Chapter 4

Energy and Momentum Conservation

This chapter takes the radiative transfer equation and asks a bigger question: what happens when we integrate over direction? The answer is that the transfer equation turns into conservation equations for the radiation field. First we get energy conservation. Then, by multiplying by direction before integrating, we get momentum conservation and the radiation force.

The route is direct: start from the familiar transfer equation, integrate it over solid angle, identify the physical moments, and then use the result in radiative equilibrium, optically thin cooling, flux constancy, radiative acceleration, and the vacuum wave limit.

Chapter Contents

  1. 1Energy Conservation from the Transfer Equation
  2. 2Radiative Equilibrium and Cooling
  3. 3Flux Constancy and the Linear Source Function
  4. 4Momentum Conservation and Radiation Force
  5. 5Vacuum Limit and the Wave Equation
  6. Ref.References

1. Energy Conservation from the Transfer Equation

We begin with the equation that has appeared again and again. The new operation is simple: integrate it over all directions. That turns the microscopic intensity equation into a macroscopic energy equation.

Start with the full radiative transfer equation:

\[ \frac{1}{c}\frac{\partial I_\nu}{\partial t} + \hat{\mathbf n}\cdot\nabla I_\nu = \eta_\nu-\alpha_\nu I_\nu . \]

Now integrate over all solid angles. The time derivative and the angular integral commute, so the first term becomes the time derivative of the radiation energy density.

\[ E_\nu=\frac{1}{c}\int_\Omega I_\nu\,d\Omega, \qquad \mathbf F_\nu=\int_\Omega I_\nu\hat{\mathbf n}\,d\Omega . \]

The directional derivative term becomes the divergence of the flux. The reason is that the spatial gradient acts on \(I_\nu\), while the angular integral is over directions:

\[ \int_\Omega \hat{\mathbf n}\cdot\nabla I_\nu\,d\Omega = \nabla\cdot\int_\Omega I_\nu\hat{\mathbf n}\,d\Omega = \nabla\cdot\mathbf F_\nu . \]

Using these definitions, the angle-integrated equation is

\[ \frac{\partial E_\nu}{\partial t} + \nabla\cdot\mathbf F_\nu = \int_\Omega(\eta_\nu-\alpha_\nu I_\nu)\,d\Omega . \]

To be complete, integrate over frequency as well:

\[ \frac{\partial E}{\partial t} + \nabla\cdot\mathbf F = \int_0^\infty\int_\Omega (\eta_\nu-\alpha_\nu I_\nu)\,d\Omega\,d\nu . \]

This is the energy equation of the radiation field. It has the usual conservation-law structure: time derivative of the quantity, plus divergence of its flux, equals sources minus sinks.

Energy conservation from angular integration The radiative transfer equation is integrated over solid angle to give time change of energy density plus divergence of flux equals radiative sources minus sinks. Start from transfer equation 1/c ∂Iν/∂t + n̂ · ∇Iν = ην − αν Iν integrate over dΩ and dν ∂E/∂t + ∇ · F = sources − sinks
Once we integrate over all directions, \(I_\nu\) turns into bulk radiation quantities: energy density \(E\) and flux \(\mathbf F\).

So this describes conservation of energy for the radiation field. The right-hand side tells us how matter and radiation exchange energy. Emission adds energy to the radiation field. Absorption removes energy from the radiation field.

2. Radiative Equilibrium and Cooling

Radiative equilibrium means that the only energy exchange is through radiation, and the total heating balances the total cooling.

Think about a gas with some density and temperature. The gas can emit photons because microscopic radiative and collisional processes are happening inside it. But when the gas emits, that energy is taken from the gas, so the gas cools. The opposite process is absorption: if radiation comes in and the gas absorbs it, the gas gains energy and heats up.

