Essential Radio Astronomy

Chapter 7 Spectral Lines

7.1 Introduction

Spectral lines are narrow (Δνν) emission or absorption features in the spectra of gaseous and ionized sources. Examples of radio spectral lines include recombination lines of ionized hydrogen and heavier atoms, rotational lines of polar molecules such as carbon monoxide (CO), and the λ=21 cm hyperfine line of interstellar Hi.

Spectral-line emission and absorption are intrinsically quantum phenomena. Classical particles and waves are idealized concepts like infinitesimal points or perfectly straight lines in geometry; they don’t exist in the real world. Some things are nearly waves (e.g., radio waves) and others are nearly particles (e.g., electrons), but all share characteristics of both particles and waves. Unlike idealized waves, real radio waves do not have a continuum of possible energies. Instead, electromagnetic radiation is quantized into photons whose energy is proportional to frequency: E=hν. Unlike idealized particles, real particles of momentum p are associated with waves whose De Broglie wavelength is λ=h/p. An electron’s stable orbit about the nucleus of an atom shares a property with standing waves: its circumference must equal an integer number of wavelengths. Planck’s constant h6.62607×10-27 erg s in these two equations is a quantum of action whose dimensions are (mass×length×2time-1), the same as (energy×time) or (angular momentum) or (length×momentum). Although h has dimensions of energy×time, physically acceptable solutions (the wave functions and their derivatives must be finite and continuous) to the time-independent Schrödinger equation exist only for discrete values of the total energy, so spectral lines have definite frequencies resulting from transitions between discrete energy states. A second quantum effect important to spectral lines, particularly at radio wavelengths where hνkT, is stimulated emission (Section 7.3.1). Fortunately, the fundamental characteristics of radio spectral lines from interstellar atoms and molecules can be derived from fairly simple applications of quantum mechanics and thermodynamics.

Spectral lines are powerful diagnostics of physical and chemical conditions in astronomical objects. Their rest frequencies identify the specific atoms and molecules involved, and their Doppler shifts measure radial velocities. These velocities yield the redshifts and Hubble distances of extragalactic sources, plus rotation curves and radial mass distributions for resolved galaxies. Collapse speeds, turbulent velocities, and thermal motions contribute to line broadening in Galactic sources. Temperatures, densities, and chemical compositions of Hii regions, dust-obscured dense molecular clouds, and diffuse interstellar gas are also constrained by spectral-line data. Radio spectral lines have some unique characteristics:

  1. 1.

    Their “natural” line widths are much smaller than Doppler-broadened line widths, so gas temperatures and very small changes in radial velocity can be measured.

  2. 2.

    Stimulated emission is important because hνkT. This causes line opacities to vary as T-1 and favors the formation of natural masers.

  3. 3.

    The ability of radio waves to penetrate dust in our Galaxy and in other galaxies allows the detection of line emission emerging from dusty molecular clouds, protostars, and molecular disks orbiting AGNs.

  4. 4.

    Frequency (inverse time) can be measured with much higher precision than wavelength (length), so very sensitive searches for small changes in the fundamental physical constants over cosmic timescales are possible.

Although the interstellar medium (ISM) of our Galaxy is dynamic, it tends toward a rough pressure equilibrium because mass motions with speeds up to the speed of sound try to reduce pressure gradients. Temperatures equilibrate more slowly, so there are wide ranges of temperature T and particle number density n consistent with a given pressure p and the ideal gas law

p=nkT. (7.1)

Typical ISM pressures lie in the range p/k=nT103104cm-3K [58]. Radiative cooling by spectral-line emission depends strongly on temperature, so most of the ISM exists in several distinct phases having comparable pressures but quite different temperatures:

  1. 1.

    cold (10s of K) dense molecular clouds

  2. 2.

    cool (102 K) neutral Hi gas

  3. 3.

    warm (5×103 K) neutral Hi gas

  4. 4.

    warm (104 K) ionized Hii gas

  5. 5.

    hot (106 K) low-density ionized gas (in bubbles formed by expanding supernova remnants, for example)

All but the hottest phase are sources of radio spectral lines.

7.2 Recombination Lines

7.2.1 Recombination Line Frequencies

The semiclassical Bohr atom (Figure 7.1) contains a nucleus of protons and neutrons around which one or more electrons move in circular orbits. The nuclear mass M is always much greater than the sum of the electron masses me, so the nucleus is nearly at rest in the center-of-mass frame. The wave functions of the electrons have De Broglie wavelengths

λ=hp=hmev, (7.2)

where p is the electron’s momentum and v is its speed. Only those orbits whose circumferences equal an integer number n of wavelengths correspond to standing waves and are permitted. Thus the Bohr radius an of the nth permitted electron orbit satisfies the quantization rule

2πan=nλ=nhmev, (7.3)

where the number n is called the principal quantum number. The requirement that

an=nh2πmev=nmev (7.4)

implies that the orbital angular momentum L=mevan=n is an integer multiple of the reduced Planck’s constant h/(2π). The relation between an and v is determined by the balance of Coulomb and centrifugal forces on electrons in circular orbits. For a hydrogen atom,

e2an2=mev2an. (7.5)

Equations 7.4 and 7.5 can be combined to eliminate v and solve for an in terms of n and physical constants:

an=n22mee2. (7.6)

Numerically, the Bohr radius of a hydrogen atom whose electron is in the nth electronic energy level is

an =2mee2n2=[6.63/(2π)×10-27ergs]29.11×10-28g(4.8×10-10statcoul)2n2
0.53×10-8cmn2.

The Bohr radius of a hydrogen atom in its ground electronic state (n=1) is only a10.53×10-8 cm, but the diameter of a highly excited (n100) radio-emitting hydrogen atom in the ISM can be remarkably large: 2a10010-4cm=1μm, which is bigger than most viruses!

Figure 7.1: The radius of the nth Bohr orbit is proportional to n2, so radio-emitting hydrogen atoms with n100 are 104 times larger than ordinary hydrogen atoms in the n=1 ground state.

The electron in a Bohr atom can fall from the level (n+Δn) to n, where Δn and n are any natural numbers (1,2,3,), by emitting a photon whose energy equals the energy difference ΔE between the initial and final levels. Such spectral lines are called recombination lines because formerly free electrons recombining with ions quickly cascade to the ground state by emitting such photons. Astronomers label each recombination line using the name of the element, the final level number n, and successive letters in the Greek alphabet to denote the level change Δn: α for Δn=1, β for Δn=2, γ for Δn=3, etc. For example, the recombination line produced by the transition between the n=92 and n=91 levels of a hydrogen atom is called the H91α line.

The total electronic energy En is the sum of the kinetic (T) and potential (V) energies of the electron in the nth circular orbit:

En=T+V=-T=V/2=-e22an=-e2(mee22n22)=-(mee422)1n2. (7.7)

The electronic energy change ΔE going from level (n+Δn) to level n is equal to the energy hν of the emitted photon:

ΔE=mee422[1n2-1(n+Δn)2]=hν, (7.8)

so the photon frequency is

ν=(2π2mee4h3c)c[1n2-1(n+Δn)2]. (7.9)

The factor in large parentheses is called the Rydberg constant R, where the subscript refers to the limit of infinite nuclear mass M:

R(2π2mee4h3c)=1.09737312×105cm-1. (7.10)

The dimensions of R are length-1, and the product Rc is the Rydberg frequency:

Rc=3.28984×1015Hz. (7.11)

Allowing for the relatively large but finite nuclear mass Mme and repeating the analysis above in the atomic center-of-mass frame yields the same frequency formula with R replaced by RM:

ν=RMc[1n2-1(n+Δn)2],where  RMR(1+meM)-1, (7.12)

The hydrogen nucleus is a single proton of mass mp1836.1me so M(H)1836.1me and RMc for a hydrogen atom is

RMc=3.28984×1015Hz(1+11836.1)-1=3.28805×1015Hz.

Thus the frequency of the photon produced by the H109α transition (n+Δn=110 to n=109) is

ν(H109α)=3.28805×1015Hz(11092-11102)5.0089×109Hz.

The mass of a neutron is about equal to the mass of a proton, so the 4He nucleus consisting of two protons and two neutrons has mass M(4He)4M(H), the isotope of carbon with six protons and six neutrons has M(12C)12M(H), and so on. Electrons recombining onto singly ionized atoms with any number Np of protons and (Np-1) electrons orbit in the potential produced by a net charge of one proton, so the recombination lines of heavier atoms are very similar to those of hydrogen, but at the slightly higher frequencies (Figure 7.2) given by Equation 7.12. For example, the primordial abundance of the rare helium isotope 3He is important because it depends on the photon/baryon ratio in the early universe. The abundance of 3He in Galactic Hii regions has been measured via radio recombination-line emission and indicates that baryons account for only a few percent of the critical density needed to close the universe.

