Sample Problem

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Sample Problem

Hunting a black hole. Observations of the light from star indicate that it is pan of a binary (two- star) system. This visible star has orbital speed .

SOLUTION: Some of the Key Ideas in this challenging problem are as follows:

I. The two stars are in circular orbits. not about each other, but about the center of mass of this two-star system.

2. As with the two-particle systems of , the center of mass of the two-star system must lie along a line connecting the centers of the stars-that is. at point 0 in . The visible star orbits at radius  the dark star at radius.

3. The center of mass of the system is not even approximately at the center of a central. massive object (like the Sun). Therefore, Kepler’s law of periods, , does not apply here and we cannot easily find mass m2 with it.

4. The centripetal force causing each star to move in a circle is the gravitational force due to the other star. The magnitude of the force is , where I’ is the distance between the centers of the stars.

We can solve this cubic equation for m2 with a polynomial solver on a calculator. Instead, since we are working with proximate masses anyway, we can substitute integer multiples of M, for m2 until we find one that makes nearly true.

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Thus, we can detect the presence of a black hole provided it is part of a binary system with a visible star whose mass, orbital speed. and orbital period can be measured.

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