In radiative equilibrium, these two processes balance. There may be emission and absorption locally, but the net radiative energy exchange is zero:

\[ \int_0^\infty\int_\Omega (\eta_\nu-\alpha_\nu I_\nu)\,d\Omega\,d\nu=0 . \]

Now make the simplifying assumption that \(\eta_\nu\) and \(\alpha_\nu\) are isotropic. Then the emission part is easy:

\[ \int_\Omega \eta_\nu\,d\Omega = 4\pi\eta_\nu = 4\pi\alpha_\nu S_\nu , \]

where \(S_\nu=\eta_\nu/\alpha_\nu\) is the source function. The absorption part is

\[ \int_\Omega \alpha_\nu I_\nu\,d\Omega = \alpha_\nu\int_\Omega I_\nu\,d\Omega = 4\pi\alpha_\nu J_\nu , \]

with

\[ J_\nu=\frac{1}{4\pi}\int_\Omega I_\nu\,d\Omega . \]

Therefore the radiative-equilibrium condition becomes

\[ \int_0^\infty 4\pi\alpha_\nu(S_\nu-J_\nu)\,d\nu=0, \]

or equivalently

\[ \int_0^\infty \alpha_\nu S_\nu\,d\nu = \int_0^\infty \alpha_\nu J_\nu\,d\nu . \]

The left side is the cooling side, because it comes from emission. The right side is the heating side, because it comes from absorption of the radiation field.

Temperature from radiative equilibrium

This condition is used all over astrophysics to estimate temperature structures. For a stellar atmosphere or an accretion disc, one often assumes radiative equilibrium and then asks: what temperature structure makes the heating and cooling balance?

It is convenient to write the frequency integrals using mean opacities:

\[ \alpha_J J = \alpha_S S, \]

where

\[ \alpha_J = \frac{\int_0^\infty \alpha_\nu J_\nu\,d\nu} {\int_0^\infty J_\nu\,d\nu}, \qquad \alpha_S = \frac{\int_0^\infty \alpha_\nu S_\nu\,d\nu} {\int_0^\infty S_\nu\,d\nu}. \]

If the gas is close to equilibrium, the source function is close to the Planck function. Then \(S\approx B\), and

\[ B=\frac{\sigma T^4}{\pi}. \]

Using \(\alpha_JJ=\alpha_BB\), we get

\[ T^4 = \frac{\pi\,\alpha_J J}{\alpha_B\sigma}. \]

This is the basic idea: calculate the radiation field, use it to calculate the heating, and then infer the temperature needed for radiative equilibrium.

Iterating temperature and radiation field A loop showing assume temperature, solve transfer equation, obtain mean intensity, impose radiative equilibrium, update temperature, and repeat. assume T solve transfer equation get J radiative equilibrium gives new T dI/ds = S − I ≈ B(T) − I
The temperature structure is often obtained by iteration: guess \(T\), solve transfer, get \(J\), impose radiative equilibrium, update \(T\), and repeat.

In practice this can become circular. To get \(J\), we have to solve the transfer equation over many rays and many directions. But the transfer equation itself needs the source function, and if \(S\approx B(T)\), then it needs the temperature. So one starts with an assumed temperature, solves the transfer equation, computes \(J\), uses radiative equilibrium to get a new temperature, and iterates.

The same idea appears in other systems too. A gas may be heated by some external radiation source, and then the gas responds by emitting radiation. If the system has time to relax, this can lead to radiative equilibrium.

Optically thin radiative cooling

There are also important cases where the gas is not in radiative equilibrium. Then the right-hand side of the energy equation is not zero. A very important example is optically thin radiative cooling.

In that case the gas is heated by some non-radiative process, or by a process not balanced by reabsorption, and then it emits photons. But the emitted photons escape. They are not reabsorbed locally. That is why we call it optically thin cooling.

\[ \frac{\partial E}{\partial t} +\nabla\cdot\mathbf F = \hbox{heating}-\hbox{cooling}. \]

The cooling part is essentially controlled by the emission coefficient:

\[ \hbox{cooling}\sim \int_0^\infty\int_\Omega \eta_\nu\,d\Omega\,d\nu. \]

This is crucial in galaxy formation. Gas can collect inside a dark-matter halo, but to form stars it must cool down to the low temperatures of dense molecular clouds, often around 10 to 20 K in the present-day universe. Without radiative cooling, the gas cannot collapse in the right way to form stars and galaxies.

It also matters in the solar corona. The corona is heated to millions of Kelvin and is optically thin. It emits radiation, and that emission is one of the important cooling channels.

3. Flux Constancy and the Linear Source Function

When the radiation field is static and in radiative equilibrium, the energy equation becomes a statement about flux constancy.

