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In the Cowling approximation, an axisymmetric meridional fluid displacement in a cylindrically rotating barotropic star obeys $-{\omega }^2\xi =\mathcal{F}\xi$, with $\delta \rho =-\nabla \cdot (\rho \xi )$ and $\delta p=-\xi \cdot \nabla p-\gamma p\nabla \cdot \xi$. Conservation of specific angular momentum supplies the epicyclic term, with ${\kappa }^2=R^{-3}d(R^4{\Omega }^2)/dR$. The Cowling energy principle for a rotating barotropic star gives its symmetric quadratic form.

High-temperature photodisintegration of neon-20 releases an alpha particle and oxygen-16. Capture of that alpha particle by another neon-20 nucleus produces magnesium-24. The breakup consumes energy, but the combined reactions can release net energy. Other captures modify the ashes in a real network.

Oxygen fusion in sufficiently hot evolved stellar cores has several reaction channels, including the displayed silicon-plus-alpha channel and channels producing phosphorus and protons. Its ashes feed silicon burning, while stellar neutrino energy loss strongly affects the evolution timescale.

At very high temperatures, photodisintegration and proton, neutron and alpha captures rearrange silicon-group material through networks approaching quasi-statistical equilibrium. The result is an iron-group mixture determined by electron fraction and weak-reaction timescales. Direct silicon-plus-silicon fusion is not the usual mechanism. The approach to maximal nuclear binding energy per nucleon ends sustained hydrostatic fusion power.

A thermal distribution of relative velocities weights a nuclear cross-section to determine the reaction rate per volume. Identical reactants require the denominator two to avoid counting each pair twice. Specific deposited power is $r_{12}{Q}_{\mathrm{deposited}}/\rho$. Charged-particle quantum tunnelling through a Coulomb barrier produces a Gamow peak, while resonances and plasma screening can alter the rate.

Temperature-sensitive nuclear fusion can heat material faster than cooling and expansion can regulate it, accelerating the burning. In a classical nova, hydrogen accumulated on a white dwarf ignites; initial electron degeneracy pressure helps suppress the usual expansion response. The eruption can eject the envelope without destroying the underlying white dwarf.

The stellar age is the elapsed time since a star formed. A constant number formation rate on a finite interval gives a uniform distribution of present ages if every formed object remains counted, including stellar remnants. An age distribution inferred only from surviving main sequence stars generally differs because their lifetimes depend on mass.

A growing ice-filled sphere expels liquid because ice is less dense than water. With porosity $\varphi$, $ϵ=1-{\rho }_s/\rho$, and pressure-induced freezing-point depression $T_i=T_m-\gamma p_i$, the Darcy velocity gives $p_i=\mu \varphi ϵa\dot{a}/\Pi$. Quasi-steady spherical thermal conduction gives $\varphi {\rho }_{s}La\dot{a}=k(T_i-T_{\infty })$. Hence $a\dot{a}=k(T_m-T_{\infty })/[\varphi {\rho }_{s}L(1+\Lambda )]$, where $\Lambda =k\gamma \mu ϵ/(\Pi {\rho }_{s}L)$. The interface approaches the far-field temperature at low permeability of a porous medium, while its pressure tends to a finite equilibrium value.

At zero liquid superheat, dense solid released by viscous overturn of a dense crust can sink and remain solid. For a thin crust on an asteroid of fixed radius $R$, efficient deposition and the equal-density volume approximation give $b^3-b_0^3=3R^2{J}_s(t-t_0)$. If surface gravity scales as $R$, the crustal flux scales as $R^{1/3}$ and the core radius as $R^{7/9}{t}^{1/3}$. The approximation ends when remaining liquid is exhausted or the renewed crust is no longer thin.

With constant kinematic viscosity and zero inner viscous torque in an accretion disk, $\nu \Sigma \sqrt{r}=(6\pi {)}^{-1}{\int }_0^r{F}_{\mathrm{acc}}(r^{\prime })/\sqrt{r^{\prime }}d{r}^{\prime }$. For the steady photoevaporating-disk mass flux, the density is $({\dot{M}}_d-{\dot{M}}_w)/(3\pi \nu )$ inside $r_g$ and the displayed logarithmic profile outside. At equal feeding and wind rates the inner density vanishes, and the outer normalized profile rises from zero with zero slope and tends to one. Its near-edge expansion is $(r/r_g-1{)}^2/8$.

A fixed effect represents an unknown nonrandom coefficient in the specified mean model. In a Gaussian linear mixed model, $X\beta$ is the population mean, while random effects are latent draws from a distribution whose parameters are estimated. Treating a group coefficient as fixed or random concerns the modelling and inferential target, not whether its estimate changes across samples.

A pairwise comparison model specifies the probability that one alternative beats another. Outcomes can be used to infer latent abilities through maximum likelihood estimation or another statistical method.

A process is nonstationary if some finite-dimensional distribution changes under a common time shift. With finite second moments, failure of a constant mean or variance, or failure of lag-only covariance, already rules out a weakly stationary process. A periodic mean or periodic variance can instead give a periodically correlated process.

Every finite-dimensional distribution of a strictly stationary process is invariant under a common shift of its time indices. If second moments are finite, strict stationarity implies a weakly stationary process. A constant mean and lag-dependent covariance alone do not imply strict stationarity.

For $|\varphi |<1$, initialize the state by $S_0={\sum }_{j\ge 0}{\varphi }^j{Z}_{-j}$, with variance ${\sigma }_z^2/(1-{\varphi }^2)$. It is orthogonal to future state noise and to observation noise orthogonal to all state noise. Under a Gaussian specification, take the initial state Gaussian with this variance and independent of future noises. Starting at zero instead gives a transient model.

The solar radius is the conventional unit of length based on the radius of the Sun. The nominal value is $R_{\odot }=6.957\times 10^8\mathrm{m}$.

A low-mass star maintained by sustained hydrogen burning, rather than the continuing cooling of a brown dwarf. Near the hydrogen-burning minimum mass its radius is of order a Jovian radius. Stars below roughly $0.35M_{\odot }$ are approximately fully convective, with a radiative photosphere; mass, composition and age affect the precise transition.

A generation groups corresponding quarks and leptons with the same gauge quantum numbers as their counterparts in other generations. The Standard Model has three such families. Yukawa matrices assign their different masses, and the mismatch of up-type and down-type mass bases produces the CKM matrix.

A solution of Stackelberg competition in which the follower chooses a best response after each observed leader action and the leader optimizes given those responses. Indifference may require a specified response or leader-selection convention; it cannot be silently treated as a strict inequality. In a sequential private-value all-pay contest, the leader's objective is $vF(b)-b$.