Star Planet Systems and Eclipsing Binary Models

In EB models or programs we need to characterize planets by those parameters usually used to describe stars. The fundamental parameters are mass, radius, and temperature. A star-planet (or other low-luminosity object) system, with transits and radial velocities for the star only, is analog to a single-lined spectroscopic and detached EB. The orbital period, P, can be obtained from either radial velocities or light curves of the system and is usually the most precisely determined quantity. The radial velocity curve provides the eccentricity, e, and the radial velocity amplitude, K of the parent star. From the transit light curves one can derive the inclination, i, and relative radii r+ and rp of star and planet with respect to the semi-major axis, a.

5.4.3.1 Comparing Stars, Brown Dwarfs, and Planets

To estimate reasonable initial values of the planet's parameters we start by comparing stars, brown dwarfs, and planets. Deuterium burning begins at a mass ~ 13Mj, making this a convenient dividing line for planets and low-mass stellar objects (Saumon et al. 1996). Objects of mass greater than this are assumed to be brown dwarfs or stars; objects with masses greater than ~ 75MJ (or 0.072 M0), with solar composition, stars. For stars with no metals, this limit increases to 90MJ. One may distinguish among these three types of objects, namely a planet from a brown dwarf, or a brown dwarf from a star, at least partly through spectral characteristics. One criterion to distinguish between a brown dwarf and a star is the presence of lithium (Li), which is easily destroyed in stars through large-scale convection. Basri (2000, p. 494) argues that any object with spectral class later than M7, in which Li is detected, must be substellar. Chabrier & Baraffe (2000) summarize the characteristic spectral features with temperature for the cool end of the sequence as follows (the temperature limits are approximate only).

• <4000 K: M dwarfs. Most of the hydrogen is in the form of H2; and most of the carbon in CO. O is bound mainly in TiO, VO, and H2O, some in OH and in monoatomic O, and metal oxides. Metal hydrides (e.g., CaH, FeH, MgH) are also present. In optical spectra, TiO and VO dominate; in the IR, H2O and CO features are seen.

• <2800 K: O-rich compounds condense in the atmosphere; possibly perovskite (CaTiO3) may be present.

• <2000 K: L dwarfs. [Example: GD 165B]. Some TiO remains, but metal oxides and hydrides disappear from the spectra. Alkali metals are present in atomic form. Some methane may be seen.

• <1800 K: Refractory elements (e.g., Al, Ca, Ti, Fe, V) condense into grains. Corundum (Al2O3), perovskite condense. Depending on the pressure, rock-forming elements such as Mg, Si, Fe may condense as metallic iron, forsterite (Mg2SiO4), or enstatite (MgSiO4).

• -1700K (to -1000K): Cross-over to methane or T dwarfs. [Example: Gliese 229B]. Methane absorption strong in H (1.7:m), K (2.4 ^m), and L (3.3 ^m), giving rise to steep spectrum at shorter wavelengths, with J - K < 0, but with I - J > 5.

It appears that objects with temperatures below about 1300 K or so may qualify as planets, but such limits are not without controversy.

Consequently it is more prudent to use the mass range rather than spectral classification to distinguish planets from brown dwarfs.

We can conclude that we could model a star-planet pair in an EB program by a cool secondary putting its temperature T2 to, say, 500-1,000 K, a small mass ratio, say, 0.01 < q < 0.1, and a small ratio, r2/r1, of radii. For optical light curves of a transit, the planetary color is effectively black relative to the star, so the planetary temperature is not critical. This is not true, however, if thermal infrared light curves are available. Indeed, occultations (eclipse of the planet by the star) have now been observed in the thermal infrared, thanks to the Spitzer Space Telescope.

5.4.3.2 Transit Geometry and Modeling Approaches

The transit geometry is identical to that of a small star transiting a large one. The plan and elevation views of the geometry can be seen in Figs. 5.1 and 5.2 taken from Williams (2001). The orbital radius is a, the planetary and stellar radii are, respectively, r and R, and the angles subtended by the planet and by the star are a1 and a2, respectively. The star's distance from the Sun is d. From Fig. 5.2, it is seen that the quantity a = a1 + a2 is related to the planet's longitude at first contact, 0, by the expression a - d - a (5.4.1)

sin a sin 0 As d > a we get for sufficiently small angles 0 < 10

Fig. 5.1 The top part of the figure, Fig. 1.14 in Williams (2001), shows the path of the planet as it transits the disk of the star. The bottom part of the figure is the corresponding light curve for the transit. Courtesy Mike D. Williams, University of Calgary, Calgary, AB, Canada
Fig. 5.2 The star has radius R. The planet's orbit has radius a and period P. The observer is located at distance d away, and the star has an angular radius a2. Courtesy Mike D. Williams, University of Calgary, Calgary, AB, Canada

