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For an adiabatic cosmological perturbation, the baryon and photon fractional density perturbations obey ${\delta }_b=3{\delta }_{\gamma }/4$. Since the pressure of the photon-baryon fluid is supplied by the photons,The photon-baryon sound speed is therefore

Use ${\bar{\rho }}_b\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$ in the Comoving Jeans lengthWell before matter-radiation equality, photon inertia dominates, $c_s\simeq c/\sqrt{3}$, andOnce baryon loading dominates while tight coupling still holds, $c_s\propto a^{-1/2}$, so ${\lambda }_{J,\mathrm{com}}$ is approximately constant. At cosmological recombination, photon pressure support disappears and the baryonic Jeans scale drops sharply. For a subsequently adiabatic monatomic gas, $T_b\propto a^{-2}$ and $c_s\propto a^{-1}$, givingin an Einstein-de Sitter universe. The requested graph therefore rises as $t^{1/4}$, flattens before recombination, jumps downward there, and then decreases as $t^{-1/3}$.

For collisionless matter, the same instantaneous estimate uses its one-dimensional velocity dispersion ${\sigma }_v$ instead of $c_s$; more precisely, suppression is described by collisionless free streaming. ThusWhile the particles are relativistic, ${\sigma }_v\simeq c$ and ${\lambda }_{J,\mathrm{com}}\propto a^{1/2}$. Once nonrelativistic but thermally coupled to radiation, $T_{\mathrm{cdm}}\propto a^{-1}$, so ${\sigma }_v\propto a^{-1/2}$ and the scale is constant. After kinetic decoupling, momentum redshifts as $a^{-1}$, so ${\sigma }_v\propto a^{-1}$ and ${\lambda }_{J,\mathrm{com}}\propto a^{-1/2}$. The second graph joins these three power laws at $t_{\mathrm{NR}}$ and $t_{\mathrm{dec}}$; unlike the baryonic graph, its final decline begins at dark-matter kinetic decoupling rather than recombination.

If $c_{s,\infty }\gg \sigma$ and $M_{\mathrm{enc}}\to M$, expansion of the denominator givesthe stated model's pressure-dominated Bondi accretion limit. Its order-unity coefficient differs from the $\pi e^{3/2}$ found for the exactly isothermal critical solution because such coefficients depend on the adopted equation of state and interpolation.

If instead $\sigma \gg c_{s,\infty }$,The gas's thermal motion is then negligible beside the galactic velocity dispersion, and the enclosed galactic mass, rather than the black hole alone, focuses the gas. For a singular isothermal sphere, $M_{\mathrm{enc}}(R)=2{\sigma }^{2}R/G$, so this becomes $\dot{M}\simeq 4\pi {\rho }_{\infty }\sigma R^2$. This second limit describes capture controlled by the host potential and is therefore not a genuinely spherical black-hole Bondi solution.

A galactic distribution function $f(\mathbf{x},\mathbf{v},t)$ is the stellar mass or number per six-dimensional phase space volume,Its velocity moments give the spatial density, mean velocity, and velocity dispersion; integrating those quantities along the line of sight and weighting by luminosity produces surface-brightness and line-of-sight-velocity observables. A model is compared with data only after the same projection, selection function, and instrumental convolution have been applied.

Jeans theorem states that every steady solution of the Collisionless Boltzmann equation depends on phase-space coordinates only through integrals of motion. Conversely, every nonnegative function of isolating integrals is a steady collisionless distribution function on the region where those integrals are defined.

For the stated power law in relative energy, isotropy givesWith the requested change of variables $v=\sqrt{2\Psi }\cos \theta$, the density becomeswhere the last equality uses the Beta function and Gamma function. Thusfor $n>1/2$.

The normalized second velocity moment isThe same substitution and the Beta-function recurrence giveConsequently the one-dimensional isotropic velocity dispersion is ${\sigma }^2=\overline{v^2}/3=\Psi /(n+1)$, proving the required linear dependence on the relative potential.