We now seek to derive the complete stellar structure equations. This includes mass conservation, hydrostatic equilibrium so far. We want to then derive the radiative transfer equation and the energy equation. With the equation of state this allows us to solve for the structure of the star.
Consider a box of length \(ds\). And in it is a bunch of stuff which can interact with light coming through it. Coming into this box is light with intensity \(I_{\nu}\) and leaving is \(I_{\nu} - dI_{\nu}\). Some bits of light will be absorbed. Each absorber has some cross section of interaction \(\sigma_{\nu}\) and some density \(n\). The number of absorbers is \(n \sigma_{\nu} ds\). \[\frac{dI_\nu}{I_\nu}= -n\sigma ds\]. This term can be seen as an optical depth \(d\tau_{\nu} = n\sigma ds\). We can write this interms of a density instead of number density \(d\tau = \rho \kappa_{\nu} ds\). where \(\kappa_{\nu}\) is the opacity. However this only captures the absorption of light. We also need an emission term (source term). \[\frac{dI_\nu}{I_\nu}= -n\sigma ds + \frac{j_\nu}{\alpha_\nu}\] Which is called a source function. \(S_\nu =\frac{j_\nu}{\alpha_\nu} \). For a black body we have that \(j_\nu = B_\nu(T)\). We say that \(\frac{j_\nu}{\rho \kappa_\nu} \) and get the transfer: \[\boxed{\frac{dI_\nu}{ds}=-\kappa_\nu \rho I_\nu + j_\nu}\] Now let us consider the case when we have a source term and it is blackbody. The condition for blackbody is that thermal equilibrium, isotropic, must absorb all light shown on it be optically thick. \[\frac{dI_\nu}{ds}=-\kappa_\nu \rho (I_\nu + B_\nu(T))\] \[B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/kT}-1}\] If we care about it radiating outwards in a spherical shell, we can write the transfer equation as: \[I(\theta, z) = B_\nu - \frac{\cos \theta}{\kappa \rho} \frac{dI_\nu}{dz}\] We then do first order perturbation of the equation from blackbody. \[I = B_\nu + \delta I^1_\nu\] Which gives \[B_\nu + \delta I^1_\nu = B_\nu - \frac{\cos \theta}{\kappa \rho} \frac{d}{dz}[B_\nu + \delta I^1_\nu]\] \[\delta I^1_\nu = - \frac{\cos \theta}{\kappa \rho} \frac{d}{dz}[B_\nu + \delta I^1_\nu]\] Drop the derivatives of higher order terms \[\delta I^1_\nu = - \frac{\cos \theta}{\kappa \rho} \frac{d}{dz}[B_\nu ]\] Next we want to find the flux through the shell. \[F_\nu = \int I \cos \theta d\Omega = \int B_\nu - \frac{\cos \theta}{\kappa \rho} \frac{dI_\nu}{dz} d\Omega\] \[F_\nu = \int B_\nu \cos \theta d\Omega - \int\frac{\cos^2 \theta}{\kappa \rho} \frac{dI_\nu}{dz} d\Omega\] The first term is isotropic and drops to zero integrating over all angles. \[F_\nu = - \int\frac{\cos^2 \theta}{\kappa \rho} \frac{dB_\nu}{dz} d\Omega\] We also see there is no angle dependence since its isotropic \[F_\nu = - \frac{dB_\nu}{dz} \int\frac{\cos^2 \theta}{\kappa \rho} d\Omega\] \[F_\nu \sim - \frac{dB_\nu}{dz}\] We can do chain rule to get \[F_\nu \sim - \frac{dB_\nu}{dT} \frac{dT}{dz}\] We integrate through all frequencies \[F \sim \int \frac{1}{\kappa_\nu}\frac{dB_\nu}{dT} \frac{dT}{dz} d\nu \] \[F \sim\frac{dT}{dz} \overbrace{ \int \frac{1}{\kappa_\nu}\frac{dB_\nu}{dT} d\nu}^{Rosseland Mean Opacity} \] \[F = \frac{dT}{dz} \frac{T^3 \sigma }{ \rho \kappa_{R}} \] To get the total energy we convert this to a luminosity: \[\boxed{\frac{dT}{dr} = (-\frac{L}{4\pi r^2}) \frac{\rho \kappa_{R}}{T^3} \frac{-3}{16 \sigma}} \] This is the radiative transfer equation for a star.
Radiative diffusion assumes that energy is transported down a negative temperature gradient (\(dT/dr < 0\)), a process microscopically driven by the random motion of photons. This transport relies on a small deviation from perfect isotropic blackbody radiation (anisotropy), which is sufficient to carry the star's entire luminosity. The magnitude of this temperature gradient is proportional to the local density \(\rho(r)\) and the energy flux (\(L/4\pi r^2\)) passing through the shell. A star is considered to be in "radiative equilibrium" if this radiative gradient alone is capable of transporting the total luminosity. However, this approximation is valid only in regimes of Local Thermodynamic Equilibrium (LTE) where the photon mean free path is much smaller than the stellar radius (\(l_\nu \ll R_*\)); therefore, while it applies to stellar interiors, it is not valid for the surface, photosphere, or atmosphere.
We can write the radiative transport in dimensionless quantities: \[\nabla_{rad} = \frac{\partial \ln T}{\partial \ln r} = \frac{3\kappa_r}{16G\pi ac}\frac{LP}{mT^4}\] We can actually compute the rosseland mean opacity and how it depends on temperature: \[\kappa_{R} \propto \rho T^{-7/2}\] This takes the form of Kramers opacity. This opacity is used for both bound-bound and bound free emission. As it gets hotter the opacity rapidly decreases while the density linear increases with opacity.
