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This gives a measure independent of the number m of free parameters and the number of data points, n. Thus, the inverse problem of light curve analysis is reduced to the following problem: Given a light curve model and a parameter vector x = (x\, x2,xm)T, we seek a solution in multidimensional parameter space minimizing the quadratic form RTR. Equivalently, as a measure for the quality of the fit, we minimize afit, which is normalized by the number of observed data points and free parameters.

3 GauB showed that the weighted sum of squares gives the most probably correct results. In modern terminology, the least-squares procedures, under the assumptions described in Appendix A.3, provide a maximum likelihood estimator (Brandt, 1976, Chap. 7).

Light curve analysis leads to a typical situation for fitting model functions (here, the light curve model) to data. In this case, it is a nonlinear parameter estimation, which in this form is a special case of unconstrained optimization. However, strictly speaking, we are confronted with a nonlinear constrained optimization problem. The optimization is subject to bounds li < Xi < Ui. (4.1.14)

An example is li = 0 and ui = 1 for the eccentricity e. Another constraint, seldomly mentioned explicitly, is a nonlinear and implicit relation among the mass ratio q, the Roche potentials ^ and and the rotation factors F1 and F2. This relation (note that f1 and f2 denote the filling (or "fill-out") factors; they should not be confused with f (x))

"f1,2 < 0" v "0 < f1 = f2 < 1" v "sign(f1) + sign(f2) = -1" (4.1.15)

guarantees that a feasible binary configuration is produced. These configurations are

• detached: two stars inside their critical lobes (f1 < 0, f2 < 0 ^ sign(f1) + sign(f2) = -2); or

• semi-detached: one star filling its critical lobe whereas the other one is well inside its critical lobe (f1 < 0, f2 = 0 or f1 = 0, f2 < 0 ^ sign(f1) + sign(f2) = -1); or

• over-contact: both stars establish an over-contact binary in the synchronous case (0 < f1 = f2 < 1); or finally,

• double-contact: both stars filling their critical lobes in the asynchronous case (f1 = f2 = 1).

The necessary and sufficient conditions for the existence of a minimal point x* of (4.1.12) can be expressed in terms of the first and second derivatives and some Lagrange multipliers (see Appendix A.2). The nonlinear structure of the problem requires an iterative solution algorithm, and it forces us to distinguish between global and local minima. x* is called global minimum if for all x e S the relation f (x*) < f (x) (4.1.16)

holds. In contrast, x* is called local minimum, if x* satisfies (4.1.16) only for a local sphere Be (x*)

around x* with a suitable e > 0. Finally, an algorithm for solving our least-squares problem is said to converge globally if from an arbitrary initial point x0 e S it converges to a local (or global) minimum x*. However, unless f (x) is a convex function, there is no algorithm available which could be proven to converge to the global minimum x*. So, in solving nonlinear least-squares problems, at best we can prove that we have reached a local minimum. We might have reached the global minimum but we are usually not able to prove it formally.

4.1.1 The Inverse Problem from the Astronomer's Perspective 4.1.1.1 The Input Database

The analysis of a photometric light curve alone cannot provide absolute dimensions of the stars or the orbit. The reason for this is a scaling property: if all geometric properties of a binary system are doubled, original light curves can be reproduced by shifting to a larger distance. The distance and surface brightnesses, which would either individually or in combination determine the scale of the model, are known only under exceptional circumstances, e.g., if the binary is a member of a well-studied star cluster.

Light curves can provide relative quantities (radii in terms of the semi-major axis a, information on temperature, relative luminosities, perhaps the photometric mass ratio, the shapes of the stars) and the orientation and eccentricity of the orbit (inclination i, argument of the periastron, «). Radial velocity curves can provide a sin i, i.e., the scaling factor a in physical units if i is known. When a and the mass ratio q are known, the masses can be found unambiguously from (4.4.14) and (4.4.16). Note that in order to derive definite masses and orbital dimensions [viz. (4.4.16) and (4.4.17)] from a radial velocity curve, the inclination needs to be known. Thus, the absolute determination of EB parameters requires at least one light curve and radial velocity curves for both components. The following tables list combinations of observables needed to derive certain binary parameters in favorable cases:

1 = at least one photometric light curve;

2 = only one radial velocity curve;

3 = both radial velocity curves, but no light curve;

4 = at least one photometric light and one radial velocity curve; and

5 = at least one photometric light curve and both radial velocity curves:

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