Observing in astrophysics is generally related to a received electromagnetic signal from the celestial object we want to detect

Observing in astrophysics is generally related to a received electromagnetic signal from the celestial object we want to detect. As in what we are concerned, i.e. galaxy clusters, this signal originates from the physical content and processes happening inside the clusters and its variation reflects their evolution in time. Here we describe different observables emanating from galaxy clusters and their physical origins.

subsection{Cluster, gastrophysics and cosmology}
hspace*{0.75in}
We have seen that the formation of a galaxy cluster is mainly driven by gravity from its dark matter mass content. But that’s not the whole picture, clusters also contain gas and radiation. To elaborate more, we say that the hierarchical model of large scale structures formation suppose that virialized systems of all
masses form via a sequence of DM component gravitational collapse and mergers of smaller objects followed by accretions of gas that settles
in hydrostatic equilibrium within the cluster potential well. Falling gas then cools through radiation and collision processes, its pressure decrease and becomes less resistant and collapse under the combined gravity of DM and gas citep{1902RSPTA.199….1J}
and form stars grouped in galaxies members of the cluster. Two process though will eventually limit further star formation. First the intergalactic gas is heated to high, X-ray emitting temperatures by adiabatic compression and shocks. Second, the formation of stars could evolve later to massive black holes which result in feedback due to supernovae or active galactic nuclei (gas accretion around black holes), that can inject substantial amounts of heat into the intergalactic medium and quench galaxy formation (see cite{2011IAUS..277..273S} for a quick review on the subject).\

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We see from this model
That means also if we see it in a different way that clusters’ mass could be inferred by studying its galaxies and gas content. The good thing also coming from this complicated formation history and evolution of clusters is that it offers many proxies the observers can probe to detect clusters and infer their mass.

Gravity being scale invariant, it results in self-similar models citep{1986MNRAS.222..323K,1995MNRAS.275..720N} where simple
relations relates basic cluster properties to the total mass so that passing from halo abundance to cluster counts could be considered as a scaling operation citep{1986MNRAS.222..323K}. We shall review in the coming sections the physics behind different mass observable connections and the resulting scaling relations.
\

subsubsection{Cluster gas physics}
label{sect:clustgas}

Here we justify the model under which clusters and their intracluster gas are treated, i.e., virialized massive halos of dark matter containing a high ratio, with respect to it galaxies, of a hot gas isotropically distributed in a hydrostatic equilibrium. The most relevant processes constituting the model will be determined by examining the different time scales involved, each related to a different physical origin (cf. Fig~
ef{fig:clustertimescales} for quantitative values)\

Beginning with the dynamical time scale or free fall collapse time:
egin{equation}
t_{dyn} = frac{1}{sqrt{G
ho}}sim10^9left( frac{1}{200
ho_{c0}Omega_m}
ight )^{-1/2} , yr,
label{eq:tdyn}
end{equation}
where $
ho_{c0}$ is present day critical density, which turns to be an order of magnitude shorter than Hubble time scale given by $t_Hsim (H_0)^{-1}$. This means that clusters are gravitationally bound and decoupled from universe expansion.\
Moreover, the calculated velocity using the virialized theorem is consistent with observed clusters velocities indicating that clusters are virialized objects.\
The gas particle follow a Maxwellian distribution via coulomb collisions yielding an equilibrium time scale
egin{equation}
t_{equ} = frac{l_e}{left}sim3 imes 10^5left( frac{T_e}{10^8 K}
ight)^{3/2} left(frac{n_e}{10^{-3} cm^{-3}}
ight)^{-1} , yr
label{eq:tcoll}
end{equation}
where $l_e$ is electron mean free path and $left$ is rms electron velocity, $T_e$ electron temperature and $n_e$ electron number density
which happens to be shorter than the dynamical time scale justifying that the gas is in thermal equilibrium
at the observed temperature of the clusters $T sim 10^7$ the gas is fully ionised and emit X-ray from thermal free free bremsstrahlung yielding a cooling time scale
egin{equation}
t_{cool} = frac{n_ek_BT}{j_
u}sim4 imes 10^{10}left( frac{T}{10^8 K}
ight)^{1/2} left(frac{n_e}{10^{-3} cm^{-3}}
ight)^{-1} , yr
label{eq:tcool}
end{equation}
where $j_
u$ is the thermal bremsstrahlung emission and $k_B$ Boltzmann constant $T$ relatif to the gas
The cooling time scale is usually longer than the Hubble time scale for density ($