Essential Radio Astronomy

Chapter 4 Free–Free Radiation

4.1 Thermal and Nonthermal Emission

Larmor’s formula (Equation 2.143)

P=2q2v˙23c3 (4.1)

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.

4.2 Hii Regions

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 λD1m (Equation 4.46). As an electron (charge -e-4.8×10-10 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

|v˙|=Fme=Ze2mer2, (4.2)

where me9.1×10-28 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 (H2), 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?

Figure 4.1: A Strömgren sphere of ionized hydrogen (Hii) with Strömgren radius RS surrounded by a thin shell of partially ionized hydrogen (Hi + Hii) surrounded by neutral hydrogen (Hi).

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,, 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,; and Larmor’s equation fairly accurately predicts the radiative lifetimes of excited hydrogen atoms. The orbital radius an of an electron in the nth energy level is an=n2a0, where a05.29×10-9 cm is called the Bohr radius. Applying Larmor’s equation shows that the radiative lifetime τ is proportional to an3 and hence to n6. Thus the (incorrect) classical result τ5.5×10-11 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 τ265.5×10-11s3×10-9 s, in reasonable agreement with the accurate quantum-mechanical result τ2×10-9 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 E13.6 eV (1 electron Volt 1.60×10-12 erg). Such energetic photons have frequencies higher than the Rydberg frequency Rc=E/h3.29×1015Hz (Equation 7.11) and wavelengths shorter than λ=912 Å =912×10-10 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 T3×104 K. The rate QH at which a star with spectral luminosity Lν produces photons that can ionize hydrogen atoms in the ground state is

QH=Rc(Lνhν)𝑑ν. (4.3)

If a star emits QH Lyman continuum photons per second, it will photoionize hydrogen atoms with number density nH throughout some volume V surrounding the star. The helium mixed with the hydrogen can often be ignored because its ionization potential is so high, E24.5 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, σ10-17cm2, 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 ΔRS of the partially ionized shell surrounding a Strömgren sphere (Figure 4.1) is

ΔRS(nHσ)-1. (4.4)

For example, if the neutral hydrogen density is nH=103atomscm-3, then

ΔRS(103cm-3×10-17cm2)-11014cm1pc. (4.5)

Light travels 1014 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 n˙H that can be written as

n˙HαHnenp, (4.6)

where n˙H is the number of recombinations per unit time per unit volume (e.g., cm-3s-1),

αH3×10-13cm3s-1 (4.7)

is the recombination coefficient for hydrogen, and the collision rate per unit volume is proportional to the product nenp of the electron and proton densities. For example, if ne=np=103cm-3,

n˙H3×10-13cm3s-1×103cm-3×103cm-33×10-7cm-3s-1. (4.8)

This example shows that the recombination time

τnen˙H3.3×109s102yr (4.9)

is usually much shorter than the >106 year lifetime of an ionizing star. (A useful relation to remember is 1 year 107.5 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,

QH=n˙HV=αHnenp43πRS3 (4.10)

yields a Strömgren radius

RS(3QH4παHne2)1/3. (4.11)

For example, an O5 star (very hot and luminous) emits QH6×1049 ionizing photons per second. If ne103cm-3,

RS[36×1049s-14π3×10-13cm3s-1(103cm-3)2]1/33.6×1018cm1.2pc.

This example illustrates that RSΔRS; 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 (M15M) short-lived (lifetimes 107 yr) main-sequence stars are big enough (R10R) and hot enough (T3×104 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 (1<M/M<8) stars whose main-sequence lifetimes are less than the age of our Galaxy (1010 yr) eventually become red giants and finally white dwarfs. Young white dwarfs are small (R10-2R) 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 λ=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 Hα at λ=6563 Å and Hβ at λ=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 T104 K [77] 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 1 pc, electron number densities 103cm-3, and masses up to 104M.

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 QH 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 2RS0.03pc) but dense (ne>104cm-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 [59] 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 T105 K) white-dwarf cores of low-mass (1<M<8) 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 0.1<M/M<1. 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.

4.3 Free–Free Radio Emission from Hii Regions

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 104 K feel? How big is a parsec compared with the height of a tree? How much is M2×1033 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 [98] 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 v˙ is inversely proportional to the particle mass m. The lightest ion is the hydrogen ion. Its mass is the proton mass mp1.66×10-24 g, which is about 2×103 times the electron mass me9.11×10-28 g. In any electron–ion collision the electron will therefore radiate at least (mp/me)24×106 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: v1˙=-v2˙. 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.

