Determination of Eclipsing Binary Parameters

The determination or estimation of physical parameters from EB light curves and/or radial velocity curves is an inverse problem and can be formulated as a nonlinear least-squares problem. It is solved by optimizing the agreement between the calculated light and the observed light curve. The parameter vector x corresponding to minimum deviation is the system solution, and the calculated light curve produced from it is said to be the best fit to the data. A measure of the deviation is the weighted sum of the squared residuals.

In this chapter, we formulate the inverse problem and describe numerical methods used in the EB community to solve the least-squares problem. Intentionally, we give a rather formal approach closer to today's education of physicists and mathematicians. In addition, Appendix A provides a more general background on least-squares problems. Regarding the analysis of light curve data, we suggest some rules for estimating initial parameters and for interpreting and checking the consistency of light curve solutions; In addition, we discuss the interpretation of errors associated with the derived parameters. Of all these issues, the interpretation of the solution is of greatest importance because it transforms the mathematical results into useful physical information.

4.1 Mathematical Formulation of the Inverse Problem

Per aspera ad astra (Through arduous labors, to the stars)

For a set {tck | 1 < k < nc} of given timelike quantities and a given m-dimensional parameter vector x e JRm, an observable Oc of curve-type c, 1 < c < C

may be calculated at nc points tck in time based on a light curve model, in the way described in Chap. 3. This problem, the mapping ocal(x):= {(tCk, ock1) I o;

J. Kallrath, E.F. Milone, Eclipsing Binary Stars: Modeling and Analysis, Astronomy and Astrophysics Library, DOI 10.1007/978-1-4419-0699-1_4, (9 Springer Science+Business Media, LLC 2009

is called the direct problem. Note that we consider C observables simultaneously. For present purposes, it is not essential that different observables in simultaneous light curve fitting are measured at the same time; in general we have tClk = tcik.

The problem of finding the parameter vector x which best fits a set of C given observed light curves {O°bs,..., O°bs,..., OCbs}

Oobs := {(c, oCks) | 1 < k < nc}, 1 < c < C, (4.1.3)

with each of them consisting of nc data points is called the inverse problem. The inverse problem yields the mapping {Oobs, •••, Oobs,O^bs| ^ x, of course, only in the sense of a statistical estimation.

In order to formulate the inverse problem, we introduce the column vectors o and c containing the observables ock of the sets and O^x);

._ /^,obs ^-.obs ^obs ^obs ^-.obs ^obs ^obs ^-.obs ^obs \T

o •— (o11 , o12 , •••, o1m, •••, 0c1 , 0c2 , •••, 0cnc, •••, 0C1 , 0C 2 , •••, 0CnC) '

c •— (oc1, o12l, •••, ocn, •••, oS1, oc2l, •••, od, •••, oC1, oC2, •••, ocnC)T, (4.1.5)

and the vector d(x) of light residuals (observed minus calculated):1

d(x) •— o - c(x), dck(x) •— oc°bs - od. (4.1.6)

Note that the total number of data points involved in the problem is n •— n1 +-----+ nc +-----+ nC• (4.1.7)

In addition, let w •— (w1, •••, wn)T — 0 be a column vector, the components of which represent the nonnegative weights of the individual observations. Furthermore, it is convenient to define a matrix W which, in the case of uncorrelated data,2 is a diagonal matrix

W •— diag(W11, W12, •••, W1m , •••, Wc1, Wc2, •••, Wcnc, •••, WC1, WC2, •••, WCnC)•

In addition, we define the weighted residual vector R(x)

Weighting is an important part of the data analysis, so in Sect. 4.1.1.5 we comment further on it.

1 The overdetermined system d(x) = 0 is also called equations of conditions.

2 Unlike the analysis of photographic plates in astrometry where errors in x and y are strongly correlated, eclipsing binary light curve and radial velocity curve data are not correlated with each other.

To solve the inverse problem and to derive the unknown parameter vector x describing the physics of the binary from the observed curves O0^, it appears reasonable to vary x until the deviation between the calculated observable Ocal(x) and Oobs attains a minimum in a well-defined sense. In data analysis several criteria are used. Mathematically they can be expressed as the problem to minimize the norm ||R||p with p e {1, 2, to}. In this book at most places, we will use p = 2. For the case p = 1, we refer the reader to Branham (1990, Chap. 6). An alternative approach to data analysis is robust estimation [cf. Press et al. (1992, pp. 694- 698)].

Since the days of GauB (see Sect. 4.1.4), it has been customary3 to use the weighted sum of squared residuals (p = 2)

f (x) := RtR = J2 R (x) = dTWd = J^ (o°°bs - of)2 (4.1.11)

as a measure of this deviation if we are convinced that the distribution of the errors can be described by a normal distribution. A parameter vector x in a light curve model is called the light curve solution or system solution, x,, if f, = f (x,) = min{ f (x) | x e S c IRm}. (4.1.12)

The subset S c IRm, called the feasible region, is implicitly determined by a set of constraints as in (A.2.2), i.e., functional relations (often equations or inequalities) applied to components of the parameter vector x. Whereas f, is unique, x, is very often not. In nonlinear models, this might be caused by the structure of the model; in linear models, correlations between parameters undermine the uniqueness of x,. Concerning the minimization of the quadratic form or function f(x), it is equivalent to minimize the corresponding function f (x) defined as the standard deviation afit

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