A method for the direct determination of the surface gravities of
  transiting extrasolar planets
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 29 October 2018 (MN LATEX style file v2.2)
A method for the direct determination of the surface
gravities of transiting extrasolar planets
John Southworth⋆, Peter J Wheatley and Giles Sams
Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
29 October 2018
ABSTRACT
We show that the surface gravity of a transiting extrasolar planet can be calculated
from only the spectroscopic orbit of its parent star and the analysis of its transit
light curve. This does not require additional constraints, such as are often inferred
from theoretical stellar models or model atmospheres. The planet’s surface gravity
can therefore be measured precisely and from only directly observable quantities. We
outline the method and apply it to the case of the first known transiting extrasolar
planet, HD 209458b. We find a surface gravity of gp = 9.28± 0.15m s
−2, which is an
order of magnitude more precise than the best available measurements of its mass,
radius and density. This confirms that the planet has a much lower surface gravity
that that predicted by published theoretical models of gas giant planets. We apply
our method to all fourteen known transiting extrasolar planets and find a significant
correlation between surface gravity and orbital period, which is related to the known
correlation between mass and period. This correlation may be the underlying effect
as surface gravity is a fundamental parameter in the evaporation of planetary atmo-
spheres.
Key words: stars: planetary systems — stars: individual: HD 209458 — stars: bina-
ries: eclipsing — stars: binaries: spectroscopic — methods: data analysis
1 INTRODUCTION
Since the discovery that the star HD209458 is eclipsed
by a planet in a short-period orbit (Henry et al. 2000;
Charbonneau et al. 2000) it has become possible to derive
the basic astrophysical properties of extrasolar planets and
compare these quantities with theoretical predictions (e.g.
Baraffe et al. 2003). However, the absolute masses, radii and
density of the transiting planet cannot be calculated directly
from the transit light curve and the velocity variation of
the parent star, so extra information is required in order
to obtain them. Additional constraints can be found from
spectral analysis of the parent star or by imposing a theo-
retical stellar mass–radius relation (Cody & Sasselov 2002),
but this causes a dependence on theoretical stellar models or
model atmospheres. The uncertainties in these constraints
dominate the overall errors in mass, radius and density (e.g.
Konacki et al. 2004), limiting the accuracy with which prop-
erties of the star and planet can be measured.
In this work we show that the surface gravity of the
planet can be measured directly using only the transit light
curve and the radial velocity amplitude of the parent star.
No additional information is required and so accurate and
⋆ E-mail: j.k.taylor@warwick.ac.uk
precise surface gravity values can be obtained. As theoretical
studies often supply predicted values for the surface gravities
of planetary objects (e.g., Baraffe et al. 2003) we propose
that this quantity is very well suited for comparing observa-
tion with theory. In addition, the surface gravity is an im-
portant parameter in constructing theoretical models of the
atmospheres of planets (Marley et al. 1999; Hubbard et al.
2001).
After deriving an equation for surface gravity in terms
of directly observed parameters, we illustrate this concept by
studying HD209458. We then apply it to the other known
transiting extradsolar planets using results available in the
literature. The planet HD209458 b is known to be over-
sized for its mass and to be strongly irradiated by the star
it orbits. An excellent transit light curve was obtained for
HD209458 by Brown et al. (2001), who used the HST/STIS
spectrograph to obtain high-precision photometry cover-
ing several different transit events. Precise radial velocity
studies of HD209458 are also available (Henry et al. 2000;
Mazeh et al. 2000; Naef et al. 2004).
c© 0000 RAS
http://arxiv.org/abs/0704.1570v1
2 Southworth, Wheatley & Sams
Table 1. Results of the modelling of the HST/STIS light curve of HD209458. The upper part of the table gives the optimised parameters
and the lower part gives quantities calculated from these parameters. The midpoint of the transit, TMin I, is expressed in HJD − 2 400 000.
Limb darkening Linear Quadratic Square-root Adopted parameters
r⋆+rp 0.12889± 0.00042 0.12771± 0.00049 0.12799± 0.00050
k 0.12260± 0.00011 0.12097± 0.00025 0.12051± 0.00037
i (deg.) 86.472± 0.040 86.665± 0.054 86.689± 0.060 86.677± 0.060
T0 51659.936716± 0.000021 51659.936712± 0.000021 51659.936711± 0.000021
u⋆ 0.494± 0.004 0.297± 0.027 −0.312± 0.127
v⋆ 0.338± 0.047 1.356± 0.218
r⋆ 0.11481± 0.00036 0.11393± 0.00042 0.11418± 0.00042 0.11405± 0.00042
rp 0.01408± 0.00006 0.01378± 0.00007 0.01376± 0.00008 0.01377± 0.00008
1.146 1.056 1.054
2 SURFACE GRAVITY MEASUREMENT
The fractional radii of the star and the planet in the system
are defined as
where a is the orbital semi-major axis, and R⋆ and Rp are
the absolute radii of the star and planet, respectively. r⋆ and
rp can be directly determined from a transit light curve.
