Gravitational clustering

The Vlasov-Newton equation has many steady solutions describing a self-gravitating cluster. This is easy to show in the spherically symmetric case (the situation we shall restrict in this work, except for a few remarks at the end of this section). If one assumes a given r(r) in the steady state, the general steady solution of Eq. (4) is a somewhat arbitrary function of the constants of the motion of a single mass in this given external held, namely a funchon/(E, I ) where niE is the total energy of a star in a potenhal (r) such that r(r) = —(r/r) [d r)/dr] and where — (r.v) is the square of the... 

The consequences of the virial theorem are important and differentiate the behavior of a gravitating cluster from a normal gas. Because the total energy is negative, a loss of mass causes the cluster to contract. This is familiar... 

Transport Disengaging Height. When the drag and buoyancy forces exerted by the gas on a particle exceed the gravitational and interparticle forces at the surface of the bed, particles ate thrown into the freeboard. The ejected particles can be coarser and more numerous than the saturation carrying capacity of the gas, and some coarse particles and clusters of fines particles fall back into the bed. Some particles also coUect near the wall and fall back into the fluidized bed. 

The existence of dark matter (either baryonic or non-baryonic) is inferred from its gravitational effects on galactic rotation curves, the velocity dispersions and hydrostatic equilibrium of hot (X-ray) gas in clusters and groups of galaxies, gravitational lensing and departures from the smooth Hubble flow described by Eq. (4.1). This dark matter resides at least partly in the halos of galaxies such as our... 

We have found now an equation for the evolution of the density inside the cluster without any uncontrolled parameter, except for the dimensionless number C. Below we shall do two things. First, in Section V, we shall find the steady solutions for the density, that turns out to transform into a quite simple problem, mathematically equivalent to the equilibrium of self-gravitating atmosphere. Then, in Section VI we shall look at the possible existence of finite time singularities in the dynamical problem. 

By identifying Cepheid variables in the globular clusters which gravitate around our own Galaxy, Harlow Shapley was able to measure their distance. He thus located their common centre and found it to be a considerable distance from us. It was clear that human beings inhabit the neighbourhood of a nondescript star, very far from the centre of the Milky Way. We are not even at the heart of our own stellar republic A second assault was thus made on human vanity, after the eviction of the Earth from the centre of the Universe. 

Eventually, the helium of the star will be exhausted, leading to further gravitational collapse with a temperature increase to 6 x 108-2 x 109 K (kT 100-200 keV). At this point the fusion reactions of the a-cluster nuclei are possible. For example,... 

Clusters form from large volumes. For a Universe with a mean mass density of 3 x 10 3° gm cm-3 (30% of closure density pc = 3 //HirG, with Ho=70 km s-1), a rich cluster with a mass of 1015M forms from a sphere with a radius of 20 Mpc. Since the dominant process in the formation of the cluster whose mass consists of cold dark matter and baryons is gravitation and the formation of collapsed objects by gravity alone should not affect the ratio of the mass components, it is believed that the mass components of today s clusters are representative of the Universe (e.g., White et al. 1993). 

X-ray observations detect deep gravitational potentials. The X-ray emission from galaxy clusters is optically thin thermal bremsstrahlung. The luminosity of a cluster is given as ... 

We begin with an introduction to a simple model first proposed by Cavaliere and Fusco-Femiano (1976). This model assumes that the gas and galaxies are in equilibrium within the same gravitational potential. Through a measure of the X-ray surface brightness, the model relates the gas temperature to the cluster velocity dispersion, measured from the galaxies. 

Independent of any model, one can use the X-ray observables, gas density and temperature, to derive the total gravitating mass of a cluster. Thus, the elusive dark matter that dominates the cluster gravitational potential can be mapped from X-ray observations. 

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