Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://lnfm1.sai.msu.ru/~marat/articles/eng/HD_209458_eng.pdf Äàòà èçìåíåíèÿ: Wed Nov 17 21:22:28 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 23:28:47 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 63

ISSN 1063-7729, Astronomy Reports, 2010, Vol. 54, No. 12, pp. 1105­1124. c Pleiades Publishing, Ltd., 2010. Original Russian Text c M.K. Abubekerov, N.Yu. Gostev, A.M. Cherepashchuk, 2010, published in Astronomicheski Zhurnal, 2010, Vol. 87, No. 12, pp. 1199­1220. i

Light Curve Analysis for Eclipsing Systems with Exoplanets. The System HD 209458
M. K. Abubekerov, N. Yu. Gostev, and A. M. Cherepashchuk
Sternberg Astronomical Institute, Moscow State University, Universitetskii pr. 13, Moscow, 119992 Russia
Received May 31, 2010; in final form, July 1, 2010

Abstract--We have analyzed precision light curves for HD 209458, a binary with an exoplanet. The parameters obtained at different epochs and different wavelengths are in good mutual agreement when confidence regions are used to calculate the uncertainty intervals. We demonstrate the effectiveness and reliability of our new method for estimating the uncertainty intervals. Reliable estimates are provided for the linear and quadratic limb-darkening coefficients of the star and their confidence intervals (uncertainties). ° We find that the wavelength dependence for the limb-darkening coefficients at = 3201-9708 A differs significantly from the corresponding theoretical relation based on thin model stellar atmospheres. DOI: 10.1134/S1063772910120048

1. INTRODUCTION Light curves with unique precision for exoplanet transits across stars have been obtained during recent years (cf., for instance, [1­4]) thanks to various space missions (HST, CoRoT, Kepler). Brown et al. [1] presented an analysis of an HST light curve obtained for HD 209458 in 2000. An analysis of multicolor HST light curves for this object obtained in 2003 was performed by Knutson et al [2]. The radii of the exoplanet and star, orbital inclination, and the stellar limb-darkening coefficients were determined in both studies. The most detailed study of a series of HST observations is that of Southworth [5], who derived these same parameters for various limbdarkening laws. As often happens in practical cases, the very high precision of the observations, on one hand, enabled Southworth [5] to derive the most reliable parameters for the binary, but, on the other hand, led to certain difficulties in interpreting the observations. First, the central values of the geometrical parameters of the model found for the different wavelengths, , demonstrated a considerable scatter, appreciably larger than the parameter uncertainties estimated using the Monte Carlo technique. Second, the geometrical parameters derived via light curve analyses for different epochs do not quite agree within the errors. These difficulties are not due to the specific properties of the applied model, since the results obtained from the high-accuracy light curves of HD 209458 using different models (two spherical stars [6] and two biaxial ellipsoids [5, 7, 8]) agree well.

These difficulties are most likely due to the fact that Southworth [5] estimated the uncertainties of the resulting parameters using the Monte Carlo method, which is known (see, for instance, [9­12]) to provide only "internal" uncertainties (those reflected by the statistics of the normal distribution of the obtained central parameter values under the strict assumption that the model is perfectly correct), which can be too low by a factor of three to five [9]. For this reason, we undertook a new analysis of the high-precision light curves of HD 209458 from [1, 2]. We estimated the parameter uncertainties using both the differential-correction and confidence-region methods (see, for example, [13]). This latter method uses the statistics of the "external" distribution of the observed data points for the light curve generated by these normally distributed measurements--the residual functional, which depends on the square of the differences between the observed and theoretical values (a statistic distributed as 2 , where M is the number of observed data M points in the light curve). In contrast to the Monte Carlo and differentialcorrection methods, which are equivalent [11, 12], the searches for the central parameter values and their uncertainties using confidence regions are both carried out using the same statistics (for example, a 2 distribution). In this case, we estimate the "exterM nal" uncertainties of the parameters, independent of the particular distribution for the derived parameters' central values. In addition, using statistics with a 2 M distribution makes it possible to avoid the artificial assumption that the applied model is perfect.

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By using the parameter uncertainties based on confidence regions (the "external" uncertainties), we are able to reconcile the fitting results for the multicolor light curves of HD 209458 and the light curves obtained for different epochs. We also confirm the existence of a significant discrepancy between the observed and theoretical wavelength dependences for the star's limb-darkening coefficients found in [5]. 2. FITTING TECHNIQUE We used a model with two spherical stars in a circular orbit, without reflection and ellipsoidal effects. According to data available for the exoplanet CoRoT-1b [3], whose orbital period (Porb = 1.509d ) is half that for the HD 209458 system (Porb = 3.52474859d ), the total observed amplitude of the reflection effect from the planet should not exceed 0.0001m , with the amplitude of the ellipsoidal effect for the optical star probably being appreciably smaller still. Consequently, brightness variations due to reflection and ellipsoidal effects will be no larger than 10-5 mag within the eclipse (whose duration for CoRoT-1b is 0.07 of the orbital period), which is negligible. A spherical approximation for the planet seems quite satisfactory, given the uncertainty in its shape, related to the possible existence of a semitransparent atmosphere, which could be as large as 5% of the planet's radius [14], possible rapid axial rotation of the planet and rotational deformation, and the small fraction of the planet that fills its Roche lobe ( < 0.5). When calculating the light curves, we described the brightness distribution across the stellar disk using a linear limb-darkening law with the linear limbdarkening coefficient x: I () = I0 2 1-x+x 1- 2 r (1)

component 2 (the planet) and the brightness at any point of the disk of this component are assumed to be zero. Component 2 (the planet) eclipses component 1 (the star) at orbital phase = . The unit of length in our model is the distance between the centers of the star and planet a = 1, and the orbit is assumed to be circular. There is no "third light" in the model. The fitted model parameters are the radii of the star and planet r1 and r2 , the orbital inclination i, the limb-darkening coefficient x1 , and, in the case of the quadratic limb-darkening law, the quadratic limbdarkening coefficient y1 . Let us introduce the following new variables: X0
(1)

= I0 (1 - x1 ), X0
(1) (1) (1)

(1)

X1

(1)

= I0 x1

(1)

(3)

for the linear limb-darkening law and = I0 (1 - x1 - 2y1 ), X2
(1)

(4)
(1) 1

X1

(1)

= I0 (x1 +2y1 ),

= I0 y

for the quadratic limb-darkening law. We can then write the brightness in the linear limb-darkening law as I
(1)

() =

X0 + X1

(1)

(1)

1-

2 2 r1

,

(5)

and in the quadratic law as I = X0 + X1
(1) (1) (1)

()
2 2 (1) + X2 2 2 r1 r1

(6) .

1-

With these variables, the brightness at a point on (1) (1) the stellar disk depends linearly on X0 , X1 , and X2 , while the corresponding brightness for the nonlinear limb-darkening law differs in a single term, with (1) the coefficient X2 . Component 2 eclipses component 1 at orbital phase = . The total brightness of the star (component 1), equal to the total out-of-eclipse brightness of the system, is
r
1

(1)

and a quadratic limb-darkening law that differs from the linear law in an additional term containing the quadratic limb-darkening coefficient y : I () = I0 1 - x 1 - -y 1 - 1- 2 r2 1-
2



2 r2

(2) L
full

= 2
0

I

(1)

()d
(1)

(7) 1- x1 3

.

2 (1) (1) 2 = r1 X0 + X1 3
r
1

2 = r1 I0

Here, is the polar distance from the center of the stellar disk, I0 the brightness at the disk center, and r the radius of the stellar disk. Below, we denote the brightness at the center of the disk of component 1 (1) (2) (the star) I0 . The brightness I0 at the center of

for the linear limb-darkening law and L
full

= 2
0

I

(1)

()d

(8)

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2 (1) 1 (1) (1) 2 = r1 X0 + X1 + X2 3 2 x1 y1 2 (1) - = r1 I0 1 - 3 6 for the quadratic limb-darkening law. The total brightness of the star for the quadratic limb-darkening law is
r

and also introduce polar coordinates with the origin at the center of the eclipsed star's disk and the polar angle increasing from the disk center of the eclipsed component, f , towards the disk center of the eclipsing component, n.Then, L
dec

(,rf ,rn ,X0 ,X1 ,X2 )
2 rf

(f )

(f )

(f )

(11)

L

(s)

= 2
0

I

(s)

()d =
(f ) X0

=
0

(, , rn )I

(f )

( )d
dec 1

2 (s) 2 1 (s) 2 (s) 2 = X0 rs + X1 rs + X2 rs , 3 2 s = 1, 2. To make the form of the formulas used to calculate the minima of the light curve universal and reduce the number of equations, we will use the index n for the eclipsing component (nearer to the observer) and the index f for the eclipsed component (further from the observer). When actually computing the lightcurve minima at orbital phases -/2 < < /2 (or for cos > 0), the variable rn should be changed to r1 and rf to r2 . The opposite change is needed for orbital phases with cos < 0: rf should be changed to r1 and rn to r2 . In this new notation, the brightness reduction during the eclipse is L
dec

L

dec 0

(,rf ,rn )+ X1 L
(f ) X2

(f )

(,rf ,rn )

+

L

dec 2

(,rf ,rn ).

