across the primary stellar disk such that the orbital and sky-projected velocities are equal.

From this light-curve model, it is impossible to determine the absolute scale (e.g., semimajor axis) of the system; rather, only dimensionless quantities, such as δ, $\tau /P$, T/P, and T_c/P, are measurable. However, one can infer

$$\begin{eqnarray}&&a/{R}_{1}=P/(2\pi {\tau }_{0})\end{eqnarray} \tag{ 7 }$$

for a circular orbit (Seager & Mallén-Ornelas 2003). With this scaling, one can match the observables with eclipsing systems of arbitrary size, whether or not the inferred parameters are physically plausible. This scaling, combined with Kepler's third law of planetary motion, enables one to measure a combination of the densities (Seager & Mallén-Ornelas 2003),

$$\begin{eqnarray}&&{\rho }_{1}+{k}^{3}{\rho }_{2}=\displaystyle \frac{3\pi }{{{GP}}^{2}}{\left(\displaystyle \frac{a}{{R}_{1}}\right)}^{3},\end{eqnarray} \tag{ 8 }$$

where G is the gravitational constant. If $k\ll 1$, then one can infer the density of the primary from the light curve. Hence, if one can measure a different combination of the primary's mass and radius, then it is possible to break these degeneracies and determine the absolute scale of the system. We discuss various ways of doing this below.

From a single-lined spectroscopic binary (SB1), the RV observables are the period P, primary star's RV semi-amplitude K₁, eccentricity e, argument of periastron ω, and time of periastron (or some other reference time in the orbit, such as the time of primary eclipse). As is well known, these observables do not allow for the unique determination of individual masses; rather, they enable one to infer the mass function of the system,

$$\begin{eqnarray}&&{ \mathcal M }=\displaystyle \frac{{M}_{2}\sin i}{{({M}_{1}+{M}_{2})}^{2/3}},\end{eqnarray} \tag{ 9 }$$

since

$$\begin{eqnarray}&&{K}_{1}={(2\pi G)}^{1/3}{P}^{-1/3}{ \mathcal M }.\end{eqnarray} \tag{ 10 }$$

If $q\ll 1$ and one assumes a value for M₁, then one can deduce ${M}_{2}\sin i$—the minimum mass of the secondary. The companion mass is then given by

$$\begin{eqnarray}&&{M}_{2}={(2\pi G)}^{-1/3}\displaystyle \frac{{K}_{1}{P}^{1/3}}{\sin i}{M}_{1}^{2/3}.\end{eqnarray} \tag{ 11 }$$

If the system is also eclipsing, then $\sin i\approx 1$, and one can infer the true mass of the secondary, given the assumption about M₁. Hence, even for an eclipsing SB1, one cannot determine the absolute scale of the system, and combinations of the primary mass and radius from other measurements are necessary.

It is worth noting that, with only eclipse and RV measurements, one can determine the surface gravity of the secondary using only directly observable quantities:

$$\begin{eqnarray}&&{g}_{2}=\displaystyle \frac{2\pi }{P}\displaystyle \frac{{K}_{1}}{{({R}_{2}/a)}^{2}\sin i}.\end{eqnarray} \tag{ 12 }$$

Since ${10}^{\mathrm{log}({g}_{1})}={{GM}}_{1}/{R}_{1}^{2}$ and ρ₁ = 3 M₁/(4π R₁)³, if we can estimate $\mathrm{log}({g}_{1})$ (via, e.g., high-resolution spectra or flicker), we immediately recover the primary's mass and radius:

$$\begin{eqnarray}&&{M}_{1}=\displaystyle \frac{9}{16{\pi }^{2}{G}^{3}}\displaystyle \frac{{g}_{1}^{3}}{{\rho }_{1}^{2}}\end{eqnarray} \tag{ 13 }$$

and

$$\begin{eqnarray}&&{R}_{1}=\displaystyle \frac{3}{4\pi G}\displaystyle \frac{{g}_{1}}{{\rho }_{1}}.\end{eqnarray} \tag{ 14 }$$

The companion's radius also follows:

$$\begin{eqnarray}&&{R}_{2}=\displaystyle \frac{3}{4\pi G}\displaystyle \frac{{g}_{1}}{{\rho }_{1}}{\delta }^{1/2}.\end{eqnarray} \tag{ 15 }$$

The RV measurements give us, via Equation (10),

$$\begin{eqnarray}&&{M}_{2}={\left(\displaystyle \frac{81}{512}\right)}^{1/3}{\pi }^{-5/3}{G}^{-7/3}\displaystyle \frac{{K}_{1}{P}^{1/3}}{\sin i}\displaystyle \frac{{g}_{1}^{2}}{{\rho }_{1}^{4/3}}.\end{eqnarray} \tag{ 16 }$$

From the above, we see that the inclination only explicitly affects the secondary's mass.

The primary density ρ₁ can be written in terms of light-curve observables by combining Equations (7) and (8):

$$\begin{eqnarray}&&{\rho }_{1}\approx \displaystyle \frac{3P}{8{\pi }^{2}G}{\tau }_{0}^{-3}=\displaystyle \frac{3P}{{\pi }^{2}G}{(T\tau )}^{-3/2}{\delta }^{3/4}.\end{eqnarray} \tag{ 17 }$$

We can then express the masses and radii in terms of observables:

$$\begin{eqnarray}&&{M}_{1}=\displaystyle \frac{{\pi }^{2}}{16G}\displaystyle \frac{{({g}_{1}T\tau )}^{3}}{{P}^{2}{\delta }^{3/2}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{R}_{1}=\displaystyle \frac{\pi }{4}\displaystyle \frac{{g}_{1}{(T\tau )}^{3/2}}{P{\delta }^{3/4}},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{M}_{2}=\displaystyle \frac{\pi }{{512}^{1/3}G}\displaystyle \frac{{K}_{1}{g}_{1}^{2}}{P\delta }{(T\tau )}^{2},\end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray}&&{R}_{2}=\displaystyle \frac{\pi }{4}\displaystyle \frac{{g}_{1}{(T\tau )}^{3/2}}{P{\delta }^{1/4}}.\end{eqnarray} \tag{ 21 }$$

From these relations, we see that the error on these four parameters will depend strongly on the primary's surface gravity, orbital period, FWHM eclipse duration, and ingress/egress duration. Section 3.1.1 details our analytic error estimates.

