Up: Simultaneous leastsquares adjustment of
Subsections
Suppose we have three sets of measurements, namely
 1.
 : a set of
measured relative rectangular coordinates of the fainter
(B) with respect to the brighter (A) component
 2.
 : a set
of radial velocity measurements for the spectroscopic component A
 3.
 : a set of radial
velocity measurements for the spectroscopic component B
for a doublelined spectroscopic visual binary. We reckon
with radial velocities expressed in km s^{1} and positions in
arcsec ('').
For the visual orbit, we have
 
(1) 
 (2) 
 (3) 
 (4) 
where X and Y (x and y) are the angular rectangular coordinates,
in the orbital (tangential) plane, of the fainter component with respect
to the brighter one; A, B, F and G are the ThieleInnes constants,
expressed in terms of a(''), i, and as
 
(5) 
 (6) 
 (7) 
 (8) 
e is the eccentricity and E is eccentric anomaly, determined unambiguously by Kepler's equation
 
(9) 
P is the period and T is the epoch of periastron passage. Seven parameters
(a(''), i, , , e, P and T) are necessary.
For a spectroscopic orbit j (j=A or j=B) , we have
 
(10) 
 (11) 
 (12) 
 (13) 
where again the symbols have their usual meaning.
We switch from the angular separation to the linear one using
 
(14) 
where is the parallax of the system. By introducing , the ratio of a_{A} to a_{A}+a_{B}, we can now write
 
(15) 
 (16) 
The system as a whole thus needs the following ten parameters for its complete specification:
 a(''): the semimajor axis of the relative orbit of the faintest component around the brightest star;
 i: the inclination of the orbital plane with respect to the plane orthogonal to the sight direction;
 : the argument of the periastron;
 : the longitude of the ascending node;
 e: the eccentricity;
 P: the period;
 T: the periastron epoch (one of them);
 V_{0}: the radial velocity of the system's center of mass;
 : the parallax of the system;
 : the ratio of the semimajor axis (relative to the brighter component) to the sum of the two semimajor axes.
With these parameters, we face a problem of constrained minimization because while and .
Other parameter sets are actually possible (e.g., by introducing K_{A} and K_{B}, the amplitude of the radial velocity curves, the mass ratio, ...). However, we do not believe one is systematically better than another. Starting from the orbital parameters of visual binaries, the extension with V_{0}, and seems natural to us.
One could be tempted to use the variable transformation proposed by
Soulié (1986) to convert a constrained minimization problem into an unconstrained one (or, at least, a less constrained one). Soulié's variable transformation is:
where x is a variable in [0,1[ and . This could be applied to e, and . Unfortunately, this makes the system even more nonlinear than it already is (the condition number of the objective function increases). We noticed that starting from a uniform distribution of x', we do not get a uniform distribution of x. We therefore decided to keep the problem in its constrained version. We managed the constraints by returning a huge value for the objective function when one of the constraints is not satisfied.
We are looking for a function that attains its minimum at the best estimate for the orbital parameters. This function has to allow one to combine visual measurements (in the current state of our work, only Cartesian coordinates (in the tangential plane)) and radial velocities. We assume that x and y are observed at the same time, but we assume nothing about the observation time of the three data sets. In fact, we treat these three sets independently.
Assume that none of the observations are correlated and choose the objective function as
 

 
 (17) 
where the hat (super) stands for the adjusted (observed) quantity.
D satisfies the mathematical and statistical requirements
for a leastsquares adjustment Eichhorn (1993).
Equation (17) can be rewritten as
where is a 2N_{v}+N_{sA}+N_{sB}vector composed of
, , and
and is a diagonal matrix composed of , , , , the variances of the observed quantities in the appropriate order.
Up: Simultaneous leastsquares adjustment of
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