Figure 7.2: Observed recombination-line spectra from the 91α and 92α transitions of hydrogen, helium, and carbon observed in an Hii region [85].

The strongest radio recombination lines are produced by transitions with Δnn, so the approximation

[1n2-1(n+Δn)2](n+Δn)2-n2n2(n+Δn)2=n2+2nΔn+(Δn)2-n2n2[n2+2nΔn+(Δn)2]2nΔnn4=2Δnn3 (7.13)

yields simpler (but not extremely accurate) approximations for radio recombination line frequencies

ν2(RMc)Δnn3, (7.14)

and for the frequency separation Δν=ν(n)-ν(n+1) between adjacent lines

Δνν3n. (7.15)

Adjacent high-n (low-ν) radio recombination lines have such small fractional frequency separations (Figure 7.3) that two or more transitions can often be observed simultaneously and averaged, to reduce the observing time needed to reach a given signal-to-noise ratio.

Figure 7.3: The Δn=1 radio recombination lines of singly ionized atoms, shown here as vertical bars, are closely spaced in frequency.

The H109α line was first detected by P. Mezger in 1965, despite (incorrect) theoretical predictions that pressure broadening would smear out the lines in frequency and make them undetectable. It is true that atomic collisions in the interstellar medium significantly disturb the energy levels of large atoms, but this disturbance is about the same for adjacent energy levels, so the differential disturbance that alters the line frequency is actually much smaller. His advice: “Don’t abandon an observation just because you have been told that it will fail.”

7.2.2 Recombination Line Strengths

The spontaneous emission rate is the average rate at which an isolated atom emits photons. Rigorous quantum-mechanical calculations of spontaneous emission rates are complicated, but a fairly good classical approximation can be derived by noting that radio photons are emitted by atoms with n1 and invoking the correspondence principle, Bohr’s hypothesis that systems with large quantum numbers behave almost classically. The time-averaged radiated power P for classical transitions is given by Larmor’s formula (Equation 2.143) for an electric dipole with dipole moment ean:

P=2e23c3(ω2an)2cos2(ωt)=2e23c3(2πν)4an22. (7.16)

The photon emission rate (s-1) equals the average power emitted by one atom divided by the energy of each photon. The spontaneous emission rate for transitions from level n to level (n-1) is denoted by An,n-1:

An,n-1=Phν, (7.17)

where

ν2RcΔnn3 (7.18)

in the limit Δnn (Equation 7.14). In that limit, An+1,nAn,n-1 also.

The atomic radius (Equation 7.6) is

ann2h24π2mee2, (7.19)

so

An+1,n Phν2e23c3(16π4ν3h)an22 (7.20)
16π43e2c3h(2Rcn3)3(n2h24π2mee2)2 (7.21)
16π43e2c3h(4π2mee4h3)3(h24π2mee2)21n5, (7.22)
An+1,n(64π6mee103c3h6)1n5. (7.23)

Evaluating the constants yields the spontaneous emission rate of hydrogen atoms:

An+1,n[64π69.11×10-28g(4.8×10-10statcoul)103(3×1010cms-1)3(6.63×10-27ergs)6]1n5, (7.24)
An+1,n5.3×109(1n5)s-1. (7.25)

For example, the 5.0089 GHz H109α transition rate is A110,1090.3s-1.

The associated natural line width or intrinsic line width follows from the uncertainty principle: ΔEΔt. Substituting hΔν for ΔE and An+1,n-1 for Δt for each energy level involved in the transition and summing these two uncertainties yields

ΔνAn+1,n/π0.1Hz. (7.26)

Natural broadening is negligibly small at the large n that produce radio-frequency photons. Collisions of the emitting atoms cause collisional broadening, where the amount of collisional broadening is a small fraction of the collision rate when Δnn. Except for very large n, collisional broadening is also small and the actual line profile (normalized intensity as a function of frequency) is primarily determined by Doppler shifts reflecting the radial velocities vr of the emitting atoms. The sign convention is vr>0 for sources moving away from the observer. Radial velocities may be microscopic (from the thermal motions of individual atoms) or macroscopic (from large-scale turbulence, flows, or rotation). In the nonrelativistic limit vrc, the Doppler equation (Equation 5.142) relating the observed frequency ν to the line rest frequency ν0 reduces to

νν0(1-vrc), (7.27)

so nonrelativistic radial velocities can be estimated from

vrc(ν0-ν)ν0. (7.28)

The thermal component of the line profile from a recombination-line source in LTE is determined by the Maxwellian speed distribution (Equation B.49) of atoms with mass M and temperature T. The speed in any one coordinate of an isotropic distribution is 3-1/2 of the total speed in three dimensions, so the Gaussian

f(vr)=(M2πkT)1/2exp(-Mvr22kT) (7.29)

is the normalized (f(vr)𝑑vr=1) radial velocity distribution. The normalized line profile (Figure 7.4) ϕ(ν) for thermal emission is

|ϕ(ν)dν| =f(vr)dvr, (7.30)
ϕ(ν) =(M2πkT)1/2exp[-M2kTc2(ν-ν0)2ν02]|dvrdν|, (7.31)
ϕ(ν)=cν0(M2πkT)1/2exp[-Mc22kT(ν-ν0)2ν02]. (7.32)
Figure 7.4: The parameters of the normalized (ϕ(ν)𝑑ν=1) line profile ϕ(ν) are the center frequency ν0, the FWHM line width Δν, and the profile height ϕ(ν0) at the center frequency.

This is a Gaussian line profile. Its full width between half-maximum points (FWHM) Δν is the solution of

exp[-Mc22kT(Δν/2)2ν02]=12, (7.33)
Mc22kTΔν24ν02=ln2, (7.34)
Δν=[8ln(2)kc2]1/2(TM)1/2ν0. (7.35)

For example, the FWHM of the H109α line (ν0=5.0089 GHz) in a quiescent (no macroscopic motions) Hii region with temperature T104 K is

Δν [8ln2 1.38×10-16ergK-1(3×1010cms-1)2]1/2(104K18369.11×10-28g)1/2
×5.0089×109Hz
3.6×105Hz.

Notice that the thermal line width Δν is much larger than the natural line width A110,1090.3 Hz.

Normalization (requiring ϕ(ν)𝑑ν=1) implies that the value of ϕ at the line center (ν=ν0) is

ϕ(ν0)=cν0(M2πkT)1/2=cΔν(8ln2kTMc2M2πkT)1/2, (7.36)
ϕ(ν0)=(ln2π)1/22Δν. (7.37)

For a given integrated (over frequency) line strength, the line strength per unit frequency at any one frequency (e.g., at ν0) is inversely proportional to the line width Δν. Integrated line strengths are frequently specified in the astronomically convenient units of Jykms-1, where 1kms-1ν0/3.00×105.

7.3 Line Radiative Transfer

7.3.1 Einstein Coefficients

The spontaneous emission coefficient AUL is the average photon emission rate (s-1) for an “undisturbed” atom or molecule transitioning from an upper (U) to a lower (L) energy state. The spectral-line radiative transfer problem also involves the absorption coefficient BLU and the stimulated emission coefficient BUL (Figure 7.5). Einstein showed that both the absorption and stimulated emission coefficients can be calculated from the spontaneous emission coefficient.

Figure 7.5: The three Einstein coefficients for a two-level system: AUL for spontaneous emission, BLU for absorption, and BUL for stimulated emission.

Consider any two energy levels EU and EL of a quantum system such as a single atom or molecule. The photon emitted or absorbed during a transition between the upper and lower states will have energy

E=EU-EL (7.38)

and contribute to a spectral line with rest frequency ν0=E/h. The energy levels actually have small but finite widths, so the spectral line has some narrow line profile ϕ(ν) centered on ν=ν0 and conventionally normalized such that 0ϕ(ν)𝑑ν=1. A system in the lower energy state may absorb a photon of frequency νν0 and transition to the upper state. The rate (s-1) for this process is proportional to the profile-weighted mean radiation energy density

u¯0uν(ν)ϕ(ν)𝑑ν (7.39)

of the surrounding radiation field, so the Einstein absorption coefficient BLU is defined to make the product

BLUu¯ (7.40)

equal the average rate (s-1) at which photons are absorbed by a single atomic or molecular system in its lower energy state.