4.3.1 Radio Radiation From a Single Electron–Ion Interaction

Figure 4.2 shows an electron passing by a far more massive ion of charge Ze, where Z=1 for a singly ionized atom such as hydrogen. Each electron–ion interaction will generate a single pulse of radiation. The total energy emitted and the approximate frequency spectrum of the pulse will be derived in this section. The exact spectrum of an individual pulse is not needed because the broad distribution of electron energies and impact parameters smears out spectral details in the total spectrum of an Hii region.

Figure 4.2: A light, fast electron passing by a slow, heavy ion. Low-energy radio photons are produced by weak scattering in which the velocity vector v of the electron changes little. The distance of closest approach b is called the impact parameter and the interval τ=b/v is called the collision time.

Radio photons are produced by weak interactions because the energy E=hν of a radio photon is much smaller than the average kinetic energy Ee of an electron in an Hii region. The numerical comparison below is an example of how astrophysical “intuition” can simplify the electron–ion scattering problem.

The mean electron energy in a plasma of temperature T is

Ee=3kT2. (4.12)

For an Hii region with T104 K,

Ee31.38×10-16ergK-1104K22×10-12erg1eV. (4.13)

(This is another useful conversion factor to remember: 1 eV is the typical particle kinetic energy associated with the temperature T104 K.) By comparison, the energy of a radio photon of frequency ν=10 GHz is only

E=hν6.63×10-27ergs1010Hz6.63×10-17erg4×10-5eV. (4.14)

The weak interactions that produce radio photons cause the trajectory of the electron to deflect by only a small angle (1 radian). As shown in Figure 4.2, the electron’s path can be approximated by a straight line.

During the interaction, the electron will be accelerated electrostatically both parallel to and perpendicular to its nearly straight path:

F =mev˙=-Ze2r2sinψ=-Ze2sinψcos2ψb2, (4.15)
F =mev˙=Ze2r2cosψ=Ze2cos3ψb2, (4.16)

where cosψ=b/r and b is the impact parameter of the interaction, the minimum value of the distance r between the electron and the ion.

For any impact parameter b, these two equations can be solved to show that the maximum of v˙ is a nonnegligible 38% of the maximum of v˙. Even so, the radio radiation arising from v˙ is completely negligible. Plotting the variation with time of v˙ and v˙ during the interaction shows pulses with quite different shapes (Figure 4.3).

Figure 4.3: The acceleration of an electron by an ion may be resolved into components perpendicular () to and parallel () to the electron’s velocity. The perpendicular acceleration (Equation 4.16) yields a roughly Gaussian pulse whose power spectrum extends to low (radio) frequencies. The parallel acceleration (Equation 4.15) gives a roughly sinusoidal pulse with no “DC” component, so the resulting radiation is strongest at higher (infrared) frequencies and very weak at radio frequencies.

The pulse duration is comparable with the collision time τb/v. The v˙ pulse is roughly a sine wave of angular frequency ωτ-1=v/b, which is much higher than radio frequencies for all relevant impact parameters b (Equation 4.43). The parallel acceleration produces some infrared radiation but very little radio radiation. The v˙ pulse is a single peak whose frequency spectrum extends from zero up to v/b because the Fourier transform of a Gaussian is also a Gaussian (Appendix B.4), so it is stronger at radio frequencies.

Inserting v˙ from Equation 4.16 into Larmor’s formula (Equation 2.143) gives the instantaneous power emitted by the acceleration perpendicular to the electron velocity:

P=23e2v˙2c3=2e23c3Z2e4me2(cos3ψb2)2. (4.17)

The total energy W emitted by the pulse is

W=-P𝑑t. (4.18)

Because (ΔEe)/Ee=Eγ/Ee1 for radio photons, the electron velocity is nearly constant. The interaction diagram in Figure 4.2 shows that

v=dxdt  and  tanψ=xb, (4.19)

so

v=bdtanψdt=bsec2ψdψdt=bdψcos2ψdt, (4.20)

and

dt=bvdψcos2ψ. (4.21)

Inserting Equations 4.17 and 4.21 into 4.18 gives

W=23Z2e6c3me2b4-π/2π/2bvcos6ψcos2ψ𝑑ψ=43Z2e6c3me2b3v0π/2cos4ψdψ. (4.22)