A measurement of the star's parallax angle in radians, ${\pi }_{p}$, gives us the distance to the star in au, ${d}_{p}\approx 1\,\mathrm{au}/{\pi }_{p}$. If the primary star's SED is sufficiently sampled to estimate its bolometric flux ${F}_{\mathrm{bol}}$, and if we have an estimate of the primary's effective temperature ${T}_{\mathrm{eff},1}$ (from, for example, fitting a template spectrum to a high-resolution spectrum or from the SED itself), we can calculate R₁,

$$\begin{eqnarray}&&{R}_{1}=\displaystyle \frac{1\ \mathrm{au}}{{\pi }_{p}}\displaystyle \frac{{F}_{\mathrm{bol}}^{1/2}}{{\sigma }_{\mathrm{SB}}^{1/2}{T}_{\mathrm{eff},1}^{2}},\end{eqnarray} \tag{ 22 }$$

where ${\sigma }_{\mathrm{SB}}$ is the Stefan–Boltzmann constant. It then follows from the eclipse photometry that

$$\begin{eqnarray}&&{M}_{1}=\displaystyle \frac{4\pi }{3}{\rho }_{1}{R}_{1}^{3}=\displaystyle \frac{4\pi }{3}{\rho }_{1}{\left(\displaystyle \frac{1\mathrm{au}}{{\pi }_{p}}\right)}^{3}\displaystyle \frac{{F}_{\mathrm{bol}}^{3/2}}{{\sigma }_{\mathrm{SB}}^{3/2}{T}_{\mathrm{eff},1}^{6}}\end{eqnarray} \tag{ 23 }$$

and

$$\begin{eqnarray}&&{R}_{2}={{kR}}_{1}=\displaystyle \frac{1\ \mathrm{au}}{{\pi }_{p}}{\delta }^{1/2}\displaystyle \frac{{F}_{\mathrm{bol}}^{1/2}}{{\sigma }_{\mathrm{SB}}^{3/2}{T}_{\mathrm{eff},1}^{2}}.\end{eqnarray} \tag{ 24 }$$

Finally, via RV observations and Equation (11),

$$\begin{eqnarray}&&{M}_{2}={\left(\displaystyle \frac{8\pi }{9G}\right)}^{1/3}\displaystyle \frac{{K}_{1}}{\sin i}{P}^{1/3}{\rho }_{1}^{2/3}{\left(\displaystyle \frac{1\mathrm{au}}{{\pi }_{p}}\right)}^{2}\displaystyle \frac{{F}_{\mathrm{bol}}}{{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff},1}^{4}}.\end{eqnarray} \tag{ 25 }$$

In terms of eclipse observables, Equations (23) and (25) can be expressed as

$$\begin{eqnarray}&&{M}_{1}=\displaystyle \frac{4}{\pi G}P{\left(\displaystyle \frac{{F}_{\mathrm{bol}}}{{\sigma }_{\mathrm{SB}}T\tau }\right)}^{3/2}{\delta }^{3/4}{\left(\displaystyle \frac{1\mathrm{au}}{{\pi }_{p}}\right)}^{3}{T}_{\mathrm{eff},1}^{-6}\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}&&{M}_{2}=\displaystyle \frac{2}{\pi G}\displaystyle \frac{{K}_{1}P}{\sin i}\displaystyle \frac{{F}_{\mathrm{bol}}}{{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff},1}^{4}}{\left(\displaystyle \frac{1\mathrm{au}}{{\pi }_{p}}\right)}^{2}\displaystyle \frac{{\delta }^{1/2}}{T\tau }.\end{eqnarray} \tag{ 27 }$$

This analysis assumes that the measured parallax is the true parallax, but this is not necessarily true: for example, Lutz–Kelker bias (Lutz & Kelker 1973) would make one more likely to underestimate the distance to a particular star. This would result in an underestimate of the stellar luminosity, as well as of the stellar radius for fixed ${T}_{{\rm{e}}{\rm{f}}{\rm{f}}.}$

For some stars, it is possible to measure their angular diameters ${{\rm{\Theta }}}_{D}\equiv 2{R}_{1}/{d}_{p}$ interferometrically. For those interferometric stars that have trigonometric parallaxes (and hence distances), these two measurements give the primary radius with no need for SED fitting. In this case, linear error propagation, ignoring covariances, gives ${({\sigma }_{{R}_{1}}/{R}_{1})}^{2}\approx {({\sigma }_{{{\rm{\Theta }}}_{D}}/{{\rm{\Theta }}}_{D})}^{2}+{({\sigma }_{{\pi }_{p}}/{\pi }_{p})}^{2}$, since ${{\rm{\Theta }}}_{D}\equiv 2\left[\tfrac{{R}_{1}}{\mathrm{au}}\right]{\pi }_{p}$. Sub-percent precision on the angular size is achievable, even when accounting for uncertainties in the limb-darkening coefficients (von Braun et al. 2014), but only for relatively bright (typically, $V\lesssim 10$) stars.

Asteroseismology provides several ways to determine the stellar density and surface gravity, which can be combined to find the stellar mass and radius. For stars with low signal-to-noise ratio photometry (resulting in a low-resolution frequency spectrum), one can measure the average large-frequency spacing $\langle {\rm{\Delta }}\nu \rangle$ and the frequency of maximum oscillation power ν_max for oscillations driven by stellar surface convection. For solar-like stars on the main sequence, empirical scaling relations from Kjeldsen & Bedding (1995) map these observables to physical parameters:

$$\begin{eqnarray}&&\displaystyle \frac{\rho }{{\rho }_{\odot }}={\left(\displaystyle \frac{\langle {\rm{\Delta }}\nu \rangle }{\langle {\rm{\Delta }}\nu {\rangle }_{\odot }}\right)}^{2},\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{g}_{1}}{{g}_{\odot }}=\left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right){\left(\displaystyle \frac{{T}_{\mathrm{eff},1}}{{T}_{\mathrm{eff},\odot }}\right)}^{1/2}.\end{eqnarray} \tag{ 29 }$$

We note that the above scaling relations, as written, may not be accurate for some stars: see the White et al. (2011) effective temperature correction for dwarfs and subgiants (which introduces a factor of order ${ \mathcal O }({T}_{\mathrm{eff},1})$ into Equation (28)), the metallicity-induced deviations from these relations found in evolved stars by Epstein et al. (2014), and the nonlinear effective temperature and metallicity correction to the large-frequency spacing reference value (${\rm{\Delta }}{\nu }_{\odot }$ in the above equations) from Guggenberger et al. (2016). The solar reference values are not inherent to the Sun but instead differ from one pipeline to the next, so we do not scale these relations by specific numerical solar values. Typical reference values are approximately ${\rm{\Delta }}{\nu }_{\odot }\sim 135$ and ${\nu }_{\max ,\odot }\sim 3100$ μHz, per Table 1 of Pinsonneault et al. (2018), who discussed this in more detail.