Einstein realized that there must be a third process in addition to spontaneous emission and absorption. It is stimulated emission, in which a photon of energy E=hν0 stimulates the system in the upper energy state to emit a second photon with the same energy and direction. The rate for this process is also proportional to u¯, so by analogy with Equation 7.40, the Einstein stimulated-emission coefficient BUL is defined to make the product

BULu¯ (7.41)

equal the average rate (s-1) of stimulated photon emission by a single quantum system in its upper energy state. Beware that some authors use I¯ instead of u¯ in Equation 7.41 to define a stimulated emission coefficient that is 4π/c times the BUL given by Equation 7.41 [98].

Stimulated emission is sometimes called negative absorption. Negative absorption is not familiar in everyday life because it is much weaker in room-temperature objects at visible wavelengths, where hν/(kT)1, but negative absorption competes effectively with ordinary absorption at radio wavelengths where hν/(kT)1.

Consider a macroscopic collection of many atoms or molecules in full thermodynamic equilibrium (TE) with the surrounding radiation field. TE is a stationary state. If there are (nU,nL) atoms or molecules per unit volume in the (upper, lower) energy states, then the average rate of photon creation by both spontaneous emission and stimulated emission must balance the average rate of photon destruction by absorption:

nUAUL+nUBULu¯=nLBLUu¯. (7.42)

In TE, the ratio of nU to nL is fixed by the Boltzmann equation

nUnL=gUgLexp[-(EU-EL)kT]=gUgLexp(-hν0kT), (7.43)

where gU and gL are the numbers of distinct physical states having energies EU and EL, respectively. The quantities gU and gL are called the statistical weights of those energy states. Examples of statistical weights include the following:

  1. 1.

    Hydrogen atoms have gn=2n2, where n=1,2,3, is the principal quantum number. The number 2n2 is the product of the 2 electron spin states and n2 orbital angular momentum states in the nth electronic energy level.

  2. 2.

    Rotating linear molecules (e.g., carbon monoxide, CO) have g=2J+1, where J=0,1,2, is the angular-momentum quantum number. For each J, there are 2J+1 possible values of the z-component of the angular momentum: Jz=-J,-(J-1),,-1,0,1,,(J-1),J.

  3. 3.

    Hydrogen atoms have two hyperfine energy levels whose difference yields the λ=21 cm (ν0=1420.406 MHz) Hi line: gU=3 and gL=1.

Solving Equation 7.42 for the profile-weighted mean energy density u¯ of blackbody radiation connects the properties of the quantum system (atom or molecule) to the radiation, just as Kirchhoff’s law (Equation 2.30) did for continuum radiation:

u¯=nUAULnLBLU-nUBUL=AUL(nL/nU)BLU-BUL. (7.44)

For full TE at temperature T, Equations 7.43 and 7.44 imply

u¯=AUL[gLgUexp(hν0kT)BLU-BUL]-1 (7.45)

for matter and Equation 2.92 implies

u¯=4πc0Bν(T)ϕ(ν)𝑑ν (7.46)

for radiation. Inserting the Planck radiation law (Equation 2.86) for Bν(T) near ν=ν0 gives

u¯4πc2hν03c2[exp(hν0kT)-1]-1. (7.47)

Equations 7.45 and 7.47 for u¯ must agree:

AUL[gLgUexp(hν0kT)BLU-BUL]-1=4πc2hν03c2[exp(hν0kT)-1]-1 (7.48)

for all temperatures T, so the equation

AULBUL[gLgUBLUBULexp(hν0kT)-1]-1=8πhν03c3[exp(hν0kT)-1]-1 (7.49)

implies both

gLgUBLUBUL=1 (7.50)

and

AULBUL=8πhν03c3. (7.51)

Equations 7.50 and 7.51 are called the equations of detailed balance. The two equations relate the three quantities AUL, BLU, and BUL, so all three can be computed if only one (e.g., the spontaneous emission coefficient AUL) is known. Equations 7.50 and 7.51 also prove that BUL cannot be zero; that is, stimulated emission must occur.

Note that these Equations 7.50 and 7.51 are valid for any microscopic physical system because they relate the coefficients AUL, BUL, and BLU, characteristic of individual atoms or molecules for which the macroscopic statistical concepts of TE or LTE are meaningless. Although TE was assumed for their derivation, the dependences on temperature T and frequency ν dropped out for a line at a single frequency ν0. Thus Equations 7.50 and 7.51 also apply to all macroscopic systems, whether or not they are in TE or even LTE. (Recall the derivation of Kirchhoff’s law jν(T)/κ(T)=Bν(T) (Equation 2.30), which also made use of full TE but also relates the emission and absorption coefficients of any matter in LTE at temperature T, independent of the actual ambient radiation field.)

7.3.2 Radiative Transfer and Detailed Balance

The two Equations 7.50 and 7.51 for detailed balance relate the three Einstein coefficients and allow the spectral-line radiative transfer problem to be solved in terms of the spontaneous emission coefficient AUL alone. The radiative transfer equation (2.27) is

dIνds=-κIν+jν, (7.52)

where Iν is the specific intensity, κ is the net fraction of photons absorbed (the difference between ordinary absorption and negative absorption) per unit length, and jν is the volume emission coefficient.

The ordinary opacity coefficient is the fraction of spectral brightness removed per unit length by absorption from the lower level to the upper level. At frequencies near the line center frequency ν0, the photon energy is hν0, the number of absorbers per unit volume is nL, the number of absorptions per unit area per unit time is nLBLU, the fraction of photons absorbed per unit area per unit length is nLBLU/c, and the photon energy loss per unit length at frequency ν is

dIνds=-κIν=-(hν0c)nLBLUϕ(ν)Iν. (7.53)

Stimulated emission is best treated as negative absorption because, like ordinary absorption and unlike spontaneous emission, its strength is proportional to Iν. The derivation of Equation 7.53 can be repeated to give

dIνds=-κIν=(hν0c)nUBULϕ(ν)Iν. (7.54)

Adding Equations 7.53 and 7.54 yields the net absorption coefficient

κ=(hν0c)(nLBLU-nUBUL)ϕ(ν). (7.55)

The spontaneous emission coefficient is the spectral brightness (power per unit frequency per steradian) added per unit volume by spontaneous transitions from the upper to lower energy levels. As above, the line photon energy is approximately hν0, the number density in the upper energy level is nU, and the photon emission rate per unit volume is nUAUL. These photons are emitted isotropically over 4πsr, so

dIνds=jν=(hν04π)nUAULϕ(ν). (7.56)

Inserting the net absorption coefficient (Equation 7.55) and the spontaneous emission coefficient (Equation 7.56) into Equation 7.52 specifies the full spectral-line equation of radiative transfer:

dIνds=-(hν0c)(nLBLU-nUBUL)ϕ(ν)Iν+(hν04π)nUAULϕ(ν). (7.57)

Equation 7.50 can be used to eliminate the stimulated emission coefficient BUL in Equation 7.55 and yield

κ=(hν0c)nLBLU(1-nUnLgLgU)ϕ(ν). (7.58)

The ratio of the emission coefficient to the net absorption coefficient is

jνκ=cnUAUL4πnLBLU(1-nUnLgLgU)-1. (7.59)

Equation 7.50 can be used to eliminate AUL from this ratio also:

jνκ=nU(8πhν03/c2)BUL4πnLBLU(1-nUnLgLgU)-1=2hν03c2BULBLU(nLnU-gLgU)-1. (7.60)

Finally, Equation 7.51 can be used to eliminate both BUL and BLU:

jνκ=2hν03c2(gUgLnLnU-1)-1. (7.61)

In LTE, Kirchhoff’s law independently implies

jνκ=Bν(T)=2hν3c2[exp(hνkT)-1]-1, (7.62)

so

gUgLnLnU=exp(hν0kT), (7.63)

recovering the Boltzmann equation (Equation 7.43) for LTE (not just for full TE):

nUnL=gUgLexp(-hν0kT). (7.64)

Equation 7.58 and the assumption of LTE allows the substitution of

BLU=gUgLBUL=gUgLAULc38πhν03 (7.65)

and

nUgLnLgU=exp(-hν0kT) (7.66)

to yield the net line opacity coefficient in LTE:

κ(ν)=c28πν02gUgLnLAUL[1-exp(-hν0kT)]ϕ(ν) (7.67)

in terms of the spontaneous emission rate AUL only; the stimulated emission coefficient BUL and absorption coefficient BLU have been eliminated.

