Introduction
An orbital fit by least-squares is a technique for computing the set of six Keplerian elements which best describe an orbit with respect to a *whole* set of observations.
As the set of Keplerian elements we choose semimajor axis a, eccentricity e, time of perigee passage τ (i.e., how much time has elapsed since the last perigee), as well as the standard Euler angles which describe the orientation of the orbital plane with respecto to the inertial frame: argument of pericenter ω, inclination I and longitude of ascending node Ω. We denote by X the set of these six Keplerian elements; that is, X = (a, e, τ, ω, I, Ω).
A least-squares fit is simply to look the minimum of a scalar (i.e., real-valued) „cost“ function Q(X). The Q function is simply a measure of how well our theoretical predictions, for a given set X of orbital elements, depart from observations. In an ideal world, we could find a set of orbital elements such that it perfectly matched the observations and thus Q(X)=0. Nevertheless, real-world measures carry uncertainties, and we can only hope to find orbital elements X which make Q attain its minimum possible value with respect to all the observations we have made.
The values of the cost function Q(X) for a least-squared problem are constructed the following way: take *all* the values you observed (call them
v_i), and substract from each of those the values you were expecting to measure for a given set of orbital elements X (say, V_i(X) ). Call each of these differences the residuals, xi_i(X) = v_i-V_i(X) . Then, square the residuals, and sum all the squared residuals. The value you obtain following this recipe, is the value of Q(X).In practice, the least-squares method finds a local minimum of the cost function by performing an iterative procedure, which needs a good initial „guess“ solution
X0; otherwise the procedure may converge to non-feasible results. Hence, it is necessary that the „good“ initial guess be provided by a preliminary orbit determination method.
