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Analysis and Interpretation of Astronomical Spectra 91

19.4 The Estimation of some Orbital Parameters

Based on purely spectroscopic observation some of the orbital parameters of the binary
system can be estimated. First, the measured radial velocities and are plotted as a

function of the time . The graph shows an orbital period of Mizar (ζ Ursae Majoris, A2V),

one of the showpieces for spectroscopic binaries with double lines (SB2 systems). Another
similar example is β Aurigae with an orbital period of about 4 days. The more these curves
show a sinusoidal shape, the lower is the eccentricity of the orbit ellipse [179].

K1

K2

Source: Uni Jena [170]

The Orbital Period

The orbital period T can directly be determined from the course of the velocity curves. As
the only variable it remains largely unaffected by perspective effects and is thus relatively
accurately determinable.

Simplifying to circular orbits

Since we are mostly confronted with randomly oriented, el- M1 rM1 B rM2 VM2
liptical orbits, a reasonably accurate determination of orbital M2
parameters is very complex. For this purpose, in addition to VM1
the spectroscopic, complementary astrometric measure-
ment data would be needed. Only for eclipsing binaries,
such as Algol, already a priori, a probable inclination of

can be assumed. For the rough estimation of other
parameters various sources suggest the simplification of the
elliptical to circular orbits. Thus, the radii and thus the
orbital velocities become constant. The mostly unknown
inclination is expressed in the formulas with the term .

The Orbital Velocity

To determine the orbital velocity we need from the velocity diagram, the maximum

values for both components. They correspond by definition, to the maximum amplitudes
and . For the circular orbit velocity follows:

s s
Determination of the orbital radii

With the orbital period and the velocity s for a circular orbit the corresponding

radius can be calculated. From geometric reasons follows generally:

s