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Diffusive shock acceleration repeatedly returns particles to a shock wave by scattering them in the upstream and downstream plasma. For fluid speeds $u_1>u_2$, particle speed $v\gg u_1$, and shock compression ratio $r=u_1/u_2$, the cycle gain is $⟨\Delta p⟩/p=4(u_1-u_2)/(3v)$ and the cycle escape probability is $4u_2/v$. Their ratio gives $f(p)\propto p^{-3r/(r-1)}$ for the isotropic phase-space one-particle distribution function. For $r=4$ this is $p^{-4}$, corresponding to $dN/dE\propto E^{-2}$ in the ultrarelativistic regime.

Let $n(\mathbf{x})=\int f{d}^{3}v$ and $\overline{A}=n^{-1}\int Af{d}^{3}v$. Assume the one-particle distribution function and its weighted tails decay enough that all required velocity integrals converge and boundary terms at infinite velocity vanish. Multiply the steady Collisionless Boltzmann equation by $v_j$ and integrate over velocity. In Cartesian coordinates, spatial differentiation commutes with this integration, while integration by parts givesConsequently the three force-balance equations areExplicitly the $x$ component is ${\partial }_x(n\overline{v_x^2})+{\partial }_y(n\overline{v_x{v}_y})+{\partial }_z(n\overline{v_x{v}_z})=-n{\Phi }_x$, with cyclic counterparts for $y,z$. For equal stellar masses, multiplying by the common mass replaces $n$ by the mass density $\rho$; for multiple stellar masses, use a mass-weighted one-particle distribution function or derive a separate equation for each tracer population.

The Jeans equations express local momentum conservation for a collisionless stellar system, just as the Euler equations do for a gas. The mean streaming velocity plays the role of fluid flow, the gravitational force is a body force, and random stellar motions supply a pressure tensor through $\rho {\sigma }_{ij}^2$. Taking moments smooths over individual stellar orbits and produces fields of density, mean velocity and random-motion stress; this is the sense of “stellar hydrodynamics”.

Ordinary gas pressure is usually nearly isotropic because collisions relax its local velocity distribution. A collisionless stellar system can retain an anisotropic velocity ellipsoid, so its pressure tensor is not generally a scalar times the identity. There are more unknown second moments than momentum equations, and their evolution introduces higher moments. The Jeans equations therefore require further dynamical assumptions or a specified one-particle distribution function for closure; they do not by themselves establish local thermal equilibrium or a gas equation of state.

The one-particle distribution function is a scalar function on the future mass shell. Choose its normalization so that, in a local orthonormal tetrad, $f{d}^{3}x{d}^{3}p$ is the expected photon number in a phase-space cell on a constant-time slice. Spin degeneracy and any phase-space normalization factors are included in $f$; with an occupation-number convention they would instead appear explicitly in the measure.

More covariantly, the photon number crossing a spacelike element is $f{p}^{\mu }d{\Sigma }_{\mu }{d}^{3}p/E$. The factor $p^{\mu }d{\Sigma }_{\mu }$ reduces to $E{d}^{3}x$ on a local constant-time slice, recovering the cell interpretation. The measure $d^{3}p/E$ is Lorentz invariant on the mass shell.

A photon carries energy $E$ and momentum $p^i$, while its velocity is $p^i/E$. Energy density, momentum density and momentum flux are thereforeThey assemble into the stress-energy tensorThe tetrad momenta are related to coordinate components by the tetrad basis. For an isotropic distribution, angular averaging gives $T^{\hat{i}\hat{j}}={\rho }_{\gamma }{\delta }^{ij}/3$ and $T^{\hat{0}\hat{i}}=0$. Thus the photon gas has pressure ${\rho }_{\gamma }/3$, consistent with its massless equation of state.

In the ultrarelativistic regime of a, retaining a constant mean fractional energy gain and ignoring losses gives, after $N$ individual crossings,After $n$ complete return cycles, $N=2n$ and $E_n\simeq E_0(1+u/c{)}^n$, to the same first-order accuracy. These products approximate the mean amplification; individual electron crossing angles give a distribution around it.

The downstream electron distribution is nearly isotropic. Its crossing particle flux per unit area is $n_{\mathrm{a}}c/4$, where $n_{\mathrm{a}}$ is the accelerated-particle number density. The downstream flow permanently advects particles away at particle flux $n_{\mathrm{a}}{u}_2$. Their ratio estimates the escape probability after a downstream visit:Upstream particles are swept back towards the shock wave in this idealized infinite-upstream model. Consequently this return probability applies once per complete cycle, not once per individual crossing or local scattering event.

Let $A\simeq 1+u/c$ be the amplification per complete cycle and $P=P_{\mathrm{ret}}$. A steady injection of ${\mathcal{N}}_0$ particles leaves ${\mathcal{N}}_0{P}^n$ surviving to reach $E=E_0{A}^n$. Eliminating $n$ gives the cumulative power law and differential energy distributionThe escape distribution has the same differential exponent when $P$ is independent of energy. Thus the standard Strong-shock Rankine-Hugoniot conditions giveEquivalently the phase-space one-particle distribution function is $f(p)\propto p^{-4}$ for ultrarelativistic particles. Equal logarithmic intervals then contain equal particle energy. A finite source age, upstream escape, and radiative losses eventually introduce cutoffs or breaks.

A particle exponent is not automatically the radiation spectral index. For optically thin uncooled synchrotron radiation from $d\mathcal{N}/dE\propto E^{-s}$, the characteristic frequency scales as $\nu \propto E^2$ and the emitted power as $E^2$. Hence $L_{\nu }\propto E^{-s}{E}^2/(d\nu /dE)\propto {\nu }^{-(s-1)/2}$. With the convention $L_{\nu }\propto {\nu }^{-\alpha }$, this gives $\boxed{\alpha \simeq 1/2}$ and $\nu L_{\nu }\propto {\nu }^{1/2}$. Strong synchrotron cooling steepens a continuously injected particle distribution by one exponent, giving $\alpha \simeq 1$ above its cooling break. A photon number distribution $d{\dot{N}}_{\gamma }/d\epsilon \propto {\epsilon }^{-\Gamma }$ has $\Gamma =\alpha +1$, or $3/2$ in the uncooled case.

Convention qualification. If the printed $P=1-u/c$ is instead imposed independently at every individual crossing, its combination with $A_{\mathrm{crossing}}=1+u/(2c)$ gives $s\simeq 3$ and uncooled $\alpha \simeq 1$. For example $u/c=0.01$ gives $s=1-\ln (0.99)/\ln (1.005)\simeq 3.015$, whereas the consistent cycle model gives $s\simeq 2.010$. The standard $s=2$ estimate requires matching the probability and amplification to the same complete cycle.

The temporal-temporal component of the photon stress-energy tensor in a local orthonormal tetrad. For an isotropic one-particle distribution function expressed using comoving energy $ϵ=aE$, it is ${\bar{\rho }}_{\gamma }=4\pi a^{-4}\int dϵ{ϵ}^3\bar{f}(ϵ)$. A thermal photon temperature perturbation gives a first-order directional energy perturbation four times its fractional temperature change. Boundary conditions on the distribution justify this factor through integration by parts.