The Mass and Radius of the Unseen M-Dwarf Companion in the Single-Lined
  Eclipsing Binary HAT-TR-205-013
THE MASS AND RADIUS OF THE UNSEEN M-DWARF
COMPANION IN THE SINGLE-LINED ECLIPSING BINARY
HAT-TR-205-013
Thomas G. Beatty1, José M. Fernández2,3, David W. Latham2, Gáspár Á.
Bakos2,4, Géza Kovács5, Robert W. Noyes2, Robert P. Stefanik2, Guillermo
Torres2, Mark E. Everett6, Carl W. Hergenrother7,2
dlatham@cfa.harvard.edu
ABSTRACT
We derive masses and radii for both components in the single-lined eclipsing
binary HAT-TR-205-013, which consists of a F7V primary and a late M-dwarf
secondary. The system’s period is short, P = 2.230736 ± 0.000010 days, with
an orbit indistinguishable from circular, e = 0.012 ± 0.021. We demonstrate
generally that the surface gravity of the secondary star in a single-lined binary
undergoing total eclipses can be derived from characteristics of the light curve
and spectroscopic orbit. This constrains the secondary to a unique line in the
mass-radius diagram with M/R2 = constant. For HAT-TR-205-013, we assume
the orbit has been tidally circularized, and that the primary’s rotation has been
synchronized and aligned with the orbital axis. Our observed line broadening,
Vrot sin irot = 28.9± 1.0 km s−1, gives a primary radius of RA = 1.28 ± 0.04 R⊙.
Our light curve analysis leads to the radius of the secondary, RB = 0.167± 0.006
R⊙, and the semimajor axis of the orbit, a = 7.54± 0.30 R⊙ = 0.0351± 0.0014
AU. Our single-lined spectroscopic orbit and the semimajor axis then yield the
individual masses, MB = 0.124 ± 0.010 M⊙ and MA = 1.04 ± 0.13 M⊙. Our
result for HAT-TR-205-013 B lies above the theoretical mass-radius models from
1Department of Astronomy, Harvard University, 60 Garden Street, Cambridge, MA 02138
2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
3Department of Astronomy, Pontificia Universidad Católica, Casilla 306, Santiago 22, Chile
4Hubble Fellow
5Konkoly Observatory, Budapest, P.O. Box 67, H-1125, Hungary
6Planetary Science Institute, 1700 East Fort Lowell Road, Suite 106, Tucson, AZ 85719
7Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85719
http://arxiv.org/abs/0704.0059v2
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the Lyon group, consistent with results from double-lined eclipsing binaries. The
method we describe offers the opportunity to study the very low end of the stellar
mass-radius relation.
Subject headings: binaries: eclipsing — binaries: spectroscopic — stars: funda-
mental parameters — stars: low-mass — stars: rotation
1. INTRODUCTION
Solving for the masses and radii of stars has traditionally been accomplished through the
analysis of double-lined eclipsing binaries, where the light of both components is detected.
Masses and radii determined this way are fundamental and can be very accurate, because
they rely only on Newton’s laws and geometry for the analysis of the spectroscopic orbit and
light curve, and not on models of stellar structure and evolution. In particular, analysis of the
eclipse light curve yields the orbital inclination, and when combined with the double-lined
spectroscopic orbit, this yields individual masses for both stars.
There are dozens of double-lined eclipsing binaries with very accurate mass and ra-
dius determinations (e.g. see Andersen 1991, for a review), but only 10 M dwarfs (in
5 systems) with accuracies better than 3 percent. In order of increasing mass, the 5
systems are: CM Draconis (Lacy 1977; Metcalfe et al. 1996), CU Cancri (Ribas 2003),
NSVS01031772 (López-Morales et al. 2006), YY Geminorum (Torres & Ribas 2002), and
GU Boötis (López-Morales & Ribas 2005). Figure 1 shows the observational results for these
10 M dwarfs on the mass-radius diagram, along with the predicted theoretical models from
Baraffe et al. (1998). All of the observed radii are larger than the theoretical predictions,
typically by 5 to 10 percent. Another striking feature of Figure 1 is the lack of accurate
mass-radius determinations in the lower half of the M-dwarf mass range, from CM Dra B
(0.214 M⊙), all the way down to the substellar limit near 0.075 M⊙. Recently, however, the
growing number of short-period single-lined eclipsing binaries with F- and G-star primaries
and M-dwarf secondaries identified by wide-angle photometric surveys for transiting planet
promises to provide a way to fill in this gap in the mass-radius diagram (Bouchy et al. 2005;
Pont et al. 2005a,b, 2006).
One approach to using a single-lined eclipsing binary to solve for the mass and radius of
the unseen companion is to use stellar models together with spectroscopic and photometric
observations of the primary to estimate its mass and radius. Then the radius ratio from the
light curve yields the radius of the secondary, and the mass function from the spectroscopic
orbit with the orbital inclination from the light curve can be combined to yield the mass of
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the secondary. However, this approach is not fundamental, because it relies on stellar models
to characterize the primary. Tests of the mass-radius relation with the low-mass secondaries
in such systems are no better than the validity of the models for the primaries. In particular,
the theoretical isochrones depend on metallicity, and significant errors can result if the wrong
metallicity is adopted.
An alternative, and more fundamental, approach relies on the expectation that short-
period binaries whose orbits have been circularized by tidal mechanisms must also have the
axial rotation of both stars synchronized to the orbital period, and both rotational axes
aligned with the normal to the orbit, so that the two inclinations are equal, irot = iorb (e.g.
see Hut 1981; Zahn 1989). In this case, a measurement of the spectral line broadening due to
rotation, Vrot sin irot, combined with a value for iorb from an analysis of the light curve, yields
the actual equatorial rotational velocity, Vrot. If the rotation is synchronized with the orbit,
then Prot = Porb, and the radius of the primary can be solved. The key to this approach
is the ability to derive accurate values for the rotational broadening of the spectral lines,
because the primary radius can be no more accurate than the value derived for Vrot. The
radius of the primary then sets the scale for the rest of the system, yielding the radius of the
secondary and the semi-major axis of the orbit in actual length units, as well as the orbital
inclination. Then Newton’s revised version of Kepler’s Third Law can be used to derive the
sum of the masses, and the mass function from the single-lined spectroscopic orbit allows the
individual masses to be determined. This approach depends on the predictions from stellar
models only in minor ways: limb darkening coefficients are needed for the detailed analysis
of the eclipse light curve, and the rotational broadening that is derived from the observed
spectra can depend weakly on the metallicity that is adopted.