The expressions for calculating Ldec and Ldec were 0 1 derived in [11]. Analogous to the procedure used in [11] to derive the formula for Ldec ,we find for Ldec : 0 2 rf 2 3 (,,rn )d (12) Ldec (,rf ,rn ) = 2 2 rf
0

= (,rf ,rn ) -

2 rf

2

+

2 rn 2 2rf

2 22 + rn (,rn ,rf )

(,rf ,rn ,X0 ,X1 ,X2 ) =
S ()

(f )

(f )

(f )

1 2 2 2 2 +5rn + rf Q (,rf ,rn ) . 8rf
dec 2

(9)

The partial derivatives of L L
dec 2

are (13)

I

(f )

(S )dS,

(,rf ,rn ) 2 2 = 2 rn (,rn ,rf ) rf -
2 2 2 + rn + rf 2 2rf

where is the distance between the disk centers and S () is the overlap area of the disks. Following [11], we introduce the following functions to calculate the integral (9): , x < -1, Ax arccos x, -1 x 1, (10) 0, x>1 and Qx

Q(,rf ,rn ),

L

dec 2

(,rf ,rn ) = 2rf (,rf ,rn ) rf 2 - Ldec (,rf ,rn ), rf 2

(14)

x, x 0, 0, x < 0, x2 +2 - y 2x
2

L

(,x,y ) A

,

(,rf ,rn ) 2 2 = 2 rn 2 + rn rn rf 2 â (,rn ,rf ) - 2 rn Q(,rf ,rn ). rf

dec 2

(15)

Q (,rf ,rn ) Q
2 rf - ( - rn )2 2 ( + rn )2 - rf

,

For a circular orbit, the distance between the stellar-disk centers depends on the phase and orbital inclination i as (, i) = cos2 i +sin2 i sin2 . (16)

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The binary's light curve for the model with the linear limb-darkening law is described by the function
(1) (1) (1) L(, r1 ,r2 ,i,X0 ,X1 ) (1) (1) (1) 0 (, r1 ,r2 ,i)+ X1 L1 (, r1

respectively. This yields X0
(1)

(17) ,r2 ,i)

=

3Lfull (1 - x1 ) 2 r1 (3 - x1 )

(23)

= X0 L

for the linear limb-darkening law and X0
(1)

and with the quadratic limb-darkening law by the function L(, r1 ,r2 ,i,X0 ,X1 ,X2 ) = X0 L
(1) (1) 0 (1) (1) (1)

=

6Lfull (1 - x1 - 2y1 ) , 2 r1 (6 - 2x1 - y1 ) 6Lfull (x1 +2y1 ) 2 r1 (6 - 2x1 - y1 )

(24)

(18)

(, r1 ,r2 ,i)+ X1 L
(1) X2

(1)

(1) 1

(, r1 ,r2 ,i)

X1

(1)

=

+

L

(1) 2

(, r1 ,r2 ,i).
(1)

for the quadratic limb-darkening law. The inverse expressions for (23) and (24) are x1 = and x1 =
2 r1 (12X0 +11X1 ) - 12L 2 3r1 (X0 + X1 ) (1) (1) (1) (1) full

Expressions needed to calculate L0,1,2 were derived in [11, 12]. Note that Eq. (16) for Ldec in [12] 2 2 contains an error: the coefficient 1/rf in front of the integral is missing. Thus, Eqs. (12)­(15) here rather than (16)­(19) from [12] should be used to calculate Ldec and its derivatives. 2 Our models assume the total brightness Lfull to be known (normalized to unity in the computations). Using (7), we exclude the parameter X1
(1)

6L
(1)

full full

2 r1 X0 - 3L

(25)

,
(1)

(26)

y1 = When minimize rameters tion can
(1)

6L

full

2 - r1 (6X0 +4X1 ) (1) (1)

(1)

2 3r1 (X0 + X1 )

.

=

3Lfull 3 (1) 2 - 2 X0 2r1

(19)

in the model with the linear limb-darkening law and, using (8), we exclude the parameter
(1) X2

searching for the residual minima, we first the residual functional in the linear pa(1) (1) X0 and X1 , since this linear minimizabe carried out analytically, yielding ana(1)

2Lfull 4 (1) (1) = 2 - 2X0 - 3 X1 r1

(20)

lytic expressions for the parameters X0 (r1 ,r2 ,i) and X1 (r1 ,r2 ,i) and their derivatives corresponding to the minimum residual for r1 , r2 , and i fixed. The character of the statistical distribution of the minimum residuals remains unchanged in the case of a linear minimization, and only the number of degrees of freedom changes (decreases) for this distribution. Further minimization is then performed for a nonlinear function of three variables: r1 , r2 , and i. The differential-correction method directly finds the cen(1) (1) tral values of r1 , r2 , i, X0 , X1 , and their covariations. We then turn to the parameters r1 , r2 , i, x1 , and y1 using (25) and (26). The dispersion estimates in the differential-correction method are derived in the same way as for the model obtained with the corresponding change of variables [12]. Note that a non-linear minimization changes the statistical distribution of the minimum residuals [10], so that it only asymptotically approaches the 2 disM tribution (as M ). Since the number of data points in the HD 209458 light curves is large in our case (M 500), the minimization of the residuals between the observed and theoretical light curves should yield reliable asymptotic confidence regions for the fitted model parameters. We will designate
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in the model with the quadratic limb-darkening law. Substituting (19) and (20) into (17) and (18), respectively, we obtain equations for the light curves, with the total brightness of the system being fixed: L(, r1 ,r2 ,i,X0 ) = + X0
(1) (1)

(21)

3Lfull 2L 2r1

(1) 1

(, r1 ,r2 ,i)
(1) 1

L

(1) 0

3 (, r1 ,r2 ,i) - L 2
(1)

(, r1 ,r2 ,i) , (22) (, r1 ,r2 ,i)

L(, r1 ,r2 ,i,X0 ,X1 ) = 2L 2L r1 - 2L
(1) 2 full (1) 2

(1)

(, r1 ,r2 ,i)+ X0 (L
(1) (1)

(1)

(1) 0

(, r1 ,r2 ,i)) + X1 (L1 (, r1 ,r2 ,i) 4 (1) - L2 (, r1 ,r2 ,i)). 3
(1)

We can also exclude I0 from (3) and (4) by expressing it using the right-hand sides of (7) and (8),


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Table 1. Fitting of the observed light curve of HD 209458 from [1] using the linear limb-darkening law. The parameter uncertainties were estimated using the differential-correction and confidence-region methods assuming 2 (where P is P the number of parameters) and 2 (where M is the number of data points in the light curve) statistics M Parameter rs ,R rp ,R i, degrees x
2 red

Differentialcorrection method (2 ) 0.11469 ± 0.000759 0.014057 ± 0.0001149 86.48 ± 0.083 0.49452 ± 0.009306

Confidence-region method, 2 (95%) P 0.1147 ± 0.001186 0.01406 ± 0.0001808 86.48 ± 0.1318 0.4944 ± 0.01406 1.103

Confidence-region method, 2 (95%) M 0.1147 ± 0.00079 0.01406 ± 0.0001211 86.48 ± 0.13 0.4945 ± 0.00954