The integral in Equation 4.22 is evaluated in Appendix B.7; it is

0π/2cos4ψdψ=3π16. (4.23)

Thus the pulse energy W radiated by a single electron–ion interaction characterized by impact parameter b and velocity v is

W=πZ2e64c3me2(1b3v). (4.24)

This energy is emitted in a single pulse of duration τb/v, so the pulse power spectrum (Appendix A.4) is nearly flat over all frequencies ν<νmax(2πτ)-1v/(2πb) and falls rapidly at higher frequencies. It is possible to calculate the actual Fourier transform of the pulse shape (solid curve in Figure 4.4), but doing so would only add an unnecessary complication to an already complicated calculation. The ranges of velocities v and impact parameters b characterizing electron–ion interactions in an Hii region are so wide that averaging over all collision parameters will wash out fine details in the spectrum associated with any particular v and b.

Figure 4.4: The actual power spectrum of the electromagnetic pulse generated by one electron–ion interaction is nearly flat up to frequency νv/(2πb), where v is the electron speed and b is the impact parameter, and declines at higher frequencies. The approximation Pν=1 for all ν<v/(2πb) and Pν=0 at higher frequencies (dashed line) is quite good at radio frequencies νv/(2πb).

The typical electron speed in a T104 K Hii region is v7×107cms-1 and the minimum impact parameter is bmin10-7 cm, so νmax1014 Hz, much higher than radio frequencies. In the approximation that the power spectrum is flat out to ν=νmax and zero at higher frequencies (dashed line in Figure 4.4), the average energy per unit frequency emitted during a single interaction is approximately

WνWνmax=(πZ2e64c3me2b3v)(2πbv), (4.25)

which simplifies to

Wνπ22Z2e6c3me2(1b2v2),ν<νmaxv2πb1014Hz. (4.26)

4.3.2 Radio Radiation From an Hii Region

The strength and spectrum of radio emission from an Hii region depends on the distributions of electron velocities v and collision impact parameters b (Figure 4.2). The distribution of v depends on the electron temperature T. The distribution of b depends on the electron number density ne (cm-3) and the ion number density ni (cm-3).

In LTE, the average kinetic energies of electrons and ions are equal. The electrons are much less massive, so their speeds are much higher and the ions can be considered nearly stationary during an interaction (Figure 4.5).

Figure 4.5: The number of electrons with speeds v to v+dv passing by a stationary ion and having impact parameters in the range b to b+db during the time interval t equals the number of electrons with speeds v to v+dv in the cylindrical shell shown here.

The number of electrons passing any ion per unit time with impact parameter b to b+db and speed range v to v+dv is

ne(2πbdb)vf(v)dv, (4.27)

where f(v) is the normalized (f(v)𝑑v=1) speed distribution of the electrons. The number n˙c(v,b) of such collisions per unit volume per unit time is

n˙c(v,b)=(2πb)vf(v)neni. (4.28)

The spectral power at frequency ν emitted isotropically per unit volume is 4πjν, where jν is the emission coefficient defined by Equation 2.26. Thus

4πjν=b=0v=0Wν(v,b)n˙c(v,b)𝑑v𝑑b. (4.29)

Substituting the results for Wν(v,b) (Equation 4.26) and n˙c(v,b) (Equation 4.28) into Equation 4.29 gives

4πjν =b=0v=0(π2Z2e62c3me2b2v2)2πb𝑑bnenivf(v)𝑑v (4.30)
=π3Z2e6nenic3me2v=0f(v)v𝑑vb=0dbb. (4.31)

Equation 4.31 exposes a problem: the integral

b=0dbb (4.32)

diverges logarithmically. There must be finite physical limits bmin and bmax (to be determined) on the range of the impact parameter b that prevent this divergence:

4πjν=π3Z2e6nenic3me2v=0f(v)v𝑑vbminbmaxdbb. (4.33)

The distribution f(v) of electron speeds in LTE is the nonrelativistic Maxwellian distribution (see Appendix B.8 for its derivation):

Figure 4.6: The nonrelativistic Maxwellian distribution of particle speeds in LTE (Equation 4.34), where vrms=(3kT/m)1/2 is the rms speed of particles with mass m at temperature T.
f(v)=4v2π(me2kT)3/2exp(-mev22kT). (4.34)