The advantage of using single-lined eclipsing binaries is that it dramatically increases
the number of low-mass stars whose masses and radii can be determined. Indeed, over the
past 30 years only seven double-lined eclipsing systems composed of low-mass stars have been
identified (the aforementioned CM Dra, CU Cnc, NSVS01031772, YY Gem, and GU Boo,
plus OGLE-BW3-V38 (Maceroni & Montalbán 2004), and TrES-Her0-07621 (Creevey et al.
2005)). Meanwhile, more than 75 single-lined eclipsing binaries with M-dwarf secondaries
have been discovered by wide-angle photometric transiting planet surveys such as Vulcan,
TrES, and HAT in just the last five years (Latham 2007).
Increasing the number of low-mass stars with fundamental determinations of masses and
radii is worthwhile because of the insights these systems can yield into stellar structure. Low-
mass stars near the hydrogen-burning limit are cool enough that their interior temperatures
are on the order of the electron Fermi temperature (Chabrier & Baraffe 1997), causing parts
of the stellar interior to be in the state of a partially degenerate electron gas, which means
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a classical Maxwell-Boltzmann description of the interior does not apply. Furthermore, the
electron number density is such that the mean inter-ionic distance is itself on the order of the
Thomas-Fermi screening length, meaning that the electron gas is polarized by the external-
ionic field (Chabrier & Baraffe 1997). Add to this the further complexity that magnetic
fields may play a role in the inner workings of low mass stars (Mullan & MacDonald 2001),
and it becomes clear that any attempt to describe the interior of low mass stars must take
into account the physics of both magnetic fields and partially degenerate polarized plasmas.
Not only is it difficult to model the interiors of low-mass stars, but the usual grey
model atmospheres are no longer applicable. The relatively low temperatures of the stellar
atmospheres allow for the recombination of molecular hydrogen and other molecules, such
as TiO. Therefore, accurate non-grey model atmospheres that take into account the effects
of molecules must be derived and matched to the interior models (Baraffe et al. 2002).
Thus stars near the bottom of the main sequence pose a challenge to dense matter physi-
cists and stellar astronomers. One of the few methods of testing models for low-mass stars
is by confronting theoretical predictions of the mass-radius relation with observations. Up
until now there has been little data with which to constrain the possible theories. Measuring
the masses and radii of M-dwarfs in single-lined eclipsing binaries therefore provides a new
opportunity to test the mass-radius relation near the bottom of the main sequence.
In this paper, we determine masses and radii for the components of HAT-TR-205-
013, an eclipsing single-lined binary identified by the HATnet (Hungarian-made Automated
Telescope Network, see Bakos et al. (2004)). The unseen M-dwarf secondary has a mass and
radius of 0.124 M⊙ and 0.167 R⊙, with 1-σ errors of 9 and 4 percent, respectively. This
result places the M dwarf about 10 percent above the radius predicted by the Baraffe et al.
(1998) models. In this first paper we describe our analysis techniques in detail. In future
papers we will present the results for additional M dwarf secondaries.
2. OBSERVATIONS AND DATA REDUCTION
2.1. HAT Photometry
The HATnet project,1 initiated in 2003 by G. Á. B, is a wide-field survey that aims for
the discovery of transiting planets around bright stars. It currently comprises 6 small wide-
field automated telescopes, each of which monitors 8◦ × 8◦ of sky, typically containing 5000
1www.hatnet.hu
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stars bright enough to permit detection of planetary transits via the typical 1% photometric
dips they induce on their parent stars. The instruments are deployed in a two-station,
longitude-distributed network, with four telescopes at the Fred L. Whipple Observatory in
Arizona, and two telescopes at the Submillimeter Array at Hawaii. For a more detailed
description of HAT’s instrumentation, observations, and data flow, see Bakos et al. (2002,
2004).
The HAT-TR-205-013 system lies in HATnet survey field G205, centered at α = 22h55m,
and δ = +37◦30′. 3357 observations were made of this field by the HATnet telescopes between
5 October 2003 and 30 January 2004. Exposure times were 5 min at a cadence of 5.5 min.
Light curves were derived by aperture photometry for the 6400 stars in G205 bright enough
to yield photometric precision of better than 2% (reaching 0.3% in some cases). In deriving
the light curves, we made use of the Trend Filtering Algorithm (Kovács, Bakos, & Noyes
2005) to correct for spurious trends in the data. We then searched all the light curves for
characteristic transit signals, using the Box-fitting Least Squares (Kovács, Zucker, & Mazeh
2002) algorithm, which searches for box-shaped dips in the parameter space of frequency,
transit duration, and phase of ingress. Candidate transit signals with the highest detection
significance were then examined individually, to isolate those with the best combination of
stellar type (preferably main-sequence stars of spectral type mid-F or later), and depth,
shape, and duration of transit. One of these was HAT-TR-205-013, for which we identified
a prominent periodicity with period 2.2307 days and transit depth of 0.02 mag. Figure 2
shows the phase-folded and flux-normalized data for HAT-TR-205-013.
Following our determination of HAT-TR-205-013 as a potential planetary system, we
identified the star in the 2MASS catalog as 2MASS 23080834+3338039, which yielded J
and K magnitudes of J = 9.691 and K = 9.408. We also found HAT-TR-205-013 in the
Tycho catalog, as TYC 2755-36-1, with magnitudes BT = 11.355 and VT = 10.729. We
then scheduled HAT-TR-205-013 for follow-up spectroscopic observations to determine if
the photometric dip was caused by a stellar companion.