The quantity

[1-exp(-hν0kT)] (7.68)

in Equation 7.67 is the sum of two terms, the first representing ordinary absorption and the second accounting for the negative absorption of stimulated emission. In the Rayleigh–Jeans limit hν0kT,

[1-exp(-hν0kT)]hν0kT1. (7.69)

Thus stimulated emission nearly cancels pure absorption and significantly reduces the net line opacity at radio frequencies. Because κT-1 and BνT, the product κBν is independent of temperature. The brightness of an optically thin (τ1) radio emission line is proportional to the column density of emitting gas but can be nearly independent of the gas temperature. Thus the Hi line flux (Jykms-1) of an optically thin galaxy is proportional to the total mass of neutral hydrogen in the galaxy but says nothing about its temperature.

7.4 Excitation Temperature

Even if a macroscopic two-level system is not in LTE, its excitation temperature Tx can be defined by

nUnLgUgLexp(-hν0kTx). (7.70)

The excitation temperature is not a real temperature; it only measures the ratio of nU to nL. In a two-level system, the excitation temperature is determined by a balance between radiative and collisional excitations and de-excitations. If collisions cause nLCLU excitations per unit volume per unit time from the lower level to the upper level and nUCUL de-excitations per unit volume per unit time from the upper level to the lower level, then Equation 7.42 becomes

nU(AUL+BULu¯+CUL)=nL(BLUu¯+CLU) (7.71)

and detailed balance requires

nLCLU=nUCUL. (7.72)

Thus

nUnL=BLUu¯+CLUAUL+BULu¯+CUL. (7.73)

Equations 7.47 and 7.51 can be combined to eliminate

BULu¯=c3AUL8πhν038πhν03c3[exp(hν0kTb)-1]-1=AUL[exp(hν0kTb)-1]-1, (7.74)

where Tb is the ambient radiation brightness temperature, in favor of AUL. Equation 7.50 eliminates

BLUu¯=BULu¯gUgL=AULgUgL[exp(hν0kTb)-1]-1. (7.75)

Finally, Equations 7.64 and 7.72 allow the replacement

CLU=nUnLCUL=gUgLexp(-hν0kTk)CUL, (7.76)

where Tk is the kinetic temperature of the gas. The numerator of Equation 7.73 becomes

AULgUgL[exp(hν0kTb)-1]-1+gUgLexp(-hν0kTk)CUL (7.77)

and the denominator is

AUL+AUL[exp(hν0kTb)-1]-1+CUL. (7.78)

Thus

nUgLnLgU =AUL+CULexp(-hν0kTk)[exp(hν0kTb)-1]AULexp(hν0kTb)+CUL[exp(hν0kTb)-1], (7.79)
exp(-hν0kTx) =exp(-hν0kTb)AUL+CULexp(-hν0kTk)[exp(hν0kTb)-1]AUL+CUL[1-exp(-hν0kTb)]. (7.80)

If the spontaneous emission rate is much larger than the collision rate, Equation 7.80 yields TxTb; if the collision rate is much higher than the spontaneous emission rate, TxTk. For any AUL and CUL, Tx lies between Tk and Tb.

7.5 Masers

If the upper energy level is overpopulated, that is,

nUnL>gUgL, (7.81)

then Tx is actually negative,

[1-exp(-hν0kTx)] (7.82)

is negative, and Equation 7.67 gives a negative net opacity coefficient κ. Negative net opacity implies brightness gain instead of loss; the intensity of a background source at frequency ν0 will be amplified. At radio wavelengths this phenomenon is called maser (an acronym for microwave amplification by stimulated emission of radiation) amplification. Astronomical masers are common at radio frequencies because hνkT and hence nU/nLgU/gL even in TE. These sources can have line brightness temperatures as high as 1015 K, which is much higher than the kinetic temperature of the masing gas. For a clear presentation covering the basics of astronomical masers, written by Reid & Moran, see Verschuur and Kellermann [110, Chapter 6].

Our model for an astronomical maser starts with the radiative transfer equation for a two-level system. Assume for simplicity that gU=gL so that Equation 7.50 implies BLU=BULB and Equation 7.51 implies AUL=8πhν03B/c3A. Then Equation 7.57 simplifies to

dIνds=-(hν0c)(nL-nU)Bϕ(ν)Iν+(hν04π)nUAϕ(ν). (7.83)

Next assume that the line profile ϕ(ν) is Gaussian with FWHM Δν (Figure 7.4) so that Equation 7.37 applies and make the numerical approximation

ϕ(ν0)=(ln 2π)1/22Δν=0.939Δν1Δν. (7.84)

Then at the line center frequency ν0,

dIνds=-hν0(nL-nU)BIνcΔν+hν0nUA4πΔν. (7.85)

The maser optical depth

τ=κ𝑑s=dIνIν=hν0BcΔν(nU-nL)𝑑s (7.86)

is called the maser gain over the path of integration, and the maser amplifies the intensity of background radiation by the factor exp(|τ|). In a laboratory maser, radiation is trapped in a high-Q resonant cavity to create effective path lengths up to 109 times the cavity length. In an astronomical maser there is no cavity, so the radiation makes only a single pass and the physical path must be much longer (s>1013cm) for significant gain to occur.

Maser emission quickly depopulates the upper energy level, so masers have to be “pumped” to emit continuously. Typically one or more higher energy levels absorb radiation from a pump source (e.g., infrared continuum from a star or an AGN), and radiative decays preferentially repopulate the upper energy level. This radiative pumping process produces no more than one maser photon per pump photon, so the pump energy required is proportional to the frequency ν=E/h of the pump photon. If the maser photon emission rate is limited by the pump luminosity, the maser is described as being saturated; if the pump power is more than adequate, the maser is unsaturated.

Strong, compact maser sources are powerful tools for high-resolution imaging and precision astrometry. They are being used to measure accurate trigonometric distances to individual stars in our Galaxy, the size and structure of our Galaxy, black-hole masses in AGNs, and distances to galaxies up to 150Mpc [89].

The most spectacular example is the circumnuclear 22 GHz H2O megamaser disk surrounding the Seyfert nucleus of NGC 4258 [74] and illustrated in Figure 7.6. The observed maser spectrum has hundreds of narrow lines in three groups clustered around the systemic recession velocity v450kms-1 of NGC 4258 and at velocities further redshifted and blueshifted by the vrot900kms-1 rotation velocities in the disk.

Figure 7.6: A simple ring model illustrating the geometry and kinematics of the edge-on 22 GHz water-maser disk orbiting the nucleus of NGC 4258 [74].

The nearly edge-on maser disk was imaged with high spectral and angular resolution by the VLBA, and its rotation curve is shown in Figure 7.7. The systemic lines are concentrated within a region <1mas0.03pc wide and centered on the line of sight to the supermassive black hole (SMBH) at the center of the disk. Photons from the central radio continuum source are amplified and beamed in our direction, so systemic maser clouds are visible only when they are within a small angle |θ|<0.07rad from the line of sight. For a constant vrot, their vz=(GM/R3)1/2b yields the sloped straight line in Figure 7.7. This slope corresponds to the gravitational acceleration az=GM/R2 of the SMBH, and long-term monitoring the velocities of individual systemic-maser lines independently yields a8kms-1yr-1. The redshifted and blueshifted maser clouds are visible only near the tangent points of the disk near θ=±π/2rad, which have the longest gain paths at nearly constant velocities vz=(GM/R)1/2 that follow the perfectly Keplerian curves in Figure 7.7. Combining the acceleration, velocity, and angular-size measurements yields a unique solution for both the SMBH mass M3.8×107M and the distance D=7.2±0.4Mpc to NGC 4258.

Figure 7.7: The NGC 4258 rotation curve is perfectly Keplerian, indicating that the mass interior to the maser disk is dominated by a single compact object whose density is too high (>5×1012Mpc-3) for it to be a compact star cluster; it must be an SMBH [74].

7.6 Recombination Line Sources

Astronomical sources of radio recombination lines are often in local thermodynamic equilibrium (LTE). The spontaneous emission rate from atomic physics and the laws of radiative transfer for spectral lines can be combined to model sources in LTE. However, LTE does not apply to all recombination lines, so departures from LTE must be recognized and treated differently.

