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For constant positive $\lambda =p_r/p_g$ and molecular gas constant $\mathcal{R}=k/({\mu }_m{m}_p)$, the gas–radiation equation of state gives $\rho =4\sigma T^3/(3c\lambda \mathcal{R})$. Vertical hydrostatic equilibrium in harmonic central gravity therefore gives $T=T_0(1-z^2/H^2)$ with $T_0={\Omega }^2{H}^2/[8(1+\lambda )\mathcal{R}]$. The mass density and pressure powers are three and four. Integrating both faces yields $\Sigma =(32/35){\rho }_{0}H$ and $\int pdz=\Sigma {\Omega }^2{H}^2/9$. With constant opacity, radiative diffusion produces a linear flux and height-independent dynamic viscosity $\mu =4c\lambda /[9\kappa (1+\lambda )]$. This is an interior formal profile; its low-density surface requires a photospheric treatment beyond diffusion.

For homologous gas-pressure-supported stars, hydrostatic equilibrium gives $T_c\propto \mu M/R$ and ${\rho }_c\propto M/R^3$. Radiative diffusion gives $L\propto R{T}_c^4/(\kappa {\rho }_c)$, so the radius cancels and yields the displayed cubic mass scaling. Composition and opacity are held fixed. Strong radiation pressure alters this gas-pressure relation towards the Eddington luminosity scaling.

Assume a thin disk has height-independent opacity ${\kappa }_{\mathrm{op}}$ and radiation-to-gas pressure ratio $\beta$, with vertical gravity $-{\Omega }_z^{2}z$. Hydrostatic equilibrium and radiative diffusion implyBalancing its derivative against viscous heating $\rho \nu q^2{\Omega }^2$ fixes the dynamic viscosityIt is independent of height under these assumptions. For a spherically symmetric potential, ${\Omega }_z=\Omega$. Integrating $\rho \nu$ through the full surface-to-surface thickness $2H$ gives $\bar{\nu }\Sigma =2H\rho \nu$, which determines the thickness from the radial accretion rate solution.

A deep irradiated layer can remain stable against convection when its thermal opacity and outward radiative flux are constant. Specify a plane-parallel atmosphere in hydrostatic equilibrium, with constant gravity $g>0$, positive constant Rosseland mean opacity ${\kappa }_R$ and upward radiative flux $F>0$. At depths where incident stellar light has already been absorbed, radiative diffusion givesConsequentlywhere $T_0>0$ is the extrapolated boundary temperature of this simplified diffusion solution. The radiative temperature gradient obeysFor a homogeneous diatomic ideal gas with fixed rotational degrees of freedom, $\gamma =7/5$ gives adiabatic temperature gradient ${\nabla }_{\mathrm{ad}}=2/7>1/4$. The Schwarzschild criterion therefore predicts stability throughout this model. Variable opacity, changing thermodynamics or varying radiative flux can change that conclusion. The diffusion approximation also needs large optical depth; it does not establish stability of the entire optically thin irradiated atmosphere.

Transport of stellar energy from the interior to escaping radiation, principally by radiative diffusion and convection in ordinary stellar models. Carrying a nonzero luminosity generally requires a nonzero temperature gradient.

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.

For an optically thick alpha disk dominated by gas pressure, vertical hydrostatic equilibrium gives $H\propto T^{1/2}/\Omega$ and $\rho \propto \Sigma \Omega T^{-1/2}$. The specified opacity therefore scales as $\kappa \propto {\Sigma }^{-3/2}{\Omega }^{-3/2}{T}^2$. Viscous heating scales as $Q^{+}\propto \alpha \Sigma T\Omega$, while radiative diffusion gives $Q^{-}\propto T^4/(\kappa \Sigma )\propto {\Sigma }^{1/2}{\Omega }^{3/2}{T}^2$. Their balance gives $T\propto \alpha {\Sigma }^{1/2}{\Omega }^{-1/2}$, hence $\nu \propto {\alpha }^2{\Sigma }^{1/2}{\Omega }^{-3/2}$. Fixed $\alpha$ and Keplerian rotation yield the displayed law.