2.2. Follow-up Spectroscopy
Our usual strategy for following up transiting-planet candidates identified by wide-field
photometric surveys is to start with an initial spectroscopic reconnaissance, to see if there
is evidence for a stellar companion that is responsible for the observed light curve. We have
used the CfA Digital Speedometers (Latham 1992) on the 1.5-m Wyeth Reflector at the
Oak Ridge Observatory in the Town of Harvard, Massachusetts and on the 1.5-m Tillinghast
Reflector at the Fred L. Whipple Observatory on Mount Hopkins, Arizona to obtain single-
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order echelle spectra in a wavelength window of 45 Å centered at 5187 Å with a resolution of
8.5 km s−1 and a typical signal-to-noise ratio per resolution element of 15 to 20. For slowly
rotating solar-type stars these spectra deliver radial velocities accurate to about 0.5 km s−1,
which is sufficient to detect orbital motion due to companions with masses down to about 5
or 10 MJ for orbital periods of a few days.
Spectroscopic follow-up observations of transiting-planet candidates are easier to sched-
ule than photometric observations, because the radial velocity varies throughout the entire
orbital phase, while the transit light curve has a duty cycle of only a few percent, e.g. 3
hours out of 3 days. Spectroscopy has the additional advantage over photometry that we
can use our spectra to classify the parent star, deriving effective temperature, rotational
velocity, and surface gravity by correlating the observed spectra with a library of synthetic
spectra. This information is often useful in rejecting host stars that are too hot, or are
rotating too rapidly, or are evolved (in which case we assume the giant must be the bright
star in a blended triple). Thus at CfA we usually start with a spectroscopic reconnaissance
before attempting follow-up photometry or very precise radial-velocity observations (e.g. see
O’Donovan et al. 2007).
Starting in 1999 with transiting-planet candidates provided by the Vulcan team (Borucki et al.
2001), we quickly learned that the vast majority of the planet candidates were actually eclips-
ing binaries masquerading as transiting planets (Latham 2003). One of the most common
imposters was an F- or G-star primary eclipsed by a late M-dwarf secondary. Such systems
produce light curves similar to planets transiting solar-type dwarfs, because M dwarfs near
the bottom of the main sequence have roughly the same size as giant planets. Nevertheless,
they are easy to distinguish from planets, because their masses are two orders of magnitude
larger, and the reflex orbital motion they induce in their parent stars has an amplitude of
at least several km s−1.
We determined the rotational and radial velocities of HAT-TR-205-013 by cross corre-
lation of the observed spectra against templates drawn from a library of synthetic spectra
calculated by Jon Morse for a grid of Kurucz (1992) stellar atmospheres. The library grid
has a spacing of 250 K in effective temperature, Teff ; 0.5 in log surface gravity, log g; and
0.5 in log metallicity relative to the sun, [Fe/H]. For the final radial-velocity determinations
we adopted the template with the Teff , log g, and Vrot values that gave the highest value for
the peak of the correlation coefficient, averaged over all the observed spectra, assuming solar
metallicity. For the correlation analysis we used xcsao (Kurtz & Mink 1998) running inside
the IRAF2 environment.
2IRAF (Image Reduction and Analysis Facility) is distributed by the National Optical Astronomy Ob-
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Our very first spectrum of HAT-TR-205-013 revealed that the spectral lines of the
primary were broadened by about 30 km s−1 of rotation. In our experience this is a strong
suggestion that the companion is a star with enough mass to synchronize the rotation of
the primary with the orbital period. The second exposure showed that the velocity varied
by several km s−1. Additional spectra soon revealed a spectroscopic orbit with a period
that matched the photometric period and an orbital semi-amplitude, K, of about 20 km s−1
implying that the secondary was a small M dwarf. Altogether we accumulated 23 spectra
yielding the radial velocities reported in Table 1, which were calculated using a template
with Teff = 6250 K, log g = 4.0, [Fe/H]=0.0, and Vrot = 30 kms
−1. The parameters of our
orbital solution for HAT-TR-205-013 are reported in Table 2 and the corresponding velocity
curve and observed velocities are plotted in Figure 3.
Because the value of the rotational velocity is critical for our determination of the radius
of the primary star, we evaluated both the internal precision of our determinations and
possible systematic errors resulting from uncertainties in the surface gravity and metallicity
of the primary star. Our final results for the mass and radius of the primary star imply a
surface gravity log g = 4.24, halfway between the nearest templates in our library of synthetic
spectra. Therefore we evaluated the rotational velocity at both log g = 4.0 and 4.5, and also
at two metallicities, [Fe/H] = 0.0 and −0.5. At each of the four combinations of log g and
[Fe/H] we ran correlations over a wide range of temperatures, 5500 to 7250 K, and rotational
velocities, 10 to 50 km s−1. At each gravity and metallicity we interpolated to find the
temperature and rotation that gave the highest value of the correlation coefficient averaged
over all the observed spectra weighted by the photon statistics. For these experiments we
did not use two of our observed spectra that had much lower exposure levels than the other
21 observations. The results are reported in Table 3. They show that the accuracy of our
rotational-velocity determination at a given log g and [Fe/H] is not seriously limited by the
scatter from the individual exposures, despite the relatively low signal-to-noise ratio of our
observed spectra. The uncertainty in the mean rotational velocity at each log g and [Fe/H]
due to internal scatter is less than 0.3 km s−1.
The results reported in Table 3 also show that the systematic errors due to uncertainties
in the gravity and metallicity are not serious. For example, the dependence of rotational
velocity on metallicity is rather weak, only 0.2 km s−1 for a change of 0.5 in [Fe/H]. On the
other hand, the dependence on gravity is rather strong, about 1.0 km s−1 for a change of 0.5
in log g. Fortunately, the uncertainty in the actual gravity of the primary star is quite small,
less than 0.2 in log g, so the rotational velocity interpolated between the two gravities should
servatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under
contract with the National Science Foundation.
