# Thermal Radio Emission from HII Regions

Larmor's  formula
$$P = {2 q^2 \dot{v}^2 \over 3 c^3}$$ states that electromagnetic radiation with power $P$ is produced by accelerating (or decelerating; hence the German name bremsstrahlung meaning "braking radiation") an electrical charge $q$. Charges can be accelerated by electrostatic or magnetic forces, gravitational acceleration being negligible by comparison. We will consider electrostatic bremsstrahlung first and later its magnetic counterpart magnetobremsstrahlung, or "magnetic braking radiation," synchrotron radiation for example.

The electric force is so much stronger than gravity that ionized interstellar clouds have almost no net charge on large scales; the charges of all free electrons in an ionized cloud are balanced by the charges of positively ions. An electron (charge $-e \approx -4.8 \times 10^{-10}$ statcoulombs) passing by an ion (charge $+e$ for a singly ionized atom, $+Ze$ for an ionized atom with $Z$ electrons removed) is accelerated by their Coulomb attraction
$$\dot{v} = {f \over m_{\rm e}} = {-Ze^2 \over m_{\rm e} l^2} ,$$ where $m_{\rm e} \approx 9.1 \times 10^{-28}$ g is the electron mass and $l$ is the distance between the electron and ion. Such radiation is called free-free radiation because an initially free electron is rarely captured by the ion during the interaction. If the ionized interstellar cloud is reasonably dense, the electrons and ions interact frequently enough that they come into local thermodynamic equilibrium (LTE) at some common temperature. Thus the radiation produced by these interactions is sometimes called thermal bremsstrahlung. This section will cover:

(1) astronomical sources of free-free emission

(2) the radio properties (spectrum, power, opacity) of free-free emission

(3) applications: what do we learn from radio observations of free-free sources?

Interstellar gas is primarily hydrogen plus some helium and trace amounts of heavier elements such as carbon, nitrogen, oxygen, iron, ...  Astronomers often lump all of the heavier elements into the category of metals, meaning elements that readily form positive ions, even though many are not metallic in the usual sense. 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, doubly ionized oxygen O$^{++}$ is called OIII, etc.

In 1939 the astronomer Bengt Strömgren realized that regions of diffuse interstellar gas are either (1) mostly neutral and with nearly all of the HI atoms in their ground electronic state or (2) almost completely ionized (HII much more abundant than HI), with very thin boundaries separating distinct HI and HII regions. Sometimes the HII regions surrounding stars are called Strömgren spheres after his early theoretical models. What is the microscopic physical basis for these ideas?

The ground electronic state of a hydrogen atom corresponds to an atom with the smallest (and hence most tightly bound) electron orbit around the nuclear proton that is consistent with a stationary electronic wave function, a standing wave. [See Sections 13.3 of Rohlfs & Wilson for a brief discussion of the Bohr orbits in Rydberg atoms.] The electronic energy levels permitted by quantum mechanics are characterized by their quantum numbers $n = 1,~2,~3,...$, where $n = 1$ corresponds to the ground state. While quantum mechanics forbids an electron in the ground state ($n = 1)$ from radiating according to the classical Larmor formula, it does not forbid radiative decay from higher levels ($n = 2,~3,...$), and Larmor's equation fairly accurately predicts the radiative lifetimes of excited hydrogen atoms. The orbital radius $a_{\rm n}$ of an electron in the $n$th energy level is $a_{\rm n} = n^2 a_0$, where $a_0 \approx 5.29 \times 10^{-9}$ cm is called the Bohr radius. Applying Larmor's equation, as in problem 2 of problem set 2, shows that the radiative lifetime $\tau$ is proportional to $a_{\rm n}^3$ and hence to $n^6$. Thus we can scale the [incorrect] classical result $\tau \approx 5.5 \times 10^{-11}$ s for $n = 1$ to estimate the radiative lifetimes of excited states. For example, the approximate radiative lifetime of the $n = 2$ state would be $\tau \approx 2^6 \times 5.5 \times 10^{-11} {\rm ~s~} \approx 3 \times 10^{-9}$ s, in reasonable agreement with the accurate quantum-mechanical result $\tau \approx 2 \times 10^{-9}$ s. Clearly excited hydrogen atoms will spontaneously decay very quickly to the ground state by emitting radiation. At any one time, almost all neutral atoms are in the ground state.