The mass function of a spectroscopic binary is given by
(e.g. Hilditch 2001):
f(Mp) =
(1− e2)
p sin
(M⋆ +Mp)2
whereK⋆ is the velocity amplitude of the star, e is the orbital
eccentricity, P is the orbital period, i is the orbital inclina-
tion, and M⋆ and Mp are the masses of the star and planet
respectively. Including Kepler’s Third Law and solving for
the sum of the masses of the two components gives:
(M⋆ +Mp)
2πGM 3p sin
(1− e2)
2K 3⋆ P
(2π)4a6
G2P 4
By substituting Rp = arp into the definition of surface grav-
ity and replacing a using Eq. 3 we find that the surface grav-
ity of the planet, gp is given by:
(1− e2)
r 2p sin i
Eq. 4 shows that we are able to calculate the surface
gravity of a transiting extrasolar planet from the quantities
P , K⋆, e, i and rp. This can be understood intuitively be-
cause both the radial velocity motion of the star and the
planet’s surface gravity are manifestations of the accelera-
tion due to the gravity of the planet. A discussion in the
context of eclipsing binaries is given by Southworth et al.
(2004b).
To measure gp using Eq. 4, the orbital period, P , can be
obtained from either radial velocities or light curves of the
system, and is typically determined very precisely compared
to the other measurable quantities. The radial velocities also
give e and K⋆, whilst the quantities i and rp can be obtained
directly from the transit light curve. Note that it is also pos-
sible to constrain the orbital eccentricity from observations
of the secondary eclipse of a system.
3 APPLICATION TO HD209458B
In order to measure the surface gravity for HD209458 b we
need to know rp and i. These quantities are standard param-
eters in the analysis of transit light curves. We have chosen
to obtain them by modelling the high-precision HST/STIS
light curve presented by Brown et al. (2001). We followed
Brown et al. by rejecting data from the first HST orbit of
each observed transit.
To model the photometric data we used the jktebop
code1 (Southworth et al. 2004a), which is a modified version
of the ebop program (Popper & Etzel 1981; Etzel 1981).
Giménez (2006) has shown that ebop is very well suited
to the analysis of the light curves of transiting extrasolar
planets. Importantly for this application, jktebop has been
extended to treat limb darkening (LD) using several non-
linear LD laws (Southworth et al. 2007). It also includes
Monte Carlo and bootstrapping simulation algorithms for
error analysis (Southworth et al. 2004a,b). ebop and jkte-
bop model the two components of an eclipsing system using
biaxial ellipsoids (Nelson & Davis 1972; Etzel 1975), so al-
low for the deformation of the bodies from a spherical shape.
When modelling the data we adopted the precise orbital
period of 3.52474859 days given by Knutson et al. (2007).
We fitted for the sum of the fractional radii, r⋆+rp, the
ratio of the radii, k =
, the orbital inclination,
and the midpoint of a transit, T0. We also fitted for the LD
coefficients of the star, rather than fixing them at values
calculated using model atmospheres, to avoid introducing a
dependence on theoretical predictions. The linear and non-
linear LD coefficients are denoted by u⋆ and v⋆, respectively.
We assumed that the planet contributes no light at the
optical wavelengths considered here (see Wittenmyer et al.
2005) and that the orbit is circular (see Laughlin et al. 2005;
Deming et al. 2005; Winn et al. 2005). Given suggestions
that the choice of LD law can influence the derived inclina-
tion (Winn et al. 2005), we obtained solutions for the linear,
quadratic and square-root laws (Southworth et al. 2007). A
mass ratio of 0.00056 was used (Knutson et al. 2007) but
large changes in this parameter have a negligible effect on
the solution.
We have calculated robust 1 σ error estimates us-
jktebop is written in fortran77 and the source code is avail-
able at http://www.astro.keele.ac.uk/∼jkt/
c© 0000 RAS, MNRAS 000, 000–000
http://www.astro.keele.ac.uk/~jkt/
Direct determination of surface gravities of exoplanets 3
Figure 1. Best fit to the HST transit light curve of HD209458
using the quadratic LD law. The residuals of the fit are shown
offset downwards by 0.02 in flux for clarity.
ing Monte Carlo simulations (see Press et al. 1992, p.684;
Southworth et al. 2004a), which we have previously found
to provide very reliable results (Southworth et al. 2005a,b).