Nevertheless, as our main goal is simply to illustrate how one can use asteroseismic measurements to constrain the mass and radius of the binary components, we perform our analysis using the above relations for simplicity. For fixed ν_max and ${T}_{\mathrm{eff},1}$, Equation (29) becomes a mass–radius relation where $M\propto {R}^{2};$ a density from, e.g., the asteroseismic large-frequency spacing or the eclipse photometry can break the degeneracy. Analogously to Section 2.3.1, the primary mass and radius are given by

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{1}}{{M}_{\odot }}=\displaystyle \frac{9}{16{\pi }^{2}{G}^{3}}{\left(\displaystyle \frac{{g}_{1}}{{g}_{\odot }}\right)}^{3}{\left(\displaystyle \frac{{\rho }_{1}}{{\rho }_{\odot }}\right)}^{-2}\end{eqnarray} \tag{ 30 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{1}}{{R}_{\odot }}=\displaystyle \frac{3}{4\pi G}\displaystyle \frac{{g}_{1}}{{\rho }_{1}}\displaystyle \frac{{\rho }_{\odot }}{{g}_{\odot }}.\end{eqnarray} \tag{ 31 }$$

For solar-like dwarfs (roughly spectral types F through K), the asteroseismic scaling relations have been tested theoretically to ∼2% and ∼5% and empirically to ∼4% and ∼10% (Huber et al. 2013b). More recently, Coelho et al. (2015) found these scaling relations to have a random scatter in ν_max of 1.5% for effective temperatures between ∼5600–7000 K. To see how such a scatter affects the mass and radius predicted by these relations, we solve for these parameters in the equations above:

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{1}}{{M}_{\odot }}=\displaystyle \frac{9}{16{\pi }^{2}{G}^{3}}{\left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right)}^{3}{\left(\displaystyle \frac{\langle {\rm{\Delta }}\nu \rangle }{\langle {\rm{\Delta }}\nu {\rangle }_{\odot }}\right)}^{4}{\left(\displaystyle \frac{{T}_{\mathrm{eff},1}}{{T}_{\mathrm{eff},\odot }}\right)}^{3/2},\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{1}}{{R}_{\odot }}=\displaystyle \frac{3}{4\pi G}\left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right){\left(\displaystyle \frac{\langle {\rm{\Delta }}\nu \rangle }{\langle {\rm{\Delta }}\nu {\rangle }_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{T}_{\mathrm{eff},1}}{{T}_{\mathrm{eff},\odot }}\right)}^{1/2}.\end{eqnarray} \tag{ 33 }$$

We see that a 1.5% uncertainty in ν_max is amplified by a factor of three before entering the uncertainty in the mass, either directly from the expressions above or from combining the asteroseismic radius with the density calculated from the eclipse measurements. Errors on individual measurements of ν_max and $\langle {\rm{\Delta }}\nu \rangle$ can further increase the uncertainty on M₁ and R₁. However, if the target star is sufficiently bright and the observing cadence sufficiently rapid, then the signal-to-noise is increased, and one can match the observed frequency spectrum to those predicted by asteroseismic stellar models; for Kepler stars, fitting for individual frequencies (Deheuvels et al. 2012; Metcalfe et al. 2014) and "peak-bagging" (Lund et al. 2017; Silva Aguirre et al. 2017) reduce the uncertainty on the mass and radius by a factor of two relative to those derived from grid-based models of the global stellar oscillation properties and by a factor of three relative to those predicted from scaling relations. This procedure permits more direct inferences of the stellar mass, radius, age, and metallicity than do the scaling relations.

In addition to seismology, the primary star's surface gravity can be determined from tracers of granulation—either the rms variations in the eclipse light curve on timescales longer than 8 hr (dubbed "flicker"; Bastien et al. 2016) or from isolating the granulation and oscillation signal and calculating the timescale for its autocorrelation function to fall to zero (Kallinger et al. 2016). The latter can provide gravities to 4% but requires short-cadence observations (such as Kepler short-cadence data) for dwarfs, which oscillate on sub-hour timescales (Kallinger et al. 2016); the former can provide $\mathrm{log}({g}_{1})$ to 0.1 dex but requires only long-cadence data.

With the primary's physical parameters determined from asteroseismology or one of the granulation tracers, the companion's parameters are easily deduced: M₂ comes from Equation (11) and ${R}_{2}={R}_{1}\sqrt{\delta }$.

We note here that asteroseismology alone directly gives a bulk density for the primary star that is independent of the orbital eccentricity. Thus, under our assumption that k is small, any discrepancy between the asteroseismic and eclipse densities can shed light on the eccentricity of the system (Tingley et al. 2011; Huber et al. 2013b).

We now determine what constraints the empirical relations of Torres et al. (2010) place on the mass and radius of a star, given a measurement of the star's metallicity and effective temperature and assuming a near-solar surface gravity. For this section, we assume that our stars are typical of the stars used to calibrate the empirical relations we examine. Torres et al. (2010) analyzed 95 detached binaries with precise masses and radii and derived polynomial expressions for the masses and radii of stars (in solar units) as functions of their effective temperatures, surface gravities, and metallicities:

$$\begin{eqnarray}\mathrm{log}{M}_{1} & = & {a}_{1}+{a}_{2}X+{a}_{3}{X}^{2}+{a}_{4}{X}^{3}\\ & & +\,{a}_{5}\mathrm{log}{({g}_{1})}^{2}+{a}_{6}\mathrm{log}{({g}_{1})}^{3}+{a}_{7}[\mathrm{Fe}/{\rm{H}}],\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}\mathrm{log}{R}_{1} & = & {b}_{1}+{b}_{2}X+{b}_{3}{X}^{2}+{b}_{4}{X}^{3}\\ & & +\,{b}_{5}\mathrm{log}{({g}_{1})}^{2}+{b}_{6}\mathrm{log}{({g}_{1})}^{3}+{b}_{7}[\mathrm{Fe}/{\rm{H}}],\end{eqnarray} \tag{ 35 }$$

where $X\equiv \mathrm{log}{T}_{\mathrm{eff},1}-4.1$ and the coefficients a_i and b_i are listed in Torres et al. (2010). The fundamental assumption that Torres et al. (2010) made in deriving these relations is that the mass and metallicity of a star does not change substantially as the star evolves off of the main sequence, and thus $\mathrm{log}({g}_{1})$ and T_eff can be used as a proxy for the age of the star.

To reduce the degrees of freedom in these relations and obtain a quantity whose role is analogous to $\mathrm{log}({g}_{1})$ from the preceding subsections, we perturb Equations (34) and (35) around $\mathrm{log}({g}_{1})=\mathrm{log}{g}_{\odot }$ to obtain

$$\begin{eqnarray}&&\mathrm{log}{M}_{1}\approx A+B(\mathrm{log}({g}_{1})-\mathrm{log}({g}_{\odot }))\end{eqnarray} \tag{ 36 }$$

and

$$\begin{eqnarray}&&\mathrm{log}{R}_{1}\approx C+D(\mathrm{log}({g}_{1})-\mathrm{log}({g}_{\odot })),\end{eqnarray} \tag{ 37 }$$

where $A\equiv {a}_{1}+{a}_{2}X+{a}_{3}{X}^{2}+{a}_{4}{X}^{3}+{a}_{5}\mathrm{log}{({g}_{\odot })}^{2}$ $\,+\,{a}_{6}\mathrm{log}{({g}_{\odot })}^{3}\,+{a}_{7}[\mathrm{Fe}/{\rm{H}}]$, $B\equiv 2{a}_{5}\mathrm{log}({g}_{\odot })$ $+\,3{a}_{6}\mathrm{log}{({g}_{\odot })}^{2}$, $C\equiv {b}_{1}+{b}_{2}X\,+{b}_{3}{X}^{2}\,+{b}_{4}{X}^{3}\,$ $+\,{b}_{5}\mathrm{log}{({g}_{\odot })}^{2}\,$ $+\,{b}_{6}\mathrm{log}{({g}_{\odot })}^{3}\,+{b}_{7}[\mathrm{Fe}/{\rm{H}}]$, and $D\,\equiv 2{b}_{5}\mathrm{log}{({g}_{\odot })+3{b}_{6}\mathrm{log}({g}_{\odot })}^{2}$. By substitution, $\mathrm{log}{M}_{1}\approx (B/D)\mathrm{log}{R}_{1}+A-({BC}/D);$ therefore, we write