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be good to perhaps 0.2 km s−1. Unfortunately we do not yet have a direct measure of the
metallicity of HAT-TR-205-013, but the fact that the templates with [Fe/H] = 0.0 give a
better match to the observed spectra than the templates with [Fe/H] =−0.5 suggests that the
metallicity is probably within 0.5 dex of solar. In the absence of an accurate metallicity we
adopt a value of [Fe/H] = −0.2, which is typical of the solar neighborhood (Nordström et al.
2004). The temperature corresponding to this metallicity that we derive from our spectra is
Teff = 6295 K, but with a systematic error that could exceed 200 K because of the degeneracy
between temperature and metallicity in the analysis of our spectra. The alternative approach
of deriving a temperature from photometric indices is hampered by the lack of accurate
photometry and the possibility that there may be significant reddening. The distance that
we derive from the Stefan-Boltzmann law using R = 1.27 ± 0.04 R⊙, Teff = 6295 ± 200 K,
VT = 10.729 ± 0.048 (which corresponds to V = 10.67 ± 0.05), and bolometric correction
= −0.011 (Flower 1996) is 232 ± 18 pc. From our spectroscopic temperature and log g, we
estimate the spectral type of the primary to be F7V (Cox 2000).
Thus our experiments with different gravities and metallicities suggest that the system-
atic errors due to uncertainties in the template parameters may be less than 0.4 km s−1.
When combined in quadrature with the internal precision estimate of 0.3 km s−1, this sug-
gests that the total error in our final interpolated rotational velocity could be as small as 0.5
km s−1. Previous experience however, would indicate that the actual uncertainty is higher
than this internal estimate. Therefore, to be conservative we adopt an uncertainty of 1.0
km s−1 and use Vrot sin irot = 28.9 ± 1.0 km s−1 for our determination of the radius of the
primary and subsequent analysis.
One could also determine the value of Vrot sin irot by spectroscopically measuring the
amplitude of the Rossiter-McLaughlin Effect during eclipse (Gaudi & Winn 2007). In the
case of HAT-TR-205-013 the amplitude of this effect would be about 0.5 km s−1, too small to
be measured with the CfA Digital Speedometers, but potentially observable with an accuracy
of a few percent by instruments such as HIRES on Keck 1.
Our approach relies on the assumption that the stellar rotation has been synchronized to
the orbital period. One way to test this assumption might be to use very precise photometry
to derive a rotational period for the star. We did look for sinusoidal variations in the HAT
photometry but did not find anything significant near the orbital period. We did detect a
marginally significant variation with half the orbital period and amplitude of about 1 mmag,
which is consistent with the the expected ellipsoidal deformation of the primary star. A
second, more global test would be to use stellar models to estimate the mass and radius of
the primary, but a prerequisite for such an analysis would be accurate determinations of the
temperature and metallicity, presumably from high-quality spectra, and such results are not
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yet available.
2.3. Follow-up KeplerCam Photometry
To provide a high-quality light curve for the analysis of the primary eclipse of HAT-
TR-205-013 we used KeplerCam on the 1.2-m telescope at the Fred L. Whipple Observatory
on Mount Hopkins, Arizona. KeplerCam utilizes a monolithic 4K×4K Fairchild 486 CCD
that provides a 23′ field and a pixel size of 0.34′′. We used the predicted eclipse times from
our spectroscopic orbit to schedule observations on the night of 22-23 October 2005. We
successfully observed a full eclipse, alternating between the Sloan g and i bands.
The seeing was approximately 2′′ FWHM throughout the night, but we deliberately
defocused the telescope to get images with 3′′ FWHM in order to keep the peak counts
in the images well below saturation. Over the course of the observations the focus of the
telescope, which depends weakly on temperature, was adjusted three times to keep the image
size near 3′′ FWHM. Because we used automatic guiding, the centroid of the images moved
less than 3 pixels over the duration of the observations. For readout we binned the pixels
2 × 2, which gave a total readout time including overhead of 12 seconds. Exposure times
were 30 seconds for the g band and 10 seconds for the i band. The images were obtained in
sequences of 3 g-band exposures followed by 6 i-band exposures. All told, we were able to
collect a total of 297 images in the g band and 588 in the i band. Some thin cirrus clouds
were present at the beginning of the night, with no noticeable degradation of the light curves.
A quarter Moon rose during the observations after the end of egress, and this contributed
to a slight increase in photometric scatter after egress.
All the images were reduced by applying an overscan correction and then subtracting
the two-dimensional residual bias pattern. After correcting for shutter effects, we flattened
each image using a normalized set of combined twilight images. To produce the light curve,
we used the first image from each filter in an observing sequence as our astrometric reference
for identifying the same stars in subsequent images. We then determined the relative shift
between images to relocate each star in the following images. We measured the flux of each
star in a 6.7′′ circular aperture around the position derived from the astrometric fit using
daophot/phot within IRAF (Tody 1986, 1993). We estimated the sky in an annulus around
each star with inner and outer radii of 9.4′′ and 13.4′′ using the sigma-rejection mode.
We iteratively selected comparison stars by removing any that showed unusual noise or
variability in their differential light curves, obtaining 21 comparison stars for the g band and
37 for the i band. Based on a weighted mean flux from the comparison stars, we calculated
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an extinction correction, and then applied this to each comparison star. The typical rms
residual for both bands was 1.5 mmag, a value that increased slightly toward the end of the
night due to the increased sky brightness from the rising Moon. The main contributor of noise
to the photometry from individual images was atmospheric scintillation, which accounted
for over 65% of our calculated rms (Young 1967). The individual photometric measurements
in the g and i bands are reported in Tables 4 and 5, respectively.