Equation 4.34 can be used to evaluate the integral over the electron speeds in Equation 4.33:

v=0f(v)v𝑑v=4π(me2kT)3/2v=0vexp(-mev22kT)𝑑v. (4.35)

Substituting xmev2/(2kT) so dx=mevdv/(kT) gives

v=0f(v)v𝑑v =4π(me2kT)3/2x=0kTmee-x𝑑x (4.36)
=2π(me2kT)1/2x=0e-x𝑑x (4.37)
=(2meπkT)1/2. (4.38)

In conclusion, the free–free emission coefficient can be written as

jν=π2Z2e6neni4c3me2(2meπkT)1/2ln(bmaxbmin). (4.39)

The remaining problem is to estimate the minimum and maximum impact parameters bmin and bmax. These estimates don’t have to be very precise because only their logarithms appear in Equation 4.39.

To estimate the minimum impact parameter bmin, notice that the net impulse (change in momentum) during a single electron–ion interaction

meΔv=-F𝑑t (4.40)

comes entirely from the perpendicular component F of the electric force because the contribution from E is antisymmetric about t=0 (Figure 4.3). Inserting F from Equation 4.16 into Equation 4.40 gives

meΔv=-(Ze2cosψr2)𝑑t=Ze2-cos3ψb2𝑑t. (4.41)

Using Equation 4.21 to change the variable of integration from t to ψ gives

meΔv=Ze2bv-π/2π/2cosψdψ=2Ze2bv. (4.42)

The maximum possible momentum transfer meΔv during the free–free interaction is twice the initial momentum mev of the electron, so the impact parameter of a free–free interaction cannot be smaller than

bminZe2mev2. (4.43)

This result is based on a purely classical treatment of the interaction (see also Jackson [56, section 13.1 and problem 13.1] for a more detailed discussion). The uncertainty principle (ΔxΔp) implies an independent quantum-mechanical limit

bmin=mev, (4.44)

but this lower limit is generally smaller than the classical limit in Hii regions and hence may be ignored. This claim can be tested by computing the ratio of the classical to quantum limits for the rms electron velocity v=(3kT/me)1/2:

(Ze2mev2)(mev)-1=Ze2v=Ze2(me3kT)1/2. (4.45)

In an Hii region with T104 K and Z=1, this ratio 3, so the classical limit is stronger. Only in much cooler (T<103 K) plasmas is the quantum-mechanical limit important.

There are two effects that might determine the upper limit bmax to the impact parameter. Because electrostatic forces always dominate gravity on small scales, electrons in the vicinity of a nearly stationary ion are free to rearrange themselves to neutralize, or shield, the ionic charge. The characteristic scale length of this shielding is called the Debye length. From Jackson [56], the Debye length is

λD(kT4πnee2)1/2. (4.46)

The Debye length is quite large in the low-density plasma of a typical Hii region. For example, if T104 K and ne103cm-3,

λD[1.38×10-16ergK-1104K4π103cm-3(4.8×10-10statcoulomb)2]1/222cm. (4.47)

An independent upper limit to the impact parameter is the largest value of b that can generate a significant amount of power at some relevant radio frequency ν. Recall that the pulse power per unit bandwidth is small above angular frequency ωv/b so

bmaxvω=v2πν, (4.48)

is the maximum impact parameter capable of emitting significant power at frequency ν.

The controlling upper limit bmax in any particular situation is the smaller of these two upper limits. Numerical values for the lower and upper limits in a typical Hii region observed at ν=1 GHz are derived in the boxed example, and Equation 4.48 gives the smaller bmax10-2cm.

Example. Estimate bmin and bmax for a pure Hii region (Z=1) with T104 K observed at a fairly low frequency ν=1 GHz =109 Hz: bmin Ze2mev2e23kT (4.8×10-10statcoulomb)231.38×10-16ergK-1104K5.6×10-8cm, bmax v2πν(3kTme)1/2(2πν)-1 (31.38×10-16ergK-1104K9.1×10-28g)1/2(2π×109s-1)-1 1.1×10-2cm. The maximum impact parameter capable of generating power at this frequency is much smaller than the λD22 cm Debye length in an Hii region with electron density ne103 cm-3 (Equation 4.47), so the Debye length is irrelevant. A T=104 K electron takes so long to move 22 cm that it would emit at unobservably low frequencies ν<1 MHz. The Debye length becomes relevant only in much denser plasmas such as the solar chromosphere (ne1012 cm-3).