Parameters from [5] (1 ) 0.11482 ± 0.00035 0.014076 ± 0.000055 86.472 ± 0.038 0.494 ± 0.004 1.1457

the limb-darkening coefficients in the linear limbdarkening law x, without the subscript 1, while the linear and quadratic limb-darkening coefficients in the quadratic limb-darkening law will continue to be designated x1 and y1 . In some cases, we denote the stellar radius r1 and the planetary radius r2 as rs and rp , to make the notation clearer. 3. OBSERVATIONAL MATERIAL We analyze here precision transit light curves of the exoplanet binary HD 209458 from [1, 2] obtained with the Hubble Space Telescope (HST). The observed light curve presented in [1] was obtained in April­May 2000. The spectra were obtained using the STIS spectrometer with the G750M spectral grating. The observations were performed at ° 5813-6382 A with a resolution of R = / = 5440 (see [1] for more detail). The normalized light curve for the transit of the exoplanet across the stellar disk is displayed in Fig. 1. The light curve consists of 556 individual brightness measurements. The rms uncertainties for the individual measurements are between obs obs i = 1.13 â 10-4 and i = 2.47 â 10-4 (in units of the out-of-eclipse intensity) in different parts of the light curve. The relative uncertainties (expressed as fractions of the eclipse depth) are between 7 â 10-3 and 1.5 â 10-2 . HST data obtained between May 3 and July 6, 2003 are presented in [2]. The spectra were acquired using the STIS spectrometer with the G430L (2930­ ° ° 5670 A) and G750L (5320­10 190 A) spectral gratings. Applying two spectral gratings made it possible to cover a fairly wide wavelength range, from ° ° 2900 A to 10 300 A. For their further analysis, Knutson et al. [2] subdivided this wavelength range into ten equal parts, providing light curves in ten different photometric bands. The light curves for the central ° wavelengths 3201, 3750, 4300, 4849, and 5398 A
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consist of 505 individual brightness measurements each; the light curves for the central wavelengths ° 5802, 6779, 7755, 8732, and 9708 A consist of 548 individual measurements. The observed transit light curves from [2] are displayed in Fig. 2. The rms uncertainties of the individual measurements vary between obs obs i = 1.79 â 10-4 and i = 6.09 â 10-4 (or be-2 and 3 â 10-2 in fractions of the eclipse tween 10 depth). More detailed information on these observations and their reduction can be found in [2]. We assume that the observational errors obey a normal distribution and that systematic errors are negligible. 4. FITTING OF THE HD 209458 LIGHT CURVES FOR THE LINEAR LIMB-DARKENING LAW The fitted parameters in our analysis of the observed light curves are the radius of the exoplanet rp , the radius of the star rs , the orbital inclination i, and the linear limb-darkening coefficient x. The orbital period adopted for the binary was Porb = 3.52474859d [2], the ratio of the masses of the planet and star was taken to be q = mp /ms = 0.00055 [2], the orbit of the system was assumed to be circular, and we adopted a unit radius for the relative orbit.

4.1. Light Curve from Brown et al. [1]
Table 1 presents the central values of the derived parameters and their uncertainties estimated using the differential-correction method (equivalent, as we noted above, to the Monte Carlo method [11, 12]) and the confidence-region method assuming 2 (where P P is the number of derived parameters) and 2 (where M M is the number of observed data points) statistics. We selected the = 95% confidence level, corresponding to 2 in the differential-correction method, where is the standard deviation.


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L 1.000

0.995

0.990

0.985

0.980 165 170 175 180 185 190 195°

Fig. 1. Observed (points) and theoretical (solid curve) light curves for the binary with an exoplanet HD 209458 from [1]. In the bottom, we show the deviations of the observed brightness from the theoretical light curve calculated using the non-linear (quadratic) limb-darkening law.

Our test of the consistency between the model and observations demonstrated that the ratio of the minimum deviation to M - P (distributed according to a reduced 2 law with M -P degrees of freedom) is 2 d = 2 -P 1.103. Using the results from [12], re M we conclude that our model can be rejected at the 0.05889 significance level (this confidence level corresponds to a half-interval 1.889 ). Thus, our model is rejected at a very low significance level ( = 6%); therefore, it is not very good, but also not hopelessly bad, since we can estimate the confidence intervals using a statistic distributed as 2 at the confidence M level = 0.95 in order to obtain the most conservative uncertainty estimates for the parameters. Note that, for this observed light curve, the uncertainty estimates in the confidence-region method obtained assuming 2 statistics exceed those obtained asP suming 2 statistics. According to the formula for M the distribution of the ratio of these intervals from [12] (assuming that the model is perfectly correct), the probability of this occurring is low: approximately 4%. The fact that our model is not very good (in rejecting the model, we would be incorrect in less than 6% of cases and correct in more than 94% of cases) does not seem surprising. First, we used the linear limb-darkening law, which describes the brightness distribution across the stellar disk only roughly. In addition, our model does not take into account fine structure on the stellar disk: spots, plages, and active areas, whose sizes can be comparable to that of the eclipsing planet. We have also neglected possible intrinsic micro-variations of the star. These fac-

tors could cause short-time-scale irregularities of the star's brightness variations during its eclipse by the planet, exceeding the statistical uncertainty of the observations. Such irregularities are especially noticeable in the lower part of the light curve of Brown et al. (Fig. 1), where the brightness deviations from the light curve reach 5 â 10-4 mag (sometimes larger), appreciably in excess of the statistical uncertainty of the observations. It is precisely the fact that our model was found not to be very good that leads us to use the 95% rather than the 68% confidence level, as is usual in the case of "good" models. A suitable criterion of a "good" model is 2 d = 2 -P 1+ 2t,where t is of re M P (it follows from this condition, that the the order of M model be acceptable at the 50% confidence level as M ). Table 1 presents projections of the asymptotic confidence region D in the space of the four fitted parameters onto the axes of the parameters rp , rs , i, and x (the confidence intervals). The probability of such a projection of the confidence region D (confidence interval) emcompassing the true parameter value exceeds 95%. The probability that the exact solution is emcompassed by the asymptotic confidence region D is guaranteed to be close to the adopted probability of 95% (because the number of light-curve data points is high, M > 500). The probability that the exact solution will be encompassed by all the projections of the confidence region D (corresponding to the exact solution being in the parallelepiped of parameter space enclosing the confidence region D)
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1111

1.04

1.02

1.00

L

0.98

0.96

0.94

0.92 165 170 175 180 185 190 195°

Fig. 2. The observed light curves of the exoplanet binary HD 209458 from [2] at 3201, 3750, 4300, 4849, 5398, 5802, 6779, ° 7755, 8732, and 9708 A (from bottom to top). The corresponding distributions of the residuals are shown in the lower part of the panel. The solid curves are the theoretical light curves for the non-linear (quadratic) limb darkening.

is higher than the adopted probability, 95%.Note that all these statements concerning the probabilities of encompassing the exact solution have been strictly proven mathematically [13], and also confirmed with numerical modeling [11, 12]. Thus, adopting the projections of the confidence region D onto the parameter axes (confidence intervals) as the uncertainties of these parameters, we can definitely guarantee that the probability that the exact solution is encompassed by the confidence region D coincides with the adopted probability, 95%. This justifies our conservative uncertainty estimates for the fitted parameters as projections of the confidence region D onto the parameter axes (Table 1), which can be considered "external" errors of the fitted parameters rp , rs , i,and x. Let us clarify the method used to find the projection of the confidence region D onto the parameter axes. To find the projection of the four-dimensional confidence region D onto the axis of some paraASTRONOMY REPORTS Vol. 54 No. 12 2010

meter--for example, rp --we first minimize the deviation between the observed and theoretical light curves allowing all the parameters but the given one (i.e., rp ) to vary. The intersection of this curve with the straight line (critical level) corresponding to a given significance level is then found (for the chosen statistics: 2 , 2 ). The values of rp providing deviM P ations below the critical level when the minimization is carried out over all the remaining parameters (rs , i, x) are combined into the confidence interval, which is just the projection of the four-dimensional confidence region D onto the rp axis. This confidence interval encompasses the true value of rp with a probability exceeding the adopted confidence level, = 1 - . Here, it is guaranteed that the exact solution (combining the true parameters rp , rs , i, x) is encompassed by the four-dimensional confidence region D with the adopted probability, = 1 - .


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i ­ ic 0.10 0.05

­ 0.00015

­ 0.00005 ­ 0.05 ­ 0.10

0.00005

0.00015 c rp ­ rp

Fig. 3. The projection of the confidence region D (at the 95% confidence level) onto the (rp , i) plane obtained for our fitting of the light curve from [1] using the linear limb-darkening law. The smaller ellipse (solid curve) corresponds to the confidence region found assuming 2 statistics and the larger ellipse (dashed curve) to the confidence region found assuming 2 statistics. M P The rectangle corresponds to the projection of the confidence region at the level 2est found using the differential-correction method.

Figure 3 displays the projection of the confidence region D onto the rp , i parameter plane. The projection characterizes the shape of the confidence region D. It is impossible to reproduce the whole multi-dimensional region D in a plane. The rectangle is the projection of the region defined by the parameter uncertainties (at the 2 level) found using the differential-correction method. Because the linear limb-darkening fit to the precise light curve by Brown et al. [1] was rejected at a fairly low significance level (see above), the difference in the sizes of the projections of the confidence regions found using the differential-correction and confidence-region method is not large (however, see Figs. 11 and 12 below). The probability of encompassing the true value of each of the parameters with the ±2 interval corresponds to the adopted probability, = 95%. However, the probability to simultaneously encompass the true values with all the intervals found in the differentialcorrection method is lower than this probability (by approximately a factor of 1.5). Thus, the probability of the corresponding four-dimensional uncertainty region simultaneously encompassing all the fitted parameters is appreciably reduced if we adopt the "internal" parameter errors found in the differentialcorrection method or Monte Carlo method under the assumption that the model is perfectly correct (see also Figs. 11 and 12 below). This can explain why the results of fitting observations acquired on different epochs often disagree beyond the "internal" errors and the parameter uncertainties derived in the differential-correction (or Monte Carlo) method are often unrealistically small.