7.6.1 Radiative Transfer in LTE

Equation 7.67 gives the absorption coefficient at the center frequency ν0 of the nn+1 electronic transition of hydrogen in an Hii region in local thermodynamic equilibrium (LTE) at electron temperature Te:

κ(ν)=c28πν02gn+1gnnnAn+1,n[1-exp(-hν0kTe)]ϕ(ν), (7.87)

where

ν0 =νn,n+12Rcn3=4π2mee4h3n3, (7.88)
gn =2n2, (7.89)

and nn is the number density of atoms in the nth electronic energy level. At radio frequencies, it can be assumed that n1, hν0kTe, and gn+1/gn1. Equation 7.23 gives the spontaneous emission rate

An+1,n64π6mee103c3h6n5 (7.90)

and Equation 7.37 parameterizes the normalized line profile

ϕ(ν0)(ln2π)1/22Δν. (7.91)

The number density nn of atoms in the nth electronic energy level is given by the Saha equation, a generalization of the Boltzmann equation (for a derivation of the Saha equation, see Rybicki and Lightman [98, Equation 9.47]):

nn=n2(h22πmekTe)3/2npneexp(χnkTe) (7.92)

where χn<0 is the ionization potential of the nth energy level. For large n, |χn|kTe and the exponential factor exp[χn/(kTe)]1 can be ignored. Combining the results from Equations 7.87 through 7.92 yields the opacity coefficient at the line center frequency ν0:

κ(ν0)c2n28πν02(h22πmekTe)3/2ne2(64π6mee103c3h6n5)hν0kTe[(ln2π)1/22Δν]. (7.93)

Some algebra reduces this to

κ(ν0)(ne2Te5/2Δν)(4πe6h3me3/2k5/2c)(ln22)1/2. (7.94)

Notice that the electronic energy level n has dropped out; Equation 7.94 is valid for all radio recombination lines with n1. The optical depth τL=κ𝑑s at the line center frequency ν0 can be expressed in terms of the emission measure defined by Equation 4.57:

EMpccm-6los(necm-3)2d(spc). (7.95)

In astronomically convenient units the line center opacity is

τL1.92×103(TeK)-5/2(EMpccm-6)(ΔνkHz)-1. (7.96)
Figure 7.8: A temperature-distribution model for the Orion Nebula Hii region based on the line-to-continuum ratios of hydrogen recombination lines [67].
Figure 7.9: The spiral pattern of our Galaxy in plan view, as traced by Hα (circles) and radio recombination lines (squares) [40].
Figure 7.10: Recombination-line observations show that the electron temperatures Te of Hii regions increase with distance from the Galactic center at the rate of 287±46Kkpc-1, probably because metallicity decreases [86].

Because τL1 in all known Hii regions, the brightness temperature contributed by a recombination emission line at its center frequency ν0 is

TLTeτL1.92×103(TeK)-3/2(EMpccm-6)(ΔνkHz)-1. (7.97)

At frequencies high enough that the free–free continuum is also optically thin, the peak line-to-continuum ratio (which occurs at frequency ν0) in LTE is

TLTC7.0×103(Δvkms-1)-1(νGHz)1.1(TeK)-1.15[1+N(He+)N(H+)]-1, (7.98)

where Δv is the line FWHM expressed as a velocity and the typical He+/H+ ion ratio is N(He+)/N(H+)0.08. The term in square brackets is necessary because He+ contributes to the free–free continuum emission but not to the hydrogen recombination line. The line-to-continuum ratio yields an estimate of the electron temperature Te that is independent of the emission measure so long as the frequency is high enough that the continuum optical depth is small.

7.6.2 Astronomical Applications

Recombination lines can be used to find the electron temperatures of Hii regions in LTE. Solving Equation 7.98 explicitly for Te gives the useful formula

(TeK)[7.0×103(νGHz)1.1 1.08-1(Δvkms-1)-1(TCTL)]0.87. (7.99)

By mapping the recombination line-to-continuum ratios TL/TC in a number of Hnα transitions, Lockman and Brown determined the temperature distribution in the Orion Nebula (Figure 7.8), a nearby Hii region.

Differences between the rest and observed frequencies of radio recombination lines are attributed to Doppler shifts from nonzero radial velocities. With a simple rotational model for the disk of our Galaxy, astronomers can convert radial velocities to distances, albeit with some ambiguities, and map the approximate spatial distribution of Hii regions in our Galaxy (Figure 7.9). They roughly outline the major spiral arms.

A plot showing the observed electron temperatures of Galactic Hii regions (Figure 7.10) reveals that temperature increases with distance from the Galactic center.

The explanation for this trend is the observed decrease in metallicity (relative abundance of elements heavier than helium) with galactocentric distance. Power radiated by emission lines of “metals” is the principal cause of Hii region cooling.

Radio recombination line strengths are much less affected by dust extinction than optical lines (e.g., the Hα and Hβ lines) are, so radio recombination lines are useful quantitative indicators of the ionization rates and hence star-formation rates in dusty starburst galaxies such as M82 (Figure 7.11).

Figure 7.11: M82 imaged in H92α (contours) and 8.3 GHz continuum (gray scale) [94].

7.7 Molecular Line Spectra

7.7.1 Molecular Line Frequencies

A molecule is called polar if its permanent electric dipole moment (Equation 7.120) is not zero. Symmetric molecules (e.g., the diatomic hydrogen molecule H2) have no permanent electric dipole moment, but most asymmetric molecules (e.g., the carbon monoxide molecule CO) do have asymmetric charge distributions and are polar. The electric dipole moments of polar molecules rotating with constant angular velocity ω appear to vary sinusoidally with that angular frequency, so polar molecules radiate at their rotation frequencies. The intensity of this radiation can be derived from the Larmor formula expressed in terms of dipole moments instead of charges and charge separations.

The permitted rotation rates and resulting line frequencies are determined by the quantization of angular momentum. The quantization rule for the permitted electron orbital radii

an=nmev

in the Bohr atom quantizes the orbital angular momentum L=meanv in multiples of h/(2π):

L=n. (7.100)

The rule that angular momentum is an integer multiple of is universal and applies to the angular momentum of a rotating molecule as well.

Consider a rigid diatomic molecule (Figure 7.12) whose two atoms have masses mA and mB and whose centers are separated by the equilibrium distance re. The individual atomic distances rA and rB from the center of mass must obey

re=rA+rB  and  rAmA=rBmB. (7.101)
Figure 7.12: A diatomic molecule rotating about its center of mass.

In the inertial center-of-mass frame,

L=Iω, (7.102)

where I is the moment of inertia and ω is the angular velocity of the rotation. Nearly all of the mass is in the two compact (much smaller than re) nuclei, so I=(mArA2+mBrB2) and L=(mArA2+mBrB2)ω. It is convenient to rewrite this as

L=(mAmBmA+mB)re2ω (7.103)

or

L=mre2ω, (7.104)

where

m(mAmBmA+mB) (7.105)

is the reduced mass of the molecule.

The rotational kinetic energy associated with this angular momentum is

Erot=Iω22=L22I. (7.106)

The quantization of angular momentum to integer multiples of implies that rotational energy is also quantized. The corresponding energy eigenvalues of the Schrödinger equation are

Erot=(22I)J(J+1),J=0,1,2,. (7.107)

Note the inverse relation between permitted rotational energies and the moment of inertia I. If the upper-level rotational energy is much higher than kT, few molecules will be collisionally excited to that level and the line emission from molecules in that level will be very weak. For example, the minimum rotational energy of the small and light H2 molecule is equivalent to a temperature T=Emin/k500 K, which is much higher than the actual temperature of most interstellar H2. Only relatively massive molecules are likely to be detectable radio emitters in very cold (tens of K) molecular clouds.

Quantization of rotational energy implies that changes in rotational energy are quantized. The energy change of permitted transitions is further restricted by the quantum-mechanical selection rule

ΔJ=±1. (7.108)

Going from J to J-1 releases energy

ΔErot=[J(J+1)-(J-1)J]22I=2JI. (7.109)

The frequency of the photon emitted during this rotational transition is

ν=ΔEroth=J2πI,J=1,2,, (7.110)

where J is the angular-momentum quantum number corresponding to the upper energy level. In terms of the molecular reduced mass m and equilibrium nuclear separation re,

ν=hJ4π2mre2,J=1,2,. (7.111)

Thus a plot of the radio spectrum of a particular molecular species in an interstellar cloud will look like a ladder (Figure 7.13) whose steps are all harmonics of the fundamental frequency that is determined solely by the moment of inertia I=mre2 of that species. The relative intensities of lines in the ladder depend on the temperature of the cloud. Since νm-1re-2, the lowest frequency of line emission depends on the mass and size of the molecule. Large, heavy molecules in cold clouds may be seen at centimeter wavelengths, but smaller and lighter molecules emit only at millimeter wavelengths.

