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An open migration process is a continuous-time Markov chain of colony populations with external Poisson processes, state-dependent departure rates, and fixed probabilistic routing that eventually exits the system. Its traffic equations of an open migration process determine the total arrival rates, and its equilibrium has a product-form stationary distribution of an open migration process when the normalizing sums are finite.

Effective bandwidth describes the exponential-moment traffic rate at a specified time scale and tail parameter. Independent sources have additive effective bandwidths. The Chernoff bound connects their sum to overflow bounds. For a single time unit, the small-parameter limit is the expected value and the large-parameter limit is the essential supremum, with appropriate moment or extended-value conventions.

Congestion control adjusts user transmission rates in response to load signals from shared resources. A resource can compute its price from local aggregate load, while each user sums prices along its route. Primal congestion potential and dual congestion potential formulations explain equilibrium and stability for suitable monotone response functions.

A theorem interchanging a parameter integral with an Itô integral, under appropriate measurability and integrability hypotheses. For a bounded deterministic integrand on a finite parameter-time rectangle, the square-integrability conditions hold; triangular domains can be handled by an indicator of the domain.

An Itô process is a continuous semimartingale expressible as the displayed sum, with adapted coefficients locally integrable for the time integral and locally square integrable for the Itô integral. The coefficients may depend on the whole past; an Itô diffusion usually specifies them as functions of the current state and time. A continuously differentiable deterministic process is an Itô process with zero Brownian coefficient. The Itô formula and Itô product rule apply to these processes.

In an optically thick steady spherical stellar wind of constant mean molecular weight ${\mu }_{\mathrm{mol}}$, let $v$ be the outward speed and $a_g^2=k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ the isothermal sound speed squared. Mass conservation gives ${\rho }^{\prime }/\rho =-2/r-v^{\prime }/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\mathrm{gas}}=\rho a_g^2$ and radiative diffusion to write $P_{\mathrm{rad}}^{\prime }/\rho =-g\Gamma$, where $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{Edd}}$ uses the local Eddington luminosity $L_{\mathrm{Edd}}=4\pi cG{M}_r/\kappa$ and opacity $\kappa$. Substituting into the momentum equation yields the displayed wind equation. Smooth sonic point crossing requires both its denominator and numerator to vanish. The radiation-pressure gradient already supplies radiative acceleration; adding that force separately would count it twice.

A steady spherical polytropic flow escaping a point-mass potential conserves the outward mass loss rate and the Bernoulli function $C=u^2/2+c_s^2/(\gamma -1)-GM/r$. For positive terminal speed, a finite-radius smooth sonic point requires $C>0$ and $1<\gamma <5/3$. Its sound speed obeys $c_{s,c}^2=2C(\gamma -1)/(5-3\gamma )$, and evaluating the conserved flux there selects the transonic mass loss rate. Other parameter limits need separate treatment.

For $p=K{\rho }^{1+1/n}$, the Lane-Emden mass formula gives $M\propto {\rho }_c^{(3-n)/(2n)}$ at fixed $K$, while $M/R\propto {\rho }_c^{1/n}$. For $3<n<5$, the Lane-Emden equation has a finite-radius solution, yet $M\to \infty$ as ${\rho }_c\to 0$. The relativistic Tolman–Oppenheimer–Volkoff equation approaches this limit since $p_c/{\rho }_c\to 0$. Monotone pressure and density alone therefore do not impose a finite stellar maximum mass.

The regular $n=0$ solution of the Lane-Emden equation is $\theta =1-{\xi }^2/6$, with first zero $\sqrt{6}$. Its interior density is constant and its pressure varies as $\theta$. This is the structural incompressible limit; $P=K{\rho }^{1+1/n}$ itself is singular at $n=0$, and does not assign the perturbative stellar adiabatic exponent of a thermally stratified ideal-gas model.

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.