4.2. Multi-Color Light Curves of Knutson et al. [2] The light curves for HD 209458 obtained in [2] at = 3201, 3750, 4300, 4849, 5398, 5802, 6779, 7755, ° 8732, and 9708 A are presented in Fig. 2 (the wavelength increases from bottom to top). Tables 2­4 present the results of fitting these light curves using various methods to estimate the uncertainties of the fitted parameters: the confidence-region method assuming 2 statistics (Table 2; dashes correspond M to the degeneration of the "exact" confidence region into an empty set because the model is rejected at the adopted significance level, = 5%)and 2 statistics P (Table 3) and the differential-correction method (Table 4; assuming a normal distribution for the resulting central values of the fitted parameters and that the model is perfectly correct). The 2 d value for the re ° = 5802 A light curve is especially large (2 d = re 1.299; the model is rejected at a very low significance level: = 5.168 â 10-6 ). This light curve was probably influenced by some additional sources of error. The 2 d values for the other wavelengths are within re reasonable limits: 2 d = 1.021-1.122. re Our test for consistency between the model and observations demonstrates that the minimum deviation for the reduced 2 statistics exceeds unity (Table 3). Using the 2 statistics in the fitting of the light M curves at = 3201, 3750, 4849, 5398, 6779, 7755, ° 8732 A demonstrates that our model is not rejected at the = 5% significance level (and can be accepted). In all these cases, it is possible to derive "exact" confidence regions D (i.e., encompassing the true
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Table 2. Results of fitting the observed multi-color light curves of HD 209458 from [2] using the linear limb-darkening law. The parameter uncertainties were found using the confidence-region method and 2 statistics; = 0.95 M ° , A 3201 3750 4300 4849 5398 5802 6779 7755 8732 9708 rs 0.112422 0.111096 0.113224 0.113315 0.114474 0.114536 0.115384 0.114060 0.115301 0.114814 M (rs ) 0.00909003 0.00518492 ­ 0.00215386 0.00332352 ­ 0.00210354 0.00232615 0.00451503 ­ rp 0.0136815 0.0135058 0.0138670 0.0138638 0.0140443 0.0141229 0.0141542 0.0139412 0.0141478 0.0141637 M (rp ) 0.00168653 0.000908670 ­ 0.000353872 0.000534038 ­ 0.000327598 0.000348055 0.000672640 ­ i 87.0418


M (i) 1.25441 ­ 0.257024 0.387603 ­ 0.240432 0.266790 0.515776 ­


x 0.839899 0.755113 0.702653 0.617381 0.561101 0.534661 0.436149 0.377645 0.317885 0.276457

M (x) 0.112573 0.0601379 ­ 0.0267281 0.0420762 ­ 0.0332838 0.0363709 0.0829246 ­

87.0084 86.9607 86.6501 86.5317 86.4712 86.3841 86.5030 86.3800 86.3856

0.685166

Table 3. Results of fitting the observed multi-color light curves of HD 209458 from [2] using the linear limb-darkening law. The parameter uncertainties were found using the confidence-region method and 2 statistics; = 0.95 (the last P column contains the reduced 2 d and the corresponding significance level ) re ° , A rs P (rs ) rp P (rp ) i 86.8383


P (i)

x

P (x)

3201 0.112789 0.00564078 0.0137580 0.00103572

0.721357 0.833506 0.0703594 0.412798 0.754423 0.0371808 0.241463 0.702492 0.0233387 0.193893 0.617462 0.0201586 0.198908 0.561665 0.0217482 0.143095 0.436699 0.0198472 0.181210 0.378139 0.0247354 0.226904 0.323146 0.0361643 0.301402 0.275733 0.0483915

2 - P M () M -P 1.103 (0.069) 1.067 (0.174) 1.121 (0.041) 1.084 (0.118) 1.046 (0.271)
-6

3750 0.111132 0.00318612 0.0135114 0.000556307 86.9630 4300 0.113232 0.00197803 0.0138682 0.000343468 86.6565 4849 0.113315 0.00162566 0.0138639 0.000267189 86.6468 5398 0.114449 0.00171209 0.0140408 0.000274972 86.5227 5802 0.114557 0.00138946 0.0141275 0.000228711 86.4703 6779 0.115386 0.00125488 0.0141545 0.000195295 86.3806 7755 0.114069 0.00158054 0.0139423 0.000236676 86.4989 8732 0.115295 0.00199596 0.0141482 0.000297907 86.3629 9708 0.114743 0.00268813 0.0141555 0.000389998 86.4012

0.162999 0.533942 0.0205499 1.299 (5.168 â 10 1.061 (0.186)

)

1.0739 (0.138) 1.021 (0.403) 1.122 (0.0334)

values with the adopted probability, = 95%) and to find projections of the region D onto the rp , rs , i, and x parameter axes (confidence intervals), which represent the most conservative, "external" uncertainties of the parameters (Table 2). Our model is rejected at the < 5% significance level (is "bad") ° for three light curves ( = 4300, 5802, 9708 A). For these light curves, we are able to find only the external uncertainties of the parameters (confidence intervals) for the 2 statistics, corresponding to the asymptotic P confidence region D, as well as the internal uncertainties of the parameters assuming normal statistics for the derived central parameter values using the
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differential-correction method. However, as is noted in [10], we should not forget that, in practice, we are dealing not with a complete observed random function (light curve) but a particular observational realization of this function. Thus, it is possible that the encountered difficulties with the fitting (the model is rejected at a rather low significance level, so that the confidence region D for the 2 statistics degenerates M into an empty set) are due to chance fluctuations of the observed function rather than flaws in the model. The applied model could be quite satisfactory for a different observational realization of the observed function, for example, for a light curve obtained at a


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Table 4. Results of fitting the observed multi-color light curves of HD 209458 from [2] for the linear limb-darkening law. The parameter uncertainties were found using the differential-correction method; the 2 uncertainties are given ° , A 3201 3750 4300 4849 5398 5802 6779 7755 8732 9708
c rs c 2est (rs ) c rp c 2est (r2 )

i

c

2est (ic ) 0.537230


xc 0.828541 0.753992 0.702653 0.617614 0.561705 0.534661 0.436943 0.379030 0.323969 0.276457

2est (xc ) 0.0505626 0.025672 0.0153839 0.0134061 0.0140619 0.0144226 0.0125312 0.0157685 0.0223688 0.0342494

0.113113 0.111175 0.113224 0.113302 0.114456 0.114535 0.115394 0.114035 0.115326 0.114814

0.00424010 0.00224100 0.00127895 0.00107659 0.00107583 0.00093831 0.00076714 0.00098878 0.00120960 0.00185497

0.0138328 0.0135199 0.0138670 0.0138618 0.0140421 0.0141229 0.0141560 0.0139372 0.0141535 0.0141637

0.000794048 0.000396390 0.000220652 0.000176852 0.000172042 0.000154114 0.000118926 0.000146933 0.000179209 0.000267488

86.7178

86.9333 86.6507 86.6440 86.5177 86.4712 86.3777 86.4998 86.3547 86.3856

0.292384 0.154521 0.1282050 0.1239884 0.1097888 0.0869080 0.1119664 0.1355734 0.2046620

different epoch, for which it would be rejected at a high significance level, enabling derivation of the confidence region D for the 2 statistics. This suggests M it could be meaningful to search for the asymptotic confidence region for the 2 statistics (which, by P definition, never degenerates into an empty set), even when the model is rejected at a fairly low significance level and is "bad." It is interesting to compare the geometrical parameters of the HD 209458 system derived for different epochs. Table 5 presents the parameters ° for the (5813-6382) A light curve obtained by Brown et al. [1] in April­May 2000 and the light ° curve with the central wavelength 6779 A ( = ° 6279-7279 A) obtained by Knutson et al. [2] in May­ July 2003 together with their uncertainties. The halfintervals for the largest differences in the central pa(2000) (2003) - rp | |rp = 0.00004775 , rameter values are 2 (2000) (2003) |i(2000) - i(2003) | - rs | |rs = 0.00034050 , = 2 2 . These differences are within the internal 2 0.0515 uncertainties and certainly within the M and P confidence intervals (Table 5). It is also of interest to compare the geometrical parameters (rp , rs , i) for the light curves of HD 209458 obtained by Knutson et al. [2] for the same epoch (May­July 2003) but at different wavelengths. The results of this comparison are collected in Table 6. The values of rp , rs ,and i derived for different wavelengths do not quite agree within the internal 2 uncertainties. At the same time, these geometrical parameters are in good agreement within the external P uncertainties, and especially the M uncertainties.