$$\begin{eqnarray}&&{M}_{1}\approx {{ZR}}_{1}^{\alpha },\end{eqnarray} \tag{ 38 }$$

where $Z\equiv {10}^{A-({BC}/D)}$ is a function of the primary star's metallicity and effective temperature, assuming its surface gravity is near the solar value, and α ≡ B/D. For solar values of ${T}_{\mathrm{eff},1}$ and [Fe/H], we find that mass is quite insensitive to radius ${M}_{1}\propto {R}_{1}^{0.2}$. This is not unexpected, as one of the assumptions used to derive the Torres et al. (2010) relations is that the mass of the star is constant. Figure 1 shows that this power law agrees with the Torres et al. (2010) mass–radius relation near the "Torres Sun" (${T}_{{\rm{e}}{\rm{f}}{\rm{f}},1}\,=\,5800$ K, [Fe/H] = 0.0, $\mathrm{log}({g}_{1})=4.4$), which is more massive and larger than the actual Sun.

Zoom In Zoom Out Reset image size

Not only does this power-law scaling imply that mass is rather insensitive to radius for near-solar values (for small perturbations about the solar surface gravity), but considering that a $\mathrm{log}({g}_{1})$ measurement carries the somewhat orthogonal constraint that ${M}_{1}\propto {R}_{1}^{2}$, an eclipse density in conjunction with both empirical relations and a $\mathrm{log}({g}_{1})$ measurement considerably tightens the uncertainties on the primary mass and radius.

From a measurement of Z and the eclipse density, we recover the primary parameters,

$$\begin{eqnarray}&&{M}_{1}={\left[{\left(\displaystyle \frac{3}{4\pi }\right)}^{\alpha /3}\displaystyle \frac{Z}{{\rho }_{1}^{\alpha /3}}\right]}^{\tfrac{3}{3-\alpha }}\end{eqnarray} \tag{ 39 }$$

and

$$\begin{eqnarray}&&{R}_{1}={\left(\displaystyle \frac{3}{4\pi }\right)}^{\tfrac{1}{3-\alpha }}{\left(\displaystyle \frac{Z}{{\rho }_{1}}\right)}^{\tfrac{1}{3-\alpha }},\end{eqnarray} \tag{ 40 }$$

where α ≡ B/D. From here, the secondary's parameters are easy to derive: ${R}_{2}={R}_{1}\sqrt{\delta }$, and M₂ is obtained by plugging M₁ directly into Equation (11).

These equations permit a more tractable analytic estimate of the contributions to the mass and radius uncertainties from the uncertainties on the primary stellar density ρ₁ and the scale factor Z of our approximate mass–radius relation in the neighborhood of solar surface gravity.

While the present analysis is concerned with the precision of the empirical relations, the accuracy of the relations is of tantamount importance. For example, the Torres et al. (2010) mass and radius relations have a tight, ∼1% scatter in radius but an ∼6% scatter in mass, suggesting that the polynomial fit to mass, ${T}_{\mathrm{eff},\mathrm{log}({{\rm{g}}}_{1})},$ and $[\mathrm{Fe}/{\rm{H}}]$ does not sufficiently capture the relationship between these parameters. Torres et al. (2010) argued that more $[\mathrm{Fe}/{\rm{H}}]$ determinations are needed to fit a more reliable, higher-order polynomial to these parameters: only 20 of their 95 systems have $[\mathrm{Fe}/{\rm{H}}]$ determinations, four of which are indirect (host cluster or galaxy metallicity) and none of which are M dwarfs. Additionally, the Mann et al. (2015) radius–T_eff relation produces radii for low-mass stars that are accurate to ∼10%, which is larger than their determined ∼5% offset between model radii and radii from interferometry.

Furthermore, as discussed in Mann et al. (2015), the interferometric measurements for single stars do not provide model-independent masses; in this case, deriving an M-dwarf mass–radius relation for this sample requires the inference of masses from stellar models or another empirical relation, such as the Delfosse et al. (2000) mass–luminosity relation (specifically, a relation between mass and absolute K_S-band magnitude). The Delfosse et al. (2000) calibration sample includes short-period low-mass EBs such as YY Geminorum and CM Draconis, so using this mass–luminosity relation requires the implicit assumption that the M dwarfs under consideration are typical of those in the calibration set, and thus that any physical processes responsible for the observation-versus-model radius and T_eff discrepancies in the short-period EBs are present at the same level in the M dwarfs to which this relation is applied. Boyajian et al. (2012) found evidence of a substantial offset in radius–T_eff space between single M dwarfs and M dwarfs in binaries, but more work is needed to determine if this offset is due to heterogeneous means of determining radius and T_eff for single and binary stars or if it indeed points to an intrinsic difference between these populations.

With sufficiently precise measurements given as inputs to empirical relations, it is possible to infer a formal uncertainty (precision) due to errors in the input parameters that is smaller than the scatter in the relations themselves, leading one to think that the inferred parameters are more reliable than they really are (when in reality, the quality of the inferred parameters is no better than the scatter). This is particularly pertinent for analyses of single-lined EBs, wherein one may consider circumventing the difficulty of inferring accurate M-dwarf parameters by applying empirical relations (e.g., the Torres relations) to the host star and then calculating the M-dwarf parameters from the inferred host-star parameters and the transit and RV data. Any inaccuracies in the host-star parameters thus propagate to the companion parameters and must be taken into account and quantified.

As such, precise, model-independent parameter measurements of low-mass EBs—and particularly long-period systems—are still necessary for improving the accuracy of stellar models and the fidelity of empirical relations for these stars.

In order to determine the constraints that isochrones place on mass and radius and estimate how uncertainties in the inputs to the iscohrones affect these parameters, we need to derive a mass–radius relation from isochrones. To this end, we use the MESA (Paxton et al. 2011) Isochrones and Stellar Tracks (MIST; Choi et al. 2016) via the Python isochrones package (Morton 2015). We set priors on ${T}_{\mathrm{eff},1}=5800\pm 150$ K and [Fe/H] = 0.0 ± 0.5, use the MCMC routine emcee (Foreman-Mackey et al. 2013) to sample these isochrones, and retain those samples that satisfy $\mathrm{log}(\mathrm{age}/[\mathrm{year}])=9.7\pm 0.4$, i.e., isochrones around the age of the Sun. Figure 1 shows the mass and radius posterior distributions as a heat map. We then perform a least-squares fit to the data, adopting a power-law function ${M}_{1}={{ZR}}_{1}^{\alpha }$, where M₁ and R₁ are in solar units, in analogy to the power-law function in Section 2.4.1. We find that $Z\approx 0.98$ and α ≈ 0.36, which is a slightly steeper dependence of mass on radius than we found with the Torres et al. (2010) relations in Section 2.4.1. This is not surprising: the isochrones account for the age and thus evolutionary state of the star, and so they also account for the change in the structure and thus radius of the star as it evolves (e.g., as the core helium fraction increases).