The local thermal stability of an accretion disk follows because, at fixed surface density and radius, cooling has logarithmic temperature slope two while heating has slope one. The viscous stability of an accretion disk follows from ${\partial }_{\Sigma }(\nu \Sigma )>0$ on the thermal-equilibrium branch. Both conclusions apply within this pressure, opacity and transport closure.

Let $\mathcal{R}=k/({\mu }_m{m}_H)$, take constant opacity $\kappa$, and prescribe the dynamic viscosity $\mu =\alpha p_g/\Omega =\alpha \mathcal{R}\rho T/\Omega$ in a Keplerian disk. The pressure supporting the disk may contain both gas pressure and radiation pressure. Combining viscous dissipation and radiative diffusion gives ${\partial }_m^2{T}^4+A\Omega T=0$ in the vertical mass coordinate of a disk, where $A=27\alpha \kappa \mathcal{R}/(16\sigma )$. Set $\zeta =2m/\Sigma$, $T=T_{\ast }t(\zeta )$ and $T_{\ast }^3=A\Omega {\Sigma }^2/4$. With a radiative-zero disk surface, $t$ satisfies $(t^4{)}^{\prime \prime }+t=0$, $t^{\prime }(0)=t(1)=0$. Writing $I_t={\int }_0^{1}td\zeta$, the density-weighted mean kinematic viscosity is $\bar{\nu }=\alpha \mathcal{R}{T}_{\ast }{I}_t/\Omega$, giving the displayed formula. This calculation does not assume gas-pressure domination: the closure depends on gas pressure while hydrostatic support can be mixed.

Let $\mathcal{R}=k/({\mu }_m{m}_p)$ and choose $h^6=3\alpha \kappa {\Sigma }^2{\mathcal{R}}^4/(16\sigma {\Omega }^5)$. Set $z=h\zeta$, $\rho =(\Sigma /h)D$, $p=\Sigma {\Omega }^{2}hP$, $T=({\Omega }^2{h}^2/\mathcal{R})\Theta$ and $F=\alpha \Sigma {\Omega }^3{h}^{2}Q$. Vertical hydrostatic equilibrium, viscous dissipation, radiative diffusion and the ideal gas law reduce to $P^{\prime }=-D\zeta$, $Q^{\prime }=9P/4$, ${\Theta }^{\prime }=-DQ/{\Theta }^3$, $P=D\Theta$. Midplane symmetry gives $Q(0)=0$; a regular radiative-zero surface has $P=\Theta =0$ at a universal ${\zeta }_s$, with $2{\int }_0^{{\zeta }_s}Dd\zeta =1$. All physical parameters have disappeared. Uniqueness of this normalized boundary-value problem makes every physical profile a rescaling of one dimensionless solution.

// Target: astrophysics.bigb

An imposed spherical profile $\rho ={\rho }_c(1-r/R)$ gives $M=\pi {\rho }_c{R}^3/3$ in hydrostatic equilibrium. For an ideal gas with zero surface pressure, $P=P_c(1-x{)}^2(1+2x-9x^2/5)$ and $T=T_c(1-x)(1+2x-9x^2/5)$, where $x=r/R$, $P_c=5G{M}^2/(4\pi R^4)$ and $T_c=5GM/(12\mathcal{R}R)$. The density has a central cusp and temperature initially rises outward, so positive outward radiative diffusion with regular positive nuclear heating cannot satisfy this imposed profile near the centre.

An approximate relation between the mass and radius of main-sequence stars in a specified composition and mass range. It is not determined by hydrostatic equilibrium alone. For gas-pressure stellar homology with Kramers' opacity law, radiative diffusion gives $L\propto {\mu }^{15/2}{M}^{11/2}{R}_{\ast }^{-1/2}$. Substitution of $R_{\ast }\propto M^q$ gives $L\propto M^{(11-q)/2}$ at fixed composition, explaining precisely which extra radius assumption produces any claimed exponent.

