Chapter 4 Free–Free Radiation

Larmor’s formula (Equation 2.143)

states that electromagnetic radiation with power $P$ is produced by accelerating (or decelerating; hence the German name bremsstrahlung meaning “braking radiation”) an electrical charge $q$. Free charged particles can be accelerated by electrostatic or magnetic forces, gravitational acceleration being negligible by comparison. Electrostatic bremsstrahlung is the subject of this chapter, and its magnetic counterpart magnetobremsstrahlung or “magnetic braking radiation” (e.g., synchrotron radiation) is covered in Chapter 5.

Thermal emission is produced by a source whose emitting particles are in local thermodynamic equilibrium (LTE) (Section 2.2.2); otherwise nonthermal emission is produced. Most astronomical sources of electrostatic bremsstrahlung are thermal because the radiating electrons have the Maxwellian velocity distribution (Appendix B.8) of particles in LTE. The relativistic electrons in most astronomical sources of magnetobremsstrahlung have power-law energy distributions and hence are not in LTE, so synchrotron sources are often called nonthermal sources. However, electrostatic and magnetic bremsstrahlung are not synonymous with thermal and nonthermal radiation, respectively. For example, electrons with a relativistic Maxwellian energy distribution are in LTE and can emit thermal synchrotron radiation. Remember also that a thermal source does not have a blackbody spectrum if the source opacity is small and the emission coefficient depends on frequency.

The electrostatic force is so much stronger than gravity that free charges in interstellar gas quickly rearrange themselves so that the negative charges of free electrons in an ionized cloud neutralize the positive charges of ions on all scales larger than the Debye length ${\lambda }_{\mathrm{D}}\le 1\mathrm{m}$ (Equation 4.46). As an electron (charge $-e\approx -4.8\times {10}^{-10}$ statcoulomb) passes by an ion (charge $+Ze$ for an atom with $Z$ electrons removed), the Coulomb force (Equation 2.134) causes an acceleration of magnitude

where $m_{\mathrm{e}}\approx 9.1\times {10}^{-28}$ g is the electron mass and $r$ is the distance between the electron and the ion. The resulting emission is called free–free radiation because the electron is free both before and after the interaction; it is not captured by the ion. If the ionized interstellar cloud is reasonably dense, the electrons and ions interact often enough that they quickly come into LTE at some common temperature, so free–free radiation is usually thermal emission.

Interstellar gas is primarily hydrogen and helium, plus trace amounts of heavier elements such as carbon, nitrogen, oxygen, neon, silicon, and iron. Astronomers call all of these heavier elements metals, meaning elements that readily form positive ions, even though most are not metallic in the usual sense of being solid, malleable, ductile, and electrically conducting solids at room temperature. Much of the interstellar hydrogen is in the form of neutral atoms (called Hi in astronomical terminology) or diatomic molecules (H${}_2$), but some is ionized. The singly ionized hydrogen atom H${}^{+}$ is referred to as Hii by astronomers, doubly ionized oxygen O${}^{++}$ is called Oiii, triply ionized carbon C${}^{+++}$ is called Civ, etc.

In 1939 the astronomer Bengt Strömgren realized that the interstellar medium can be divided into distinct regions in which hydrogen is either (1) mostly atomic or molecular, with nearly all of the hydrogen atoms in the ground electronic state or (2) almost completely ionized. Furthermore, the boundaries separating these Hi and Hii regions are very thin. Sometimes the Hii regions surrounding stars are called Strömgren spheres (Figure 4.1) after his early theoretical models. What is the microscopic physical basis for this picture?