Figure 7.13: The rotational spectrum of 12C16O looks like a ladder whose rungs indicate the J levels and line frequencies. The 12C16O molecule has a relatively small moment of inertia, so the lowest rung of this ladder is at ν115 GHz (λ2.6 mm); it has no cm-wavelength lines.

For example, the laboratory spectrum of the 12C16O carbon-monoxide molecule shows that the fundamental J=10 transition emits a photon at ν=115.27120 GHz. (See the online spectral-line catalog called Splatalogue11 1 http://www.splatalogue.net/. for accurate frequencies of radio spectral lines.) The distance re between the C and O nuclei can be estimated from

re=12π(hJmν)1/2, (7.112)

where the reduced mass is

m=mCmOmC+mOmH(121612+16)1.67×10-24g6.861.15×10-23g.

Thus the equilibrium distance between the C and O nuclei is

re=12π(6.63×10-27ergs1115.27120×109Hz1.15×10-23g)1/21.13×10-8cm.

The centrifugal forces acting on the nuclei increase as the molecule spins more rapidly, so a nonrigid bond will stretch and re will increase slightly with J. Spectral lines emitted by more rapidly rotating 12C16O molecules will have frequencies slightly lower than the harmonics 2ν10,3ν10, of the J=10 line: 2ν10=230.542416 GHz and the actual J=21 frequency is ν21=230.538000 GHz. Chemists use these line frequencies to determine re, and the difference between 2ν10 and ν21 is a measure of the stiffness of the carbon–oxygen chemical bond. Since the actual frequency is only slightly less than the harmonic frequency, the stiffness of the “spring” connecting the atoms is quite high. Consequently the fundamental vibrational frequency of the CO molecule is much higher than the fundamental rotational frequency, and CO emits vibrational lines at mid-infrared wavelengths λ5μm.

Equation 7.111 can be used to calculate frequencies for molecules containing rare isotopes (e.g., 13C16O) that might be more difficult to measure in the lab:

m(13C16O)m(12C16O)=1316/(13+16)1216/(12+16)1.0460, (7.113)

so we expect

ν10(13C16O)ν10(12C16O)[m(13C16O)m(12C16O)]-1, (7.114)
ν10(13C16O)115.27120/1.0460110.20GHz. (7.115)

The actual 13C16O J=10 frequency is 110.201354 GHz.

Polar diatomic molecules emit a harmonic series of radio spectral lines at millimeter wavelengths. Bigger and heavier linear polyatomic molecules have ladders of lines starting at somewhat lower frequencies. Nonlinear molecules such as the symmetric-top ammonia (NH3) with two distinct rotational axes have more complex spectra consisting of many parallel ladders (Figure 7.14).

Figure 7.14: Energy levels of ammonia (NH3) in the lowest vibrational state [115]. On the abscissa, K is the quantum number corresponding to the z-component of the angular momentum. Transitions between the two spin states of the nitrogen atom cause the line splitting shown and yield emission at frequencies near 24 GHz. NH3 is a very useful thermometer for molecular clouds [52].

7.7.2 Molecular Excitation

Molecules are excited into Erot>0 states by ambient radiation and by collisions in a dense gas. The minimum gas temperature Tmin needed for significant collisional excitation is

TminErotk. (7.116)

From Equations 7.107 and 7.111,

Erot=J(J+1)22I  and  ν=hJ4π2I, (7.117)

so

TminJ(J+1)h224π2Ik=hJ4π2Ih(J+1)2k=EUk, (7.118)

where EU is the rotational energy of the upper energy level for the transition. Thus a minimum gas kinetic temperature

Tminνh(J+1)2k (7.119)

is required to excite the JJ-1 transition at frequency ν. For example, the minimum gas temperature needed for significant excitation of the 12C16O J=21 line at ν230.5 GHz (Figure 7.15) is

Tminνh(J+1)2k230.5×109Hz6.63×10-27ergs(2+1)21.38×10-16ergK-116.6K.
Figure 7.15: The upper-level energies EU for 12C16O JJ-1 transitions are proportional to J(J+1). The corresponding minimum temperatures Tmin=EU/k required for collisions to excite the molecules are also proportional to J(J+1), so high-J lines will be weak in cold molecular clouds.

The Tmin=EU/k values for many molecular lines may be found in the online spectral-line catalog Splatalogue. If Tmin2.7 K, then radiative excitation by the cosmic microwave background is ineffective.

7.7.3 Molecular Line Strengths

Larmor’s formula for a time-varying dipole can be applied to estimate the average power radiated by a rotating polar molecule. The electric dipole moment p of any charge distribution ρ(x) is defined as the integral

pxρ(v)𝑑v, (7.120)

over the volume v containing the charges. In the case of two point charges +q and -q with separation re,

|p|=qre. (7.121)

When a molecule rotates with angular velocity ω, the projection of the dipole moment perpendicular to the line of sight varies with time as qreexp(-iωt). Equation 7.101 states that

rAmA=rBmB, (7.122)

so

v˙A=r¨A=ω2rA  and  v˙B=r¨B=ω2rB. (7.123)

Equation 2.133 from the derivation of Larmor’s formula states that each charge contributes

E=qv˙sinθrc2 (7.124)

to the radiated electric field at distance r from the source. The fields from both charges add in phase because reλ, so the total radiated field E is

E=q(ω2rA+ω2rB)sinθrc2exp(-iωt). (7.125)

Thus the instantaneous power emitted is

P=2q23c3ω4|reexp(-iωt)|2 (7.126)

and the time-averaged power is

P=2q23c3(2πν)4re22=64π43c3ν4(qre2)2. (7.127)

This can be expressed as

P=64π43c3|μ|2ν4, (7.128)

where

|μ|2(qre2)2 (7.129)

defines the mean electric dipole moment. For the radiative transition between upper and lower energy levels U and L, the spontaneous emission coefficient is

AUL=Phν, (7.130)
AUL=64π43hc3|μUL|2ν3. (7.131)

The value of μUL for the JJ-1 transition of a linear rotating molecule with dipole moment μ is

|μJJ-1|2=μ2J2J+1. (7.132)

(This equation reflects the complex angular wave functions involved, and we won’t derive it here.)

Dipole moments are often expressed in debye units defined by 1debye=1D10-18statcoulcm=10-10statcoul×10-8cm, where 10-10statcoul0.2e is a typical charge imbalance for a polar molecule and 10-8cm is a typical value for the interatomic separation re. For example, the CO molecule has dipole moment μ0.11×10-18statcoulcm=0.11D.

Combining Equations 7.131 and 7.132 yields, in convenient units,

(AJJ-1s-1)1.165×10-11|μD|2(J2J+1)(νGHz)3. (7.133)

For example, the spontaneous emission coefficient A10 for the CO J=10 line at ν115 GHz is

A101.165×10-110.112(13)11537.1×10-8s-1. (7.134)

This is close to the more accurate Splatalogue value, A107.202×10-8 s-1.

The typical time A10-1107 s for a CO molecule to emit a photon spontaneously may be longer than the average time between molecular collisions in an interstellar molecular cloud, so CO can approach LTE with the excitation temperature of the J=10 line being nearly equal to the kinetic temperature T of the molecular cloud. For any molecular transition, there is a critical density defined by

n*AULσv (7.135)

at which the radiating molecule suffers collisions at the rate n*σvn(H2)σv equal to the spontaneous emission rate AUL. Typical collision cross sections are σ10-15 cm2, and the average velocity of the abundant H2 molecules is v(3kT/m)1/25×104 cm s-1 if T20 K. Thus the critical density of the CO J=10 transition is

n*AULσv7×10-8s-110-15cm25×104cms-11.4×103cm-3.

Many Galactic molecular clouds have higher densities than this, so Galactic CO J=10 emission is strong and widespread. Also, photons from a particular transition may be repeatedly absorbed and reemitted within the molecular cloud. Such line trapping lowers the effective emission rate AUL and reduces the effective value of n* needed for LTE.