Let $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units, and let $\sigma$ be the Stefan–Boltzmann constant in the Stefan–Boltzmann law. The gas sound speed used here satisfies $c_s^2=\mathcal{R}T$, and $\Omega$ is the local orbital frequency. For the local opacity approximation $\kappa ={\kappa }_0{\rho }^{1/3}{T}^{10}$, a gas-supported optically thick alpha disk has $\rho \sim \Sigma /H$ and $\mathcal{R}T\sim {\Omega }^2{H}^2$. Integrated viscous heating is $F_{+}\sim \alpha \Sigma {\Omega }^3{H}^2$. Radiative diffusion gives $F_{-}\sim (\sigma /{\kappa }_0)T^{-6}{\Sigma }^{-4/3}{H}^{1/3}\sim (\sigma /{\kappa }_0){\mathcal{R}}^6{\Omega }^{-12}{\Sigma }^{-4/3}{H}^{-35/3}$. Equating the fluxes proves the thickness scaling. At fixed composition and $\alpha$, it gives $H/r\propto r^{53/82}{\Sigma }^{-7/41}$ and $c_s\propto r^{6/41}{\Sigma }^{-7/41}$ in Keplerian rotation. Radial powers therefore require a surface-density profile; they are not determined by opacity alone. The formula is local to the stated opacity regime and small disk aspect ratio.

An optically thick medium has optical depth $\tau \gg 1$ along the relevant path. Photons undergo many interactions before escaping. Radiative diffusion applies under suitable near-isotropy conditions, but a large scattering depth alone does not guarantee thermalization when true absorption is weak.

In local thermodynamic equilibrium, write the one-ray radiative transfer equation asLarge total optical depth removes the boundary contribution, but an additional local condition is essential: the temperature and opacity vary little over an absorption length ${\kappa }_{\nu }^{-1}$. Iterating once in that small gradient givesThe integrated Planck function and its derivative areUsing the specified Rosseland mean opacity, with ${\kappa }_R$ measured here in inverse length, the source's definition $F_{\mathrm{ray}}=\int I_{\nu }d\nu$ therefore givesThe printed leading term $\sigma T^4$ lacks a factor $1/\pi$ under that definition. An isothermal optically thick medium is a concrete counterexample: $I_{\nu }=B_{\nu }$ gives $F_{\mathrm{ray}}=\sigma T^4/\pi$ exactly, while the printed formula gives $\sigma T^4$.

A physical radiative flux requires an angular integral. For direction cosine $\mu$, the same gradient expansion gives $I_{\nu }(\mu )=B_{\nu }-\mu {\kappa }_{\nu }^{-1}{\partial }_z{B}_{\nu }$. Integration over all directions gives the net radiative diffusion fluxThe isotropic equilibrium contribution cancels between opposite directions. For comparison the outward-hemisphere flux alone is $\sigma T^4-8\sigma T^3(dT/dz)/(3{\kappa }_R)$ at this order. None of these consistent definitions yields the printed mixture of coefficients.

Stellar rotation is a problem of coupled force balance, heat transport and angular momentum transport. A star need not rotate as a rigid body: in the Sun, the convective envelope rotates faster near the equator than near the poles, while much of the radiative interior rotates more nearly uniformly. The tachocline is the thin transition near the base of the convection zone. Surface tracking and Doppler effect measurements constrain the exterior flow; solar rotational mode splitting in helioseismology constrains interior stellar rotation through integrals of the form $\delta {\omega }_{n\ell m}\simeq m\int K_{n\ell m}\Omega dV$. Multiple kernels are needed, so surface rotation is not a measurement of every depth.

