M

or in dimensionless units as log g = log — = M/M° with g0 = 2.74 ■ 102m ■ s-2. (4.4.22)

Note that log g for both components is required if stellar atmospheres are to be incorporated in a light curve model. Note that log gi and log g2 are related by lo^ = M2/M12 = _L_ (4.4.23)

If a light curve program provides the polar surface brightness, it may be used to compute monochromatic luminosities and, by applying a bolometric correction, eventually the bolometric luminosity. However, from the standpoint of calculating the uncertainty, we prefer effective temperatures. Thus, for spherical stars, the bolo-metric luminosity of each component is

and for the more general case we have

The absolute bolometric magnitude follows:

= MQo1 - 5log1o(—/rQ) - 10log1o(T/T0), from which the absolute V magnitude is derived by subtraction of the bolometric correction11 B.C., i.e.,

Alternatively, with a light curve program available, it might be more consistent to start with the computation of the monochromatic luminosity

in absolute units. Then, MV is computed according to

a bolometric correction is applied

and, finally, the bolometric luminosity follows from

The color indices of the individual components and of the system may be used to find the interstellar reddening E, especially if the spectral types are sufficiently well known. From this the unreddened distance modulus (m - M)0 is derived. In the Johnson system introduced in Sect. 2.1.1, we have the relations

where r is the distance in parsecs, and R = AV/EB-V is the ratio of the attenuation, AV, of the V light by the interstellar medium to the color excess EB-V. (N.B. : m v = v ) Although exceptions among determined values of R are well documented, typically, 3.0 < R < 3.4.

If masses have been determined, the evolutionary state of the components may be explored. Evolutionary tracks through the two components of known mass are

11 We use the definition B.C. = Mbo1 — MV. Since solar-like stars have their radiation maximum in the visual region of the spectrum, and due to the definition of the zero point of B.C., most stars have negative bolometric corrections. Some care is necessary in the use of B.C. from tables because sometimes the definition B.C. = MV — Mbo' is used.

assumed to be coeval, and theoretical models should predict correctly the sizes and luminosities of these components in straight-forward situations. Two examples out of numerous such studies are

• The analysis of the system DS Andromedae (Schiller & Milone 1988) in the open cluster NGC752 showed that the radius of the hotter component agreed with the age of the cluster and that the Roche radius has not yet been reached.

• The determination of the masses and luminosities of the double-lined spectroscopic and EB HD27130 in the Hyades star cluster by Schiller & Milone (1987) suggests a mass-luminosity relation for these two stars which differs from that of the local field of the Sun.

4.4.4.2 The Incomplete Data Case

If only one component with mass M1 has observed radial velocities, the system is called a single-lined spectroscopic binary. In this case, besides the orbital period P the radial velocity amplitude,12 K1, is known and a useful quantity known as the mass function f (Mi, M2, i) can still be obtained:

Note that f (M1, M2, i) can be calculated based on quantities which can be derived from the spectroscopy only: The period,13 P, and the eccentricity, e, of the orbit [see Aitken (1964) for details]. If the mass ratio q and the inclination i are known from photometry, then M2 follows as

sin3 i and M1 as

1 (1+ q )2 1 M1 = -M2 = --^--^ f (M1, M2, i). (4.4.36)

q q3 sin3 i

12 The radial velocity amplitudes, K1 and K2, are only useful quantities if proximity effects are absent. In that case, the mass ratio is just q = K1 /K2. So, if they are used, they should be considered with great care.

13 The period is often more accurately derived from photometry because eclipses make good timing ticks. Also the number of photometric observations tends to be much larger than the number of radial velocity data points. Furthermore, radial velocity observations tend to be taken at the quadratures also, where they have limited use in defining the phases of conjunction.

If we have no photometric light curves (so that the inclination i and mass ratio q are not available), we still can get some information: Combining formulas (4.4.35) and (4.4.36), the sum of masses is

sin3i

For known f, the right-hand side of (4.4.37) takes its minimum value for q = to and i = 90°. Thus the mass function provides a weak lower bound on the sum of masses, and because q = to implies M1 = 0, we get

i.e., the mass of the star without radial velocities.

If we are sure that q < 1, the minimum of M1 + M2 occurs for q = 1 which leads to the tighter bound

4.5 Suggestions for Improving Performance

Et respice finem (Consider the end)

Sirach 7, 40; Gesta Romanorum, c. 103

The determination of light curve parameters from EB light curves may require substantial14 computing time. However, there are several mathematically well-defined alternatives with lower computational cost while still controlling accuracy:

1. Symmetries in the light curve model.

2. Local interpolation of total light between phase grid points, as is done in Linnell's program (Linnell 1989).

3. Choice of lower grid density on the stellar surfaces. When a direct search method is used as the optimizing tool, the required precision (and hence the grid density) does not need to be as high for the calculation of the theoretical light curve by numerical quadrature of the flux over the stellar surface as for a derivative-based method.

4. Use of analytical partial derivatives dl/dxi.

5. Accurate finite difference approximations.

14 We might argue that hardware improvements make these considerations less and less important as time goes on. That is certainly true for existing programs, but models get more and more sophisticated and include more and more details. Also, because the points considered in the next subsections have a sort of general character, it is worthwhile to keep them in mind.

Use of both coarse and fine grids, with the fine grids used only where needed. Only parameters whose variation changes the geometry need fine grids. The future might see adaptive grids over the surface.

4.5.1 Utilizing Symmetry Properties

Two symmetries can reduce computational time significantly: Orbital symmetry and surface symmetry. Orbital symmetry leads to symmetry of the light curve w.r.t. either minimum. Expressed w.r.t. the secondary minimum we have lcal(0.5 - ) = lcal(0.5 + ^0). (4.5.1)

For the computation of light values, lcal, this symmetry can be utilized in the form lcal(0), 0 < 0 < 0.5, lcal(1 - 0), 0.5 < 0 < 1.

Exploiting the orbital symmetry reduces the computational needs by a factor of 2 in simple cases without complicating structures, e.g., circular orbit models without spots. Models including spots or eccentric orbits do not have orbital symmetry in general. Since observations fall wherever they fall, only the solution of the direct problem benefits from this symmetry. Solution of the inverse problem also benefits from the symmetry in the use of interpolation techniques described in Sect. 4.5.2.

Symmetry on the stellar surface is present if a given function f (0, y), such as the flux emitted from one of the components, is symmetric with respect to the stellar latitude 0 and/or longitude y. Models which are based on spherical stars have the symmetry property

: f (0, n + ¿y) = f (0, n - V), * e{1, 2}, ¿y e n

whereas the Roche model with circular orbits has only the symmetry properties S0 and S2. If the orbital and rotational axes are parallel, the x-y and x-z planes are symmetry planes for each component.

4.5.2 Interpolation Techniques

An efficient method to reduce computation is to use local interpolating polynomials p(0) for determining light curve parameters with derivative-free procedures. Let n + 1 be the number of equidistant grid points X; = <Pi = iy = f (Xi) = lcal(0), i = 0,..., n; y = f (x) = lcal(0), (4.5.5)

with phase grid size

. , 1 i 1, no special orbital symmetry is available,

The interpolating polynomial may be chosen according to specific needs. Here we choose cubic polynomials based locally on four grid points and evaluated by the Bessel method with central differences [see, e.g., Scarborough (1930)]. This method uses

h := xi+1 — x; = —, u :=-, v := u — 1 (4.5.7)

and is based on the scheme

X—1

y—1

x0

y0

x1

y1

x2

Local interpolation of the function f (x) by a cubic polynomial p3(x) is to be under-

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