Whether or not a molecular cloud is in LTE, Equation 7.67 with the temperature T replaced by excitation temperature Tx can be used to calculate the line opacity coefficient:

κ(ν)=c28πν02gUgLnLAUL[1-exp(-hν0kTx)]ϕ(ν). (7.136)

At the line center frequency ν0 , Equation 7.37 gives

ϕ(ν0)=(ln 2π)1/22Δν=(ln 2π)1/22cν0Δv,

where Δν is the FWHM line width expressed as a frequency and Δv is the line width in velocity units (e.g., km s-1). The line-center optical depth is τ0=κ0𝑑s along the line of sight, and the column density is NL=nL𝑑s along the line of sight, so

τ0=(ln 2)1/24π3/2c3ν03gUgLAULΔνNL[1-exp(hνkTx)]. (7.137)

In the Rayleigh–Jeans approximation hνkTx, the brightness-temperature difference between the line center and the nearby off-line continuum is

ΔTb=(Tx-Tc)[1-exp(-τ0)], (7.138)

where Tc is the brightness temperature of any background continuum emission (e.g., Tc2.73K from the CMB). In the limit of low line optical depth, the line brightness

ΔTb(Tx-Tc)τ0=(Tx-TcTx)NL(ln 2)1/24π3/2hc3kν02gUgLAULΔv (7.139)

is proportional to the column density NL. The spontaneous emission coefficient AUL is proportional to ν3 (Equation 7.131), so spectral lines tend to have higher brightness temperatures and become more prominent at higher radio frequencies (Figure 7.16).

The hydrogen molecule H2 is by far the most abundant molecule in interstellar space. Unfortunately, it is symmetric so its dipole moment is zero. Observable but comparatively rare polar molecules such as CO are only tracers, and total column densities of molecular gas must be estimated indirectly by the use of a fairly uncertain CO to H2 conversion factor XCO relating H2 column density in cm-2 to CO velocity-integrated line brightness in Kkms-1. The best current value for our Galaxy is

XCO=(2±0.6)×1020cm-2(Kkms-1)-1. (7.140)

XCO is probably higher in galaxies with low metallicity and lower in starburst galaxies [14]. Figure 8.12 shows the distribution of CO emission tracing molecular gas and obscured star formation in the interacting starburst galaxies NGC 4038/9.

Isotopologues are molecules that differ only in isotopic composition; that is, only in the numbers of neutrons in their component atoms. Lines of the most abundant isotopologue of carbon monoxide, C1612O, are often optically thick, so the C1612O line brightness temperature approaches the molecular gas kinetic temperature and is nearly independent of column density. Lines of rarer isotopologues such as C1613O or C1812O are usually optically thin and can be used to measure the column densities needed to estimate the total mass of molecular gas in a source. Intensity ratios of optically thin lines from different J levels can be used to measure excitation temperature, which is close to the kinetic temperature in LTE.

Transitions with high emission coefficients (e.g., the HCN (hydrogen cyanide) J=10 line at ν88.63 GHz has AUL2.0×10-5 s-1) are collisionally excited only at very high densities (n*105 cm-3 for HCN J=10). They are valuable for highlighting only the very dense gas directly associated with the formation of individual stars.

The discovery of ammonia (NH3) in the direction of the Galactic center by Cheung et al. [22] immediately led to the realization that the interstellar medium must contain regions much denser than previously expected because the critical density needed to excite the NH3 line is n*103 cm-3.

Figure 7.16: This λ1.3 mm spectrum of the molecular cloud SgrB2(N) near the Galactic center is completely dominated by molecular lines from known and unknown (U) species [119]. More than 140 different molecules containing up to 13 atoms (HC11N) have been identified in space.

7.8 The Hi 21-cm Line

Hydrogen is the most abundant element in the interstellar medium (ISM), but the symmetric H2 molecule has no permanent dipole moment and does not emit detectable spectral lines at radio frequencies. Neutral hydrogen (Hi) atoms are abundant and ubiquitous in low-density regions of the ISM. They are detectable in the λ21 cm (ν10=1420.405751 MHz) hyperfine line. Two energy levels result from the magnetic interaction between the quantized electron and proton spins. When the relative spins change from parallel to antiparallel, a photon is emitted.

The Hi line center frequency is

ν10=83gI(memp)α2(RMc)1420.405751MHz, (7.141)

where gI5.58569 is the nuclear g-factor for a proton, αe2/(c)1/137.036 is the dimensionless fine-structure constant, and RMc is the hydrogen Rydberg frequency (Equation 7.12).

By analogy with the emission coefficient of radiation by an electric dipole

AUL64π43hc3νUL3|μUL|2, (7.142)

the emission coefficient of this magnetic dipole is

AUL64π43hc3νUL3|μB|2, (7.143)

where μB is the mean magnetic dipole moment for Hi in the ground electronic state (n=1). The magnitude |μB| is called the Bohr magneton, and its value is

|μB|=e2mec9.27401×10-21erggauss-1. (7.144)

Thus the emission coefficient of the 21-cm line is only

A1064π4(1.42×109Hz)3(9.27×10-21erggauss-1)236.63×10-27ergs(3×1010cms-1)3, (7.145)
A102.85×10-15s-1, (7.146)

so the radiative half-life of this transition is very long:

τ1/2=A10-13.5×1014s11millionyears. (7.147)

Such a low emission coefficient implies an extremely low critical density (Equation 7.135) n*1cm-3, so collisions can easily maintain this transition in LTE, even in the outermost regions of a normal spiral galaxy and in tidal tails of interacting galaxies.

Regardless of whether the Hi is in LTE or not, we can define the Hi spin temperature Ts (the Hi analog of the molecular excitation temperature Tx defined by Equation 7.70) by

n1n0g1g0exp(-hν10kTs), (7.148)

where the statistical weights of the upper and lower spin states are g1=3 and g0=1, respectively. Note that

hν10kTs6.63×10-27ergs1.42×109Hz1.38×10-16ergK-1150K5×10-41 (7.149)

is very small for gas in LTE at TTs150 K, so in the ISM

n1n0g1g0=3  and  nH=n0+n14n0. (7.150)

Inserting these weights into Equation 7.67 gives the opacity coefficient of the λ=21cm line:

κ(ν) =c28πν102g1g0n0A10[1-exp(-hν10kTs)]ϕ(ν) (7.151)
c28πν1023nH4A10(hν10kTs)ϕ(ν), (7.152)
κ(ν)3c232πA10nHν10hkTsϕ(ν), (7.153)

where nH is the number of neutral hydrogen atoms per cm3. The neutral hydrogen column density along any line of sight is defined as

ηHlosnH(s)ds. (7.154)

The total opacity τ of isothermal Hi is proportional to the column density. If τ1, then the integrated Hi emission-line brightness Tb is proportional to the column density of Hi and is independent of the spin temperature Ts because TbTsτ and τTs-1 in the radio limit hν10/(kTs)1. Thus ηH can be determined directly from the integrated line brightness when τ1. In astronomically convenient units it can be written as

(ηHcm-2)1.82×1018[Tb(v)K]d(vkms-1), (7.155)

where Tb is the observed 21-cm-line brightness temperature at radial velocity v and the velocity integration extends over the entire 21-cm-line profile. Note that absorption by Hi in front of a continuum source with continuum brightness temperature >Ts, on the other hand, is weighted in favor of colder gas (Figure 7.17).

Figure 7.17: The Hi absorption and emission spectra toward the source 1714-397 [35].

The equilibrium temperature of cool interstellar Hi is determined by the balance of heating and cooling. The primary heat sources are cosmic rays and ionizing photons from hot stars. The main coolant in the cool atomic ISM is radiation from the fine-structure line of singly ionized carbon, Cii, at λ=157.7μm. This line is strong only when the temperature is at least

kThν=hcλ, (7.156)

so the cooling rate increases exponentially above

Thckλ6.63×10-27ergs3×1010cms-11.38×10-16ergK-1157.7×10-4cm91K. (7.157)

The actual kinetic temperature of Hi in our Galaxy can be estimated from the Hi line brightness temperatures in directions where the line is optically thick (τ1) and the brightness temperature approaches the excitation temperature, which is close to the kinetic temperature in LTE. Many lines of sight near the Galactic plane have brightness temperatures as high as 100–150 K, values consistent with the temperature-dependent cooling rate.