Conduction: In Conduction is caused by the thermal motion of electrons. In a classical gas like the Sun, electrons interact frequently (short mean free path), making conduction inefficient compared to radiative diffusion In degenerate cores, electrons move at relativistic speeds and have long mean free paths (due to filled quantum states). Here, electron conduction becomes very efficient, leading to nearly isothermal cores.
Convection occurs when a blob of fluid is hot it rises and then cools and then sinks back down and in the process it transports energy. This is caused by the buoyancy force. The buoyancy force is caused by the difference in density between the hot and cold blob. Lets try to make this concrete:
Consider a blob of fluid with distance \(r\) and \(r+dr\) out. There is two blobs of fluids. The lower one denoted by \(1\) and the upper one denoted by \(2\). There is also the properties of ambient outside denoted as \(*\) and ones of the blob denoted \(blob\). \[\]
We wish to write the properties of the higher blob in terms of the lower blob. \[\rho_{2blob} = \rho_{1blob}+ \frac{d\rho_{1blob}}{dr} \cdot dr\] But we realise this blob is adiabatic and does not exchange heat outside: \[\frac{d\rho_{1blob}}{dr} = \frac{1}{\gamma}\frac{\rho_{1blob}}{P_{1blob}} \frac{dP_{1blob}}{dr}\] \[\boxed{\rho_{2blob} = \rho_{1blob}+ \frac{1}{\gamma}\frac{\rho_{1blob}}{P_{1blob}} \frac{dP_{1blob}}{dr}}\] We then look at the abmient environment \[\boxed{\rho_{2*} = \rho_{1*}+ \frac{d\rho_{1*}}{dr} \cdot dr}\] To then calculate the net force density we look at \(\rho_{1*} = \rho_{1blob}\) \[f_{net} = -g(\rho_{2blob}-\rho_{2*}) = g( \rho_{1blob}+ \frac{1}{\gamma}\frac{\rho_{1blob}}{P_{1blob}} \frac{dP_{1blob}}{dr} -\rho_{1*}- \frac{d\rho_{1*}}{dr} \cdot dr) \]
\[F_{net} = g( \frac{d\rho_{1*}}{dr} \cdot dr - \frac{1}{\gamma}\frac{\rho_{1blob}}{P_{1blob}} \frac{dP_{1blob}}{dr})\]
We get a differential equation where there is a coefficent \[A = \frac{d\rho}{dr} - \frac{1}{\gamma}\frac{\rho}{P} \frac{dP}{dr}\] \[N = \sqrt{-Ag}\] if \(N\) complex then it will oscillate and remain stable and NOT trigger convection. If \(N\) is real then it will grow and trigger convection.
We can make this even more explicit for condition to trigger convection. If we are classical gas \[P = \frac{\rho}{\mu m_p}kT\] We can then write the differential equation as: \[A = \frac{d\rho}{dr} - \frac{1}{\gamma}\frac{\rho}{P} \frac{dP}{dr}\] \[\frac{dP}{dr} = \frac{P}{T}\frac{dT}{dr} + \frac{P}{\rho} \frac{d\rho}{dr}\] \[\frac{1}{T}\frac{dT}{dr} = \frac{1}{P}\frac{dP}{dr} - \frac{1}{\rho} \frac{d\rho}{dr}\] if we are in adiabatic condition then we have that \[\frac{dT}{dr}_{ad} = \frac{1}{\gamma}\frac{\rho}{P} \frac{dP}{dr}\] We can write \[A = \frac{1}{T}[\frac{dT}{dr}_{ad} - \frac{dT}{dr}]\] \[\boxed{\frac{dT}{dr}_{ad} >\frac{dT}{dr} \Rightarrow \text{trigger convection} }\]
Dominant energy transport: defined by if convection ever gets triggered. If it ever gets triggered it always wins. \[\boxed{\frac{dT}{dr}_{ad} >\frac{dT}{dr} \Rightarrow \text{ convection wins} }\] \[\boxed{\frac{dT}{dr}_{ad} <\frac{dT}{dr} \Rightarrow \text{ radiation wins} }\]
We can now write down the stellar structure equations. \[\frac{dM}{dr} = 4\pi r^2 \rho \tag{mass conservation}\] \[\frac{dP}{dr} = -\frac{GM}{r^2} \rho \tag{hydrostatic equilibrium}\] \[\frac{dT}{dr} = -\frac{3}{16\sigma} \frac{L}{4\pi r^2} \frac{\rho \kappa_{R}}{T^3} \tag{radiative diffusion}\] \[\frac{dL}{dr} = 4\pi r^2 \rho \epsilon \tag{energy production}\]
We also have equation of state which is relates pressure with density and temperature.
Model for convection. MLT treats convection like a field of rising bubbles. It assumes a "blob" (or convective cell) of gas is displaced from its original position
There is a scale height distance: \[H_p = |\frac{dr}{d\ln P}| = |P \frac{dr}{dP}| = \frac{P}{\rho g} = l_m = \text{mixing length}\]
Total energy transported out is the temperature difference of the blob and the ambient environment. \[ u \sim c_p\Delta T \Rightarrow \Delta T = (T_{blob} - T_{*}) = (\frac{dT}{dr}_{ad} - \frac{dT}{dr} )l_m\] We then \[\Delta T = \frac{T}{H_p}l_m (\nabla_{rad} - \nabla_{ad})\]