Our simple estimate of the ratio

bmaxbmin(3kTme)1/2(2πν)-1(3kTZe2)(3kTme)3/2me2πZe2ν (4.49)

is very close to the result of the very detailed derivation in Oster [76]. The ratio (bmax/bmin) is of order 105, which is much greater than the fractional velocity range σv/v1 in the Maxwellian velocity distribution (Figure 4.6). Also note that

ln(bmaxbmin)12 (4.50)

varies slowly with changes in either bmax or bmin, so small uncertainties in these limits have very little effect on the calculated emission coefficient of an Hii region.

Because the Hii region is in local thermodynamic equilibrium (LTE) at some temperature T, Kirchhoff’s law (Equation 2.30) immediately yields the absorption coefficient κ (Equation 2.18) in terms of the emission coefficient and the blackbody brightness Bν(T):

κ=jνBν(T)jνc22kTν2 (4.51)

in the Rayleigh–Jeans limit. Thus

κ=1ν2T3/2[Z2e6cneni12π(mek)3]π24ln(bmaxbmin). (4.52)

The limit bmax (Equation 4.48) is inversely proportional to frequency so the absorption coefficient is not exactly proportional to ν-2. A good numerical approximation is κ(ν)ν-2.1.

The total opacity τ of an Hii region is the integral of -κ along the line of sight, as illustrated in Figure 4.7:

Figure 4.7: Astronomers often approximate Hii regions by uniform cylinders whose axis is the line of sight because this gross oversimplification finesses the radiative-transfer problem. It is for good reason that astronomers often feature in jokes beginning “Consider a spherical cow….”
τ=-losκ𝑑sneniν2.1T3/2𝑑sne2ν2.1T3/2𝑑s. (4.53)

At frequencies low enough that τ1, the Hii region becomes opaque, its spectrum approaches that of a blackbody with brightness temperature approaching the electron temperature (TbT104 K), and its flux density obeys the Rayleigh–Jeans approximation Sν2. At very high frequencies, τ1, the Hii region is nearly transparent, and

S2kTν2c2τ(ν)ν-0.1. (4.54)

On a log-log plot, the overall spectrum of a uniform Hii region looks like Figure 4.8, with the spectral break corresponding to the frequency at which τ1.

Figure 4.8: The radio spectrum of an Hii region. It is a blackbody at low frequencies, with slope 2 if a uniform cylinder as shown in Figure 4.7 and <2 otherwise. At some frequency ν the optical depth τ=1, and at much higher frequencies the spectral slope becomes -0.1 because the opacity coefficient κ(ν)ν-2.1. The source brightness at low frequencies equals the electron temperature. The brightness at high frequencies is proportional to the emission measure (Equation 4.57) of the Hii region.

The spectral slope on a log-log plot is often called the spectral index and denoted by α, whose sign is defined ambiguously:

α±dlogSdlogν. (4.55)

Beware the ± sign! Unfortunately both sign conventions are found in the literature, and you have to look carefully at each paper to find out which one is being used. With the + sign convention, the low-frequency spectral index of a uniform Hii region would be α=+2. The - sign convention was introduced in the early days of radio astronomy because most sources discovered at low frequencies are stronger at low frequencies than at high frequencies. Thus α=+0.7 might mean

dlogSdlogν=-0.7. (4.56)

The (+) spectral index of any inhomogeneous Hii region will be α-0.1 well above the break frequency, but the break will be more gradual and the low-frequency slope will be somewhat less than +2 just below the break. For example, ionized winds from stars are quite inhomogeneous. Mass conservation in a constant-velocity, isothermal spherical wind implies that the electron density is inversely proportional to the square of the distance from the star: ner-2. The low-frequency spectral index of free–free emission by such a wind is closer to +0.6 than to +2.

The emission measure (EM) of an Hii region is defined by the integral of ne2 along the line of sight expressed in astronomically convenient units:

EMpccm-6los(necm-3)2d(spc). (4.57)

Because κ is proportional to nenine2, the optical depth τ is proportional to the emission measure. The emission measure is commonly used to parameterize τ in astronomically convenient units:

τ3.014×10-2(TK)-3/2(νGHz)-2(EMpccm-6)gff, (4.58)

where the free–free Gaunt factor [18] gff is a parameter that absorbs the weak frequency dependence associated with the logarithmic term in κ:

gffln[4.955×10-2(νGHz)-1]+1.5ln(TK). (4.59)

Mezger and Henderson [73] found a very good approximation for the free–free opacity τ that is easy to evaluate numerically:

τ3.28×10-7(T104K)-1.35(νGHz)-2.1(EMpccm-6). (4.60)

Mezger and Henderson [73, Table 6] lists the errors introduced by this approximation over wide ranges of temperature and frequency.