With this functional form, the primary mass and radius are given by Equations (39) and (40).

We first note that the spectroscopic surface gravity and RV semi-amplitude measurements are independent of each other, as well as of the photometric density and eclipse depth. Performing the first-order propagation of uncertainty analysis on Equations (18)–(21) and keeping only the lowest-order term in θ yields

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2}\approx 9{\left(\displaystyle \frac{{\sigma }_{{g}_{1}}}{{g}_{1}}\right)}^{2}+9{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{g}_{1}}}{{g}_{1}}\right)}^{2}+\displaystyle \frac{9}{4}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{g}_{1}}}{{g}_{1}}\right)}^{2}+\displaystyle \frac{9}{4}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 46 }$$

and

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2}\approx 4{\left(\displaystyle \frac{{\sigma }_{{g}_{1}}}{{g}_{1}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}+4{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{\sin i}}{\sin i}\right)}^{2}.\end{eqnarray} \tag{ 47 }$$

We note that the uncertainty on the ingress/egress time dominates any effect that the uncertainty on the depth would contribute to the precision on R₂, so we forthwith ignore the depth uncertainty. From here on, we assess the impact of inclination uncertainties on M₂. It can be shown that ${\sigma }_{\sin i}/\sin i\approx {\cot }^{2}i({\sigma }_{\cos i}/\cos i)$. Since $\cos i=2\pi {\tau }_{0}b/P$ for a circular orbit,

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\sigma }_{\cos i}}{\cos i}\right)\approx \displaystyle \frac{1}{2}\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right),\end{eqnarray} \tag{ 48 }$$

so

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\sigma }_{\sin i}}{\sin i}\right)\approx \displaystyle \frac{{\cot }^{2}i}{2}\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right).\end{eqnarray} \tag{ 49 }$$

Thus, the contribution from the uncertainty on $\sin i$ becomes comparable to the contribution from the uncertainty on ρ₁ when ${\cot }^{2}i\approx 4$, which occurs when $| i| \lesssim 26\buildrel{\circ}\over{.} 57$. In this case, for a transiting system with $i\approx 90$°, the uncertainty on $\sin i$ is negligible and thus contributes negligibly to the uncertainties on the transit observables. However, we note that, for a high-inclination orbit causing near-grazing or grazing eclipses, the uncertainty on the inclination can significantly affect both the precision of the transit observables and the accuracy of the piecewise-linear light-curve model assumed for this analysis.

Figure 2 shows the allowable uncertainties needed to achieve 3% uncertainty on the masses and radii. The radii are less sensitive to the uncertainty on surface gravity and ingress/egress time, and they are (as expected) independent of the uncertainty on the RV semi-amplitude. The companion mass is less sensitive to uncertainties in these observables, but unlike M₁, M₂ obviously also depends on the primary star's RV semi-amplitude K₁. Also note that the R₁ and R₂ lines overlap completely: R₂ ∝ R₁, since R₂ is determined from R₁ and the transit depth, so any parameter that contributes to the uncertainty on R₁ contributes to the uncertainty on R₂ by the same amount.

Zoom In Zoom Out Reset image size

If we impose a uniform fractional uncertainty on all four quantities (e.g., ≤3% precision), then we see immediately that the uncertainty in M₁ places the tightest constraints on the requisite precision of the observables; these constraints require that g₁ be known to better than 1% or, equivalently, that $\mathrm{log}({g}_{1})$ be known to better than 0.0043 in dex, since ${\sigma }_{\mathrm{log}g}\approx {\sigma }_{g}/(g\mathrm{ln}(10))$. Spectroscopic $\mathrm{log}g$ uncertainties are typically an order of magnitude larger, such as the ∼0.03 dex uncertainties achieved for F, G, and K stars by Brewer et al. (2016).

First, we note that the parallax measurement is independent of all other parameters, and we assume that the bolometric flux and effective temperature are as well.⁴ Hence, the only covariances are again between $T,\tau ,$ and δ, and these are small. As before, we assume that the parallax angle in radians is sufficiently small such that $\sin {\pi }_{p}\approx {\pi }_{p}$. Proceeding as before, we take Equations (22), (24), (26), and (27) and obtain the following expressions for the fractional uncertainties:

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2} & \approx & 36{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+9{\left(\displaystyle \frac{{\sigma }_{{\pi }_{p}}}{{\pi }_{p}}\right)}^{2}+\displaystyle \frac{9}{4}{\left(\displaystyle \frac{{\sigma }_{{F}_{\mathrm{bol}}}}{{F}_{\mathrm{bol}}}\right)}^{2}\\ & & +\,\displaystyle \frac{9}{4}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}\approx 4{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{\pi }_{p}}}{{\pi }_{p}}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{{F}_{\mathrm{bol}}}}{{F}_{\mathrm{bol}}}\right)}^{2},\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2} & \approx & 16{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+4{\left(\displaystyle \frac{{\sigma }_{{\pi }_{p}}}{{\pi }_{p}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{F}_{\mathrm{bol}}}}{{F}_{\mathrm{bol}}}\right)}^{2}\\ & & +\,{\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 52 }$$

and

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2} & \approx & 4{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{\pi }_{p}}}{{\pi }_{p}}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{{F}_{\mathrm{bol}}}}{{F}_{\mathrm{bol}}}\right)}^{2}\\ & & +\,\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{\delta }}{\delta }\right)}^{2}.\end{eqnarray} \tag{ 53 }$$

Figure 3 illustrates how the precision of the masses and radii scales with the uncertainty on the observables. In this case, the biggest contributors to the error budgets are not from the photometry or RV; rather, uncertainties on the primary's effective temperature dominate, and uncertainties on the parallax angle also contribute substantially. Indeed, one would need to know ${T}_{\mathrm{eff},1}$ to better than half a percent to hope to know the primary's mass to within 3%; even though the companion mass is proportional to ${M}_{1}^{2/3}$ (so uncertainties on ${T}_{\mathrm{eff},1}$ contribute less to M₂ than to M₁ by a factor of 2/3), ${T}_{\mathrm{eff},1}$ still needs to be known to better than 0.75% for 3% precision on M₂.

Zoom In Zoom Out Reset image size

Such precise effective temperatures are elusive, although some groups have demonstrated effective temperatures to relative precisions of 25 K (Brewer et al. 2016). We stress that, at such high precision, the accuracy of the T_eff measurement and/or the underlying T_eff scale is critical. As an example, Casagrande et al. (2014) discussed zero-point offsets at the tens of K level between different T_eff scales and the difficulty in determining a "true" T_eff scale. For a well-calibrated, internally consistent T_eff scale whose values differ from another T_eff scale by some fixed amount (a zero-point offset), comparative analyses of different stars using the same T_eff scale may be more informative than one that uses different T_eff determinations.