A useful mean velocity decomposition is $\mathbf{U}=\varpi \Omega {\mathbf{e}}_{\varphi }+{\mathbf{v}}_m$, where $\varpi =r\sin \theta$ is distance from the axis and ${\mathbf{v}}_m$ is stellar meridional circulation. In a frame of constant angular velocity ${\mathbf{\Omega }}_0$, use the relative velocity $\mathbf{V}=\mathbf{U}-{\mathbf{\Omega }}_0\times \mathbf{r}$. Its momentum equation includesThe Coriolis force changes the direction of a parcel's motion but does no work. Thermal driving, release of differential rotational kinetic energy, contraction, or external torques supply the energy of the flows. Solid-body stellar rotation in a suitable hydrostatic equilibrium does not by itself require a meridional flow.

For slowly circulating, approximately axisymmetric material, the leading mechanical balance in an inertial frame is $\nabla p/\rho =-\nabla \Phi +\varpi {\Omega }^2{\mathbf{e}}_{\varpi }$. Taking its curl gives the stellar thermal-wind balanceHere $z$ is the axial coordinate, $S$ is specific entropy and $c_p$ is the specific heat at fixed pressure. The last form uses approximately radial hydrostatic equilibrium and uniform composition. If the material is barotropic, the right-hand side vanishes and angular velocity is constant on cylinders. Small latitudinal entropy differences can instead support the observed noncylindrical differential rotation. Stresses and inertial terms modify this leading balance. The equation relates shear to baroclinicity; it does not give the circulation speed or explain how the shear is maintained.

That second question requires an angular-momentum budget. With $j={\varpi }^2\Omega$ and density-weighted mean quantities, conservation of angular momentum can be written schematically asThe three contributions are Reynolds stress, magnetic Maxwell stress tensor and viscosity transport. Appropriate density weighting replaces the simple correlations in a strongly compressible average. Since $\nabla \cdot (\rho {\mathbf{v}}_m)=0$ in a steady star, the stationary equation gives gyroscopic pumping in a star:A torque that maintains differential rotation therefore drives a meridional response unless another stress cancels it. Conversely the circulation transports angular momentum and feeds back on the rotation. An assumed meridional pattern without its driving torque is not a closed dynamical model.

In the convection zone, outward heat transport drives convection. Rotation deflects the rising and falling parcels and makes their correlations anisotropic. Nondiffusive convective angular momentum transport can drive shear even from an initially uniform rotation rate; treating every Reynolds stress as a positive eddy viscosity would only smooth shear and miss its source. A limiting parcel that conserves $j$ when it moves outwards slows relative to uniform rotation, whereas rapid rotational constraint and correlated horizontal motions change the sign and direction of the net transport. Thus the solar equator-fast pattern requires the actual convective correlations, not a universal parcel argument. The Rossby number $u_c/(2\Omega {\ell }_c)$ compares convective inertia with rotational deflection: deep, slower, larger-scale convection is more rotationally constrained than the fastest photospheric motions. Turnover times ${\ell }_c/u_c$ range from minutes near the visible surface to weeks or longer at greater depth. On these times the Reynolds stresses can redistribute angular momentum and drive circulation, while the star's total spin changes much more slowly.

In a radiative region, rotation distorts equipotential surfaces. Under uniform rotation and uniform composition, leading hydrostatic equilibrium makes pressure, density and temperature constant on each such surface, but effective gravity varies along it. Radiative diffusion then produces a flux $\mathbf{F}=f({\Phi }_{\mathrm{eff}})\nabla {\Phi }_{\mathrm{eff}}$. Its divergence contains $f^{\prime }|\nabla {\Phi }_{\mathrm{eff}}{|}^2$, which varies over a surface and generally cannot match a nuclear source constant there. This is the radiative-equilibrium obstruction in a rotating barotropic star. Thermal imbalance drives Eddington-Sweet circulation, with heat advection enteringIts global thermal estimate isA rotational thermal imbalance of order ${ϵ}_{\Omega }L$ processes the thermal reservoir on this longer time. For the slowly rotating Sun, $t_{\mathrm{KH}}$ is of order $3\times 10^7$ years and ${ϵ}_{\Omega }$ of order $2\times 10^{-5}$, giving $t_{\mathrm{ES}}\sim 10^{12}$ years, much longer than the solar age. Stable composition gradients impede the flow further. Thus classical radiative Eddington-Sweet circulation cannot by itself establish the Sun's interior rotation rapidly. In a rapidly rotating star the same estimate is shorter and rotational mixing can compete with its evolutionary lifetime. Torque-driven convection-zone circulation and radiative thermal circulation have different controlling balances.