3. LIGHT-CURVE ANALYSIS
Our KeplerCam light curves for HAT-TR-205-013 give good coverage of the eclipse
centered at HJD 2453666.747±0.001, with more than two hours of coverage both before the
start of ingress and after the end of egress. The eclipse itself lasts about 3 hours between
first and fourth contact and is about 2 percent deep, in agreement with the discovery light
curve from HAT. Our KeplerCam light curves clearly show the effects of limb darkening on
eclipse shape as a function of wavelength: the i-band light curve is slightly shallower and
possesses a flatter bottom than the g-band curve. The portions of the light curves before
and after eclipse showed a slight drift, which we removed with a linear fit. The resulting
light curves are plotted in Figure 4.
From our fits to the KeplerCam light curves we derive transit centers of 2453666.7465±
0.0005 and 2453666.7473 ± 0.0005 in the g and i bands, respectively. When we com-
bine these transit times with the results from the HAT photometry, we get the ephemeris
2453666.74748± 0.00018 +N × 2.230736± 0.000010.
Knowing the duration and depth of the eclipse, together with the orbital period, we
were able to derive rough values for the ratio of semimajor axis and radius of the secondary
to the radius of the primary star using the relations
δ (2)
where a is the orbital semimajor axis, RA and RB are the primary and secondary stars’ radii,
P is the orbital period, ∆ttr is the transit length, and δ is the transit depth in relative flux.
For our values of P ∼ 2.23 days, ∆ttr ∼ 2.5 hours and δ ∼ 0.02, we obtained a/RA ∼ 5.93
and RB/RA ∼ 0.133 from this first approximation.
We then built a two dimensional grid of light-curve fits in a/RA and RB/RA, centered
on the rough values obtained above. The impact parameter b was introduced as a third
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dimension in the grid, varying from b = 0 (central transit) to b = 1 + RB/RA (grazing
transit). The grid steps were 0.002 in a/RA, 0.00001 in RB/RA, and 0.001 in b. We generated
synthetic light-curves for each combination of parameters using the routines provided by
Mandel & Agol (2002), together with the quadratic limb-darkening coefficients derived by
Claret (2004), using the temperature and surface gravity for the primary we had derived
spectroscopically. We adopted solar values for the metallicity and surface turbulence of the
primary star. The exact coefficients we used were u1 = 0.4238 and u2 = 0.3250 in the g,
and u1 = 0.1814 and u2 = 0.3723 in the i band. Uncertainties in our values of the limb-
darkening coefficients, largely a result of the uncertainty in the spectroscopic temperature
measurement, had a negligible effect on our final results.
To identify the synthetic light curve that gave the best fit to the actual data, we looked
for the fit with the smallest value of χ2, inspecting the χ2 contours to ensure that we had
found the global minimum. We also inspected plots of the fits to the data as a qualitative
check. We applied this procedure to both the g and i light curves. The results are summarized
in Table 6.
To calculate the uncertainties for our values of a/RA, RB/RA, and b, we performed
additional fits wherein we fixed one of the parameters at a value differing slightly from the
best-fit value, and fit for the other two parameters. We changed this fixed parameter until
the eventual fit achieved a χ2 that was 1-σ away from our best fit parameters’ χ2. We
then used the difference between this 1-σ value and our best fit value as the uncertainty
for that particular parameter. We examined the correlation between the three different
parameters by making a final fit to the data using the Levenberg-Marquardt method – a
combination of the Inverse-Hessian and Steepest Descent methods (Press et al. 1992). This
allowed us to compute the covariance matrix for the three fit parameters, and thus the
correlation coefficients for the parameters. For all three, the correlation with any of the
other two was always smaller than 0.5 in both bands. While this is not negligible, we found
that the contribution to the total estimated error was minor. Of course, the Levenberg-
Marquardt method can also be used to calculate the uncertainties in a/RA, RB/RA, and b.
However, the uncertainties that we derived from our grid analysis were substantially larger.
To be conservative, we adopted the grid uncertainties instead of the Levenberg-Marquardt
uncertainties. Figure 5 shows contour plots of χ2 for our fits. Table 6 reports the final fitting
results and errors, and Table 7 lists the correlation coefficients. The KeplerCam data and
best-fit light curves are shown in Figure 4. Simultaneous fitting of the light curves in both
bands produced fits with inferior χ2 statistics, a result of the slight difference in RB/RA in
the two bands. We therefore use a simple average of the light curve results as our adopted
values for the parameters in Table 6.
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3.1. MASSES AND RADII FOR HAT-TR-205-013
Using the values that we have measured from the eclipse light curves, as well as the
observed spectroscopic orbit parameters, we may restrict the location of HAT-TR-205-013
A and B on the mass-radius diagram to unique curves described completely by these observ-
ables. To emphasize this, we have written the observable values in brackets in the following
derivation.
To begin, we use Newton’s revised version of Kepler’s Third Law and the spectroscopic
mass function from the orbital solution:
[P ]2 =
G (MA +MB)
a3 (3)
sin iorb
(MA +MB)
1− [e]2 (4)
where [KA] is the semi-amplitude of the spectroscopic orbit for the primary in km s
−1, [e]
the eccentricity of the orbit, and G is the Gravitational Constant. With these two equations,
the two unknowns (MA and MB) may be found:
[P ]2
[P ] [KA]
1− [e]2
2πa sin iorb
sin iorb
1− [e]2 (6)
Both the orbital inclination and the semimajor axis can be expressed in terms of the
observables [a/RA], [RB/RA] and [b], and the unknown stellar radii RA and RB:
By taking our value for the impact parameter [b], which measures the projected separa-
tion of the primary and secondary stars at the mid-point of the eclipse, the orbital inclination
can be expressed in terms of the observables [a/RA] and [b]:
[b] = [a/RA] cos iorb (7)
sin iorb =
[a/RA]2
Similarly, the semimajor axis may be written in terms of the observables [a/RA], [RB/RA]
and [b], and the unknown stellar radii RA and RB:
a = [a/RA] RA =
[a/RA]
[RB/RA]
RB (9)
– 13 –
Substituting these values into Eq.(5) and (6) gives expressions for the mass of each
component as functions of only the observables and the respective stellar radii:
G[P ]2
[a/RA]
[P ] [KA]
1− [e]2
2π(1− [b]2/[a/RA]2)1/2 [a/RA] RA
R3A (10)
G[P ]
[a/RA]
[RB/RA]
1− [e]2
(1− [b]2/[a/RA]2)
R2B (11)
Therefore, solely from what we are able to measure using the eclipse light curves and
spectroscopy, we may confine HAT-TR-205-013 B to a single curve on the mass-radius dia-
gram that goes as MB/R
B = constant. This constant is directly proportional to the surface
gravity of the object, which has been noted previously by Southworth et al. (2004). In our
case, the quality of our photometry allows us to measure this constant extremely accurately;
the uncertainty region around this surface gravity line is on the order of the line width itself
when it is plotted.