This procedure assumes that systematic errors are negligi-
ble. We find no reason to suspect that significant systematic
errors remain in the HST light curve after the processing of
this data described by Brown et al. (2001) (see Fig. 1).
The best-fitting parameters of the transit light curve are
given in Table 1. The best-fitting model using the quadratic
LD law is shown in Fig. 1 along with the residuals of that fit.
The solution using linear LD can be rejected as its reduced
2 is substantially larger than for the other two solutions.
The quadratic and square-root LD laws give very similar
solutions with reduced χ2 values close to one. For our final
results we adopt the mean for each parameter along with
uncertainties from the Monte Carlo simulations (Table 1).
These are in good agreement with the light curve solutions
obtained by Giménez (2006) and Mandel & Agol (2002),
both of which used the approximation that the planet is
spherical.
With the orbital period given by Knutson et al. (2007),
the stellar velocity ampitude K⋆ = 85.1 ± 1.0m s
−2 from
Naef et al. (2004), and the results of our light curve analysis
(Table 1) we find the surface gravity of HD209458 b to be
gp = 9.28±0.15m s
−2. In this case the total uncertainty in gp
is due to almost equal contributions from the uncertainties
in K⋆ and rp.
4 APPLICATION TO ALL KNOWN
TRANSITING EXTRASOLAR PLANETS
We have calculated the surface gravity values for each of the
known transiting extrasolar planets (apart from HD209458),
using data taken from the literature (Table 2). In several
cases (marked with asterisks in Table 2) it was not possible
Table 2. Surface gravity values for the known transiting extra-
solar planets. These have been calculated using Eq. 4 with input
parameters taken from the literature.
System Surface gravity Literature references
m s−2 rp and i K⋆
HD 189733 21.5± 3.5 1 2
HD 209458 9.28± 0.15 3 4
OGLE-TR-10 4.5± 2.1 5 6
OGLE-TR-56 17.9± 1.9 5 2
OGLE-TR-111 13.3± 4.2 7 8
TrES-1 16.1± 1.0 9 10
WASP-1 10.6± 1.7 11 12
∗ HAT-P-1 7.1± 1.1 13 13
∗ XO-1 13.3± 2.5 14 14
∗ HD 149026 16.4± 2.5 15 15
∗ OGLE-TR-113 28.3± 4.4 16 17
∗ OGLE-TR-132 18.0± 6.0 18 17
∗ TrES-2 20.7± 2.6 19 19
∗ WASP-2 20.1± 2.7 20 12
∗ The surface gravity values for these objects have larger error
estimates than are needed, because their fractional radii are not
available in the literature. In these cases we have had to calculate
them from Rp and a, which are less certain than rp because of
the need to adopt an additional constraints to calculate them
(see text).
References: (1) Winn et al. (2007c); (2) Bouchy et al. (2005);
(3) This work; (4) Naef et al. (2004); (5) Pont et al. (2007); (6)
Konacki et al. (2005); (7) Winn et al. (2007a); (8) Pont et al.
(2004); (9) Winn et al. (2007b); (10) Alonso et al. (2004);
(11) Shporer et al. (2007); (12) Cameron et al. (2007); (13)
Bakos et al. (2007); (14) McCullough et al. (2006); (15)
Sato et al. (2005); (16) Gillon et al. (2006); (17) Bouchy et al.
(2004); (18) Gillon et al. (2007); (19) O’Donovan et al. (2006);
(20) Charbonneau et al. (2007).
to obtain rp directly from the results available in the litera-
ture. In these cases it had to be calculated from Rp and a,
resulting in an increased uncertainty. This is because rp is
a parameter obtainable directly from a transit light curve,
whereas additional constraints (for example using theoreti-
cal stellar models) are needed to calculate a and Rp.
The orbital periods and surface gravities of all fourteen
transiting extrasolar planets are plotted in Fig. 2, and show
that these quantities are correlated. The linear Pearson cor-
relation coefficient of these data is r = −0.70, indicating
that the correlation is significant at better than the 0.5%
level. This correlation is certainly related to that found by
Mazeh et al. (2005) between the orbital periods and masses
of the six transiting extrasolar planets then known. How-
ever, it may be that surface gravity, rather than mass or
radius, is the main parameter correlated with orbital period
for these objects. Theoretical calculations have shown that
surface gravity is a fundamental parameter in the evapora-
tion rates of the atmospheres of irradiated gas giant planets
(Lammer et al. 2003).