Based on the transit geometry with its accompanying requirement of sphericity, one could get approximate solutions by exploiting the analytic formulae, as outlined, for instance, by Seager & Mallen-Ornelas (2003) or Kipping (2008). Although, analytic transit models cannot cover more complicated physics such as distortion of the star due to its rotation, or distortion of the planet if it is close to the star, nonlinear limb darkening, or stellar atmospheres to name a few, their solutions can provide initial parameter estimations to light curve programs.

The next step could be to use a simple light curve program such as EBOP, which simulates the components of an EB using biaxial ellipsoids. As EBOP can also be restricted to spherical objects, so systematic effects arising from the assumption of a physical shape can be easily quantified. The EBOP model has been shown to work well for transiting extrasolar planetary systems by a number of authors, including Gimenez (2006), Wilson et al. (2006), Shporer et al. (2007), and Southworth et al. (2007b). Southworth (2008) provided homogeneous studies of 14 well-observed transiting extrasolar planets, among them also HD 209458, based on a modified version of EBOP. His study focused on the effect of limb darkening on the accuracy and error limits of the solution. To include more detailed physics of the star, the WD program is an appropriate candidate. As it uses Roche geometry, it very accurately reproduces distorted surfaces if the Roche surface grid are carefully generated. In addition it can model the star's radiation properly.

5.4.3.3 Representing Planets in the WD Model

In this section we review a few literature cases in which a star-planet system has been modeled with EB models and the WD program, see also Milone et al. (2004a). The first problem to be resolved in modeling planetary transits using EB models is to find an appropriate discretization or resolution of time, or equivalently, phase. The relative sizes of the transiting and transited object determine the scale. For the transit of a Jupiter-sized object across the disk of the Sun, 0.1 is a good approximation to rp. Adequate phase sampling in such a case requires longitudinal grid elements of order 30. This is quite similar to that usually used in stellar eclipse modeling, and holds for both occultation and transit eclipses. One may model the transit of a white dwarf, with characteristic radius of the Earth, across the disk of a red dwarf (i.e., a red main sequence star), with characteristic radius of, say, 0.5R0, appropriate for an M2V star. But in this case, one would need a resolution of 0.006 or about 165 grid elements to be able to sample ingress and egress adequately. The occultation eclipse is not observable in the visible (we will discuss the infrared situation in the next section), so the transit eclipse alone must determine the period and the geometry sufficiently well to yield the planet's parameters.

The second problem arises from the relative surface brightnesses. In visible light the contrast between the stellar surface brightness of the two objects determines the relative eclipse depths. In the case of a white dwarf transiting a red dwarf, visible light curves may not even register a dip, because the eclipsed surface brightness of the red star may be too low. In such a case, infrared photometry is necessary to observe a sufficient depth for analysis. In the case of a cataclysmic variable (CV), the red star has filled its inner lobe so the infrared light curve may reveal sufficient curvature in the light curve to demonstrate the shape of the cool secondary. But in this case, the occultation eclipse will be very deep in the visible. For the transit of a much cooler star across a solar-type star, however, the cooler star's contribution to system light may yield a negligible dip at primary minimum. This is the case for planetary transits, and for most purposes, at least for optical light curves, we may assume zero surface brightness of the planet during the transit.

5.4.3.4 HD 209458b: Transit Analysis of an ExtraSolar Planet

This planet was the first extrasolar planet to have its radius determined from transit analysis and the first to have had a constituent of its atmosphere detected. The HST observations were so precise that both extensive rings and satellites of this object can be ruled out (Brown et al. 2004). The HST and previous light curves were analyzed with new light curve analysis programs in use at the University of Calgary. Williams (2001) assumed the values M/M0 = 1.09±0.01 and R/RQ = 1.145±0.003 for the star HD 209458a, and a = 10.06R0. From a simultaneous analysis of transit light curve (Fig. 5.3) and radial velocity curve (Fig. 5.4) he derived the following best-fit parameters: Mp/Mj = 0.69 ± 0.01, Rp/Rj = 1.37 ± 0.01, P/days = 3.52478 ± 0.00005, E0 = 2451254.587 ± 0.002, e = 0.0000 ± 0.0001, and i = 86.54 ± 0.02. These results were more precise than previous published results.