A hydrogen atom in the ground state has the smallest and most tightly bound electronic orbit around the nuclear proton that is consistent with a stationary electronic wave function. (See Section 7.2.1 and Figure 7.1 to review the Bohr model of hydrogen atoms.) The permitted electronic energy levels are characterized by their principal quantum numbers $n=1,2,3,\mathrm{\dots },$ where $n=1$ corresponds to the ground state. Although quantum mechanics forbids an electron in the ground state from radiating according to the classical Larmor formula, it permits radiative decay from higher levels $n=2,3,\mathrm{\dots };$ and Larmor’s equation fairly accurately predicts the radiative lifetimes of excited hydrogen atoms. The orbital radius $a_n$ of an electron in the $n$th energy level is $a_n=n^2{a}_0$, where $a_0\approx 5.29\times {10}^{-9}$ cm is called the Bohr radius. Applying Larmor’s equation shows that the radiative lifetime $\tau$ is proportional to $a_n^3$ and hence to $n^6$. Thus the (incorrect) classical result $\tau \approx 5.5\times {10}^{-11}$ s for radiation from the $n=1$ ground state can be scaled to estimate the radiative lifetimes of excited states. For example, the approximate radiative lifetime of the $n=2$ state would be $\tau \approx 2^6\cdot 5.5\times {10}^{-11}\mathrm{s}\approx 3\times {10}^{-9}$ s, in reasonable agreement with the accurate quantum-mechanical result $\tau \approx 2\times {10}^{-9}$ s. Excited hydrogen atoms spontaneously decay very quickly to the ground state by emitting radiation so, at any instant, almost all neutral atoms are in the ground state.

Hydrogen atoms in the ground state can be ionized by photons with energy $E\ge 13.6$ eV (1 electron Volt $\approx 1.60\times {10}^{-12}$ erg). Such energetic photons have frequencies higher than the Rydberg frequency $R_{\mathrm{\infty }}c=E/h\approx 3.29\times {10}^{15}\mathrm{Hz}$ (Equation 7.11) and wavelengths shorter than $\lambda =912$ Å $=912\times {10}^{-10}$ m, a far-ultraviolet (UV) wavelength. These Lyman continuum photons are produced in significant numbers by the Wien tail of blackbody radiation from stars hotter than $T\sim 3\times {10}^4$ K. The rate $Q_{\mathrm{H}}$ at which a star with spectral luminosity $L_{\nu }$ produces photons that can ionize hydrogen atoms in the ground state is

If a star emits $Q_{\mathrm{H}}$ Lyman continuum photons per second, it will photoionize hydrogen atoms with number density $n_{\mathrm{H}}$ throughout some volume $V$ surrounding the star. The helium mixed with the hydrogen can often be ignored because its ionization potential is so high, $E\approx 24.5$ eV, that only exceptionally hot stars can ionize significant amounts of helium.

The absorption cross section of a neutral hydrogen atom to photons with energies just above 13.6 eV is large enough, $\sigma \approx {10}^{-17}{\mathrm{cm}}^2$, that each ionizing photon is absorbed and produces a new ion shortly after it passes from the ionized Strömgren sphere into the surrounding Hi region. The thickness $\mathrm{\Delta }{R}_{\mathrm{S}}$ of the partially ionized shell surrounding a Strömgren sphere (Figure 4.1) is

For example, if the neutral hydrogen density is $n_{\mathrm{H}}={10}^3\mathrm{atoms}{\mathrm{cm}}^{-3}$, then

Light travels $\approx {10}^{14}$ cm per hour, so an ionizing photon typically survives only about an hour in such an Hi cloud before being absorbed.

Once ionized from Hi into free protons (H${}^{+}$ ions) and electrons, the Hii region has a much lower opacity to ionizing photons. Thus a new ionizing star turning on in a uniform-density Hi cloud will fully ionize a sphere whose Strömgren radius (Figure 4.1) grows with time until equilibrium between ionization and recombination is reached. This is sometimes called an ionization bounded Hii region. If the surrounding Hi cloud is small enough that the star can ionize it completely, the Hii region is said to be matter bounded or density bounded.

Inside the Hii region, electrons and protons occasionally collide and recombine at a volumetric recombination rate ${\dot{n}}_{\mathrm{H}}$ that can be written as

where ${\dot{n}}_{\mathrm{H}}$ is the number of recombinations per unit time per unit volume (e.g., ${\mathrm{cm}}^{-3}{\mathrm{s}}^{-1}$),

is the recombination coefficient for hydrogen, and the collision rate per unit volume is proportional to the product $n_{\mathrm{e}}{n}_{\mathrm{p}}$ of the electron and proton densities. For example, if $n_{\mathrm{e}}=n_{\mathrm{p}}={10}^3{\mathrm{cm}}^{-3}$,

