For a spherical stellar polytrope, the axial moment of inertia is $I=(8\pi /3){\int }_0^R\rho (r)r^{4}dr$. With Lane-Emden variables for a stellar polytrope, first zero ${\xi }_1$ and Lane-Emden surface mass constant ${\omega }_n$, its dimensionless form isA polytrope of index zero gives $2/5$, while a polytrope of index one gives $(2/3)(1-6/{\pi }^2)$. The latter is smaller because more of the mass lies near the center. The scalar second mass moment $\int r^{2}dm$ is $3I/2$, not the axial moment of inertia.

For a radiative envelope, $d{P}_{\mathrm{rad}}/dP=\kappa L_r/(4\pi cGm)$. Exactly uniform stellar gas-pressure fraction instead requires this derivative to be $1-\beta$. Constant opacity therefore requires $L_r$ to increase in proportion to enclosed mass. A positive-density finite-mass envelope with no local energy generation has constant $L_r$ and cannot satisfy all these conditions pointwise. The Eddington-model convective-core mass fraction uses constant $\beta$ as a global approximation, rather than an exact detailed envelope constraint.

In a dilute electron gas, thermal kinetic energies become relativistic near $k_{B}T=m_e{c}^2$, corresponding to $T_{\ast }\simeq 5.93\times 10^9\mathrm{K}$. This is a crossover in the momentum distribution, not an abrupt temperature transition. Degenerate electrons can instead be relativistic at much lower temperature if their Fermi momentum exceeds $m_{e}c$.

For a nondegenerate fully ionized gas, equating $P_{\mathrm{gas}}=\rho k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ with $P_{\gamma }=a_{\mathrm{rad}}{T}^4/3$ gives $T^3=3k_B\rho /(a_{\mathrm{rad}}{\mu }_{\mathrm{mol}}{m}_u)$. Radiation dominates at higher temperature for fixed density. A degenerate electron gas requires its degeneracy pressure instead of extrapolating this ideal-gas formula.

Equating radiation pressure to cold electron degeneracy pressure gives $T=[3P_e(\rho )/a_{\mathrm{rad}}{]}^{1/4}$. Because the two electron pressure limits scale as ${\rho }^{5/3}$ and ${\rho }^{4/3}$, this boundary has logarithmic slopes $5/12$ and $1/3$, respectively. Above it, photon pressure can dominate even when the net electrons remain degenerate; degeneracy of one component and dominance of the total pressure are different criteria. Thermal pairs modify the high-temperature pressure comparison.

For one ionic species of number density $n_i$, the ion-sphere radius is $a_i=(3/(4\pi n_i){)}^{1/3}$. The ionic Coulomb coupling parameter compares the interaction energy from Coulomb's law at that separation with thermal energy $k_{B}T$. At fixed composition, $\Gamma \propto {\rho }^{1/3}/T$. Values near one mark significant corrections to the ideal ion gas; strong coupling can lead to an ordered solid. Screening, mixtures and quantum motion modify detailed transitions, so this estimate is not a universal crystallization boundary. It is separate from the electron degeneracy criterion based on the electron Fermi temperature.

This term accounts for thermal storage and compression or expansion in the stellar energy balance equation. At fixed composition the first law of thermodynamics makes it $-TDS/Dt$. Kelvin-Helmholtz contraction can supply positive luminosity, while thermal adjustment may locally absorb energy. It is not simply the full change in gravitational potential energy: the stellar virial theorem also requires changes in internal energy.

Escaping neutrinos remove energy produced by nuclear reactions or drawn from thermal reservoirs. Pair annihilation, plasma excitations, photoneutrino emission and electron-ion bremsstrahlung provide thermal losses. This reduces photon luminosity or increases the required nuclear and gravothermal supply. At sufficiently extreme collapse densities, neutrino transport must replace a free-escape loss approximation.

The stellar crossing time is a characteristic time to traverse a system, $t_{\mathrm{dyn}}\sim R/v$. In a virialized self-gravitating system, $v^2\sim GM/R$, so $t_{\mathrm{dyn}}\sim \sqrt{R^3/(GM)}$. It is distinct from the much longer spherical stellar relaxation time. Radius versus diameter and characteristic-speed conventions change order-one factors.

Gravitational softening regularizes a simulation particle's short-distance force, for example by replacing $-Gm/r$ by $-Gm/\sqrt{r^2+{ϵ}^2}$. It suppresses artificial close encounters and effectively raises the diffusion cutoff to order $\max (ϵ,Gm/u^2)$, but alters the resolved force below $ϵ$. Particle number, softening and duration must be chosen together; a point-particle relaxation estimate alone does not specify a simulation's accuracy.

An N-body simulation approximates a dynamical system by evolving $N$ interacting particles. In a collisionless galaxy calculation these particles sample a smooth mass distribution rather than represent individual physical stars. Their larger individual masses enhance artificial two-body relaxation; softening, force accuracy and convergence with particle number must be checked over the intended duration.

