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Stars

Radiative Processes in Stellar Interiors

Four processes are principally responsible for creating, thermalizing, and impeding the flow of radiation in the interior of a star. From those that are dominant at high temperature to those dominant at low, these processes are Compton scattering, bremsstrahlung emission and absorption, photo-ionization and recombination, and atomic line emission and absorption. Each of these interactions is described by a probability.

Each possible interaction can be thought of as one of a pair of interactions, one forward, the other reverse; this pairing permits the interaction of radiation with matter to obey the laws of thermodynamics. A process that creates a photon is paired with an inverse process that destroys a photon of the same energy, and the probability of creating a photon of a given density is precisely tied to the probability of its destruction. These properties of the interaction between matter and radiation are the reasons that hot thermal matter will modify the spectrum of electromagnetic radiation until the spectrum is black-body. When the photons acquire a black-body spectrum, the creation and destruction of photons of a given energy are in balance. The scattering process does not create or destroy photons; rather, it allows the exchange of energy between radiation and matter. The forward and reverse scattering processes, where the inverse process is the forward process reversed in time, have probabilities that are precisely related, with the forward and reverse processes occurring at the same rate when the radiation has a black-body spectrum.

Compton Scattering

A photon can scatter, exchanging energy and momentum, with a free electron—an electron that is not bound to an atom. This process is called Compton scattering. While the process does not destroy or create photons, it does keep the photons in thermal equilibrium with the electrons of a star, and it slows the diffusion of radiation from the core of a star. Compton scattering is the dominant radiative process for photons that are hard x-rays (energies above several keV) and gamma-rays. It is the dominant process for the thermalization and transport of radiation when the temperature is above several tens of millions of degrees, where a large fraction of the photons in a black-body spectrum are x-rays.

The probability that a photon Compton scatters with an electron is proportional to the electron density. The probability is independent of photon energy as long as the energy is well below the electron rest-mass energy. When the energy exceeds the electron rest-mass energy, the probability of a scattering between a photon and electron decreases almost inverse-proportionally with photon energy; this effect is unimportant in stars, which have black-body photons that are far below the electron rest-mass energy.

Bremsstrahlung

A free electron moves along a hyperbolic path past an ion, curving towards the ion. As the electron accelerates through this path, it emits electromagnetic radiation, and if electromagnetic wave are passing by at the time, the electron can absorb electromagnetic radiation. The radiation created in this way is called bremsstrahlung, or braking radiation. The absorption of radiation through this process is often called free-free absorption, referring to the state of the electron before and after the event.

The rate per unit volume at which radiation is created through bremsstrahlung is proportional to the free-electron density times the ion density. It is inversely-proportional to the square-root of the temperature, because slow electrons follow sharper paths than do fast electrons, producing more electromagnetic radiation. This means that this mechanism becomes less efficient as the temperature rises. On the other hand, as the density increase, the rate at which electromagnetic energy is release rapidly rises.

Photo-ionization

Electromagnetic radiation can free an electron that is bound within an atom; the only requirement is that the photon must carry an energy at least equal to the binding energy of the electron. The absorption of a photon through photo-ionization is often called bound-free absorption. The probability of this interaction occurring is greatest for photons carrying the binding energy of an electron. As the energy of the photon increases above the binding energy, the probability of it freeing the electron decreases.

Ionization and its inverse, recombination, are important for hydrogen and helium over a narrow range of temperatures. At high temperatures, the atoms quickly dissociate, so that in equilibrium virtually no hydrogen and helium with bound electrons exists. On the other hand, at low temperatures, there are no photons that can dissociate a hydrogen or helium atom. It is only in the narrow ranges of temperatures that keep the density of neutral hydrogen , neutral helium, and partially-ionized helium about equal to the density of free electrons that ionization and recombination of hydrogen and helium dominate the diffusion of radiation.

While the ionization of hydrogen and helium occur at temperatures characterized by ultraviolet radiation, many other elements within a star ensure that ionization and recombination play an important role in hinder the diffusion of radiation at higher temperatures. The ionization of the most tightly-bound electrons of iron provides a particularly important role in the diffusion of x-rays.

Atomic Lines

At low temperatures, most of the electrons are bound within atoms and the average energy of the photons is too low to ionize an atom. Under these condition, the radiation interacts with atoms by forcing the bound electrons to change their orbits within atoms. These interactions are resonant, meaning that they occur at very particular photon energies, where the frequency of the electrons motion within the atom matches the frequency of the light. In fact, the energies at which the interactions occur are the energies that separate pairs of quantized electron orbits within an atom. Unlike the orbit of a planet around the Sun, an electron can only orbit the nucleus of an atom at very specific energies.

For an electron to move from one state to the next, it must acquire or lose the amount of energy that separates the two states. This can be done through a collision between two atoms, and it can be done through the absorption or emission of a photon. When the second happens, the result is the creation and destruction of photons at very specific energies. In practice, these interactions occur over narrow ranges of energies, partly from the Doppler shift of the line from the random motion of the ions, and partly from the property of quantum mechanics that an electron energy state becomes a narrow continuum of values when an electron spends a finite amount of time in that state. Atoms therefore emit and absorb photons over narrow ranges of energies that have widths associated with the widths of the electron energy states.

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