5. WAVELENGTH DEPENDENCE OF THE LINEAR LIMB-DARKENING COEFFICIENT Since a simple model with linear limb-darkening applied to the light curves of Brown et al. [1] and Knutson et al. [2] is not hopelessly "bad" and is rejected only at a significance level of several percent ° (except for the = 5802 A light curve), it is reasonable to analyze the dependence of the limb-darkening coefficient x in the linear limb-darkening law on the wavelength and compare this dependence to the corresponding theoretical relation that follows from the theory of thin stellar atmospheres. Table 7 presents our values for x and their uncertainties (2 , P , M ) as functions of . We minimized the residual functional for each in four parameters: rp , rs , i, x. The dashes correspond to cases when the model is rejected at the significance level < 5%. Table 7 shows that, with the most conservative estimates of the external uncertainties in x (for the 2 statistics), the relative uncertainties in x are M ° 8%-13% in the blue ( = 3201-3750 A), 4%-8% in ° the visible ( = 4849-6779 A), and 8%-26% in the ° red ( = 7755-8732 A). The dependence of the linear coefficient x (in the linear limb-darkening law) on the wavelength is presented in Figs. 4 and 5. We used the results of Claret [15] for the ugr iz and UB V RI J photometric systems for the theoretical wavelength dependence of x. Using different photometric systems indicates the extent to which the x() relation is determined by the choice of photometric system (the need for such a check was emphasized in [5]).
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Table 5. Differences between geometrical parameters derived from light curves for different epochs: internal and external uncertainties Largest parameter differences |rp
(2000)

Differences in central parameter values 0.000048 0.00034 0.051


Differential-correction method (2 ) 0.00012 0.00077 0.087


Confidence-region method, 2 (95%) P 0.00018 0.0012 0.13


Confidence-region method, 2 (95%) M 0.00033 0.0021 0.24


(2000) |rs

|i

(2000)

- rp | 2 (2003) - rs | 2 - i(2003) | 2

(2003)

Table 6. Scatter of the derived central values of the fitted geometrical parameters found from 10 light curves for various wavelengths (the ratios of the (max­min)/2 semi-interval to the corresponding mean values of the uncertainty intervals from Tables 2, 3, 4 are given in brackets) Largest scatter of the central values (min­max) 0.111096­0.115394 0.121609­0.123362 86.3547­87.0400 Differential-correction method (2 ) Confidence-region method, 2 (95%) P Confidence-region method, 2 (95%) M

Parameter rs rp k = rp /rs i, degrees

0.112373­0.115507 (1.37) 0.112076­0.115984 (1.10) 0.109620­0.117820 (0.52) 0.121876­0.123264 (1.26) 86.36­86.74 (1.82) 0.12157­0.12357 (0.88) 86.31­86.77 (1.49) 0.12077­0.124219 (0.47) 86.13­87.15 (0.67)

0.0135058­0.0141542 0.0137074­0.0142326 (1.23) 0.013667­0.014293 (1.04) 0.013220­0.014600 (0.47)

Table 7. Limb-darkening coefficients derived for the linear limb-darkening law. The central values and indicated 2 uncertainty intervals were found using the differential-correction method and the confidence-region method assuming 2 and 2 statistics ( = 0.95) P M ° , A 3201 3750 4300 4849 5398 5802 6779 7755 8732 9708 xc 0.828541 0.753992 0.702653 0.617614 0.561705 0.534661 0.436943 0.379030 0.323969 0.276457 2est (xc ) 0.0505626 0.0256720 0.0153839 0.0134061 0.0140619 0.0144226 0.0125312 0.0157685 0.0223688 0.0340324 x 0.833506 0.754423 0.702492 0.617462 0.561665 0.533942 0.436699 0.378139 0.323146 0.275733 P (x) 0.0703594 0.0371808 0.0233387 0.0201586 0.0217482 0.0205499 0.0198472 0.0247354 0.0361643 0.0483915 x 0.839899 0.755113 ­ 0.617381 0.561101 ­ 0.436149 0.377645 0.317885 ­ M (x) 0.112573 0.0601379 ­ 0.0267281 0.0420762 ­ 0.0332838 0.0363709 0.0829246 ­

Figure 4 shows the central values of x derived using the differential-correction method together with their internal 2 uncertainties. In general, our results are in good agreement with those of Southworth [5]. Figure 4 shows that the linear limb-darkening coeffiASTRONOMY REPORTS Vol. 54 No. 12 2010

cients x derived from the observed light curves do not agree with the theoretical values. Figure 5 displays the central values of x and the most conservative estimates of the external uncertainties obtained using the confidence-region method and 2 statistics. Note that we adopted = 95%. M


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x 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 2000 4000 6000 8000 10 000 12 000 , å ugriz UBVRIJ

Fig. 4. Wavelength dependence of the limb-darkening coefficient x for HD 209458 for the linear limb-darkening law. The limb-darkening coefficients were derived from the light curves of [2]. The quoted 2 uncertainties in the limb-darkening coefficients were determined using the differential-correction method. The theoretical limbdarkening coefficients in the ug r iz and UB V RI J photometric systems are plotted according to [15].

x 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 2000 4000 6000 8000 10 000 12 000 , å ugriz UBVRIJ

This result is very important for improving current models of thin stellar atmospheres. Tables 2­4 contain the x() values and their uncertainties obtained by minimizing the residuals between the observed and theoretical light curves in the four parameters rp , rs , i, and x. In this case, each x value for a given wavelength corresponds to its own set of the geometrical parameters rp , rs , and i. At the same time, it is clear that i should not depend on . The planet's radius may depend on because it has an atmosphere, but this effect is small and can be detected only in very precise eclipse light curves (for instance, cf. [16]). Therefore, we determined the limb-darkening coefficients x() for the fixed geometrical parameters rp , rs , and i, common for all . We adopted the arithmetic means of all the values of rp , rs , i found for the different for these fixed values: r p = 0.0139657, rs = 0.113937, i = 86.55 . The limb-darkening coefficients x() and their uncertainties derived via minimization of the residuals with fixed rp , rs ,and i are collected in Table 8. We believe that these x() values are the most reliable representation of the wavelength dependence of x,shown in Fig. 6. The presented external uncertainties are the P values found assuming 2 statistics. When P 2 statistics are used, the model is rejected at the M < 5% significance level in most cases, because the more rigid model assumption that the geometrical parameters are the same for all wavelengths are applied in this case. This model can be accepted at the level = 5% for only three wavelengths (4849, 5398, ° 8732 A). Table 8 presents the corresponding central values, x(), and the most conservative estimates of the external uncertainties obtained assuming 2 M statistics. 6. INTERPRETATION OF THE HD 209458 LIGHT CURVES USING THE QUADRATIC LIMB-DARKENING LAW Southworth [5] also presents the quadratic limbdarkening coefficients derived from the multi-color light curves of HD 209458 [2]. Southworth [5] notes that the linear limb-darkening model can be rejected for the most accurate light curve obtained by Brown et al. [1], at a high significance level (exceeding 99.99%). This conclusion qualitatively agrees with our own results (see above), but there remain quantitative differences. Usually, the significance level of a statistical test (see, for example, [13]) is understood as the number of errors of the first kind we make when applying the test. An error of the first kind is a situation when a correct model is rejected by the statistical test. According to [5], if the model can be rejected at a significance level exceeding 99.99%, this means that,
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Fig. 5. Same as Fig. 4 for the uncertainties in the limb-darkening coefficients derived using the confidenceregion method assuming 2 statistics; the confidence M level is = 0.95.

This figure also shows the corresponding theoretical x() relations of Claret [15]. Although the uncertainties in x increased by a factor of two to four compared to the internal 2 uncertainties, the discrepancy between the observed and theoretical limb-darkening coefficients, x(), remains significant. The observed x() values are systematically below the theoretical ones, with the difference increasing with wavelength.