Through the assumed synchronization of the primary’s rotation to the orbital period we
may locate HAT-TR-205-013 B on this gravity determination curve, but it is important to
note that the line itself is defined without having made any assumptions about the system.
Indeed, it is possible to calculate a similar gravity curve for any eclipsing system, or for a
system containing a transiting planet. All that is required is a good-quality light curve and
spectroscopic orbit. This has previously been done for the case of a transiting planet by
Winn et al. (2006).
To place HAT-TR-205-013 on the mass-radius diagram more specifically, we note that
that the measured eccentricity of our spectroscopic orbit for HAT-TR-205-013 is indistin-
guishable from circular, and therefore, for the reasons described in the Introduction, we
assume that the spin axes of both stars have been aligned with the orbital normal and that
the rotation of both stars has been synchronized to the orbital period. This allows us to use
the observed rotational line broadening of the primary to solve for the radius of the primary
in linear units, which in turn allows us to convert the orbital size and secondary radius into
linear units from the values of [a/RA] and [RB/RA] derived from the light curves.
Using the assumption of synchronization, and that iorb = irot, we see by inspection that
[Vrot sin irot]
sin iorb
[Vrot sin irot]
sin iorb
[RB/RA] (13)
where [Vrot sin irot] is the projected rotational broadening of the primary derived from its
observed spectra. We may now substitute in Eq.(8) for sin iorb to get both radii in terms of
– 14 –
our observables:
2π (1− [b]2/[a/RA]2)
[Vrot sin irot] (14)
2π (1− [b]2/[a/RA]2)1/2
[RB/RA] [Vrot sin irot] (15)
By combining these two statements with Eq.(9) and (10), we arrive at expressions for
the masses of each component in terms of just the observable quantities:
[a/RA]
(1− [b]2/[a/RA]2)3/2
1− [e]2
[a/RA][Vrot sin irot]
[Vrot sin irot]
3 (16)
[a/RA]
(1− [b]2/[a/RA]2)3/2
1− [e]2 [Vrot sin irot]2 (17)
The results for the masses and radii for both components of HAT-TR-205-013 are pre-
sented in Table 8. The errors were estimated using Monte-Carlo simulations and were com-
pared with the results of formal error propagation, including the correlation coefficients
derived from the light-curve fits. Both approaches delivered similar results. The mass and
radius obtained for the primary star are essentially the same for both the g and i light curves,
but the mass and radius for the secondary differ by 0.8 and 3 percent, respectively. This
radius difference between the two light curves is close to 1-σ, and may be due to uncertainties
in the limb-darkening coefficients. Our adopted values are based on the average values of
the light curve parameters.
4. DISCUSSION
In Figure 6 we plot our mass and radius for the M-dwarf secondary HAT-TR-205-
013 B on a mass-radius diagram, together with isochrones for ages of 0.5 and 5 Gyr from
Baraffe et al. (1998). We also plot the results for 11 M dwarf secondaries from the sample
of OGLE planetary candidates analyzed by Bouchy et al. (2005); Pont et al. (2005a,b, 2006)
and listed in Table 9. For the systems OGLE-TR-34 (Bouchy et al. 2005), OGLE-TR-120
(Pont et al. 2005b), and the low mass systems OGLE-TR-122 (Pont et al. 2005a) and OGLE-
TR-123 (Pont et al. 2006) the authors had to use stellar models to to estimate the masses
and radii of the primaries without the assumption of synchronization, as synchronization
implied masses and radii that were inconsistent with the spectroscopic observations. For the
other seven systems, they were able to assume synchronization and to derive the radius of the
primary from the observed rotational line broadening. In general the agreement between the
– 15 –
OGLE results and the Baraffe et al. (1998) isochrones looks promising, but the observational
uncertainties are still too large to allow a critical test of the theoretical models. The OGLE
systems are all much fainter than HAT-TR-205-013, which presents significant challenges
for both the spectroscopic and photometric follow-up observations. Spectroscopy with the
resolution and signal-to-noise ratio suitable for determining accurate values for rotational
broadening requires time on large telescopes, and photometry for high-quality light curves
also requires large telescopes to achieve the needed photon statistics. Eclipsing binaries
identified by wide-angle surveys are much brighter and therefore less challenging on both
counts.
Our value for the radius of the M-dwarf secondary in HAT-TR-205-013 lies 11 percent,
about 3-σ, above the theoretical isochrones. This divergence is further reinforced by Eq.(11),
which, as has been previously noted, restricts the position of HAT-TR-205-013 B to lie on a
single line that is determined by the surface gravity of the object. This gravity curve does
not rely on any prior assumptions about the HAT-TR-205-013 system, nor does it depend
upon our measured value of Vrot sin irot, which is the biggest contributor of uncertainty to our
final results. We use the assumption of synchronization and the spectroscopically measured
Vrot sin irot to place HAT-TR-205-013 B at a specific location along the curve, but it is
important to note that in the region that we find HAT-TR-205-013 B, the curve of allowable
locations runs nearly parallel to the theoretical models. This is illustrated in Figure 6 by
the red line that passes through our point for HAT-TR-205-013 B. Thus the conclusion that
the theoretical models predict a radius for HAT-TR-205-013 B that is too small by about 10
percent is on much firmer ground than the error bar in the observed radius might suggest.
Indeed, it would require a 6-σ difference in Vrot sin irot to place HAT-TR-205-013 B onto the
Baraffe models.