Using Equation (49), the contribution to the uncertainty on M₂ from the uncertainty on $\sin i$ becomes comparable to the contribution from the uncertainty on ρ₁ when ${\cot }^{2}i\approx 2$ or $| i| \lesssim 35\buildrel{\circ}\over{.} 26$. Again, for a transiting system with $i\approx 90$°, the contribution from $\sin i$ is small, and so we ignore it. As we cautioned in the previous subsection, we warn again that the uncertainty on the inclination can significantly affect both the precision of the observable transit quantities (e.g., ingress/egress duration) and the accuracy of the light-curve model we adopt in this paper.

With the primary's parameters M₁ and R₁ determined from asteroseismology, flicker, and/or the granulation timescale, we can easily express the fractional uncertainties in M₂ and R₂ as

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}+\displaystyle \frac{4}{9}{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2}\end{eqnarray} \tag{ 54 }$$

and

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{\delta }}{\delta }\right)}^{2}.\end{eqnarray} \tag{ 55 }$$

In this case, asteroseismology obviates the need for precision ingress and egress measurements, the dominant sources of uncertainty in eclipse modeling. Alongside the precision on the RV parameters, only the eclipse signal-to-noise ratio Q matters, since ${\sigma }_{\delta }/\delta \sim 1/Q$.

For frequency spectrum matching, we cannot determine analytically which errors propagate into the uncertainties on the primary's mass and radius; we can, however, do so for the scaling relations. Assuming negligible covariance between the large-frequency spacings and the frequency of maximum power, first-order propagation of error on Equations (32) and (33) reveals that

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2}\approx 16{\left(\displaystyle \frac{{\sigma }_{\langle {\rm{\Delta }}\nu \rangle }}{\langle {\rm{\Delta }}\nu \rangle }\right)}^{2}+9{\left(\displaystyle \frac{{\sigma }_{{\nu }_{\max }}}{{\nu }_{\max }}\right)}^{2}+\displaystyle \frac{9}{4}{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}\end{eqnarray} \tag{ 56 }$$

and

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}\approx 4{\left(\displaystyle \frac{{\sigma }_{\langle {\rm{\Delta }}\nu \rangle }}{\langle {\rm{\Delta }}\nu \rangle }\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{\nu }_{\max }}}{{\nu }_{\max }}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2},\end{eqnarray} \tag{ 57 }$$

which translate to, through Equations (54) and (55),

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2}\approx \displaystyle \frac{64}{9}{\left(\displaystyle \frac{{\sigma }_{\langle {\rm{\Delta }}\nu \rangle }}{\langle {\rm{\Delta }}\nu \rangle }\right)}^{2}+4{\left(\displaystyle \frac{{\sigma }_{{\nu }_{\max }}}{{\nu }_{\max }}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}\end{eqnarray} \tag{ 58 }$$

and

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2}\approx 4{\left(\displaystyle \frac{{\sigma }_{\langle {\rm{\Delta }}\nu \rangle }}{\langle {\rm{\Delta }}\nu \rangle }\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{\nu }_{\max }}}{{\nu }_{\max }}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{\delta }}{\delta }\right)}^{2}.\end{eqnarray} \tag{ 59 }$$

The above equations illustrate clearly that the dominant sources of error on all four physical parameters are the asteroseismic quantities—particularly the large-frequency spacing, which contributes a factor of ∼2–4 more uncertainty than does ν_max. This is also shown in Figure 4. It is typically easier to measure $\langle {\rm{\Delta }}\nu \rangle$ than ν_max in noisy data (Huber et al. 2013b), so the uncertainty in the former should generally be smaller than the uncertainty in the latter. With Kepler data, it is possible to measure ${\rm{\Delta }}\nu$ and ν_max to better than 1%, as in the case of the Kepler-56 planetary system (Huber et al. 2013a).

Zoom In Zoom Out Reset image size

To make the analytic estimates more tractable, we assert that the exponent α in Equation (38) is more or less constant for near-solar surface gravity over a small range of effective temperature and metallicity centered on solar values. Figure 5 shows that, for an ∼2.6% fractional deviation in ${T}_{\mathrm{eff},1}$ around 5800 K and a 0.05 dex deviation in [Fe/H] about 0, the slope of the mass–radius relation changes nearly imperceptibly. Hence, for sufficiently precise measurements of effective temperature and metallicity, we can ignore the contribution of the uncertainty in α to the uncertainties on the masses and radii, which will almost certainly be significantly more affected by uncertainties in Z, ρ₁, δ, and K₁.

Zoom In Zoom Out Reset image size

We first note that Z is independent of the transit and RV parameters and propagate errors through Equations (39) and (40):

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2}\approx {\left(\displaystyle \frac{3}{3-\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2}+{\left(\displaystyle \frac{3\alpha }{6-2\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 60 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}\approx {\left(\displaystyle \frac{3}{6-2\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}+{\left(\displaystyle \frac{1}{3-\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2},\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}+{\left(\displaystyle \frac{2}{3-\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2}+{\left(\displaystyle \frac{\alpha }{3-\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 62 }$$

and

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{3}{6-2\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}+{\left(\displaystyle \frac{1}{3-\alpha }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2},\end{eqnarray} \tag{ 63 }$$

where we again assume that the uncertainties on the ingress/egress duration dominate the uncertainties on the primary eclipse depth. For α = 0.2, these equations become

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2}\approx 1.15{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2}+0.01{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 64 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2}\approx 0.29{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}+0.13{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2},\end{eqnarray} \tag{ 65 }$$

and

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}+0.51{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2}+0.01{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}.\end{eqnarray} \tag{ 66 }$$

Thus, while the ingress/egress duration dominates the radius error budget, the uncertainty in Z dominates the primary mass uncertainty, and the RV semi-amplitude error dominates the secondary mass uncertainty. Figure 6 illustrates the contributions of ${\sigma }_{Z}/Z$ and ${\sigma }_{\tau }/\tau$ to the error budgets of the four physical parameters of interest.

Zoom In Zoom Out Reset image size

To see if the requisite precision on Z is achievable, we write the uncertainty in Z as a function of ${T}_{\mathrm{eff},1}$ and [Fe/H]:

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2} & \approx & {\mathrm{ln}}^{2}(10)\left[\left(3\left({a}_{4}-\displaystyle \frac{B}{D}{b}_{4}\right){\mathrm{log}}^{2}{T}_{\mathrm{eff},1}\right.\right.\\ & & +\,\left[2\left({a}_{3}-\displaystyle \frac{B}{D}{b}_{3}\right)-24.6\left({a}_{4}-\displaystyle \frac{B}{D}{b}_{4}\right)\right]\mathrm{log}{T}_{\mathrm{eff},1}\\ & & +\,({a}_{2}-{b}_{2})-8.2\left({a}_{3}-\displaystyle \frac{B}{D}{b}_{3}\right)\\ & & {\left.+50.43\left({a}_{4}-\displaystyle \frac{B}{D}{b}_{4}\right)\right)}^{2}\\ & & \times \,{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}\left.+{\left({a}_{7}-\displaystyle \frac{B}{D}{b}_{7}\right)}^{2}{\sigma }_{[\mathrm{Fe}/{\rm{H}}]}^{2}\right].\end{eqnarray} \tag{ 67 }$$

