Solution

(10 yr)(3.16 X 107 s/yr)[(10 + 20)(1 X 105 cm/s)]3 2^(6.67 X 10-8 dyn cm2/g2)(2 X 1033 g)(sin3i)

We find the ratio of the masses from the ratio of the radial velocities m1/m2 = v2/v1 = 2.0

This means that m1 = 2m2, so giving m1 = 6.8 M0 and m2 = 3.4 M0. If i = 45°, 1/sin3i = 2.8. The ratio of the masses does not change, since the sin i drops out of the ratio of the radial velocities. This means that we can just multiply each mass by 2.8 to give 19.2 and 9.5 M0, respectively.

It is often the case that only one Doppler shift can be observed. Let's assume that we measure v1 but not v2. We must therefore eliminate v2 from our equations. We can write v2 as

The sum of the velocities then becomes

If we now substitute into equation (5.24), we find

The quantity on the right-hand side of equation (5.25) is called the mass function. If we can measure only one Doppler shift, we cannot determine either of the masses. We can only measure the value of the mass function. We can, however, obtain information on the masses of various spectral types through extensive statistical studies.

We can also look at energy of a binary system with circular orbits. It is the sum of the kinetic energies of the two stars plus the gravitational potential energy, which we take to be zero when the two objects are infinitely far apart. The energy is

E = (1/2) m1 v2 + (1/2) m2 v2 - Gm1 m2/R From equation (5.15), we have

w and, using equation (5.12), we obtain m2V2

m1m2

Substituting these into equation (5.26) and simplifying gives

Compare this with equation (3.4), which has the energy for circular orbits with electrical (rather than gravitational) forces.

The negative energy means that the system is bound. We would have to add at least (1/2)Gm1m2/R to break up the binary system. (We can think of this as being analogous to the binding energy of a hydrogen atom.) For any pair of masses, as you make the orbits of the binary system smaller, the energy becomes more negative.

Example 5.4 Binding energy of a binary system. What is the binding energy of a binary system with two 1 M0 stars orbiting with each 100 AU from the center of mass?

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