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Suppose an isothermal gas with ${\rho }_g\propto r^{-2}$ is supported in a spherical halo with $M(r)=Mr/R$. Then hydrostatic equilibrium and $P={\rho }_g{k}_{B}T/\bar{m}$ give $-2{\rho }_g{k}_{B}T/(\bar{m}r)=-{\rho }_{g}GM/(Rr)$, hence $k_{B}T=\bar{m}GM/(2R)$. Here $\bar{m}$ is the dimensionful mean particle mass. The factor two follows from the density slope and isothermal pressure response; it is not the uniform-sphere virial coefficient. The singular profile and finite boundary are idealizations.

The mean particle mass is the mass density divided by the total number density of freely moving particles. It enters $P=\rho k_{B}T/\bar{m}$. ionization changes the particle count without significantly changing the mass, so it changes $\bar{m}$. The dimensionless mean molecular weight is $\bar{m}$ divided by a stated reference mass, commonly the atomic mass unit; it must not be substituted as a dimensionless number into a formula requiring a mass.

Original source: physical PDF pages 471–473; paper pages 1–3. All optional notes and all three questions are answered. The temperature estimate in Solution 3 retains an unspecified mean particle mass.

inside-out collapse of a singular isothermal sphere starts from an idealized static envelope,with constant isothermal sound speed $a_s$. Collapse begins centrally after formation of a central mass. An expansion wave moves outward through the initially static envelope at the sound speed:“Expansion” describes the outward-moving information front; the gas behind it moves inward. The exterior remains close to its original $r^{-2}$ equilibrium until the wave reaches it.

In the inner region, the central mass dominates and pressure becomes less important:The self-similar accretion scale is ${\dot{M}}_{\ast }=m_0{a}_s^3/G$, where the dimensionless coefficient $m_0$ follows from the dynamical similarity solution, not from counting material swept by the front. Indeed, the initial mass enclosed by the front grows at $2a_s^3/G$, and some of that material remains in the infalling envelope. For fixed mean particle mass, ${\dot{M}}_{\ast }\propto T^{3/2}$. The original similarity calculation establishes the static outer and free-fall inner regimes.^[ref]

The model assumes spherical symmetry, isothermality and negligible rotation or magnetic stresses. An asymmetric optically thick spectral line alone does not establish infall: an optically thin systemic-velocity tracer and spatial maps help distinguish inward motion from rotation, foreground absorption and outflow.

The normalized cycloid hasThe initially expanding perturbation reaches maximum radius at $\eta =\pi$, $t=t_m$, then formally collapses to zero radius at $\eta =2\pi$, $t=2t_m$. A finite virialized object is assigned the conventional formation epoch $t_f=2t_m$; random motions replace the pressureless point-collapse singularity. The cycloid alone does not describe that relaxation.

For the standard top-hat virialization estimate, write $U=-C_{G}G{M}^2/R$ and retain the same gravitational structure factor $C_G$ at turnaround and final equilibrium. The coherent kinetic energy vanishes at turnaround, whereas the virial theorem gives $2K_v+U_v=0$ in the final isolated equilibrium. conservation of energy givesThe final density is eight times the turnaround density. Meanwhile $\bar{\rho }\propto a^{-3}\propto t^{-2}$, so $\bar{\rho }(t_f)=\bar{\rho }(t_m)/4$. The Einstein-de Sitter virial overdensity follows by combining these factors with the supplied turnaround contrast:The value 200 is a convenient rounded overdensity convention, not exactly the Einstein-de Sitter result.

The virial radius of a dark-matter halo follows from $M_v=4\pi R_v^3{\rho }_v/3$:The hydrostatic temperature of an inverse-square halo follows from hydrostatic equilibrium and the ideal gas law:Since ${\rho }_g\propto r^{-2}$ and $M(r)=M_{v}r/R_v$, cancellation of ${\rho }_g/r$ gives $2k_B{T}_v/\mu =G{M}_v/R_v$. ThereforeHere $\mu$ is a dimensionful mean particle mass, as specified in the paper; a dimensionless mean molecular weight would need multiplication by a particle-mass unit. The isothermal interior is a further halo idealization, not an exact continuation of the uniform top-hat profile used in the virial-overdensity estimate. Changing its gravitational structure factor in a literal energy comparison would change the turnaround-radius relation.

The formation redshift from halo mean density uses the cluster data, which imply ${\rho }_v=3M_v/(4\pi R_v^3)=5.37148\times 10^{13}{M}_{\odot }{\mathrm{Mpc}}^{-3}$. Thus, using the requested rounded convention,Using $18{\pi }^2$ instead gives $z_f\simeq 1.00$. In the Einstein-de Sitter time scaling the rounded-convention result is $t_f/t_0=(1+z_f{)}^{-3/2}\simeq 0.376$. Directly from the mass and radius,For the estimate, take mean particle mass $\mu =0.6m_p$, appropriate approximately to a hydrogen-helium mixture with complete ionization. This gives $T_v\simeq 1.41\times 10^8\mathrm{K}$, or $k_B{T}_v\simeq 12.1\mathrm{keV}$. The composition choice is additional input, not a value supplied by the examination; the boxed expression retains the result for any $\mu$.