Principles of Astrophysics by Charles Keeton

Author:Charles Keeton
Language: eng
Format: epub, pdf
Publisher: Springer New York, New York, NY

The effective potential is useful because we can think of it as a surface and use our intuition to understand what would happen to a ball on that surface. Some examples are shown by the dashed curves in Fig. 10.3. For ℓ = 0 the ball would roll all the way down to r = 0. For any nonzero value of ℓ, however, the centrifugal term causes an upturn at small radius. This creates a stationary point that corresponds to a constant radius and hence a circular orbit. In the Newtonian effective potential, the stationary point is a minimum of Φ, so the orbit is stable: if you put the ball near the minimum but give it a little kick, it will oscillate around the minimum but remain confined.

In general relativity the equation of motion can be written in the form of (10.31) but with a different effective potential. To find that potential, recall that special relativity has a relation between energy, momentum, and mass: (Eq. 10.26). With the Schwarzschild metric the analogous relation has factors of :

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