The radius of this planet is larger than expected for a planet less massive than Jupiter because of its proximity to the star, which increases its equilibrium temperature and therefore the pressure scale height of the atmosphere:

Mmug

Phase

Fig. 5.3 This figure, Fig. 4.3 in Williams (2001), shows the HST transit light curve and best-fit model obtained by Williams. Courtesy Mike D. Williams, University of Calgary, Calgary, AB, Canada

Phase

Fig. 5.3 This figure, Fig. 4.3 in Williams (2001), shows the HST transit light curve and best-fit model obtained by Williams. Courtesy Mike D. Williams, University of Calgary, Calgary, AB, Canada

Phase

Fig. 5.4 This figure, Fig. 4.1 in Williams (2001), shows the radial velocity curve of HD 209458 and best-fit model obtained by Williams. Courtesy Mike D. Williams, University of Calgary, Calgary, AB, Canada

Phase

Fig. 5.4 This figure, Fig. 4.1 in Williams (2001), shows the radial velocity curve of HD 209458 and best-fit model obtained by Williams. Courtesy Mike D. Williams, University of Calgary, Calgary, AB, Canada where H is the pressure scale height, k is the Boltzmann constant, p is the mean molecular weight, mu is the atomic mass unit, basically the mass of the proton, and g is the acceleration of gravity.

The mean density of HD 209458b from Williams's work is found to be 330 ± 10 kg/m3, less than half that of Saturn. The atmosphere is clearly distended and a reasonable explanation has been advanced to explain it. Observations (Vidal-Madjar et al. 2004) suggest the existence of a trailing cloud of hydrogen, carbon, and oxygen, indicating hydrodynamic loss of atmosphere from this planet. If the observations are fully confirmed, nothing could better illustrate the dynamic character of such a planet. These authors suggest that planets older and closer to their parent stars than HD 209458b may have been deprived of their atmospheric envelopes and become a new class of planets (chthonian). The confirmed OGLE planets do constitute a closer and therefore hotter class of hot Jupiters, but their sizes appear to be smaller than that of HD 209458b. The system TrES-1 similarly appears to be smaller. Recent planetary modeling also suggests that HD 209458b is anomalously large, but it is not alone. It is being joined by a growing number of low-density extrasolar planets. HAT P-1b and WASP-1b, from two of the many surveys now being undertaken, are two examples.

5.4.3.5 The OGLE-TR-56 Star Planet System

Vaccaro & Van Hamme (2005) simultaneously fitted light and velocity data for the star-planet system OGLE-TR-56 with the WD program. They solved for orbital and planet parameters, along with the ephemeris using all currently available observational data: one photometric light curve and the star's radial velocity curve. The mass, Ms, and temperature, 71, for the star (OGLE-TR-56a) were kept fixed at values derived from spectral characteristics and stellar evolutionary tracks (Cody & Sasselov, 2002; Sasselov, 2003), namely Ms = 1.04M0 and T = 5900 K. A logarithmic limb-darkening law was adopted with coefficients x, y from Van Hamme (1993). The star's rotation rate F1 was set to 0.06 corresponding to a rotational period of 20 days (Sasselov, 2003).

In the WD program, adjustable radial velocity-related parameters are the semimajor axis, a, systemic velocity, VY, and mass ratio, q. As in a single-lined binary, the mass ratio q cannot be determined from the velocity curve, they proceeded as follows: From an initially assumed planetary mass, Mp, they adopted q = Mp/Ms. An initial value of the semi-major axis, a, follows from known period, P, and Kepler's third law

follow q and the planet's mass M p = qMs.

Time instead of phase was the independent variable, and ephemeris parameters (reference epoch T0, period P, and possibly rate of period change dP/dt) were fitted together with the other parameters (inclination i, a and systemic velocity, VY, Roche potentials, and ^2). Note that q was kept fixed. The result of this fit were values for a, i, and other adjustable parameters consistent with the fixed mass ratio. However, a, P, and q in Kepler's law (5.4.4) could lead to a different stellar mass, Ms. Therefore, for given Ms from (5.4.4) a new mass ratio is computed from

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The data fitting procedure was repeated until the triple (a, P, q, Ms) converged to values yielding a stellar mass from (5.4.4) consistent with the pregiven value of Ms.

Their results are in good agreement with parameters obtained by other authors but have significantly smaller errors for i and VY, slightly smaller errors for period, and larger errors for Ms and Mp. Especially, the value and error of the stellar radius, found by fitting the light curve data, agree very well with the value found by other authors who fit evolutionary tracks. The authors found no significant change in orbital period that may be due to orbital decay.

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