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Table 8. Results of fitting the observed light curves of HD 209458 from [2] using the linear limb-darkening law, fixing r p , rs , and i to be their averages for all wavelengths. The parameter uncertainties for = 0.95 were found using the confidence-region method and 2 and 2 statistics P M ° , A 3201 3750 4300 4849 5398 5802 6779 7755 8732 9708 x 0.805194 0.732868 0.697317 0.608848 0.552146 0.552724 0.445318 0.395653 0.352846 0.329928 P (x) 0.0196023 0.0117864 0.00718819 0.00705112 0.00778080 0.00662294 0.00680517 0.00922067 0.0125393 0.0181280 ­ ­ ­ 0.608828 0.552016 ­ ­ ­ 0.352893 ­ x M (x) ­ ­ ­ 0.00992982 0.0194529 ­ ­ ­ 0.00567959 ­
2 red

1.127 1.208 1.190 1.103 1.068 1.420 1.214 1.354 1.110 1.158

when rejecting the model, we make an error of the first kind (reject the correct model) in 9999 cases of 10 000; i.e., we are correct in rejecting the model in only one case out of 10 000. Thus, we have no reason to reject the model in this case, and it can be accepted. When quoting the value 99.99%, Southworth [5] probably meant the confidence level = 1 - rather than the significance level . Correcting for this confusion, the linear limb-darkening model applied to the precise light curve of Brown et al. [1] is rejected in [5] at a very low significance level, < 0.01%.Note that, according to [5], the minimum residual for the reduced 2 for the linear limb-darkening law (and the light curve of Brown et al. [1]) is 2 d = 1.1457, re corresponding, as follows from [12], to rejecting the model at the level = 1% (not < 0.01%, as in [5]). According to our fitting, the linear limb-darkening model applied to the precise light curve of Brown et al. [1] is rejected at a low significance level ( 6%), in agreement with the minimum reduced 2 value, 2 d = 2 -P /(M - P ) 1.103 [12]. Thus, as is M re noted above, the linear limb-darkening model is not very good, but also not hopelessly poor, since, in this case, we are able to estimate conservative parameter uncertainties at the = 95% confidence level. We also fitted the precise light curve of Brown et al. [1] using the quadratic limb-darkening law. We minimized the residual functional in the five parameters rp , rs , i, x1 , and y1 , where x1 , y1 are the coefficients in the limb-darkening law: I () = I0 1 - x1 (1 - ) - y1 (1 - )
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2

Here, = cos ( is the angle between the normal to the stellar surface and the line of sight). The results are presented in Table 9, which contains the central values of the parameters rp , rs , i, x1 , and y1 and their uncertainties found using the differentialcorrections and confidence-region methods for = 95%. The reduced 2 is also given (2 d = 1.01340), re and is much lower than the reduced 2 for the linear limb-darkening model (2 d = 1.103). Our quadratic re limb-darkening model is rejected at the significance
x 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 2000 4000 6000 8000 10 000 12 000 , å ugriz UBVRIJ

.

Fig. 6. Same as Fig. 5 for fixed r p = 0.0139657, r s = 0.113937, and i = 86 .555243, equal to their average values for all wavelength, and assuming 2 statistics. P

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Table 9. Results of fitting the observed light curves of HD 209458 from [1] using the non-linear (quadratic) limbdarkening law. The parameter uncertainties were found using the differential-correction method and confidence-region method, assuming 2 (P is the number of fitted parameters) and 2 (M is the number of data points in the light curve) P M statistics. The bottom line contains the reduced 2 d values re Parameter r1 (r1 ) r2 (r2 ) ic ,deg (i),deg x1 (x1 ) y
1

Differential-correction method (2 ) 0.113836 0.000853921 0.0137654 0.000144763 86.6756 0.108695 0.294517 0.0546122 0.344130 0.0955230 1.01340

Confidence-region method, 2 (95%) P 0.113845 0.00145205 0.0137679 0.000250444 86.6789 0.189625 0.296029 0.0927646 0.343221 0.163519

Confidence-region method, 2 (95%) M 0.113776 0.00324496 0.0137456 0.000562880 86.7196 0.429125 0.297905 0.207913 0.352285 0.366437

(y1 )
2 red

Table 10. Results of fitting the observed light curves of HD 209458 from [1] using the non-linear (quadratic) limbdarkening law. The parameter uncertainties were found using the differential-correction method and confidence-region method, assuming 2 (P is the number of fitted parameters) and 2 (M is the number of data points in the light curve) P M statistics. The chosen confidence level is = 0.68 Parameter r1 (r1 ) r2 (r2 ) i,deg (i),deg x1 (x1 ) y
1

Differential-correction method ( ) 0.113836 0.000426960 0.0137654 0.0000723814 86.6756 0.0543473 0.294517 0.0273061 0.344130 0.0477615

Confidence-region method, 2 (68%) P 0.113840 0.00104013 0.0137677 0.000179204 86.6765 0.135610 0.295082 0.0662467 0.343627 0.116747

Confidence-region method, 2 (68%) M 0.113844 0.00155419 0.0137676 0.000268077 86.6799 0.203029 0.296223 0.0993592 0.343281 0.175127

(y1 )

level > 46%, so that we are able to estimate the external parameter uncertainties at the 95% confidence level for both 2 and 2 statistics. Thus, P M the model with the quadratic limb-darkening law is

"good" when applied to the precise light curve of Brown et al. [1] (in rejecting the model, we would make an error of the first kind, i.e. reject the correct model, in less than 46% of all cases). This indicates
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Table 11. Results of fitting the observed light curves of HD 209458 from [2] using the non-linear (quadratic) limbdarkening law. The 2 parameter uncertainties were found using the differential-correction method (the last two columns contain the reduced 2 values and the corresponding significance levels ) ° , A
c r1 c 2est (r1 ) c r2 c 2est (r2 )

i

c

2est (ic ) 0.46


xc 1 1.024

2est (xc ) 1

y

c 1

c 2est (y1 )

2 red

0.205 0.181 0.0409 0.118 0.410
-6

3201 0.1128 0.003574 0.01407 0.0007801 86.59

0.1965 -0.3706

0.3530 1.062

3750 0.1113 0.002223 0.01360 0.0004571 86.88 4300 0.1133 0.001312 0.01388 0.0002658 86.64 4849 0.1133 0.001122 0.01385 0.0002147 86.65 5398 0.1138 0.001222 0.01381 0.0002279 86.67 5802 0.1142 0.001034 0.01400 0.0001969 86.55 6779 0.1149 0.0008532 0.01399 0.0001520 86.49 7755 0.1134 0.001130 0.01374 0.0001909 86.64 8732 0.1144 0.001424 0.01388 0.0002411 86.54 9708 0.1136 0.002296 0.01380 0.0003744 86.64

0.31 0.18 0.15 0.16 0.14 0.11 0.15 0.19

0.7978 0.1074 -0.08720 0.2086 1.068 0.7105 0.07557 -0.01514 0.1408 1.123 0.6119 0.07144 0.01035 0.1274 1.086 0.4286 0.07723 0.2419 0.4551 0.08225 0.1389 0.3066 0.07529 0.2183 0.2035 0.09479 0.2899 0.07225 0.1369 0.4119 0.5815 0.1426 1.023

0.1418 1.293 8.55 â 10 0.1250 1.042 0.1580 1.048 0.2265 0.9966 0.3519 1.100 0.292 0.257 0.574 0.0702

0.30 -0.07901 0.2096

Table 12. Results of fitting the observed light curves of HD 209458 from [2] usingr the quadratic limb-darkening law. The parameter uncertainties were found using the confidence-region method assuming 2 statistics; the adopted confidence P level is 95% ° , A 3201 3750 4300 4849 5398 5802 6779 7755 8732 9708 r1 0.1128 0.1112 0.1133 0.1133 0.1138 0.1142 0.1149 0.1134 0.1144 0.1133 p (r1 ) 0.0053 0.0034 0.0023 0.0018 0.0021 0.0016 0.0014 0.0018 0.0023 0.0038 r2 0.01439 0.01359 0.01388 0.01385 0.01380 0.01402 0.01399 0.01374 0.01388 0.01374 p (r2 ) 0.00140 0.00068 0.00051 0.00035 0.00039 0.00031 0.00026 0.00030 0.00039 0.00063 i 86.53


p (i) 0.81


x1 1.04 0.80 0.72 0.62 0.43 0.46 0.31 0.21 0.07 -0.08

p (x1 ) 0.30 0.16 0.13 0.12 0.13 0.13 0.13 0.15 0.22 0.34

y

1

p (y1 ) 0.67 0.32 0.26 0.21 0.24 0.22 0.21 0.25 0.37 0.57

-0.52 -0.07 -0.02 0.01 0.25 0.13 0.22 0.29 0.42 0.60

86.93 86.66 86.66 86.69 86.55 86.49 86.64 86.56 86.73

0.49 0.34 0.25 0.29 0.22 0.19 0.24 0.31 0.53

that the non-linear (quadratic) limb-darkening law is preferable to the linear law. Moreover, the quadratic limb-darkening model is sufficiently good that we can estimate the conservative external uncertainties of x1 and y1 at the = 1 - = 68% confidence level, collected in Table 10.
ASTRONOMY REPORTS Vol. 54 No. 12 2010

Our conservative estimate of the external uncertainties of the parameters x1 , y1 at the 95% confidence level shows that the derived values of x1 , y1 are at the detection limit (Table 9; see also Table 13). If we use only the internal uncertainties in x1 and y1 (Table 9) or the conservative external uncertainties at the 68%