Since $B\approx -0.11$ and $D\approx -0.55$ for $\mathrm{log}{g}_{\odot }=4.4$, plugging in these values and the values of the a_i and b_i coefficients gives

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2} & \approx & \left[97.5{\left(\displaystyle \frac{\mathrm{log}{T}_{\mathrm{eff},1}}{\mathrm{log}5800{\rm{K}}}\right)}^{2}\right.\\ & & {\left.-206\left(\displaystyle \frac{\mathrm{log}{T}_{\mathrm{eff},1}}{\mathrm{log}5800{\rm{K}}}\right)+111\right]}^{2}\\ & & \times \,{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+0.01{\sigma }_{[\mathrm{Fe}/{\rm{H}}]}^{2}.\end{eqnarray} \tag{ 68 }$$

For ${T}_{\mathrm{eff},1}=5800$ K, ${\sigma }_{{T}_{\mathrm{eff},1}}/{T}_{\mathrm{eff},1}$ contributes to the fractional uncertainty on Z by a factor of 2.5. Meanwhile, a 0.1 dex spread in metallicity yields ${\sigma }_{Z}/Z\approx 0.01 \%$. Figure 7 illustrates these contributions for ${T}_{\mathrm{eff},1}=5800$ K. Hence, Z can be calculated rather precisely for Sun-like stars.

Zoom In Zoom Out Reset image size

From our least-squares fit in Section 2.4.2, we find that a 150 K spread in effective temperature and 0.5 dex spread in metallicity produces a 2.8% uncertainty in Z and a 0.5% uncertainty in α; the latter implies that, as with the empirical relations in Section 3.1.4, relatively precise effective temperature and metallicity measurements allow us to ignore the contribution of the uncertainty in the power-law index (α) to the uncertainty on the masses and radii. As a result, Equations (60)–(63) apply for the fractional uncertainties on the masses and radii. With α ≈ 0.36, these equations become

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2}\approx 1.29{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2}+0.04{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2},\end{eqnarray} \tag{ 69 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2}\approx 0.32{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}+0.14{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2},\end{eqnarray} \tag{ 70 }$$

and

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}+0.57{\left(\displaystyle \frac{{\sigma }_{Z}}{Z}\right)}^{2}+0.02{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}.\end{eqnarray} \tag{ 71 }$$

Figure 8 shows how uncertainties on Z and the ingress/egress duration contribute to the uncertainties on the masses and radii. As with the empirical relations, Z has a more significant impact on the mass uncertainties than the ingress/egress duration does, but the latter dominates the radius uncertainties.

Zoom In Zoom Out Reset image size

We have assumed that the companion contributes negligibly to the total flux of the system, so that we can attribute the total flux solely to the primary star. In reality, the companion does contribute to the total flux, which has two effects on our analysis.

First, neglecting the companion's light leads to an underestimate of the companion-to-primary radius ratio. The transit depth is the difference between the out-of-transit flux ${F}_{\mathrm{out}}$ and in-transit flux ${F}_{\mathrm{in}}$ relative to the out-of-transit flux; for our simplified case, this is

$$\begin{eqnarray}&&\delta =({F}_{\mathrm{out}}-{F}_{\mathrm{in}})/{F}_{\mathrm{out}}.\end{eqnarray} \tag{ 72 }$$

Here ${F}_{\mathrm{out}}={F}_{1}+{F}_{2}$ is the total flux from both components, and ${F}_{\mathrm{in}}={F}_{1}+{F}_{2}-{k}_{\mathrm{true}}^{2}{F}_{1}$ is the total flux minus the amount of primary stellar flux that is obscured by the companion; ${k}_{\mathrm{true}}$ is the true radius ratio. Writing the flux ratio as $f={F}_{2}/{F}_{1}$ and solving for ${k}_{\mathrm{true}}$ in Equation (72) gives

$$\begin{eqnarray}&&{k}_{\mathrm{true}}=\sqrt{\delta (1+f)},\end{eqnarray} \tag{ 73 }$$

Performing our linear error propagation analysis gives this expression for the fractional uncertainty on R₂:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{R}_{2}}}{{R}_{2}}\right)}^{2}={\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{\delta }}{\delta }\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{(1+f)}}{(1+f)}\right)}^{2},\end{eqnarray} \tag{ 74 }$$

so the uncertainty on R₁ remains the dominant contributor to the uncertainty on R₂, while the flux ratio and depth uncertainties contribute a smaller amount.

Second, neglecting the companion's light leads to an overestimate of the primary star's bolometric flux in an SED analysis. At fixed distance and ${T}_{\mathrm{eff},}$ this results in an overestimated primary stellar radius, since ${R}_{1}\propto \sqrt{{F}_{\mathrm{bol}}};$ this then increases the other primary and companion parameters. Let ${f}_{\mathrm{bol}}$ be the companion-to-primary bolometric flux ratio. The primary's bolometric flux would be overestimated by a factor of $1+{f}_{\mathrm{bol}}$, and Equation (51) would become

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{{R}_{1}}}{{R}_{1}}\right)}^{2} & \approx & 4{\left(\displaystyle \frac{{\sigma }_{{T}_{\mathrm{eff},1}}}{{T}_{\mathrm{eff},1}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{{\pi }_{p}}}{{\pi }_{p}}\right)}^{2}\\ & & +\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{{F}_{\mathrm{bol},1}}}{{F}_{\mathrm{bol},1}}\right)}^{2}+\displaystyle \frac{1}{4}{\left(\displaystyle \frac{{\sigma }_{{f}_{\mathrm{bol}}}}{1+{f}_{\mathrm{bol}}}\right)}^{2}.\end{eqnarray} \tag{ 75 }$$

As before, the precision on ${T}_{\mathrm{eff},1}$ dominates the R₁ error budget, and the bolometric flux terms are the weakest contributors.

A physically motivated accounting of this increase depends on the wavelength range and stellar parameters in a nonanalytic way, so it is beyond the scope of this paper.

We have also assumed that the companion is small so that q ≪ 1 and k ≪ 1. We will now consider the effects of a comparably sized companion. We had assumed q ≪ 1 to obtain Equation (11) from Equations (9) and (10). If we account for a nonnegligible mass ratio, then Equation (11) becomes

$$\begin{eqnarray}&&{M}_{2}={(2\pi G)}^{-1/3}\displaystyle \frac{{K}_{1}{P}^{1/3}}{\sin i}{M}_{1}^{2/3}{q}^{2/3},\end{eqnarray} \tag{ 76 }$$

which shows that the small-q assumption leads to an underestimate of the companion mass. Linear error propagation then yields

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2}\approx {\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}+\displaystyle \frac{4}{9}{\left(\displaystyle \frac{{\sigma }_{{M}_{1}}}{{M}_{1}}\right)}^{2}+\displaystyle \frac{4}{9}{\left(\displaystyle \frac{{\sigma }_{q}}{1+q}\right)}^{2}.\end{eqnarray} \tag{ 77 }$$

The uncertainty on the RV semi-amplitude dominates the M₂ error budget, with the uncertainty on the mass ratio contributing relatively weakly.