This example shows that the recombination time

is usually much shorter than the $>{10}^6$ year lifetime of an ionizing star. (A useful relation to remember is 1 year $\approx {10}^{7.5}$ s.) The volume $V$ of an ionization-bounded Hii region grows until the total ionization and recombination rates in the Strömgren sphere are equal. In equilibrium,

yields a Strömgren radius

For example, an O5 star (very hot and luminous) emits $Q_{\mathrm{H}}\approx 6\times {10}^{49}$ ionizing photons per second. If $n_{\mathrm{e}}\approx {10}^3{\mathrm{cm}}^{-3}$,

This example illustrates that $R_{\mathrm{S}}\gg \mathrm{\Delta }{R}_{\mathrm{S}}$; that is, the radius of the fully ionized Strömgren sphere is much larger than the thickness of its partially ionized skin.

Two distinct kinds of stars produce most of the Hii regions in our Galaxy:

1. 1.

   The most massive ($M\ge 15M_{\odot }$) short-lived (lifetimes $\le {10}^7$ yr) main-sequence stars are big enough ($R\sim 10R_{\odot }$) and hot enough ($T\ge 3\times {10}^4$ K) to be very luminous sources of ionizing UV. Such stars were recently formed by gravitational collapse and fragmentation of interstellar clouds containing neutral gas and dust grains.

2. 2.

   Old lower-mass () stars whose main-sequence lifetimes are less than the age of our Galaxy ($\approx {10}^{10}$ yr) eventually become red giants and finally white dwarfs. Young white dwarfs are small ($R\sim {10}^{-2}{R}_{\odot }$) but hot enough to ionize the stellar envelope material that was ejected during the red giant stage, and these ionized regions are called planetary nebulae because many looked like planets to early astronomers using small telescopes.

Most ionizing stars are approximately blackbody emitters and their ionizing photons from the high-frequency Wien tail have energies only somewhat greater than the $E=13.6$ eV minimum needed to ionize a hydrogen atom from its ground state. Momentum conservation during ionization ensures that nearly all of the photon energy in excess of 13.6 eV is converted into kinetic energy of the ejected electron. Collisions between these hot photoelectrons, and between electrons and ions, thermalize the ionized gas and gradually bring it into local thermodynamic equilibrium (LTE). Consequently, the thermalized electrons have a Maxwellian energy distribution. Eventually this heating is balanced by radiative cooling. Collisions of electrons with “metal” ions can excite low-lying (a few eV) energy states that decay slowly via forbidden transitions and emit visible photons that may escape from the nebula. Examples of visible cooling lines include the green lines of Oiii at $\lambda =4959$ Å and $5007$ Å, first discovered in nebulae and named nebulium lines because these forbidden lines hadn’t been observed in the laboratory and were thought to be from a new element found only in nebulae (just as helium lines in the solar spectrum were once ascribed to a new element found in the Sun). The Balmer hydrogen recombination lines H$\alpha$ at $\lambda =6563$ Å and H$\beta$ at $\lambda =4861$ Å also contribute to the characteristic colors of Hii regions.

Thermal equilibrium between heating and cooling of Hii regions is usually reached at a temperature close to $T\approx {10}^4$ K [76] that is much higher than the initial temperature  K of the neutral interstellar gas. The heated gas expands, reversing any infall onto the ionizing star and sending shocks into the surrounding cold gas, thereby both inhibiting and stimulating the subsequent production of stars in the region. Typical Hii regions have sizes $\sim 1$ pc, electron number densities $\sim {10}^3{\mathrm{cm}}^{-3}$, and masses up to ${10}^4{M}_{\odot }$.

The free–free radio emission from an Hii region is a tracer of the electron temperature, electron density, and ionized volume. It constrains the production rate $Q_{\mathrm{H}}$ of ionizing photons and, with an assumption about the initial mass function (IMF) (the mass distribution of new stars), the total star-formation rate in an Hii region. The radio data are important for reliable quantitative estimates of star formation because they do not suffer from extinction by interstellar dust.