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Table 13. Results of fitting the observed light curves of HD 209458 from [2] using the quadratic limb-darkening law. The parameter uncertainties were found using the confidence-region method assuming 2 statistics; the adopted confidence M level is 95% ° , A 3201 3750 4300 4849 5398 5802 6779 7755 8732 9708 r1 0.1174 0.1109 ­ 0.1133 0.1136 ­ 0.1148 0.1133 0.1136 0.1134 M (r1 ) 0.0141 0.0051 ­ 0.0022 0.0044 ­ 0.0027 0.0033 0.0056 0.0031 r2 0.01428 0.01356 ­ 0.01385 0.01373 ­ 0.01399 0.01373 0.01380 0.01376 M (r2 ) 0.00249 0.00110 ­ 0.00043 0.00085 ­ 0.00050 0.00056 0.00105 0.00051 i 86.96


M (i) 1.61


x1 0.91 0.81 ­ 0.62 0.42 ­ 0.32 0.21 0.09 -0.08

M (x1 ) 0.43 0.26 ­ 0.14 0.26 ­ 0.24 0.28 0.58 0.28

y

1

M (y1 ) 1.40 0.50 ­ 0.25 0.52 ­ 0.40 0.46 0.95 0.47

-0.19 -0.08 ­ 0.01 0.28 ­ 0.21 0.29 0.44 0.60

87.01 ­ 86.67 86.79 ­ 86.51 86.67 86.72 86.70

0.81 ­ 0.30 0.65 ­ 0.36 0.44 0.86 0.42

confidence level (Table 10), we may consider the derived limb-darkening coefficients x1 , y1 to correspond to their real values. We also applied a model with a quadratic limbdarkening law to the multi-color light curves of Knutson et al. [2]. The corresponding fitting results are presented in Tables 11­13. It follows from Table 11
x1 1.5 1.3 1.1 0.9 0.7 0.5 0.3 0.1 ­ 0.1 ­ 0.3 ­ 0.5

ugriz UBVRIJ

that, on average, 2 d is lower than for the model with re the linear limb-darkening law (Table 3). Figures 7­ 10 display the coefficients x1 , y1 as functions of the wavelength . We also show the theoretical x1 , y1 values from Claret [15]. The non-linear coefficient y1 for HD 209458 agrees with the theoretical relation within the errors. However, the observed values of the linear coefficient x1 for the quadratic limb-darkening law significantly deviate from the theoretical values (as for the linear limb-darkening law). We show the projections of the confidence region D onto the x1 , y1 and rs , rp parameter planes in Figs. 11 and 12. Here, the rectangles show the projections of the region defined by the 2 parameter uncertainties found using the differential-correction method. 7. USING THE APPROXIMATE METHOD SUGGESTED IN [12] TO ESTIMATE THE EXTERNAL PARAMETER UNCERTAINTIES The above estimates of the external uncertainties for the parameters of the system HD 209458 were made via an exhaustive parameter search to find the corresponding surface of the residual functional in the multi-dimensional parameter space. The procedure used for the exhaustive parameter search is laborious, though it can be performed comparatively easily for our simple model (two spherical stars in a circular orbit). However, for other, more complex models the exhaustive parameter search can become difficult. Thus, we suggest in [12] an approximate method for estimating the external parameter uncertainties, which can easily be implemented practically. We present in [12] tables and corresponding
ASTRONOMY REPORTS Vol. 54 No. 12 2010

4000

6000

8000

10 000

12 000 , å

Fig. 7. Wavelength dependence of the limb-darkening coefficient x1 for HD 209458 in the quadratic limbdarkening law. The limb-darkening coefficients were derived using the light curves of [2]. The quoted 2 uncertainties in the limb-darkening coefficients were derived using the differential-correction method. The theoretical limb-darkening coefficients in the ug r iz and UB V RI J photometric systems are plotted according to [15].


LIGHT CURVE ANALYSIS
y1 1.50 1.25 1.00 0.75 0.50 0.25 0 0.25 0.50 0.75 1.00 1.25 1.50 3000
x1 1.5 1.3 1.1 0.9 0.7 0.5 0.3 0.1 ­ 0.1 ­ 0.3 ­ 0.5

1121

ugriz UBVRIJ

ugriz UBVRIJ

­ ­ ­ ­ ­ ­

5000

7000

9000 , å

4000

6000

8000

10 000

12 000 , å

Fig. 8. Same as Fig. 7 for the limb-darkening coefficient y1 .

diagrams that can be used to compute the coefficient for translation from the internal parameter uncertainties derived in the Monte Carlo or differentialcorrection method to the external uncertainties derived in the confidence-region method for 2 statisM tics (tmax for the one-dimensional problem or kp tpmax for the multi-dimensional problem) [12]. Having calculated the internal uncertainties 1 ( ) and become convinced that the model used is consistent with the observations (for which it is sufficient to determine the minimum residual and calculate the reduced 2 d ), re we can find the most conservative estimates of the external parameter uncertainties for 2 statistics by M multiplying the internal uncertainties by the coefficient kp tmax [12]. M (, ) calculated Table 14 presents values of 1 ( ) from our uncertainty intervals (and collected in Tables 2 and 4). 1 = 2est is the uncertainty found using the differential-correction method. Note that M (, ) ratio can change considerably the actual 2est for different realizations of the light curve (see the distribution density function in [12]). However, this theoretical function possesses an extremum, and the method suggested in [12] deals with the most probM (, ) . For the light-curve of [2], the able ratio 2est M (, ) should be 2.6. Based upon observed ratio 2est the suggested method of [12], we conclude that the M (, ) is 4. most probable theoretical value of 2est This exceeds the difference (derived from observations) between the uncertainties estimated using the
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Fig. 9. Same as Fig. 7 for the case when the uncertainties in the limb-darkening coefficients are found using the confidence-region method with 2 statistics; the confiP dence level is = 0.95.

differential-correction method and confidence-region method (with 2 statistics) by a factor of 1.5. M Nevertheless, using the most probable theoretical raM (, ) [12], enables tio of the intervals, tmax1 = 1 ( ) us to avoid inconsistencies between parameters derived from different observations, which often arise for the differential-correction method (or Monte Carlo method), specify confidence intervals for the fitted parameters, and achieve a considerable reduction in the time required to calculate the uncertainty intervals.

­ ­ ­ ­ ­ ­

y1 1.50 1.25 1.00 0.75 0.50 0.25 0 0.25 0.50 0.75 1.00 1.25 1.50 3000

ugriz UBVRIJ

5000

7000

9000 , å

Fig. 10. Same as Fig. 9 for the limb-darkening coefficient y1 .


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ABUBEKEROV et al.

y1 ­yc 1 0.3 0.2 0.1 ­ 0.2 ­ 0.1 ­ 0.1 ­0.2 ­ 0.3 0.1 0.2 x1 ­x c 1

Fig. 11. Projection of the confidence region D (at the 95% confidence level) onto the (x1 , y1 ) plane for the model with the quadratic limb-darkening law. The smaller ellipse (dashed curve) corresponds to the region found using 2 statistics and the P larger ellipse to the region found using 2 statistics. The rectangle corresponds to the projection of the confidence region at the M 2est level found using the differential-correction method. The projection of the confidence region D is based on the observed light curve from [1].
c p

rp ­r

0.0004 0.0002

­ 0.003

­ 0.002

­ 0.001 ­ 0.0002 ­ 0.0004 ­ 0.0006

0.001

0.002

0.003 rs ­r c s

Fig. 12. Same as Fig. 11 for the projection of the confidence region D (at the 95% confidence level) onto the (rp , rs )plane.

