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Introduce specific heat capacity $c_p$, latent heat per unit mass $L_f$, and $\Delta T=T_m-T_s>0$. Write the prescribed convective heat flux as $F=K(\overline{T}-T_m)$, whereThere is a dimensional convention to specify: with $Q$ measured in $\mathrm{W}\mathrm{k}{\mathrm{g}}^{-1}$ and dimensionless $\lambda$, this printed expression makes $K$ a heat transfer coefficient only if $\mu$ is kinematic viscosity. If $\mu$ denotes dynamic viscosity, the numerator needs ${\rho }_0^{2}g\alpha Q$ instead. The question says only “viscosity”; use the consistent kinematic interpretation, or equivalently the corrected dynamic formula. The following balances apply to either convention once $K$ has units $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$.

For a thin lava lake crust with a linear temperature profile, Fourier's law gives the outward conductive flux $k\Delta T/a$. The Stefan condition balances it against upward thermal convection and freezing, while the well-mixed liquid loses the convective flux and gains radiogenic heating. To leading order in $a/H$ the radiogenically heated lava-lake crust model isThe fixed surface temperature already incorporates radiative cooling, so no extra radiative loss is added to the interior balance. The linear conductive profile is interpreted as a quasi-steady approximation, so sensible-heat storage within the crust is not evolved. Thinness alone does not establish this approximation: rapid thermal adjustment in the crust is also needed, for example a small Stefan number during the initial conduction-driven growth. Replacing the liquid depth by $H-a$ and retaining the moving liquid-volume enthalpy would introduce thin-crust corrections: the requested constant-temperature branch refers to the leading model above. A transient thermal profile in a thick or newly nucleating crust would require the full Stefan problem, rather than this linear-profile closure.

Using the thermal diffusion time $H^2/\kappa$, defineThe three independent dimensionless parameters are the convective-conductive coefficient $B$, the heating parameter $R$, and the Stefan number $S$. For positive $B,R,S$ the evolution equations becomewhere prime denotes $d/d\tau$. Temperature has the exact solutionConsequently the unique constant-temperature initial condition isAt this temperature, the convective flux equals the column's radiogenic heat production. Starting with no crust, the constant-temperature lava-crust growth law is obtained by separating $h^{\prime }=S(1/h-R)$:The crust grows towards $h_{\ast }=1/R$, or $a_{\ast }=k\Delta T/({\rho }_{0}QH)$, where conduction carries all the radiogenic input. The logarithm makes reaching this equilibrium take infinite time. For an initial thickness $h_0>0$, subtract $[-R{h}_0-\log (1-R{h}_0)]/(S{R}^2)$ on the same side of the equilibrium; the corresponding antiderivative uses $\log |1-Rh|$ on either branch. The thin-crust description through equilibrium requires $R\gg 1$.

For general initial temperature, the full early and intermediate history is specified by the exact temperature above and one scalar ordinary differential equation. A particularly convenient integral formulation for an initially bare surface isThis removes the infinite initial derivative of $h$. The right-hand side points into $y>0$ at zero, and is decreasing in $y$ when ${\theta }_0\ge 0$; two nonnegative solutions therefore cannot separate, which gives uniqueness. Its solution, together with the temperature formula, determines the entire model evolution and can be computed without a phase-boundary search. The early expansion follows by substitution:The dimensional leading crust thickness is $a\sim \sqrt{2k\Delta Tt/({\rho }_0{L}_f)}$. Thus hotter liquid reduces the first correction to conductive crust growth, whereas colder liquid permits more rapid thickening.

If ${\theta }_0>{\theta }_{\ast }$, the temperature decreases and the instantaneous stationary lava-crust thickness $h_e(\tau )=1/[B\theta (\tau )]$ increases towards $h_{\ast }$. Starting from zero, the crust always remains below this rising curve: at a first contact $h^{\prime }=0$ but $h_e^{\prime }>0$, so it cannot cross from below. Hence hot-start crust grows monotonically. It also stays thinner than the constant-temperature solution: the early expansion puts it below, and at a proposed first crossing their derivative difference is $-SB({\theta }_0-{\theta }_{\ast })e^{-B\tau }<0$, preventing crossing from below. If $0\le {\theta }_0<{\theta }_{\ast }$, the temperature increases and the instantaneous stationary lava-crust thickness decreases. The crust stays thicker than the constant-temperature solution by scalar comparison, but it may cross the decreasing $h_e$, reach a maximum and subsequently melt back towards $h_{\ast }$. Such overshoot is possible, not inevitable; the two relevant relaxation rates are $B$ and $S{R}^2$. The numerical plot uses $B=4$, $R=20$, $S=0.1$, so crust adjustment is faster than temperature relaxation and the cool-start example does overshoot:

The curves use ${\theta }_0/{\theta }_{\ast }=0.5,1,1.5$ and initially $h=0$. They solve the complete reduced time-dependent equations, rather than replacing the moving stationary thickness by the actual crust. At late times, set $h=h_{\ast }+v$, $\delta ={\theta }_0-{\theta }_{\ast }$; linearization gives $v^{\prime }=-S{R}^{2}v-SB\delta e^{-B\tau }$. For $S{R}^2\ne B$,At equal rates the forced term is $-SB\delta \tau e^{-B\tau }$. In particular, slow warming with $B<S{R}^2$ approaches equilibrium from a thicker crust, explaining the plotted remelting. These long-time expressions describe the linearized response; nonlinear corrections can have their own faster exponential rates. All predictions are restricted to $a/H\ll 1$, nonnegative liquid superheat, and the assumed vigorous-convection flux law. If crust growth violates those conditions, the reduced model must be replaced rather than extrapolated to total freezing.