Most interstellar hydrogen atoms are in the ground electronic state ($n = 1$) because an electron orbiting in a higher orbit ($n = 2,~3,\dots$) quickly drops to the ground state by emitting radiation as predicted by the classical Larmor equation.

Suppose that a volume of interstellar space initially contains HI atoms with some number density $n_0$. Then turn on a star hot enough ($T > 3 \times 10^4$ K) that its approximately blackbody radiation includes a significant number of photons with energy $E \geq 13.6$ eV (1 electron Volt $\approx 1.60 \times 10^{-12}$ erg), the energy needed to ionize a hydrogen atom initially in the ground state. Since $E_\gamma = h \nu = h c / \lambda$, such photons have wavelengths $\lambda \leq 912$ Å $= 912 \times 10^{-10}$ m, corresponding to far-ultraviolet (UV) radiation. If these ionizing photons are generated at a rate $N_{\rm u}$ photons s$^{-1}$, they will photoionize the hydrogen in some volume $V$ surrounding the star. We ignore the helium that is mixed with the hydrogen because its ionization potential is so high, $E \approx 24.5$ eV, that only exceptionally hot stars can ionize helium at all.

The absorption cross-section of a neutral hydrogen atom to ionizing UV photons is very high: $\sigma \approx 10^{-17}$ cm$^2$, so each ionizing photon entering an HI region is quickly absorbed and produces a new ion almost as soon as it passes from an HII region to an HI region.

Example: How far will a typical ionizing UV photon travel in an HI cloud with density $n_0 = 10^3$ atoms cm$^{-3}$?
$$\Delta R_{\rm S} \approx (n_0 \sigma)^{-1} \approx (10^3 {\rm ~cm}^{-3} \times 10^{-17}{\rm ~cm}^2)^{-1} \approx 10^{14}{\rm ~cm} \ll 1{\rm ~pc} \approx 3 \times 10^{18}{\rm ~cm}$$
Light travels $10^{14}$ cm in about one hour, so in this realistic example, ionizing photons only survive about an hour in the 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 if we turn on an ionizing star in a uniform-density HI cloud, it will fully ionize a sphere whose Strömgren radius $R_{\rm S}$ grows with time until equilibrium between ionization and recombination is reached, and the partially ionized boundary of this sphere will be quite thin, only about $10^{14}$ cm thick.  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.

This Strömgren sphere of ionized hydrogen (HII) is surrounded by a thin shell of partially ionized hydrogen (HI and HII) embedded in a neutral hydrogen (HI) cloud.

Inside the HII region, electrons and protons occasionally collide and recombine at a volumetric rate $r_{\rm r}$ that can be written as
$$r_{\rm r} \approx \alpha_{\rm H} n_{\rm e} n_{\rm p} ,$$ where $r_{\rm r}$ is the number of recombinations per unit time in a unit volume (e.g., cm$^{-3}$ s$^{-1}$), $\alpha_{\rm H} \approx 3 \times 10^{-13}$ cm$^3$ s$^{-1}$ is a parameter specifying the hydrogen recombination rate, and the collision rate per unit volume is proportional to the product of the electron density $n_{\rm e}$ and the proton density $n_{\rm p}$.

Example: If $n_{\rm e} = n_{\rm p} = 10^3$ cm$^{-3}$, what is the volumetric recombination rate $r_{\rm r}$?
$$r_{\rm r} \approx \alpha_{\rm H} n_{\rm e} n_{\rm p} \approx 3 \times 10^{-13} {\rm ~cm}^3 {\rm ~s}^{-1} \times 10^3 {\rm ~cm}^{-3} \times 10^3 {\rm ~cm}^{-3} \approx 3 \times 10^{-7} {\rm ~cm}^{-3} {\rm ~s}^{-1}$$