5 SUMMARY AND DISCUSSION
We have shown that the surface gravity of transiting extraso-
lar planets can be measured from analysis of the light curve
c© 0000 RAS, MNRAS 000, 000–000
4 Southworth, Wheatley & Sams
Figure 2. Comparison between the surface gravities and orbital
periods of the known transiting exoplanets. Filled and open circles
denote the systems in the upper and lower halves of Table 2, re-
spectively. The errorbars for HD209458 (P = 3.52 d) are smaller
than the plotted symbol.
and a spectroscopic orbit of the parent star. We have anal-
ysed the HST/STIS light curve of HD209458 (Brown et al.
2001) with the jktebop code. By combining the results
of the light curve analysis with published spectroscopy
(Naef et al. 2004) we find that the planet has a surface grav-
ity of gp = 9.28 ± 0.15m s
−2. We stress that this measure-
ment does not depend on theoretical stellar evolutionary
models or model atmospheres.
In Fig. 3 we have plotted theoretical isochrones for ages
of 0.5 to 10 Gyr from Baraffe et al. (2003) against the mass
and surface gravity of HD209458 b, adopting a mass of
Mp = 0.64±0.06 MJup from Knutson et al. (2007). The dis-
crepancy between the observed and predicted surface gravity
can clearly be seen. Note that these models do not include
the effects of irradiation from the star.
The density of a transiting extrasolar planet is of-
ten used to compare observation with theory, but is typi-
cally measured with a much lower precision than its sur-
face gravity, given the same dataset. For example, the den-
sity of HD209458 b derived by Knutson et al. (2007) is
345±50 kgm−3. Using the mass and radius given by Knutson
et al. leads to gp = 9.1±0.9m s
−2, where the uncertainty has
been calculated by simple error propagation. These quanti-
ties are both much less precise and require more elaborate
calculations than using Eq. 4 to find the surface gravity:
gp = 9.28± 0.15m s
We have applied Eq. 4 to each of the known transiting
extrasolar planets (Table 2). The resulting surface gravities
show a clear correlation with orbital period (Fig. 2) which is
connected with the known correlation between orbital period
and mass for these objects. We propose that surface gravity
may be the underlying parameter of the correlation due to
its influence on the evaporation rates of the atmospheres of
short-period giant planets.
Figure 3. Plot of mass versus surface gravity for HD209458 b
compared to the theoretical model predictions of Baraffe et al.
(2003) for ages 0.5, 1.0, 5.0 and 10.0 Gyr (from lower to higher
log g).
As gp can be very precisely measured, and can be di-
rectly compared with theoretical models and used to con-
struct model atmospheres of the planet, we propose that it
is an important parameter in our understanding of short-
period extrasolar giant planets. In the near future, the high-
precision light curves obtained by the CoRoT and Kepler
satellites will allow accurate surface gravity values to be ob-
tained for the terretrial-mass transiting extrasolar planets
which these satellites should find.
6 ACKNOWLEDGEMENTS
We are grateful to David Charbonneau for making the
HST/STIS light curve of HD209458 available on his website
(http://cfa-www.harvard.edu/∼dcharbon/frames.html),
to Pierre Maxted for discussions, and to the referee whose
report contributed significantly to improvements in this
work.
JS acknowledges financial support from PPARC in the
form of a postdoctoral research assistant position. The fol-
lowing internet-based resources were used in research for
this paper: the NASA Astrophysics Data System; the SIM-
BAD database operated at CDS, Strasbourg, France; and
the arχiv scientific paper preprint service operated by Cor-
nell University.