Magnetic fields introduce a further coupling. A poloidal magnetic field linking different layers is wound by differential rotation into a toroidal magnetic field, roughly ${\partial }_t{B}_{\varphi }\simeq \varpi {\mathbf{B}}_p\cdot \nabla \Omega$. The resulting Maxwell stress tensor transport angular momentum. Communication along a field occurs on an Alfvén speed crossing time $t_A\sim \ell \sqrt{{\mu }_0\rho }/B$, which can be far shorter than molecular diffusion if a coherent connecting field exists. In steady ideal axisymmetric induction, absence of continuing winding requires ${\mathbf{B}}_p\cdot \nabla \Omega =0$; this is constancy along the field, not automatically solid-body rotation throughout the star. The topology and strength of an interior field matter. Solar dynamo action, especially the interaction of shear with poloidal-field regeneration, couples stellar rotation to the magnetic cycle. Magnetic feedback can change shear and circulation on dynamical or cycle times; an approximately eleven-year activity cycle and twenty-two-year magnetic polarity cycle are distinct from secular spin-down.

A magnetized stellar wind also supplies an external torque. It enforces approximate corotation out to an effective Alfvén radius $R_A$, giving wind-driven magnetic braking of a solar-type star $\dot{J}\sim -{\dot{M}}_w\Omega R_A^2$. Its spin-down time is $t_J\sim I/({\dot{M}}_w{R}_A^2)$, much longer than a convective turnover time. For slowly changing moment of inertia and an unsaturated field scaling that gives $\dot{J}=-K{\Omega }^3$, integration gives ${\Omega }^{-2}={\Omega }_0^{-2}+2Kt/I$, the Skumanich rotation law. A growing or contracting star also changes $I$ and can spin up or down even without torque. Differential spin-down can create shear between layers unless internal transport couples them.

Finally, shear can excite Kelvin-Helmholtz instability, and convection can launch internal gravity waves into the stable radiative interior. Wave damping deposits angular momentum; the net transport depends on excitation and selective absorption and is not fixed by a wave-crossing time alone. Such stresses, magnetic coupling and circulation are candidate agents for the nearly uniform radiative rotation and the narrow tachocline. Molecular kinematic viscosity has a diffusion time ${\ell }^2/\nu$ generally too long to couple the entire solar interior over its age, whereas a turbulent effective kinematic viscosity can act much faster but is not appropriate to every stable layer. A circulation has an advection time $\ell /U_m$, thermal diffusion has time ${\ell }^2/\chi$, and magnetic diffusion has time ${\ell }^2/{\eta }_m$; these must be compared with the local stellar rotation and evolutionary times rather than conflated with them. The solar dynamical time $\sqrt{R^3/(GM)}$ is about half an hour, its rotation period is of order a month, and secular magnetic braking takes an evolutionary time. Rapid force adjustment, convective maintenance of shear, cyclic magnetic feedback and slow global spin evolution are separate processes. Their coupled conservation equations and timescale hierarchy are the basis of a consistent account of solar rotation.

Set $\mathcal{R}=k/({\mu }_m{m}_H)$, so $p_g=\mathcal{R}\rho T$ and the prescribed dynamic viscosity is $\mu =\alpha \mathcal{R}\rho T/\Omega$. Keplerian shear and radiative diffusion then giveDifferentiating the second expression, equating it to the first and dividing by $\rho$ givesThis reduction uses the gas-pressure viscosity closure, not an assumption that gas pressure dominates the hydrostatic support.

