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"Scale height is the distance over which a quantity falls to 1/e of its maximum value.

"Scale height is the distance over which a quantity falls to 1/e of its maximum value.

and size of the Earth's orbit to tell us the mass of the Sun, we can use the orbital periods at different distances from the galactic center to tell us about the distribution of mass in the galaxy.

To see how this works, we assume that all matter follows circular orbits. At a distance R from the center, the orbital speed is v(R), and the angular speed is ^(R). The mass interior to radius R is

where p(r) is the density at radius r, and dV is a volume element. For a spherical mass distribution, the motion of an object at R depends only on M(R). Furthermore, the mass M(R) behaves as if it were all concentrated at the center. (This also works for particles in the plane of a thin disk.)

For a particle of mass m orbiting at a radius R, the gravitational force is GM(R)m/R2, and this must provide the acceleration for circular motion, so

Solving for M(R) gives v2(R)R

Therefore, if we can measure v(R), we can deduce M(R), the mass distribution in the galaxy. Equivalently, we can use ^(R), since

Substituting into equation (16.3) gives

If all of the mass is, indeed, concentrated at the center of the galaxy, then M(R) is a constant, so equation (16.5) gives Kepler's third law (mentioned in Chapter 5). (We speak of the orbits as being "Keplerian".) The function v(R), or ^(R), is called the rotation curve for the galaxy.

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