This example shows that the recombination time
$$\tau \approx { n_{\rm e} \over r_{\rm r}} \approx 3.3 \times 10^9 {\rm ~s} \approx 10^2 {\rm ~yr}$$
is very short compared with the lifetime of an ionizing star. [A useful relation to remember is: 1 year $\approx 10^{7.5}$ s.] Thus we expect that a steady state will develop in which the total ionization and recombination rates in the Strömgren sphere are equal. Let $N_{\rm Ly}$ be the number of Lyman continuum photons (that is, photons with $\lambda < 912$ Å capable of ionizing hydrogen) per second emitted by the star. In equilibrium,
$$N_{\rm Ly} = r_{\rm r} V = \alpha_{\rm H} n_{\rm e} n_{\rm p} {4 \over 3} \pi R_{\rm S}^3$$
$$R_{\rm S} \approx \biggl( { 3 N_{\rm Ly} \over 4 \pi \alpha_{\rm H} n_{\rm e}^2 } \biggr)^{1/3}$$

Example: An O5 star (very hot and luminous) emits $N_{\rm Ly} \approx 6 \times 10^{49}$ ionizing photons per second. If $n_{\rm e} \approx 10^3$ cm $^{-3}$,
$$R_{\rm S} \approx \biggl[ { 3 \times 6 \times 10^{49} {\rm ~s}^{-1} \over 4 \pi \times 3 \times 10^{-13} {\rm ~cm}^3 {\rm ~s}^{-1} (10^3 {\rm ~cm}^{-3})^2 } \biggr]^{1/3} \approx 3.6 \times 10^{18} {\rm ~cm} \approx 1.2 {\rm ~pc}$$
Note that $R_{\rm S} \gg \Delta R_{\rm S}$; that is, the radius of the fully ionized Strömgren sphere is much larger than the thickness of its partially ionized skin.

What kinds of stars produce HII regions?

(1) Massive ($M \geq 15 M_\odot$) short-lived (main-sequence lifetimes $\tau \leq 10^7$ yr) stars are hot enough ($T \geq3 \times 10^4$ K) to be very luminous in the UV. New stars are formed by gravitational collapse and fragmentation of interstellar clouds containing neutral gas and dust grains.

(2) Old low-mass ($1 < M < 8 M_\odot$) stars whose main-sequence lifetimes are less than the age of the Galaxy ($\approx 10^{10}$ years) eventually become red giants and finally white dwarfs. Young white dwarfs are hot enough to ionize the stellar envelope material that had been ejected during the red giant stage, and these ionized regions are called planetary nebulae because they looked like planets to early astronomers using small telescopes.

Since the ionizing stars have roughly blackbody spectra, most of their ionizing photons are in the high-frequency tail of the blackbody spectrum and have energies somewhat greater than the $E_\gamma = 13.6$ eV needed to ionize hydrogen from its ground state. Momentum conservation ensures that nearly all of the excess photon energy becomes kinetic energy of the new free 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. 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, emitting 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 once called 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 recombination lines H$\alpha$ at $\lambda = 6563$ Å and H$\beta$ at $\lambda = 4861$ Å also contribute to the characteristic colors of HII regions. HST image of the Keyhole Nebula. UV photons from a star off to the upper left ionize hydrogen gas seen fluorescing in the red H$\beta$ line. Image credit

Thermal equilibrium between heating and cooling of HII regions is usually reached at a temperature $T \approx 10^4$ K that is much higher than the initial temperature $T < 100$ 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 densities $\sim 10^3$ cm$^{-3}$, and masses up to $10^4M_\odot$.

We can use free-free radio emission from an HII region to estimate the electron temperature, electron density, and ionized volume. These physical parameters yield the ionization rate $N_{\rm Ly}$ (photons s$^{-1}$) and, with an assumption about 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.

Planetary nebulae are HII regions surrounding the hot (up to $T \sim 10^5$ K) white-dwarf cores of low-mass ($1 < M_\odot < 8$) stars that have ejected their envelopes as stellar winds. White dwarfs are small in size, so they are not as luminous as massive O stars, and the ionized nebular masses are only $0.1 < M_\odot < 1$. Planetary nebulae are relatively luminous indicators of the last stages in the lives of 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 discrete radio continuum sources in our Galaxy. In this HST false-color image of the planetary nebula IC 418, red shows emission from ionized nitrogen (the coolest gas in the nebula, located furthest from the hot nucleus), green shows emission from hydrogen, and blue traces the emission from ionized oxygen (the hottest gas, closest to the central star). Image credit Detections (NVSS 1.4 GHz contours) of faint and obscured planetary nebulae.  Planetary nebulae account for the majority of compact continuum radio sources in our Galaxy.