Example. The interstellar medium of our Galaxy contains a diffuse ionized component, some of which is “warm” (T104 K) and some is “hot” (T106 K). These two phases are roughly in pressure equilibrium so the hot medium is less dense by a factor of 102. The combination of high Te and low ne of the hot phase means that only the warm component contributes significantly to the free–free opacity of the ISM. The warm ionized gas is largely confined to the disk of our Galaxy, where we reside. There must be some frequency ν below which this disk becomes opaque and we cannot see out of our Galaxy, even in the direction perpendicular to the disk. From the observed brightness spectrum in the direction perpendicular to the disk, Cane [21] found that τ1 at ν2 MHz. This result can be inserted into Equation 4.60 to estimate the rms electron density in the warm ISM: 13.28×10-7(1)-1.350.002-2.1ne21000pc, ne21/2(0.0022.13.28×10-4)1/20.08cm-3.

From the optical depth τ and the electron temperature T it is possible to calculate the brightness temperature

Tb=T(1-e-τ) (4.61)

of free–free emission. The line-of-sight structure of an Hii region is not normally known, so it is common to approximate the geometry of an Hii region by a circular cylinder whose axis lies along the line of sight, and whose axis length equals its diameter. Suppose further that the temperature and density are constant throughout this volume. Then it is very easy to estimate physical parameters of the Hii region (e.g., electron density, temperature, emission measure, production rate QH of ionizing photons) from the observed radio spectrum, once the distance to the Hii region is known.

A useful approximation relating the production rate of ionizing photons to the free–free spectral luminosity Lν at the high frequencies where τ1 of an Hii region in ionization equilibrium is [95]

(QHs-1)6.3×1052(T104K)-0.45(νGHz)0.1(Lν1020WHz-1). (4.62)
Example. Suppose an idealized Hii region with temperature T=104 K at distance d=10 kpc produces the radio spectrum shown in Figure 4.8 where S is in Jy and ν is in GHz. What constraints on the Hii region are provided by this spectrum? At low frequencies ν1 GHz, the Hii region is optically thick, so it is a blackbody radiator and obeys the Rayleigh–Jeans approximation from which the source solid angle can be derived. From Figure 4.8, S0.1 Jy at ν=0.3 GHz (λ=1 m) so Ω=λ2S2kT(1m)20.1×10-26Wm-2Hz-121.38×10-23JK-1104K3.6×10-9sr. Thus the angular diameter of the Hii region is θ(4Ωπ)1/27×10-5rad and its linear diameter is about 0.7 pc. At high frequencies ν1 GHz, the opacity τ1 so the ratio of observed flux density to blackbody flux density is τ. At ν=10 GHz, the observed flux density is S0.8 Jy while the Rayleigh–Jeans extrapolated flux density is S0.1Jy(ν0.3GHz)2110Jy, so the optical depth at 10 GHz is τ0.8/1100.007.

Free–free emission accounts for about 10% of the 1 GHz continuum luminosity in most spiral galaxies. It is the strongest component in the frequency range from ν30 GHz to ν200 GHz, above which thermal emission from cool dust grains dominates (Figure 2.24). Free–free absorption flattens the low-frequency spectra of spiral galaxies, and the frequency at which τ1 is higher in galaxies with high star-formation rates, especially if the star formation is confined to a compact region near the nucleus. If the free–free and synchrotron emission from a starburst galaxy are roughly cospatial, its radio brightness temperature at frequencies ν<100 GHz is [30]

TbT[1-exp(-τ)][1+10(νGHz)0.1+α], (4.63)

where T104 K and α-0.8 is the spectral index of the synchrotron radiation. Free–free absorption of the synchrotron radiation limits the maximum brightness temperature to Tb105 K at frequencies ν1 GHz. This limit can be used to identify the energy source powering a compact radio source at the center of a galaxy: if its brightness temperature is significantly higher than 105 K, it is powered by an AGN, not a compact starburst.