By Neb Duric

Graduate textbook in astrophysics.

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**Example text**

It does not satisfy Laplace’s equation because matter is present there. We therefore need to correct the solution for the presence of the thin matter disk. Since the matter distribution is two dimensional, that is it forms a surface rather than a volume, it makes sense to use Gauss’s theorem here. Since we are interested in finding the surface density that gives rise to the discontinuity in ∇Φ ∂Φk = −k J0 (k R), ∂z ∂Φk = k J0 (k R), ∂z lim z → 0+ lim z → 0− . The integral of ∇Φk over a closed unit surface must equal 4GΣk (48) so that Σk (R) = − k J0 (k R) 2G (54) is the surface density that we need.

268–278). Combining (40) and (41), yields 4 2 (a1 sin i)3 m 32 sin3 i = . (m 1 + m 2 )2 G P2 (42) The quantity on the right can be estimated from observations. The left-hand side is referred to as the mass function. Though it contains information about the masses it is not possible to determine individual masses or, for that matter, the total mass. If two spectra are visible the ratio of the masses can also be obtained. If the binary is eclipsing (i = 90◦ ) the individual masses can also be obtained.

The local supercluster Similar measurements of the Virgo supercluster (we sit in its outskirts at d ≈ 11 h −1 0 Mpc) yield η ≈ (80 ± 30)h 0 . Thus, we see that light-to-mass ratios of all structures larger than galaxies are much greater than those of the individual galaxies. Studies such as these have been used to estimate the amount of matter in the Universe. 3.