REFERENCES
Alonso, R., et al., 2004, ApJ, 613, L153
c© 0000 RAS, MNRAS 000, 000–000
http://cfa-www.harvard.edu/~dcharbon/frames.html
Direct determination of surface gravities of exoplanets 5
Bakos, G. Á., et al., 2007, ApJ, 656, 552
Baraffe, I., Chabrier, G., Barman, T. S., Allard, F.,
Hauschildt, P. H., 2003, A&A, 402, 701
Bouchy, F., Pont, F., Santos, N. C., Melo, C., Mayor, M.,
Queloz, D., Udry, S., 2004, A&A, 421, L13
Bouchy, F., et al., 2005, A&A, 444, L15
Brown, T. M., Charbonneau, D., Gilliland, R. L., Noyes,
R. W., Burrows, A., 2001, ApJ, 552, 699
Cameron, A. C., et al., 2007, MNRAS, 375, 951
Charbonneau, D., Brown, T. M., Latham, D. W., Mayor,
M., 2000, ApJ, 529, L45
Charbonneau, D., Winn, J. N., Everett, M. E., Latham,
D.W., Holman, M. J., Esquerdo, G. A., O’Donovan, F. T.,
2007, ApJ, 658, 1322
Cody, A. M., Sasselov, D. D., 2002, ApJ, 569, 451
Deming, D., Seager, S., Richardson, L. J., Harrington, J.,
2005, Nature, 434, 740
Etzel, P. B., 1975, Masters Thesis, San Diego State Uni-
versity
Etzel, P. B., 1981, in Carling, E. B., Kopal, Z., eds., Pho-
tometric and Spectroscopic Binary Systems, p. 111
Gillon, M., Pont, F., Moutou, C., Bouchy, F., Courbin, F.,
Sohy, S., Magain, P., 2006, A&A, 459, 249
Gillon, M., et al., 2007, A&A, in press, astro-ph/0702192
Giménez, A., 2006, A&A, 450, 1231
Henry, G. W., Marcy, G. W., Butler, R. P., Vogt, S. S.,
2000, ApJ, 529, L41
Hilditch, R. W., 2001, An Introduction to Close Binary
Stars, Cambridge University Press, Cambridge, UK
Hubbard, W. B., Fortney, J. J., Lunine, J. I., Burrows, A.,
Sudarsky, D., Pinto, P., 2001, ApJ, 560, 413
Knutson, H. A., Charbonneau, D., Noyes, R. W., Brown,
T. M., Gilliland, R. L., 2007, ApJ, 655, 564
Konacki, M., Torres, G., Sasselov, D. D., Jha, S., 2005,
ApJ, 624, 372
Konacki, M., et al., 2004, ApJ, 609, L37
Lammer, H., Selsis, F., Ribas, I., Guinan, E. F., Bauer,
S. J., Weiss, W. W., 2003, ApJ, 598, L121
Laughlin, G., Marcy, G. W., Vogt, S. S., Fischer, D. A.,
Butler, R. P., 2005, ApJ, 629, L121
Mandel, K., Agol, E., 2002, ApJ, 580, L171
Marley, M. S., Gelino, C., Stephens, D., Lunine, J. I.,
Freedman, R., 1999, ApJ, 513, 879
Mazeh, T., Zucker, S., Pont, F., 2005, MNRAS, 356, 955
Mazeh, T., et al., 2000, ApJ, 532, L55
McCullough, P. R., et al., 2006, ApJ, 648, 1228
Naef, D., Mayor, M., Beuzit, J. L., Perrier, C., Queloz, D.,
Sivan, J. P., Udry, S., 2004, A&A, 414, 351
Nelson, B., Davis, W. D., 1972, ApJ, 174, 617
O’Donovan, F. T., et al., 2006, ApJ, 651, L61
Pont, F., Bouchy, F., Queloz, D., Santos, N. C., Melo, C.,
Mayor, M., Udry, S., 2004, A&A, 426, L15
Pont, F., et al., 2007, A&A, 465, 1069
Popper, D. M., Etzel, P. B., 1981, AJ, 86, 102
Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery,
B. P., 1992, Numerical recipes in FORTRAN 77. The art
of scientific computing, Cambridge: University Press, 2nd
Sato, B., et al., 2005, ApJ, 633, 465
Shporer, A., Tamuz, O., Zucker, S., Mazeh, T., 2007, MN-
RAS, in press, astro-ph/0610556
Southworth, J., Maxted, P. F. L., Smalley, B., 2004a, MN-
RAS, 351, 1277
Southworth, J., Zucker, S., Maxted, P. F. L., Smalley, B.,
2004b, MNRAS, 355, 986
Southworth, J., Maxted, P. F. L., Smalley, B., 2005a, A&A,
429, 645
Southworth, J., Smalley, B., Maxted, P. F. L., Claret, A.,
Etzel, P. B., 2005b, MNRAS, 363, 529
Southworth, J., Bruntt, H., Buzasi, D. L., 2007, A&A, in
press
Winn, J. N., Holman, M. J., Fuentes, C. I., 2007a, AJ, 133,
Winn, J. N., Holman, M. J., Roussanova, A., 2007b, ApJ,
657, 1098
Winn, J. N., et al., 2005, ApJ, 631, 1215
Winn, J. N., et al., 2007c, AJ, 133, 1828
Wittenmyer, R. A., et al., 2005, ApJ, 632, 1157
c© 0000 RAS, MNRAS 000, 000–000
http://arxiv.org/abs/astro-ph/0702192
http://arxiv.org/abs/astro-ph/0610556
	Introduction
	Surface gravity measurement
	Application to HD209458b
	Application to all known transiting extrasolar planets
	Summary and discussion
	Acknowledgements