Set \(\partial E/\partial t=0\). If the right-hand side is also zero because we are in radiative equilibrium, then

\[ \nabla\cdot\mathbf F = 0. \]

In one-dimensional Cartesian geometry, this simply means

\[ \frac{dF_z}{dz}=0, \qquad F_z=\hbox{constant}. \]

In spherical symmetry, the divergence condition becomes

\[ \frac{1}{r^2}\frac{d}{dr}(r^2F_r)=0. \]

Therefore \(r^2F_r\) is constant. Since luminosity is \(L=4\pi r^2F_r\), the radiative luminosity is constant through the spherical object:

\[ r^2F_r=\frac{L}{4\pi}=\hbox{constant}. \]
Flux constancy in Cartesian and spherical geometry One panel shows constant vertical flux through plane layers, and the other shows spherical luminosity with r squared F r constant. 1D Cartesian dFz/dz = 0 spherical symmetry r²Fr = L/(4π)
Radiative equilibrium plus stationarity gives \(\nabla\cdot\mathbf F=0\). In Cartesian geometry the flux is constant; in spherical geometry the luminosity is constant.

Now connect this to the diffusion approximation and to the Eddington-Barbier idea from the previous chapter. In diffusion we had

\[ F_z = -\frac{c}{3\alpha}\frac{dE}{dz}. \]

Since \(E=4\pi J/c\), this becomes

\[ F_z = -\frac{4\pi}{3\alpha}\frac{dJ}{dz}. \]

Use the plane-parallel optical-depth coordinate, where \(d\tau=-\alpha\,dz\). Then

\[ F_z = \frac{4\pi}{3}\frac{dJ}{d\tau}. \]

But in static radiative equilibrium \(F_z\) is constant. So integrate directly:

\[ J(\tau_1)-J(\tau_0) = \frac{3F_z}{4\pi}(\tau_1-\tau_0). \]

Put the surface at \(\tau_0=0\). Then

\[ J(\tau) = J(0)+\frac{3F_z}{4\pi}\tau . \]

Now make the grey radiative-equilibrium assumption. Grey means frequency independent opacity, so the opacity can be taken out of the frequency integrals. The radiative-equilibrium condition then reduces to \(J=S\). Therefore the source function is also linear:

\[ S(\tau)=a+b\tau. \]

This is why the linear source-function assumption used in Eddington-Barbier is not random. It appears naturally from radiative equilibrium plus diffusion plus flux constancy. It is an approximation, but it already explains a lot of the basic physics of the light emerging from a star.

This is also why limb darkening becomes understandable. If the source function increases inward, then rays through the center see deeper, hotter, brighter layers. Rays near the limb reach optical-depth unity higher up, so they see cooler, dimmer layers. In research one often uses numerical limb-darkening laws from detailed atmosphere models, but the basic reason is already contained in this simple argument.

The same idea matters for exoplanet transits. If we want precise properties of an exoplanet from a transit light curve, we need to know how the host star darkens toward the limb. The detailed model may be numerical, but the physical starting point is this source-function and optical-depth picture.

4. Momentum Conservation and Radiation Force

Now do the same kind of moment operation, but multiply the transfer equation by \(\hat{\mathbf n}\) before integrating over direction. That gives the momentum equation of the radiation field.

Start again from the transfer equation, multiply by \(\hat{\mathbf n}\), and integrate over solid angle. The first term becomes a time derivative of the flux, and the streaming term produces the radiation pressure tensor.

\[ \mathbf P_\nu = \frac{1}{c}\int_\Omega I_\nu\,\hat{\mathbf n}\hat{\mathbf n}\,d\Omega . \]

After dividing through by \(c\), the monochromatic momentum equation is

\[ \frac{1}{c^2}\frac{\partial \mathbf F_\nu}{\partial t} + \nabla\cdot\mathbf P_\nu = \frac{1}{c}\int_\Omega (\eta_\nu-\alpha_\nu I_\nu)\hat{\mathbf n}\,d\Omega . \]

Now integrate over frequency. If the emission coefficient is isotropic, then \(\int\eta_\nu\hat{\mathbf n}\,d\Omega=0\), because every direction has an opposite direction. The emission term does not produce a net momentum source if it is isotropic.