For $z\ge 0$, introduce the vertical mass coordinate of a diskLet $\Sigma ={\int }_{-z_s}^{z_s}\rho dz$ be the full surface density of a reflection-symmetric disk. The upper surface is at $m_s=\Sigma /2$. With $\zeta =2m/\Sigma$, the temperature equation becomesWrite $T=T_{\ast }t(\zeta )$ and chooseThe equation reduces exactly to $(t^4{)}^{\prime \prime }+t=0$. Midplane symmetry gives $t^{\prime }(0)=0$, and the zero surface temperature gives $t(1)=0$. The stipulated unique positive nontrivial dimensionless solution therefore applies at every radius and surface density, with all physical dependence contained in $T_{\ast }$ and the mass-coordinate scaling. A zero temperature at the ideal surface does not mean zero escaping flux: $F=-(4\sigma /(3\kappa )){\partial }_m{T}^4$ can have a finite nonzero surface value.

The required viscosity is the density-weighted mean kinematic viscosity, not an unweighted vertical mean. Since $\rho \nu =\mu$,Substituting the dimensionless solution gives, with $I_t={\int }_0^{1}t(\zeta )d\zeta$,For central mass $M_{\ast }$, $\Omega =(G{M}_{\ast }/r^3{)}^{1/2}$, so the requested explicit radius and surface-density dependence isThe universal integral need not be evaluated. No separate solution for $\rho (z)$ is needed to find this mean: the mass-coordinate transformation removes it from both the temperature equation and the viscosity integral. The result is the gas-pressure viscosity closure with mixed pressure support.

In the limiting model with negligible gas pressure, $p=p_r=4\sigma T^4/(3c)$. Its gradient is $p_z=16\sigma T^3{T}_z/(3c)$. Combining radiative diffusion and hydrostatic equilibrium givesBecause $\kappa$ and $\Omega$ are constant with height at the fixed disk radius,Equating this to the viscous heating rate $9\mu {\Omega }^2/4$ forcesThis proves constant viscosity in radiation-supported vertical balance. It fixes the local effective dynamic viscosity uniquely, independent of height, surface density and radius within this ideal limiting model. It does not fix a height-independent kinematic viscosity: $\nu (z)=4c/(9\kappa \rho (z))$ still depends on the local mass density.

The vertical hydrostatic equilibrium equation balances the gradient of the total pressure against gravity. In a thin disk with Keplerian rotation, expanding the central gravitational field about the midplane gives vertical acceleration $-{\Omega }^{2}z$, so $p_z=-\rho {\Omega }^{2}z$. Disk self-gravity is neglected in this approximation.

The flux-gradient equation is local thermal balance: the upward radiative flux increases with height because differential rotation dissipates energy. For a Newtonian effective dynamic viscosity $\mu$, the volumetric heating rate is $\mu (r{\Omega }^{\prime }{)}^2$, which becomes $9\mu {\Omega }^2/4$ for Keplerian rotation. The symbol $\mu$ here is dynamic viscosity, not kinematic viscosity; the latter is $\nu =\mu /\rho$.

The radiative diffusion equation applies to an optically thick medium in local thermodynamic equilibrium. Its diffusion coefficient is $c/(3\kappa \rho )$, and the equilibrium radiation energy density is $u_{\mathrm{rad}}=4\sigma T^4/c$. HenceThe minus sign means energy flows down the temperature gradient. Here $\sigma$ is the Stefan-Boltzmann constant, $\kappa$ the opacity per unit mass, and $c$ the speed of light. Constant Thomson scattering opacity is the adopted transport approximation; local thermodynamic equilibrium is an additional assumption behind the radiation equation of state.