Ultra-compact (UC) Hii regions [23] are small (diameter $2R_{\mathrm{S}}\le 0.03\mathrm{pc}$) but dense ($n_{\mathrm{e}}>{10}^4{\mathrm{cm}}^{-3}$) Hii regions ionized by O and B stars so young that they are still optically obscured by the dusty molecular clouds from which they formed. The dust makes them strong far-infrared sources, and free–free radio emission penetrates the dust so they can be imaged and studied at radio wavelengths. UC Hii regions are valuable tracers of the formation and early evolution of massive stars, and of their interactions with their environment. On a much larger scale, extragalactic ultra-dense (UD) Hii regions [58] are ionized by young super star clusters (SSCs) so dense and massive that they may be the progenitors of long-lived globular clusters, and so luminous that they may disrupt the ISM in dwarf galaxies.

Planetary nebulae are Hii regions surrounding the hot (up to $T\sim {10}^5$ K) white-dwarf cores of low-mass () stars that have ejected their outer envelopes as stellar winds. White dwarfs are small, about the size of the Earth, so they are much less luminous than massive O stars and the ionized nebular masses are only . Planetary nebulae are nonetheless fairly luminous indicators of the last stages in the lives of low-mass stars. They are potentially useful as a record of low-mass star formation throughout the history of our Galaxy. However, the optical selection of planetary nebulae is affected by dust extinction. Far-infrared and radio selection may avoid this limitation. Planetary nebulae are not particularly luminous radio sources, but they are the most numerous compact radio continuum sources in our Galaxy.

Thermal bremsstrahlung from ionized hydrogen is often called free–free emission because it is produced by free electrons scattering off ions without being captured—the electrons are free before the interaction and remain free afterward. What are the basic properties of free–free radio emission from an astrophysical Hii region? Despite all of the simplifications introduced in Section 4.2, this problem can’t be solved without several additional approximations. A certain amount of astrophysical “intuition” is required to distinguish important from negligible effects. For example, the energy lost by an electron when it interacts with an ion is much smaller than the initial electron energy. Radiation from electron–electron collisions and from ions can be ignored. Formally divergent integrals over impact parameters can be avoided with physical limits to the range of integration and still yield reasonably accurate, but not exact, results. Most astrophysical conditions are so far removed from personal experience (How hot does ${10}^4$ K feel? How big is a parsec compared with the height of a tree? How much is $M_{\odot }\approx 2\times {10}^{33}$ g compared with the mass of a person?) that astrophysical intuition depends on being familiar with numerical values for the relevant parameters, so it is possible to decide quickly what is important and what can be neglected. As Linus Pauling said, “The way to get good ideas is to get lots of ideas and throw the bad ones away.” The following analysis of free–free emission from an Hii region illustrates approaches and techniques used to solve “messy” astrophysical problems. For similar but alternative analyses, see the chapters on bremsstrahlung in Rybicki and Lightman [97] and in Wilson et al. [116].

Why should an Hii region emit radio radiation at all? The answer is, because charged particles are being accelerated electrostatically, and nonrelativistic free accelerated charges radiate power according to Larmor’s formula (Equation 2.143). Electrostatic interactions among many kinds of charged particles take place in an Hii region, but most do not emit significant amounts of radiation. The magnitude of the acceleration $\dot{v}$ is inversely proportional to the particle mass $m$. The lightest ion is the hydrogen ion. Its mass is the proton mass $m_{\mathrm{p}}\approx 1.66\times {10}^{-24}$ g, which is about $2\times {10}^3$ times the electron mass $m_{\mathrm{e}}\approx 9.11\times {10}^{-28}$ g. In any electron–ion collision the electron will therefore radiate at least ${(m_{\mathrm{p}}/m_{\mathrm{e}})}^2\approx 4\times {10}^6$ as much power as the ion, so all ionic radiation can be neglected. Interactions between identical particles also do not radiate significantly because the accelerations of the two particles are equal in magnitude but opposite in direction: $\dot{v_1}=-\dot{v_2}$. Their radiated electric fields are equal in magnitude but opposite in sign, so the net radiated electric field approaches zero at distances much larger than the collision impact parameter (Figure  4.2) and the radiation from electron–electron collisions can be ignored. The bottom line is that only the electron–ion collisions are important, and only the electrons radiate significantly.