Examples

Example 17.1 Show that if the stars are uniformly distributed in space and there is no interstellar extinction, the number of stars brighter than apparent magnitude m is

Let us suppose first that all stars have the same absolute magnitude M. The distance in parsecs of those stars having the apparent magnitude m is r = 10 x 10a2(m-M) .

In order to appear brighter than m, a star has to be within a sphere of radius r. Because the stellar density is constant, the number of such stars is proportional to the volume:

The result does not depend on the absolute magnitudes of the stars, so that the same result still applies when the magnitudes are not equal, as long as the luminosity function does not depend on distance. Thus the equation is generally valid under the stated conditions.

Example 17.2 The Estimation of Distances by Means of Oort's Formulas

An object in the galactic plane at longitude l = 45° has the radial velocity of 30 km s-1 with respect to the LSR. What is its distance? According to (17.10), vr = Ar sin 2l.

30 km s

A sin2l 15 km s :kpc

In practice, the peculiar velocities are so large that this method cannot be used for distance determination. Oort's formulas are mostly suitable only for statistical studies.

Example 17.3 Discussion of the Gravitational Field of a Uniform Disc

It can be shown that the gravitational field of a homogeneous infinite thin disc is constant and directed towards the plane of the disc. If the mass per unit area of the disc is a, the gravitational field is g = 2nGa .

A test particle located outside the plane will therefore get a constant acceleration towards the plane. Taking a numerical example, assume a mass of 1011 Me distributed uniformly on a circular disc 20 kpc in diameter. The mass per unit area will be a

The corresponding gravitational field is g = 2.8 x 10-10ms-2 .

Let a star be located at d = 1 kpc above the plane, initially at rest (not very near the edge of the disk, in order to keep our approximation valid). The disc will pull it towards the plane, and when the star crosses the plane, it has acquired a velocity given by v = v^gd = 130 kms-1 .

The time required to reach the plane is t = v/g = 15 x 106 a.

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