Thermal and nonthermal radiation

Temperature is one of the most basic quantities in physics and astrophysics. At its simplest, according to the approximation hv ~ kT (2.15), the frequencies of detected photons may be an indicator of the temperature of the originating bodies. For example, photons from the sun at optical frequencies tell us that the surface layers of the sun have temperature of ~6000 K (kelvin).

The infrared emission from a human being indicates a temperature of about 300 K. A pervasive radio radiation from the sky at millimeter wavelengths indicates a temperature of 3 K for the cooled radiation initially emitted in the early years of the universe. X rays of energy 1-10 keV indicate temperatures of ~107 K for the emitting plasmas in the vicinity of very compact gravitational objects, namely neutron stars and black holes. This high temperature represents the high kinetic energies demanded by the virial theorem (8) because the potential energies are large and negative.

The approximate relation between photon frequency and temperature quoted just above is generally valid only for "thermal" bodies where there is an equilibrium between photons and particles. The equilibrium is "perfect" for a blackbody in which case the energy spectrum of the radiation will have the characteristic blackbody shape we will encounter in Section 11.3.

It is possible to have non-thermal processes, such as synchrotron emission (Section 11.3), where the originating particles are not in thermal equilibrium with their surroundings. In this case, the spectrum will have a different shape, a temperature can not be defined, and the emitting particle energy can be very different than hv.

Temperature measurements

There are several ways to measure temperatures in optical astronomy. The different methods can lead to somewhat different values if the source region is not in perfect thermal equilibrium. They are color temperature, effective temperature, excitation temperature, and ionization temperature. Underlying these is the basic laboratory kinetic temperature.

Kinetic temperature

In thermal equilibrium, a gas will have a distribution of speeds that is called the Maxwell-Boltzmann distribution. The average translational kinetic energy of atoms or molecules in a gas is related to the temperature as

On earth, we sometimes measure the temperature with a mercury thermometer. Higher molecular speeds impart energy to the mercury causing it to expand and indicate a higher temperature. Unfortunately, it is not possible to directly do this with a distant star.

Color temperature

The color temperature Tc is derived from broadband UBV photometry, the measurement of fluxes in the U, B, and V bands, under the assumption that the observed object has a blackbody spectrum. The relative flux densities in these bands are a measure of the color of the object (Section 8.3). The nature of the blackbody curve is such that, for stellar temperatures, the B and V filters provide a fair measure of temperature.

The ratio of fluxes in the B and V bands, Fp B / Fp V, translates, as noted in Section 8.3, into a difference in the (logarithmic) magnitudes, B-V, called the color index,

(Kinetic temperature) (9.10)

This follows from the definition of magnitudes (8.12) where ^p,B and ^p,V are the measured photon fluxes (photons m-2 s-1) in the two bands. The difference in magnitudes is thus a measure of the color. The two colors B-V and U-B are the often quoted results of UBV photometry. If a color B-V is measured, it can be described as a color temperature, Tc in kelvin, namely the temperature of an ideal blackbody that would yield the measured value of B-V.

Effective temperature The distribution of light from normal stars roughly approximates the continuum spectrum from a blackbody wherein each square meter emits a flux of a T4 (W/m2) and a is the Stefan-Boltzmann constant, a = 5.67 x 10-8 Wm-2 K-4. The luminosity of a spherical star of radius R may therefore be expressed as

L = 4^R2a T4eff (W; defines effective temperature) (9.12)

The effective temperature Teff is used here because the spectra of stars deviate somewhat from the blackbody shape. The spectra of normal stars show absorption lines at certain frequencies which are not found in true blackbody radiation. This equation defines Teff to be the hypothetical temperature that would yield the true luminosity if the spectra were exactly blackbody in form. The definition is thus

where 4^R2 is the surface area of the star.

In other words, Teff is the temperature that a blackbody of the same luminosity and radius would have. The effective temperature of the sun is 5800 K. For more on blackbody radiation, see Section 11.3.

Excitation temperature The excitation temperature is that deduced from the populations of atomic excited states observed in stellar spectra. The higher the energy of the occupied states, the higher the temperature. For example, the greater the number of hydrogen atoms in the n = 2 state relative to the number in the n = 1 state, the higher is the excitation temperature. The expression that governs this is the Boltzmann formula, nj gj ( h vi A

ni gi \ kT / excitation temperature)

which gives the ratio of numbers (or densities) of atoms in the excitation states i (the lower) and j (the higher) as a function of the temperature T, the energy difference between the two states Ej - Ei = hvij, and the statistical weights gi and gj of the two states. The statistical weight of a state is the degeneracy of the state, i.e., the number of substates with differing quantum numbers that have the same energy. Thus, the n = 1 level has g1 = 2 (two s states) and the n = 2 level has g2 = 8 (two s states and six p states). The ratio nj/m can sometimes be inferred from the relative strengths of various spectral lines in the spectrum, in which case, the excitation temperature can be obtained from (14).

The exponential factor in (14) tells us that a high-energy state has a lower probability of being occupied than a lower-energy state with the same statistical weight and that the population of the higher state decreases with the increase in the energy difference between the two states. The ratio gj /gt tells us that the probability of a state being occupied is proportional to its statistical weight. For example, if the higher state has more substates than the lower, the probability of finding atoms in that state relative to the lower is higher proportionally by the factor gj /gl.

