By Neb Duric
Graduate textbook in astrophysics.
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Extra info for Advanced Astrophysics (2003)(en)(310s)
3 model. From: Perlmutter, S. , The Supernova Cosmology Project, 1999, ApJ, 517, pp. 565–586. Inspection of Fig. 7. The sum of the two terms is consistent with Ω = 1. Thus, it appears that we live in a flat Universe. There is still considerable uncertainty in these estimates, as shown in Fig. 5. Future projects, such as the Supernova Acceleration Probe (SNAP), are designed to bring down the uncertainties, as illustrated in Fig. 5. 2 Large-scale cosmic structure 45 3 No Big Bang 95% 42 Supernovas 90% 2 68% 1 WL SNAP SAT Target Statistical Uncertainty ve r expands fore a ll y re collapses eventu 0 0 1 d se clo flat n e op −1 Flat Λ=0 Universe WM 2 3 Fig.
Ri − r j | (67) Equating (66) and (67) yields the relation that defines the theorem 1 1 E k = − E G = |E G |. 2 2 ⇒ the Virial theorem (68) According to the Virial theorem the total energy of a stationary system is E T = Ek + E G = 1 EG . 2 A stationary system is one in which no significant dynamical evolution is taking place. 3 Clusters of galaxies Let us consider now a cluster of galaxies and the application of the Virial theorem to such a cluster. The kinetic energy can be obtained by summing over the kinetic energies of the individual galaxies in the cluster.
Now we can determine the potential associated with the surface density. Using (53) Φk (R, z) = −e−k|z| 2GΣk (R) . k (55) 30 Galaxy dynamics Keep in mind that this is a special solution corresponding to a particular surface density Σk . To get the general solution for an arbitrary surface density Σ we need to allow for all possible values of k. If we can find a function S(k) such that ∞ Σ(R) = S(k)Σk (R) dk = − 0 ∞ 1 2G S(k)J0 (k R) k dk (56) S(k)J0 (k R) e−k|z| dk (57) 0 then we will have ∞ Φ(R, z) = ∞ S(k)Φk (R, z) dk = 0 o and the equations will be in terms of an arbitrary mass density Σ(R).
Advanced Astrophysics (2003)(en)(310s) by Neb Duric