Many stars in the sky are in fact multiple systems composed by two or more stars orbiting around the center of mass of the system. For these systems having a common center of mass, it is possible to calculate some stellar characteristics, including the mass. Let’s take a look at binary star systems.
Binary star systems (made up of two stars) are classified according to observational characteristics, which vary in the way in which data is analysed:
- Optical double: not true binaries, simply in a similar line of sight. Gravitationally not interacting.
- Visual binary: Both stars can be resolved visually and their motions can be monitored (unless their orbital period is too long).
- Astrometric binary: If one of the stars is brighter, the dimmer one may not be observable. The existence of the unseen member may be deduced from the oscillating motion of the other due to gravitational interaction.
- Eclipsing binary: If the orbital planes are approximately along the line of sight of the observer, each star may periodically eclipse the other. The amount of light received from such system varies regularly. Relative effective temperatures and radii can be measured.
- Spectrum binary: System with two superimposed, independent, discernible spectra. Spectral shifts due to Doppler effect caused by the motion of the stars may be detected. If the Doppler shift cannot be detected (e.g. the orbital plane is perpendicular to the line of sight), the two sets of superimposed spectra may be detected if coming from stars of differing spectral classes.
- Spectroscopic binary: A periodic shift in the spectral lines caused by the motion of the stars is observed. Case if the orbital period is not too long, and the orbital motion has a component along the line of sight. This will be observed for both stars, if both have similar luminosity. One star’s spectral lines will shift toward red, while in the other the lines will shift toward blue. Even if one star is so much brighter than the other, so that the spectrum of the dimmer star is overwhelmed, the shift will still be observed for the brightest star, revealing a binary system. The next video is quite illustrative:
Masses can be determined for three cases:
- Visual binaries combined with parallax.
- Visual binaries for which radial velocities are available over an entire orbit.
- Eclipsing, double-line, spectroscopic binaries.
Case where the orbital plane is perpendicular to the line of sight:
Case where orbital plane is tilted with angle i with respect to the line of sight:
i can be deduced, because the focus of a tilted elipse, as seen from Earth, is not consistent with the 3rd Kepler’s Law. A tilted elipse is also seen as an elipse from the point of view of the observer, but the projected focus is not in the position where the focus of the projected elipse should be (see the picture). Moreover, the position of the projected focus depends on the tilted angle i.
The dependence on the angle i cancels out in the mass ratio anyway:
However, it is not cancelled in 3rd Kepler’s Law, and it depends on i:
Elliptical orbits can be more complexly tilted, with both the semimajor and semiminor axis at an angle with respect to the observer, but the same principles discussed above always apply albeit the formula may become more complex.
It is possible to determine the individual masses of members of visual binaries even if the distance to the system is not known, if detailed radial velocities are available.
Eclipsing, spectroscopic binaries:
Many spectroscopic binaries possess nearly circular orbits, simplifying the analysis. If this is the case, the eccentricity of the orbits is much smaller than 1, and the speeds of motion of the stars, v1 and v2 are nearly constant.
Distinguish two cases, similarly as in the visual binary star systems:
- Orbital plane lies in the line of sight of the observer, i = 90º. In this case, the radial velocity curves will produce sinusoidal curves.
- The orbital plane is tilted. i ≠ 90º. In this case, similarly as in visual binaries, the ratio of the masses is not affected, but the sum of the masses depends on i.
If the eccentricity of the orbits is significantly larger than 0, the radial velocity curves will be somehow skewed.
There are cases where one star is much brighter than the other. Such systems are single-line spectroscopic binaries. Still, we can know something called mass function:
This depends only on observable quantities (period and radial velocity of the brighter star), but cannot provide information about mass ratios, unless we know the mass of one of the two stars by some other means.
It is not possible to get exact values for the masses of the stars without knowing i. i can be however estimated statistically. We could choose an integral average of sin3 i evaluated between 0º and 90º to obtain 0.42. Or a larger value can be taken such as 2/3, since we can assume that it is easier to identify spectroscopic binary star systems if i differs significantly from 0º (the reason is that no Doppler shift will be produced at 0º for us to see).
The evaluation of masses of binaries has show a mass-luminosity relation.
What if the system is eclipsing? Then, we can assume that the tilt angle is close to 90º, unless the stars were very close to each other in comparison with their radii. Even an error in the tilt angle of 15º would result in an error of only 10% in the value of sin3 i and the determination of the sum of the masses.
There are two cases of eclipsing binaries, similar to the cases of the sun and moon:
- Total: One star completely blocks the light from the other, at a given time. The brightness of the received spectra has nearly constant minima when the total eclipse takes place.
- Partial: Neither star is ever completely eclipsed by its companion. The brightness minima are no longer constant.
In eclipsing binaries, we can estimate:
- The radii of the stars. The ts are indicated in the figure above and correspond to certain times during the eclipses.
- The ratio of effective temperatures. B stands for brightness. B0 for the brightness when both stars are fully visible. Bp for the primary minimum (the deepest), and Bs for the secondary minimum. Subindex l and s stand for large star and small star.