Our result for HAT-TR-205-013 B supports the suggestion from the results for double-
lined eclipsing binaries plotted in Figure 1 that the models predict radii for M dwarfs that
are too small by up to 10 percent. This discrepancy has been noted before, for example
by Torres & Ribas (2002) in the case of YY Gem. Torres et al. (2006) raised the issue of
whether short-period eclipsing binaries are representative of isolated field stars and wide
binaries where tidal forces are negligible. They suggested that the rapid rotation of the
stars in these systems caused by tidal synchronization might give rise to enhanced magnetic
activity, thus decreasing the efficiency of energy transport in the convective envelopes and
leading to inflated stellar radii. For low mass stars, this effect is examined in more detail by
Lopez-Morales (2007).
In the case of HAT-TR-205-013, we see no evidence in the photometry of star spots on
the primary star, which would be tell-tale indicators of enhanced stellar magnetic activity.
– 16 –
Though HAT-TR-205-013 A is rapidly rotating, the lack of magnetic ativity is not suprising,
given its spectral type (F7). The star’s outer convective layer is relatively shallow, and
it is not unusual for rapidly rotating stars of this type to lack strong magnetic activity
(Torres et al. 2006).
In some instances it may be possible to independently determine the rotational period of
the primary through high-quality light curves used to definitively identify photometric varia-
tion outside of eclipse. This would serve as a check to the assumption of tidal synchronization
in the system.
In future papers we will present the results for several additional single-lined eclipsing
binaries with circularized orbits.
We thank Joe Zajac, Perry Berlind, and Mike Calkins for obtaining some of the spectro-
scopic observations; Bob Davis for maintaining the database for the CfA Digital Speedome-
ters; and John Geary, Andy Szentgyorgyi, Emilio Falco, Ted Groner, and Wayne Peters for
their contribution to making KeplerCam such an effective instrument for obtaining high-
quality light curves. TGB thanks the Harvard University Origins of Life Initiative for sup-
port. GK thanks the support of OTKA K-60750. The HATnet project is supported by
NASA Grant NNG04GN74G. This research was supported in part by the Kepler Mission
under NASA Cooperative Agreement NCC2-1390.
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– 19 –
Table 1. Individual Radial Velocities
HJD Vrad σ(Vrad)
(days) (km s−1) (km s−1)
2453034.45642 −2.02 1.38
2453035.47574 −10.11 1.61
2453035.58018 −18.52 1.01
2453036.48778 −12.93 1.53
2453037.46565 −0.47 1.43
2453037.61215 −6.83 0.91
2453038.46413 −25.72 1.48
2453038.57874 −19.76 1.15
2453040.47360 −28.58 1.24
2453042.58686 −27.79 0.91
2453043.58338 +4.84 1.23
2453044.58422 −19.73 1.14
2453045.57911 −6.42 0.73
2453046.46373 −2.11 0.80
2453046.60000 −10.47 0.77
2453047.50881 −20.45 1.49
2453047.58731 −14.83 0.95
2453543.94910 −4.01 1.10
2453658.69572 −20.53 1.16
2453659.75967 +3.14 2.28
2453659.78398 +2.85 1.09
2453660.70213 −25.98 1.57
2453664.70202 −19.39 1.21
– 20 –
Table 2. Spectroscopic Orbital Parameters
Parameter Value
P (days) 2.23072± 0.00005
γ (km s−1) −9.83± 0.30
K (km s−1) 18.33± 0.47
e 0.012± 0.021
ω (◦) 143± 90
Epoch (HJD) 2, 453, 198.61± 0.56
Nobs 23
O − C rms (km s−1) 1.06
f(M) (M⊙3) 0.00142± 0.00023
aA sin i (Gm) 0.562± 0.030
– 21 –
Table 3. Rotational Velocity Results
log g,[Fe/H] Teff < Vrot > σ(< Vrot >) Correlation
(K) (km s−1) (km s−1) Coefficient
4.0,0.0 6340 29.4 0.25 0.826
4.5,0.0 6540 28.4 0.24 0.823
4.0,−0.5 5960 29.2 0.21 0.821
4.5,−0.5 6150 28.2 0.24 0.816
Adopted:
4.24,−0.2 6295 28.9 1.0
– 22 –
Table 4. g Band Photometry
HJD Flux
2453666.575985 1.00054
2453666.576472 0.99856
2453666.576946 1.00108
2453666.579226 1.00084
2453666.579712 1.00008
2453666.580198 0.99985
2453666.582501 0.99998
2453666.582976 0.99882
2453666.583474 1.00159
Note. — Table 4 is pre-
sented in its entirety in the
electronic edition of the As-
trophysical Journal. A por-
tion is shown here for guid-
ance regarding its form and
content.
column (1): Heliocentric
Julian Date,
column (2): Normalized in-
strumental flux.
– 23 –
Table 5. i Band Photometry
HJD Flux
2453666.574226 0.99951
2453666.574469 0.99803
2453666.574724 1.00096
2453666.574967 0.99846
2453666.575233 0.99628
2453666.575488 1.00014
2453666.577432 1.00062
2453666.577698 0.99886
2453666.577965 0.99941
2453666.578208 0.99613
2453666.578474 0.99957
2453666.578728 0.99962
2453666.580719 0.99837
2453666.580974 0.99976
2453666.581228 0.99761
2453666.581494 1.00025
2453666.581772 1.00113
2453666.582027 0.99823
Note. — Table 5 is pre-
sented in its entirety in the
electronic edition of the As-
trophysical Journal. A por-
tion is shown here for guid-
ance regarding its form and
content.
column (1): Heliocentric
Julian Date,
– 24 –
column (2): Normalized in-
strumental flux.