The Boltzmann formula follows from considerations of detailed balance when the atoms are immersed in a bath of photons in true thermodynamic equilibrium, i.e., with the blackbody spectrum for the indicated temperature. Basically, one equates the probabilities of upward and downward transitions in photon-atom interactions. Transitions due to particle-particle collisions also yield this relation as long as the system is in true thermodynamic equilibrium.

A famous example is the excitation of cyanogen molecules (CN) in interstellar space by the 2.7-K cosmic microwave background (CMB; see Section 11.3). This was the earliest (1941) detection of this radiation from the early universe. The significance of the measurement was not appreciated until the radiation was detected directly as microwave waves more than two decades later.

The Boltzmann formula is not perfectly valid if, for example, there are particles of a different temperature present. This must be taken into account in the calculation of the temperature of the CMB from the CN measurements. An extreme violation of the Boltzmann formula is the relative population of atoms in a laser or maser. In this case the atoms are externally "pumped" to a high, relatively stable ("metastable") state with applied radiation. This results in an excess population in the higher state which by definition is a violation of the Boltzmann formula. The laser emission is then due to stimulated emission from the higher state.

Ionization temperature The ionization temperature is also determined through spectral studies. It depends upon the extent to which different ionization states (as distinct from the excitation states just discussed) appear in the spectrum, e.g., the ratio of He I to He II, neutral to ionized helium. The relative numbers of ions in the several ionization states at a given temperature and density are obtained from an analysis of equilibrium interactions similar to that for the Boltzmann formula. The equation that follows from this analysis is the Saha equation. It can be used to define a temperature if the relative densities of the several ions and of free electrons can be independently known, for example, from the strengths of spectral lines.

The excitation and ionization temperatures represent the physical conditions of the atoms in a region of interest. For the quasi-equilibrium conditions that are typical of local regions of stellar atmospheres, these two temperatures and the kinetic temperature will all be nearly equal.

For a typical star, the temperature varies with height in its interior and also in its atmosphere. For the sun, the temperature is 6400 K at the altitude where the escaping photons can, for the most part, leave the sun without further scatters; this is known as the photosphere. The temperature is decreasing in this region. This leads to absorption at certain frequencies by the outer cooler gases so the total radiation observed is less than that expected from a 6400 K blackbody. This is why the effective temperature is only 5800 K. As one moves outward, the temperature arrives at a minimum of ~4200 K. It then begins to increase, eventually reaching 106 K in the corona.

Saha equation

The Saha equation allows one to obtain the ionization temperature from measurements of the relative numbers of atoms of a given species that are in adjacent ionization states (Figure 1). The equation follows from consideration of detailed balance wherein the probabilities of absorption and emission of an electron by the gr+1, gr, i ^

Atom in r ioniz. state

Atom in r+1 ioniz. state (one less electron)

Figure 9.1. Hypothetical energy levels of two states of ionization of a given species of atom, e.g., Ca. The transition between the ionization states can occur between any excitation levels in one state to any excitation level in the other. The statistical weights of the individual states g and the ionization potential xr are indicated. The latter is the energy between the respective ground states. The Saha equation relates the densities of atoms in the two ionization states to the temperature T and free electron density. [Adapted from Principles of Stellar Evolution and Nucleosynthesis, D. Clayton, McGraw Hill, 1968, p. 33]

atom are equated. These transitions are associated with the emission and absorption of a photon respectively. The derivation assumes conditions of perfect thermal equilibrium.

For a given temperature T, the Saha equation yields the quotient (nr+1ne)/nr, where nr is the density of atoms (m-3) in the lower (r) of two ionization states, nr+1 is the density of atoms in the next higher (r + 1) ionization state (with one less orbital electron), and ne is the density (m-3) of free electrons; see Fig. 1. We quote the equation without derivation, nr + ine Gr +1ge (2vmekT)3/2 ( X -=--:-exp nr Gr h3 V kT

where the quantities are:

nr+1 Density of atoms in ionization state r + 1 (m 3)

nr Density of atoms in ionization state r (m 3)

ne Density of electrons (m-3)

Gr+i Partition function of ionization state r + 1

Gr Partition function of ionization state r ge = 2 Statistical weight of the electron me Mass of the electron = 0.911 x 10-30 kg

Xr Ionization potential of state r (to reach state r + 1)

h Planck constant = 6.63 x 10-34 J s k Boltzmann constant = 1.38 x 10-23 J/K

The partition function Gr (T) is a kind of statistical weight for ionization state r taking into account all the possible bound states it can have. Under many conditions, the upper states do not play amajor role. For most astrophysical conditions, the partition functions for neutral and ionized states of hydrogen are closely approximated by Gr (T) ~ gr 0 = 2 and Gr+1(T) ~ gr+10 = 1, respectively. The ionization energy Xr for the ground state of hydrogen is, of course, 13.6 eV.

The ratio on the left of (15) is large for a high proportion of ionized atoms in the plasma. This is favored for high values of the ratio of statistical weights (first term on right) and the number of phase-space states available to the electron (second term), and if the ionization energy xr is not large compared to kT.

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