There are two other effects. As ${R}_{2}\to {R}_{1}$, the flux decrease as the companion moves across the limbs of the primary star is no longer linear in time; this is due to the occulted region of the primary star changing nonlinearly during ingress/egress. Thus, the piecewise-linear transit model adopted for our analytic estimates would no longer be valid, as the transit shape would no longer be trapezoidal, even in the absence of limb darkening. In contrast, the Mandel–Agol transit model does account for this effect, albeit while treating both the primary and companion as circular disks.

Moreover, we have used the small-companion assumption to set the companion's contribution to the left-hand side of Equation (8) to zero; however, if the small-companion assumption is false, then the primary stellar density we infer from this equation will be an overestimate.

Nonzero eccentricity complicates the equations for the density and RV uncertainties, among others. For an eccentricity e and argument of periastron ω, the stellar density uncertainty given in Equation (8) becomes (under the assumption that the companion is negligibly small)

$$\begin{eqnarray}&&{\rho }_{1}\approx {\rho }_{1,{circ}}{\left(\displaystyle \frac{\sqrt{1-{e}^{2}}}{1+e\sin \omega }\right)}^{3},\end{eqnarray} \tag{ 78 }$$

and the linear error propagation (ignoring covariances) yields

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{{\rho }_{1}}}{{\rho }_{1}}\right)}^{2} & \approx & {\left(\displaystyle \frac{{\sigma }_{{\rho }_{1}}}{{\rho }_{1}}\right)}_{\mathrm{circ}}^{2}+9{[e(1+e\sin \omega ){\sigma }_{\sin \omega })]}^{2}\\ & & +\,{\left(\displaystyle \frac{3{e}^{2}}{1-{e}^{2}}+\displaystyle \frac{3e\sin \omega }{1+e\sin \omega }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{e}}{e}\right)}^{2}.\end{eqnarray} \tag{ 79 }$$

Similarly, the RV semi-amplitude uncertainty becomes

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}^{2}={\left(\displaystyle \frac{{\sigma }_{{K}_{1}}}{{K}_{1}}\right)}_{\mathrm{circ}}^{2}+{\left(\displaystyle \frac{{e}^{2}}{1-{e}^{2}}\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{e}}{e}\right)}^{2}.\end{eqnarray} \tag{ 80 }$$

In the case of a primary stellar radius from parallax, we can combine Equation (80) with Equation (52) to get the uncertainty on the companion mass in terms of the uncertainty on the eccentricity, the argument of periastron, and the mass in the circular-orbit case:

$$\begin{eqnarray}{\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}^{2} & \approx & {\left(\displaystyle \frac{{\sigma }_{{M}_{2}}}{{M}_{2}}\right)}_{\mathrm{circ}}^{2}+4{\left(\displaystyle \frac{{\sigma }_{\sin \omega }}{1+e\sin \omega }\right)}^{2}\\ & & +\,{\left(\displaystyle \frac{3{e}^{2}}{1-{e}^{2}}+\displaystyle \frac{2e\sin \omega }{1+e\sin \omega }\right)}^{2}{\left(\displaystyle \frac{{\sigma }_{e}}{e}\right)}^{2}.\end{eqnarray} \tag{ 81 }$$

Thus, the fractional eccentricity uncertainty contributes a factor of $3{e}^{2}/(1-{e}^{2})$ to the uncertainties on the primary density and secondary mass, if the argument of periastron is zero. Figure 10 illustrates the contribution from the eccentricity term to the variances of the primary density and companion mass for different combinations of e and $\sin \omega$ values. While the contributions from the eccentricity uncertainty are small for low-eccentricity systems, they increase dramatically for highly eccentric systems, thus warranting exquisite RV phase coverage and per-point precision.

Zoom In Zoom Out Reset image size

To explore the effects of limb darkening on the achievable precision, we consider the same fiducial hot Jupiter as described in Section 3.2 in a circular, edge-on orbit. Along with the simulated RV and SED measurements, we perform EXOFASTv2 fits using two different synthetic light curves: one with quadratic limb-darkening coefficients u₁ = 0.454 and u₂ = 0.263 (corresponding to the V band) and another fit with fixed u₁ = u₂ = 0.

Figure 11 shows the model fits to the light curves, RVs, and SEDs, while Table 3 gives the uncertainties on the best-fit model parameters for both cases. As expected, the presence of limb darkening increases the uncertainties on the transit ingress/egress duration and depth, which then propagate through to the component masses and companion radius. Interestingly, the effective temperature uncertainty is lower in the presence of limb darkening: at each step in the MCMC chain, the limb-darkening coefficients are selected by interpolating the limb-darkening tables—in this case, the Claret & Bloemen (2011) tables—in ${T}_{\mathrm{eff},1}$, $\mathrm{log}({g}_{1})$, and $[\mathrm{Fe}/{\rm{H}}]$, so ${T}_{\mathrm{eff},1}$ is correlated with the limb-darkening coefficients. These results also suggest that, in the absence of limb darkening, ${\sigma }_{{\rho }_{1}}/{\rho }_{1}\approx 1.5({\sigma }_{\tau }/\tau )$, while ${\sigma }_{{\rho }_{1}}/{\rho }_{1}\approx {\sigma }_{\tau }/\tau$ in the presence of limb darkening. The latter relation is the one seen in the fiducial fits of Section 3.2. This suggests that adding free parameters for limb darkening weakens the strong dependence of the estimate of the density of the primary on the ingress/egress duration while simultaneously contributing significantly to the uncertainties on both these quantities.

Zoom In Zoom Out Reset image size

Table 3.  Uncertainties on Fiducial Transiting Hot Jupiter Parameters with and without Limb Darkening

Parameter	No Limb Darkening	Limb Darkening
Transit Parameters
${\sigma }_{\tau }/\tau$	4.6%	23%
${\sigma }_{\rho }/\rho$	6.3%	23%
${\sigma }_{\delta }/\delta$	1.3%	5.2%
RV Parameters
${\sigma }_{{K}_{1}}/{K}_{1}$	1.6%	1.5%
SED Parameters
${\sigma }_{{T}_{\mathrm{eff},1}}/{T}_{\mathrm{eff},1}$	4.7%	2.6%
System Parameters
${\sigma }_{{M}_{1}}/{M}_{1}$	12%	22%
${\sigma }_{{R}_{1}}/{R}_{1}$	3.2%	1.7%
${\sigma }_{{M}_{2}}/{M}_{2}$	8.2%	16%
${\sigma }_{{R}_{2}}/{R}_{2}$	3.3%	3.1%

Download table as:  ASCIITypeset image

We note that our analysis involves interpolating in Claret & Bloemen (2011) limb-darkening tables and assuming a quadratic limb-darkening law. As a result, any inaccuracies in the tabulated limb-darkening coefficients—or any inaccuracy of the underlying limb-darkening equation—would induce systematic errors. Espinoza & Jordán (2016) showed how different limb-darkening laws result in not only different precision for transit parameters but also systematic differences between laws at the couple-percent level, which is significant.

Hence, choosing red filters—such as TESS's 600–1000 nm bandpass—will minimize the effects of limb darkening and thus provide the highest precision on transit/eclipse observables while also improving the accuracy of the inferred transit parameters.