8. CONCLUSIONS We have determined the parameters of the system HD 209458 based on an analysis of high-precision satellite light curves of the eclipse of the star by the exoplanet. We have found the radii of the star and the exoplanet, the orbital inclination, and the linear and non-linear limb-darkening coefficients for the star. The derived parameters were analyzed in detail taking into account both internal and external parameter uncertainties, the latter being larger than the former by factors of two to four. Allowing for the external uncertainties makes it possible to reconcile

the geometrical parameters derived from observations obtained at different wavelengths. The relative radius of the exoplanet in the HD 209458 system (in units of the radius of the binary orbit) is rp = 0.01386 (the mean value from Table 12). The optical star in HD 209458 is a G0V dwarf. This star's radius is Rs = 7.98 â 1010 cm (according to [2], assuming the star and planet masses are Ms = 1.101M and Mp = 0.64MJup ). Thus, the radius of the planet is Rp = 9.70 â 109 cm. The planet's mass is [17] Mp = 1.215 â 1030 g, and its mean density is p = 0.318 g cm-3 . Using the formula relating ¯ the surface gravity of the planet, gp , and the radialASTRONOMY REPORTS Vol. 54 No. 12 2010


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1123

velocity semi-amplitude of the star, Ks [5]: gp = Ks 2 1 - , 2 sin i Porb rp
1/2 e2

and substituting e = 0, Ks = 85.1 ± 1.0 m/s [18], Porb = 3.52474859d , and i = 86.65 (Table 12), we find gp = 8.61 m/s2 , close to that at the Earth's surface. The ratio of the planet and star masses is q = mp /ms = 0.00055, and the planet's relative radius is Rp = 0.01386. To test our assumption that the rp = a planet is spherical, let us calculate the mean relative radius of the Roche lobe for the planet using the approximate formulas [19] RR = 0.49a 0.62q
2/3

Table 14. Observed ratios of the uncertainty interval M found in the confidence-region method with 2 statisM tics ( = 0.95) and the uncertainty 2est estimated using the differential-correction method: the observed ratio M (, ) is presented for k = rs /rp , i, rp ,and rs 2est ° , A 3201 3750 4300 4849 5398 5802 6779 7755 8732 k 2.13 2.30 ­ 2.00 3.10 ­ 2.75 2.36 3.74 ­ 2.63 rs 2.14 2.31 ­ 2.00 3.09 ­ 2.74 2.35 3.73 ­ 2.62 rp 2.12 2.29 ­ 2.00 3.10 ­ 2.75 2.37 3.75 ­ 2.63 i 2.13 2.30 ­ 2.00 3.10 ­ 2.75 2.36 3.74 ­ 2.63

q +ln(1+ q

2/3

1/3

) .

,

(27)

RR = 0.46a

M2 M1 + M2

1/3

(28)

9708 Mean

Table 15 presents the mean relative radii of the Roche lobe, RR /a (where a is the radius of the binary orbit) for various ratios of the planet and star masses, q = mp /ms , in the range q = 0.00055- 2. The RR /a values calculated from (27) and (28) are in good agreement, indicating that the estimate of the relative Roche-lobe radius for the planet is reliable. The radius of the planet is rp = 0.01386 and that of its Roche lobe is RR /a = 0.039785, so that the degree of filling of the planet's Roche lobe is = 0.35, much less than 0.5. Thus, our assumption that the planet is spherical is quite justified (if we neglect a small amount of oblateness due to its axial rotation). The same is true for the optical star. Information on the star's limb darkening is very valuable. Our analysis of the precise multi-color satellite light curves of Knutson et al. [2] has yielded wavelength dependences for the linear limbdarkening coefficient, x(), as well as the non-linear limb-darkening coefficients in the quadratic law, x1 () and y1 (). We confirm the difference between the observed and theoretical wavelength dependences for the linear limb-darkening coefficient x found by Southworth [5]. Our new result is that, even when the most conservative estimates of the external uncertainties in the limb-darkening coefficient x (with 2 statistics) are used, there remains a significant M discrepancy between the observed and theoretical x() relations. The observed x() are systematically below the theoretical values, with the difference increasing with wavelength, so that the observed linear limb-darkening coefficient is smaller than the
ASTRONOMY REPORTS Vol. 54 No. 12 2010

° theoretical one by a factor of about 1.5 at 9000 A. Thus, the stellar disk appears much more uniform in the red than is predicted by the theory of thin stellar atmospheres. The dependence for the limbdarkening coefficient x1 in the quadratic law also differs significantly from the corresponding theoretical relation: the observed x1 values are systematically below the theoretical ones, with difference increasing with wavelength . Note that this difference persists even when using the conservative external uncertainties of x1 derived for 2 statistics. The dependence P of the limb-darkening coefficient y1 in the quadratic law agrees with the corresponding theoretical relation within the errors. Until recently, model stellar atmospheres were tested predominantly by comparing observed and theoretical spectra for the entire stellar disk. Precise satellite multi-color observations of exoplanet transits across stars provide a new unique possibility for independently checking model stellar atmospheres based on the angular distribution of the radiation exiting the star's atmosphere at different . An important advantage of eclipses of stars by exoplanets for determining stellar limb-darkening coefficients is that reflection and ellipsoidal effects are negligible and the eclipse is annular, with a relatively small radius for the eclipsing planet. Thus, despite the fact that the relative accuracy of the eclipse light curves is not extremely high (1%-2% of the eclipse depth in the case of satellite observations), the above


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Table 15. Dependence of the mean relative radius of the Roche lobe on the component-mass ratio calculated using (27) and (28) q Equation 0.00055 0.001 0.01 0.05 0.1 RR /a (27) (28) 0.039785 0.037727 0.048366 0.045985 0.10201 0.098776 0.16833 0.16673 0.20677 0.20684 0.28103 0.28215 0.32079 0.31895 0.37892 0.36510 0.44000 0.40185 0.3 0.5 1 2

advantages make it possible to reliably determine the limb-darkening coefficients for both linear and nonlinear limb-darkening laws. Thus, such studies are important not only for determining the fundamental characteristics of exoplanets, but also for the further development of the theory of stellar atmospheres. We note that Tutukov [20, 21] was the first to suggest the possibility of detecting eclipses of stars by exoplanets. ACKNOWLEDGMENTS The authors express special thanks to H.A. Knutson for providing his observations of HD 209458. This work was supported by the Russian Foundation for Basic Research (project 08-02-01220), the Program of State Support of Leading Scientific Schools of the Russian Federation (grant NSh-7179.2010.2), a grant from the President of Russia for the Support of Young Russian PhDs (grant MK-206.2009.2), and the analytic departmental program "Development of the Scientific Potential of Higher Education" (grant RNP-2.1.1.2906). After our article was put to bed, we got acquainted with the article by A. Claret (Astron. Astrophys. 506, 1335 (2009)) where the limb-darkening coefficients of the stellar disks were recalculated on the basis of Kurucz models. The new theoretical values of coefficients in the linear and nonlinear darkening law are close to previous ones (A. Claret, Astron. Astrophys. 428, 1001 (2004)). Thereby, discrepancy between the observation of HD209458 and the theory is confirmed. Our work further shows that even using the most conservative estimates of the "external" parameters uncertainties, discrepancy between the observed and theoretical limb-darkening coefficients of star HD209458 is still significant. REFERENCES
1. T. M. Brown, D. Charbonneau, R. L. Gilliland, et al., Astrophys. J. 552, 699 (2001).

2. H. A. Knutson, D. Charbonneau, R. W. Noyes, et al., Astrophys. J. 655, 564 (2007). 3. I. A. G. Shellen, E. J. W. de Mooij, and S. Albrecht, Nature 459, 543 (2009). 4. The CoRoT Space Mission: Early Results, Ed. by C. Bertout, T. Forveille, N. Langer, and S. Shore, Astron. Astrophys. 506, 1 (2009). 5. J. Southworth, Mon. Not. R. Astron. Soc. 386, 1644 (2008). 6. A. Gimenez, Astron. Astrophys. 450, 1231 (2006). 7. D. M. Popper and P. B. Etzel, Astron. J. 86, 102 (1981). 8. J. Southworth, P.F. L.Maxted, and B.Smalley, Mon. Not. R. Astron. Soc. 351, 1277 (2004). 9. D. M. Popper, Astron. J. 89, 132 (1984). 10. A. M. Cherepashchuk, Astron. Zh. 70, 1157 (1993) [Astron. Rep. 37, 585 (1993)]. 11. M. K. Abubekerov, N. Yu. Gostev, and A. M. Cherepashchuk, Astron. Zh. 85, 121 (2008) [Astron. Rep. 52, 99 (2008)]. 12. M. K. Abubekerov, N. Yu. Gostev, and A. M. Cherepashchuk, Astron. Zh. 86, 778 (2009) [Astron. Rep. 53, 722 (2009)]. 13. S. Wilks, Mathematical Statistics (Wiley, New York, 1967; Nauka, Moscow, 1967). 14. A. Burrows, I.Hubeny, J. Budai, and W.B.Hubbard, Astrophys. J. 661, 502 (2007). 15. A. Claret, Astron. Astrophys. 428, 1001 (2004). 16. F. Pont, H. Knutson, R. L. Gilliland, et al., Montly Notices of the Royal Astron. Soc. 385, 109 (2008). 17. D. Charbonneau, T. M. Brown, D. W. Latham, and M. Mayor, Astrophys. J. 529, L45 (2000). 18. D. Naef, M. Mayor, J. L. Beurit, et al., Astron. Astrophys. 414, 351 (2004). 19. P. P. Eggleton, Astrophys. J. 268, 368 (1983). 20. A. V. Tutukov, Astron. Zh. 69, 1275 (1992) [Sov. Astron. 36, 650 (1992)]. 21. A. V. Tutukov, Astron. Zh. 72, 400 (1995) [Astron. Rep. 39, 354 (1995)].

Translated by N. Samus

ASTRONOMY REPORTS

Vol. 54 No. 12

2010