The extinction term survives because it contains the intensity:

\[ -\frac{1}{c}\int_0^\infty \alpha_\nu\mathbf F_\nu\,d\nu . \]

Define a flux-weighted mean extinction coefficient \(\alpha_F\) by

\[ \alpha_F\mathbf F = \int_0^\infty \alpha_\nu\mathbf F_\nu\,d\nu . \]

Then the momentum equation contains the term

\[ -\frac{\alpha_F\mathbf F}{c}. \]

This is the loss of momentum from the radiation field. The material receives the opposite momentum. Therefore the radiation force per unit volume on the gas is

\[ \mathbf f_{\rm rad} = \frac{\alpha_F\mathbf F}{c} = \frac{\kappa_F\rho\,\mathbf F}{c}. \]

The corresponding radiative acceleration is force per unit mass:

\[ \mathbf g_{\rm rad} = \frac{\kappa_F\mathbf F}{c}. \]
Radiation force from photon momentum A photon beam enters a gas parcel and transfers momentum, producing a radiative acceleration in the direction of the flux. gas F g_rad = κ_F F / c photons carry momentum material gets a kick
We usually notice photons by their energy, but photons also carry momentum. In very luminous systems that momentum transfer becomes dynamically important.

Check the units. \(\kappa_F\) has units \(\mathrm{cm^2\,g^{-1}}\), flux has units of energy per area per time, and \(c\) has units \(\mathrm{cm\,s^{-1}}\). The combination \(\kappa_FF/c\) gives \(\mathrm{cm\,s^{-2}}\), which is an acceleration. So the expression is dimensionally correct.

In daily life we do not normally feel radiative acceleration. A photon entering the eye gives a tiny momentum kick, but it is far too small to matter mechanically. In very luminous systems, however, this force can be critical. Around supermassive black holes, in accretion discs, and in the most massive stars, radiation pressure can push matter strongly.

This is also the starting point for understanding the upper mass limit of stars. Once we know how to compute the opacity \(\kappa_F\), we can compare the outward radiative acceleration with inward gravity. That comparison leads naturally toward the Eddington limit.

5. Vacuum Limit and the Wave Equation

Finally, take the energy and momentum equations into the vacuum limit. This gives a useful consistency check: radiation should propagate as a wave at the speed of light.

In vacuum, there is no emission and no extinction:

\[ \eta_\nu=0, \qquad \alpha_\nu=0. \]

So the right-hand sides of the energy and momentum equations vanish. In one-dimensional Cartesian geometry, write

\[ \frac{1}{c^2}\frac{\partial F}{\partial t} + \frac{\partial P}{\partial z} = 0, \]
\[ \frac{\partial E}{\partial t} + \frac{\partial F}{\partial z} = 0. \]

Here \(F=F_z\) and \(P=P_{zz}\). Now introduce the Eddington factor

\[ f=\frac{P}{E}. \]

Assume \(f\) is constant. Take a time derivative of the energy equation:

\[ \frac{\partial^2E}{\partial t^2} + \frac{\partial}{\partial z} \left(\frac{\partial F}{\partial t}\right) = 0. \]

From the momentum equation,

\[ \frac{\partial F}{\partial t} = -c^2\frac{\partial P}{\partial z} = -c^2f\frac{\partial E}{\partial z}. \]

Insert this into the previous equation:

\[ \frac{\partial^2E}{\partial t^2} - c^2f\frac{\partial^2E}{\partial z^2} = 0. \]

This is a wave equation. The wave speed is

\[ v_{\rm wave}=c\sqrt{f}. \]

In the free-streaming vacuum limit, \(P=E\), so \(f=1\). Then \(v_{\rm wave}=c\), exactly as it should be.

In an isotropic equilibrium radiation field, \(P=E/3\), so \(f=1/3\). That gives a characteristic speed

\[ v=\frac{c}{\sqrt{3}}. \]

In that equilibrium-like limit, one should not really ignore the source and sink terms; including them leads to damping. But the speed \(c/\sqrt{3}\) is still important. It can be interpreted as a radiation sound speed.

This is especially interesting for the early universe. When the universe was radiation dominated, acoustic waves were controlled by radiation pressure, and their characteristic speed was \(c/\sqrt{3}\). Later, during matter-radiation equality and recombination, that picture changes. The full cosmology story comes later, but the transport-equation view already shows why this speed appears.

The vacuum wave equation is also a useful bridge toward the next physics topics. In vacuum it connects cleanly to the electromagnetic wave description of light. That classical wave picture becomes useful when describing scattering processes, such as electron scattering and Rayleigh scattering, and later line scattering once quantum mechanics is included.

References

These references support the radiation moment equations, radiative equilibrium, radiative acceleration, and the wave-equation limit used in this chapter.