– 25 –
Table 6. Light-Curve Fit Results
Parameter g Band i Band Adopted
a/RA 5.93± 0.15 5.91± 0.16 5.92± 0.11
RB/RA 0.1330± 0.0010 0.1288± 0.0007 0.1309± 0.0006
b 0.36± 0.06 0.37± 0.07 0.365± 0.046
– 26 –
Table 7. Correlation Coefficients
Coefficient g Band i Band
(a/RA,RB/RA) 0.28 0.27
(a/RA,b) −0.21 −0.42
(RB/RA,b) 0.04 0.01
– 27 –
Table 8. Physical Parameters for HAT-TR-205-013
Parameter g Band i Band Adopted
MA (M⊙) 1.04± 0.14 1.03± 0.14 1.04± 0.13
RA (R⊙) 1.28± 0.04 1.28± 0.04 1.28± 0.04
MB (M⊙) 0.124± 0.011 0.123± 0.011 0.124± 0.010
RB (R⊙) 0.169± 0.006 0.164± 0.006 0.167± 0.006
a (AU) 0.0351± 0.0015 0.0351± 0.0015 0.0351± 0.0014
– 28 –
Table 9. Masses and Radii for Low-Mass Stars
Name M (M⊙) R (R⊙) Type Ref.
OGLE-TR-123 B 0.085± 0.011 0.133± 0.009 SB1 EB 1
OGLE-TR-122 B 0.092± 0.009 0.120± 0.018 SB1 EB 2,3
OGLE-TR-106 B 0.116± 0.021 0.181± 0.013 SB1 EB 3
HAT-TR-205-013 B 0.123± 0.011 0.167± 0.007 SB1 EB 13
OGLE-TR-125 B 0.209± 0.033 0.211± 0.027 SB1 EB 3
CM Dra B 0.2136± 0.0010 0.2347± 0.0019 SB2 EB 4,5
CM Dra A 0.2307± 0.0010 0.2516± 0.0020 SB2 EB 4,5
OGLE-TR-78 B 0.243± 0.015 0.240± 0.013 SB1 EB 3
OGLE-TR-5 B 0.271± 0.035 0.263± 0.012 SB1 EB 6
OGLE-TR-7 B 0.281± 0.029 0.282± 0.013 SB1 EB 6
OGLE-TR-6 B 0.359± 0.025 0.393± 0.018 SB1 EB 6
OGLE-TR-18 B 0.387± 0.049 0.390± 0.040 SB1 EB 6
CU Cnc B 0.3890± 0.0014 0.3908± 0.0094 SB2 EB 7
OGLE-BW3-V38 B 0.41± 0.09 0.44± 0.06 SB2 EB 8
CU Cnc A 0.4333± 0.0017 0.4317± 0.0052 SB2 EB 7
OGLE-BW3-V38 A 0.44± 0.07 0.51± 0.04 SB2 EB 8
OGLE-TR-120 B 0.47± 0.04 0.42± 0.02 SB1 EB 3
TrES-Her0-07621 B 0.489± 0.003 0.452± 0.050 SB2 EB 9
TrES-Her0-07621 A 0.493± 0.003 0.453± 0.060 SB2 EB 9
NSVS01031772 B 0.4982± 0.0025 0.5088± 0.0030 SB2 EB 10
OGLE-TR-34 B 0.509± 0.038 0.435± 0.033 SB1 EB 6
NSVS01031772 A 0.5428± 0.0027 0.5260± 0.0028 SB2 EB 10
YY Gem A & B 0.5992± 0.0047 0.6191± 0.0057 SB2 EB 11
GU Boo B 0.599± 0.006 0.620± 0.020 SB2 EB 12
GU Boo A 0.610± 0.007 0.623± 0.016 SB2 EB 12
References. — 1. Pont et al. (2006); 2. Pont et al. (2005a); 3. Pont et al.
(2005b); 4. Lacy (1977); 5. Metcalfe et al. (1996); 6. Bouchy et al. (2005);
– 29 –
7. Ribas (2003); 8. Maceroni & Montalbán (2004); 9. Creevey et al.
(2005); 10. López-Morales et al. (2006); 11. Torres & Ribas (2002); 12.
López-Morales & Ribas (2005); 13. This paper
– 30 –
Fig. 1.— The mass-radius diagram for 10 stars in 5 double-lined eclipsing binaries each
composed of two M dwarfs, and with errors better than 3 percent.
– 31 –
 9.98
 10.02
 10.04
 10.06
 10.08
 10.1
-0.4 -0.2  0  0.2  0.4
Phase
HAT-5 TFA data
HAT-8 TFA data
 10.01
 10.02
 10.03
 10.04
 10.05
 10.06
-0.1 -0.05  0  0.05  0.1
Fig. 2.— The phase-folded HATnet light curve for HAT-TR-205-013.
– 32 –
Fig. 3.— The velocity curve for our orbital solution for HAT-TR-205-013, together with the
individual observed velocities. The lower panel shows the O-C velocity residuals from the
orbital solution.
– 33 –
Fig. 4.— KeplerCam light curves for HAT-TR-205-013 in the SDSS g and i bands. Contin-
uous lines show the best fit synthetic light curves for each.
– 34 –
Fig. 5.— Contours of χ2 for the results from fits to the light curves in the g-band (left
panels) and i-band (right panels). For each band the three panels show the projections onto
the three possible planes involving b, a/RA, and RB/RA. The 1-σ, 2-σ, and 3-σ contours are
plotted.
– 35 –
Fig. 6.— The mass-radius diagram for M dwarfs in single-lined eclipsing binaries. The M
dwarfs from Pont et al. (2005a,b, 2006) are plotted as open circles. The red line passing
through the point for HAT-TR-205-013 B shows the constraint imposed on its location by
Eq.(11) and our observed quantities, without making any explicit assumptions (such as
synchronization) about the system. Assuming synchronization, the hash marks on the line
show the effect that differences of ± 1, 2, & 3 km s−1 in Vrot have on our final results.
	INTRODUCTION
	OBSERVATIONS AND DATA REDUCTION
	HAT Photometry
	Follow-up Spectroscopy
	Follow-up KeplerCam Photometry
	LIGHT-CURVE ANALYSIS
	MASSES AND RADII FOR HAT-TR-205-013
	DISCUSSION