The Hernquist model has relative potential $\psi =GM/(r+a)$ and density $\rho =Ma/[2\pi r(r+a{)}^3]$. It has enclosed mass $M{r}^2/(r+a{)}^2$, a central $r^{-1}$ cusp and an outer $r^{-4}$ tail. Since $r\rho =a{\psi }^3/(2\pi G^3{M}^2)$, the half-anisotropic distribution function is $f(E,L)=3a{E}^2/(4{\pi }^3{G}^3{M}^{2}L)$ for $0<E<GM/a$. Its moments are $⟨v_r^2⟩=\psi /4$ and each tangential moment $\psi /8$. Isotropic and other admissible anisotropic distributions can also support this same density; the gravitational potential does not select a unique velocity anisotropy.

The spherical Plummer model has $\psi =GM/\sqrt{a^2+r^2}$ and $\rho =3M{a}^2/[4\pi (a^2+r^2{)}^{5/2}]$. It is the $p=2$ hypervirial model. In unit scales its isotropic distribution is $24\sqrt{2}{E}^{7/2}/(7{\pi }^3)$ for positive relative energy.

For fixed-composition monatomic ideal gas plus equilibrium radiation pressure, specific internal energy is $u=3\mathcal{R}T/2+a_{\mathrm{r}}{T}^4/\rho$. An isentropic process satisfies $du=Pd\rho /{\rho }^2$, giving ${\Gamma }_3-1=(8-6\beta )/(24-21\beta )$ andThe stellar gas-pressure fraction must be allowed to change along the adiabat. The pure-radiation and pure-gas limits give ${\Gamma }_1={\Gamma }_2=4/3$ and $5/3$, respectively. Partial ionization or different gas heat capacities change these formulas.

The leading separation between successive high-order low-degree stellar acoustic mode frequencies is the inverse round-trip acoustic travel time, $\Delta \nu \simeq [2{\int }_0^{R}dr/c_s{]}^{-1}$. For homologous stars it scales approximately as $(M/R^3{)}^{1/2}$ and therefore measures mean density.

The Stefan-Boltzmann constant converts the fourth power of absolute temperature into blackbody radiant exitance:Here $k_B$ is the Boltzmann constant, $h$ the Planck constant and $c$ the speed of light. In the current International System of Units, these three constants have exact defining values, so $\sigma$ is exact as an expression in them and $\pi$. The displayed finite decimal is an approximation, not an exact decimal or a measured uncertainty. The 2022 recommended-constants table lists an ellipsis and an exact uncertainty for $\sigma$.^[ref]

A dimensional check uses $[k_B]=\mathrm{J}{\mathrm{K}}^{-1}$, $[h]=\mathrm{J}\mathrm{s}$ and $[c]=\mathrm{m}{\mathrm{s}}^{-1}$. Consequently $[k_B^4/(h^3{c}^2)]=\mathrm{J}{\mathrm{s}}^{-1}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}=\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$, exactly the units needed in the Stefan–Boltzmann law.

For an equilibrium photon gas, the energy density is $u=a_{\mathrm{rad}}{T}^4$ with radiation constant $a_{\mathrm{rad}}=4\sigma /c$. The radiation pressure is $p_{\mathrm{rad}}=u/3=4\sigma T^4/(3c)$. These describe the radiation filling an equilibrium cavity; $u$ has units $\mathrm{J}{\mathrm{m}}^{-3}$, whereas the outward radiant exitance through a small opening is $cu/4$. The factor $1/4$ comes from averaging propagation directions and their projected area over a radiation field with isotropy, not from reducing the opening's emissivity.

The Pearson chi-squared test of homogeneity tests whether several independent categorical samples have the same category probabilities. Under the null, expected cell counts equal the product of the corresponding row and column totals divided by the total sample size. The Pearson chi-squared statistic for contingency tables $\sum (O-E{)}^2/E$ has an asymptotic chi-squared distribution with $(r-1)(c-1)$ degrees of freedom, when expected counts are sufficiently large. The sampling design differs from a Pearson chi-squared test of independence, though the statistic has the same form.

A least-favourable configuration maximizes a test procedure’s type I error probability over allowed null hypothesis parameters. For one-sided normal comparisons with a fixed covariance matrix, coupling by common centered variables shows monotonicity in the means. For the familywise error rate, mixed true and false null configurations must also be checked.

A scan statistic searches a family of candidate locations or subsets and reports the largest local statistic. Under a null hypothesis, a union bound converts a tail estimate for each candidate into a bound for the maximum; no independence between local statistics is required. Structured alternatives often permit a chi-squared testing lower bound using the overlap geometry of two candidates.

If $q={\sigma }^{2}p$ and $q^{\prime }=2bp$, then $pLu=(q{u}_x{)}^{\prime }/2$. Insert compact cutoffs ${\chi }_R$, integrate the backward equation in time, and integrate by parts twice to move spatial derivatives onto ${\chi }_R$. The error is bounded by $t\Vert u{\Vert }_{\infty }(C{R}^{-2}\Vert q{\Vert }_1+C{R}^{-1}\Vert q^{\prime }{\Vert }_1)/2$. When these norms are finite it vanishes, proving $\int u(t)p=\int fp$ without assuming derivative behavior at infinity.