Finally the equation of state adds ideal gas gas pressure $p_g=\rho kT/({\mu }_m{m}_H)$ and isotropic radiation pressure $p_r=u_{\mathrm{rad}}/3=4\sigma T^4/(3c)$. The Boltzmann constant is $k$, and ${\mu }_m{m}_H$ is the mean mass per gas particle. Both contributions supply the total pressure entering vertical hydrostatic equilibrium.

A Type Ia supernova is a thermonuclear disruption associated with a carbon-oxygen white dwarf in a binary system. Electron degeneracy pressure supports the dense star, and initially depends much less on temperature than an ideal gas pressure does. Once carbon burning heats degenerate material, the immediate pressure response is insufficient to produce the usual expansion-and-cooling thermostat. The strongly temperature-sensitive nuclear reactions can therefore undergo a thermonuclear runaway.

One channel involves accretion toward the Chandrasekhar limit, about $1.4$ solar masses for a typical electron composition. Other proposed channels involve merging white dwarfs or an initial helium-layer detonation that triggers carbon burning below that mass. Thus the Chandrasekhar limit motivates an important model, but it is not an assumption that every Type Ia supernova has exactly the same progenitor mass or ignition history. Burning carbon and oxygen into more tightly bound nuclei releases enough nuclear binding energy to overcome the stellar gravitational binding energy. An ordinary successful event ejects the star rather than leaving the compact core characteristic of a core-collapse supernova.

The ejecta contain iron-group material, including radioactive nickel-56, and intermediate-mass elements from incomplete burning. The radioactively powered Type Ia supernova light curve is powered largely by the nickel-56 to cobalt-56 to iron-56 decay chain: its energetic products heat the ejecta, and radiative diffusion delays the emerging radiation. Near maximum light, ordinary Type Ia supernovae show prominent silicon absorption and lack the hydrogen lines defining Type II events. The spectral classification and the thermonuclear mechanism provide complementary descriptions.

The similar progenitors and radioactive power make Type Ia supernovae useful standard candles, but their peak luminosities are not identical. The relation between light curve width and peak brightness, together with color corrections, allows standardization for distance measurements. Extinction, composition and explosion diversity still introduce systematic uncertainties. A Type Ia event is a white-dwarf thermonuclear explosion with a radioactively powered light curve; its corrected luminosity makes it a valuable distance indicator.

The first equation is vertical hydrostatic equilibrium: the upward pressure gradient balances the central mass's vertical gravitational force. For $|z|\ll r$, the point-mass acceleration is $-GMz/(r^2+z^2{)}^{3/2}\simeq -{\Omega }_K^{2}z$, where Keplerian rotation gives ${\Omega }_K^2=GM/r^3$. The disc's self-gravity and vertical inertia are neglected.

The second equation is local thermal balance. The divergence of the vertical radiative flux equals viscous dissipation. In the convention of this alpha disk, $\nu =\alpha p/(\rho \Omega )$, and the dominant Keplerian shear has $rd\Omega /dr=-3\Omega /2$. ThereforeThe radiative flux increases from zero at the symmetric midplane to a positive outgoing value at the upper surface. Radial heat transport, thermal advection and rapid storage of thermal energy are excluded from this local balance.

The third equation is radiative diffusion, equivalentlyIt applies to optically thick material, with $\kappa$ the opacity per unit mass and $\sigma$ the Stefan-Boltzmann constant. Here constant electron-scattering opacity represents Thomson scattering; the formula uses a diffusive, nearly thermal radiation field. Positive upward flux requires temperature to decrease upwards.

The final equation is the ideal gas pressure law, with mean molecular weight ${\mu }_m$ in proton-mass units. Writing $\mathcal{R}=k/({\mu }_m{m}_p)$ gives $p=\mathcal{R}\rho T$. The pressure is gas pressure because radiation pressure is assumed negligible. The four equations express vertical support, local heating balance, diffusive cooling and the gas equation of state.