The mass of a star governs the physical nature of its interior, nut masses are hard to determine: they can only be measured for stars in binary systems, and only then under favourable conditions.

The two stars in a binary system orbit around their common centre of mass. The properties of the orbit are connected with the masses of the’ stars by equations that are easily derived from the law of gravitation. These equations can be used to find the masses as long as several other quantities can be measured by means of independent observations.

Visual binaries have orbits that can be observed directly as ellipses traced out relative to the background of stars. If the linear dimensions of the orbits and the individual motions of the two components are known, then it is possible to deduce the masses of both stars. In practice this only been achieved for a few dozen binary systems. If any data is missing for example, if one of the two stars is too faint to be visible I hen il may be possible to deduce the ratio of the masses, or perhaps the sum of the quantity but not the individual masses.

Spectroscopic Binary stars cannot be resolved by any telescope , but the periodic Doppler shift of the spectrum lines give away the orbital motion. In general, the angle i at which the orbital plane in inclined to the perpendicular to the line of sight is unknown and in these cases the best data on the masses that can be obtained from the motion are m1 sin3 i and m2 sin3 i ,where m1 and m2 denote the two masses. This sets lower limits on the masses as sin3 i has a maximum value of l . If only one spectrum is visible, then only quantity known as the MASS FUNCTION can be deduced. However, if a spectroscopic binary is also eclipsing the value of must be close to 90o (sin3 i about 1.0) and the values of m1, and m2 can be found

There is another method of mass determination which has been applied to a handful of white dwarfs, but it is inapplicable to ordinary stars. White dwarfs are so dense that the spectrum lines suffer a gravitational redshift. The wavelength shift, AX, is related to the mass m1 and the radius R of the star by the equation and is of order l0-2nm in the visible part of the spectrum. The results indicate that the masses of white dwarfs lie in the range 0.5 to 1.4 solar masses.

Masses are known for only a couple of hundred stars, but this data shows that their values cover a comparatively narrow range, between 0.1 and 50 M0. Stars of low luminosity and low mass are very much more common than high mass stars. Examples of stellar masses and luminosities from visual binary systems are given in table 2.3, and figure 3.2 shows the important mass-luminosity relation. It can be expressed approximately by LocM3 for main-sequence stars much more massive or much less massive than the Sun and LocM4 5 for those similar to the Sun.

The stars are much too distant to show a measurable disc in conventional telescopes, and so several indirect methods have been developed for getting information on stellar diameters. The largest body of data comes from the eclipsing binary stars, which as we saw above, are so useful for giving us star masses. In the eclipsing binaries it is possible to work out the size of the star orbit in kilometres, and to measure what fraction of the orbit is taken up by an eclipse. The relative lengths of the eclipses depend on the diameters of the two stars concerned, and therefore the diameters in kilometres can be deduced.

It is also possible to measure the angular diameter of nearby giant stars by use of stellar interferometer or speckle interferometry. For example, this has been accomplished for the red-giant star Betelgeuse, in Orion, for Aldebaran and for Sirius. In the latter case, the angular size is 0.007 arc sec, and the parallax 0.377 arc sec (2.65 pc). which gives a diameter about twice that of the Sun. The masses and radii of stars on the main sequence are related as shown in figure 2.6. which illustrates the mass-radius relation for stars in binary systems.

We will mention one final method of deducing a star’s surface area, and so being able to work out the radius. The spectrum of a star carries information on the temperature of the surface, and this temperature in turn governs the luminosity per unit area. If we can find the absolute brightness of the same star, then we know its total luminosity. Dividing this latter by the luminosity per unit area yields the area and hence the radius. Although this method is indirect it is of importance historically because it showed that giant and dwarf stars existed with roughly similar temperatures.

Comments are closed.