For the initially homogeneous sphere, $V_i=-3G{M}^2/(5R_i)$ and $K_i=0$. Conservation of energy and the virial theorem imply $E_f=V_f/2=V_i$, hence $V_f=2V_i$. With the usual homologous-sphere approximation, $V\propto -1/R$, soThese are the conventional collapse estimates. A final virial equilibrium is not, by itself, guaranteed to remain homogeneous: if $V_f=-{\alpha }_{f}G{M}^2/R_f$, the actual relation is $R_f/R_i={\alpha }_f/(2{\alpha }_i)$ with ${\alpha }_i=3/5$. Thus the factor of two applies exactly to the gravitational radius, and to the outer radius only under the stated structural approximation.

Here “non-interacting” means collisionless apart from gravity. Baryonic gas can collide, convert ordered motion to thermal energy, and radiate it. If radiative cooling is effective, the fixed-energy virial state loses pressure support and contracts further. Baryonic cooling can destabilize the collisionless virial state; baryons are not inevitably unstable if cooling is slow or heating and pressure support maintain equilibrium.

The equilibrium photon gas is isotropic, so the energy density per unit solid angle in a narrow angular-frequency interval is $u(T,\omega )d\omega /(4\pi )$. For rays making angle $\theta$ with an outward surface normal, a surface element $dA$ is crossed in time $dt$ by rays initially in a volume $cdtdA\cos \theta$. Only the outward hemisphere contributes. ThusThe hemispherical restriction supplies a factor $1/2$ and the directional projection supplies another $1/2$. A small aperture in a cavity therefore emits this spectral power per unit area. A blackbody at the same temperature emits the same amount: otherwise radiation exchange between it and the aperture would violate thermal equilibrium.

Integrating Planck's law over all positive angular frequencies gives the total emitted heat flux densitySet $x=\hslash \omega /(k_{B}T)$. ThenThe remaining integral is independent of temperature, proving $M\propto T^4$. Its value follows by expanding $(e^x-1{)}^{-1}={\sum }_{m=1}^{\infty }{e}^{-mx}$ and integrating termwise:Consequently the Stefan–Boltzmann law isThis is emitted power alone. A body in a radiation environment also absorbs incoming radiation, which must be included when calculating net radiative cooling.

core-collapse supernovae follow loss of pressure support in the evolved core of a massive star. electron capture and nuclear dissociation contribute to collapse; formation of a hot compact object launches a shock that can stall as dissociation and neutrino losses drain it. neutrino heating and multidimensional fluid motions can revive the shock, but some collapses fail to produce a normal bright explosion. The remnant may be a neutron star or black hole. The neutrino burst detected from SN 1987A provides direct evidence for the core-collapse energy channel: the water-Cherenkov signal is a particle-interaction measurement, requiring detector-response and accidental-background checks rather than an optical light-curve inference.^[ref]

Type Ia supernovae instead involve thermonuclear destruction of a degenerate carbon-oxygen white dwarf. They leave expanding ejecta without an ordinary surviving neutron-star core. The ignition channel need not have a unique total mass or binary history. Radioactive nickel/cobalt decay powers much of the later luminosity, and empirical light-curve corrections make these events useful distance indicators. Spectral type is not identical to explosion mechanism: hydrogen-rich Type II events are core collapse, but stripped-envelope Ib/c events can also be core collapse despite lacking conspicuous hydrogen.

The ejecta first expand nearly freely while swept-up mass is small. A strong shock heats ambient gas, and once swept-up material dominates, an approximately adiabatic remnant approaches the Sedov-Taylor scalingSubsequent radiative cooling makes a dense shell and ends that energy-conserving approximation. Thermal X-ray lines probe shocked gas and nucleosynthetic ejecta; radio and nonthermal X-rays trace relativistic particles and magnetic fields. shock acceleration makes remnants plausible contributors to Galactic cosmic rays. Expansion, ambient-density structure, cooling and cosmic-ray energy losses complicate energy estimates; a bright synchrotron shell alone is not proof of a specific accelerated-particle composition. Supernovae return metals and mechanical energy to the interstellar medium and therefore connect stellar death to the feedback cycle.

A shock wave with significant radiation losses in its downstream evolution. Often a narrow dissipative jump initially heats the gas according to adiabatic Rankine-Hugoniot conditions, followed by a much longer radiative cooling layer. The gas's specific entropy increases at the jump and may then fall as radiation carries entropy away. If both distant states have equal temperature, the combined mass and momentum balances give an isothermal shock; the gas's energy flux is not conserved across the whole radiating region.

For a gas-supported alpha disk with $\kappa \propto {\rho }^{1/3}{T}^{10}$, compare rates at fixed surface density of a disk, radius, $\alpha$ and composition on the thermal timescale of an accretion disk. Assume that the vertical dynamical timescale of a disk is shorter than that thermal time, and that the viscous timescale is longer; the separation is controlled for $\alpha \ll 1$ and small disk aspect ratio. Vertical hydrostatic equilibrium gives $H\propto T^{1/2}$, $\rho \propto T^{-1/2}$, so $\kappa \propto T^{59/6}$. Viscous heating satisfies $Q^{+}\propto T$ and radiative cooling satisfies $Q^{-}\propto T^4/(\kappa \Sigma )\propto T^{-35/6}$. At equilibrium a positive temperature perturbation changes the net heating by $(41/6)(Q_0/T)\delta T>0$. With positive effective heat capacity, it grows rather than relaxes. This is a local thermal statement and uses the fixed-column, rather than fixed-volume-density, comparison.