7.8.1 Galactic Hi

Neutral hydrogen gas in the disk of our Galaxy moves in nearly circular orbits around the Galactic center. Radial velocities vr measured from the Doppler shifts of Hi λ=21 cm emission lines encode information about the kinematic distances d of Hi clouds, and the spectra of Hi absorption in front of continuum sources can be used to constrain their distances also. Hi is optically thin except in a few regions near the Galactic plane, so the distribution of hydrogen maps out the large-scale structure of the whole Galaxy, most of which is hidden by dust at visible wavelengths.

Figure 7.18: In the simplest realistic model for the Galactic disk, the Sun and all Hi clouds are in circular orbits about the Galactic center, and the angular orbital velocity ω is a monotonically decreasing function of the orbital radius R. The distance of the Sun from the Galactic center is R=8.0±0.5 kpc, and the Sun’s orbital speed is ωR220 km s-1. The angle l defines the Galactic longitude. For |l|<π/2, two Hi clouds (1 and 2) can have the same radial velocity but be at different distances from the Sun.

Figure 7.18 shows a plan view of the Galactic disk. The Sun () lies in the disk and moves in a circular orbit around the Galactic center. The distance to the Galactic center R=8.0±0.5 kpc and the Sun’s orbital speed ωR220 km s-1 have been measured by a variety of means [90]. All Hi clouds at galactocentric distance R are assumed to be in circular orbits with angular velocity ω(R), where ω(R) is a monotonically decreasing function of R. For cloud 1 at galactocentric azimuth θ on the line of sight at Galactic longitude l, the observed radial velocity vr relative to the Sun is given by

vr=ωRcos[π/2-(l+θ)]-ωRcos(π/2-l). (7.158)

Using the trigonometric identities cos[π/2-(l+θ)]=sin(l+θ) and sin(l+θ)=sinθcosl+cosθsinl we obtain

vr =ωR(sinθcosl+cosθsinl)-ωRsinl (7.159)
=R(ω-ω)sinl. (7.160)

To apply this equation, we need to determine the rotation curve Rω(R). The maximum radial velocity on the line of sight at longitude l is called the “terminal velocity” vT. Since ω decreases with R, this velocity occurs at the minimum R=Rmin=Rsinl where the orbit is tangent to the line of sight:

vT=R[ω(Rmin)-ω]sinl. (7.161)

We can determine the rotation curve from measurements of vT spanning a wide range of l and thus of Rmin.

Example. At Galactic longitude l=30, the terminal velocity is observed to be vT130 km s-1. What is Rmin and the orbital speed Rminω(Rmin)? Rmin =Rsinl=8.0kpc0.5=4.0kpc, vT =R[ω(Rmin)-ω]sinl =Rminω(Rmin)-Rωsinl, Rminω(Rmin) =vT+Rωsinl =130kms-1+220kms-10.5=240kms-1. Beware that, for |l|<π/2, there is a distance ambiguity: clouds 1 and 2 have the same radial velocity but different distances d. There is no distance ambiguity for |l|>π/2.

7.8.2 Hi in External Galaxies

The 1420 MHz Hi line is an extremely useful tool for studying gas in the ISM of external galaxies and tracing the large-scale distribution of galaxies in the universe because Hi is detectable in most spiral galaxies and in some elliptical galaxies.

Because λ=21 cm is such a long wavelength, many galaxies are unresolved by single-dish radio telescopes. For example, the half-power beamwidth of the 100-m GBT is about 9 arcmin at λ=21 cm. Thus a single pointing is sufficient to obtain a spectral line representing all of the Hi in any but the nearest galaxies.

The observed center frequency of the Hi line can be used to measure the radial velocity vr of a galaxy. The radial velocity of a galaxy is the sum of the recession velocity caused by the uniform Hubble expansion of the universe and the “peculiar” velocity of the galaxy. The radial component of the peculiar velocity reflects motions caused by gravitational interactions with nearby galaxies and is typically 200 km s-1 in magnitude. The Hubble velocity is proportional to distance from the Earth, and the Hubble constant of proportionality has been measured as H0=67.8±0.9kms-1Mpc-1 [83]. If the radial velocity is significantly larger than the radial component of the peculiar velocity, the observed Hi frequency can be used to estimate the distance dvr/H0 to a galaxy.

Beware that astronomers still use inconsistent radial velocity conventions that were established when most observed radial velocities were much less than the speed of light. The approximation

vrcνe-νoνe  (vrc), (7.162)

where νe is the line frequency in the source frame and νo is the observed frequency, was used to define the radio velocity for any vr(radio) as

vr(radio)c(νe-νoνe) (7.163)

because radio astronomers measure frequencies, not wavelengths. Optical astronomers measure wavelengths, not frequencies, so the nonrelativistic approximation

vrcλo-λeλe  (vrc) (7.164)

is the basis for the optical velocity defined for any vr(optical) by

vr(optical)c(λo-λeλe)=cz, (7.165)

where z is the redshift defined by Equation 2.127. The optical and radio velocity conventions are not exactly the same, and neither agrees with the relativistically correct radial velocity calculated from Equation 5.142. Occasionally an observer confuses velocity conventions, fails to center the observing passband on the correct frequency, and ends up with only part of the Hi spectrum of a galaxy. Outside the local universe (z1) the concept of distance itself becomes more complicated. To calculate distances to astronomical objects with larger redshifts, see Hogg [53, “Distance measures in cosmology”].

Figure 7.19: This integrated Hi spectrum of UGC 11707 was obtained by Haynes et al. [47] with the 140-foot telescope (beamwidth 20 arcmin) and shows the typical two-horned profile of a spiral galaxy. The velocity axis is clearly labeled as showing the “optical” velocity.

For example, the Hi emission-line profile (Figure 7.19) of the galaxy UGC 11707 can be used to estimate its distance dvr/H0. The observed line center frequency is νo1416.2 MHz, so the “radio” and “optical” velocities are

vr(radio) c(1-νoνe)3×105kms-1(1-1416.2MHz1420.4MHz)890kms-1,
vr(optical) c(νeνo-1)3×105kms-1(1420.4MHz1416.2MHz-1)889kms-1.

Using the “optical” velocity gives

dvr(optical)H0889kms-167.8kms-1Mpc-113.1Mpc.

If the Hi emission from a galaxy is optically thin, then the integrated line flux is proportional to the mass of Hi in the galaxy, independent of the unknown Hi temperature. It is a straightforward exercise to derive from Equation 7.155 the relation

(MHM)2.36×105(dMpc)2[S(v)Jy](dvkms-1) (7.166)

for the total Hi mass MH of a galaxy. The integral S(v)𝑑v over the line is called the line flux and is usually expressed in units of Jy km s-1. For example, to estimate the Hi mass of UGC 11707, assume τ1. The single-dish Hi line profile of UGC 11707 (Figure 7.19) indicates a line flux

S(v)𝑑v0.35Jy200kms-170Jykms-1,

so

(MHM)2.36×105(12.4)2702.5×109.

Small statistical corrections for nonzero τ can be made from knowledge about the expected opacity as a function of disk inclination, galaxy mass, morphological type, etc.

A well-resolved Hi image of a galaxy yields the total mass M(r) enclosed within radius r of the center if the gas orbits in circular orbits.

If the mass distribution of every galaxy were spherically symmetric, the gravitational force at radius r would equal the gravitational force of the enclosed mass M(r). This is not a bad approximation, even for disk galaxies. Thus for gas in a circular orbit with orbital velocity vrot,

GMr2=vrot2r, (7.167)

where M is the mass enclosed within the sphere of radius r and v is the orbital velocity at radius r, so

vrot2=GMr. (7.168)

Note that the velocity vrot is the full rotational velocity, not just its radial component vrotsini, where i is the inclination angle between the galaxy disk and the line of sight. The inclination angle of a thin circular disk can be estimated from the axial ratio

cosi=θmθM, (7.169)

where θm and θM are the minor- and major-axis angular diameters, respectively. Converting from CGS to astronomically convenient units yields

[(vrotcms-1)(105cms-1kms-1)]2 =[6.67×10-8dynecm2g-2(Mg)(2×1033gM)]
×[(rcm)(3.09×1021cmkpc)]-1, (7.170)
1010(vrotkms-1)2 =[6.67×10-82×1033(MM)]
×[3.09×1021(rkpc)]-1, (7.171)

and we obtain the total galaxy mass inside radius r in units of the solar mass:

(MM)2.3×105(vrotkms-1)2(rkpc). (7.172)

Thus the total mass of UGC 11707 (Figure 7.20) can be estimated from

vrotsini Δvrotsini2(1000kms-1-800kms-1)2100kms-1,
cosi minoraxismajoraxis0.73×10-3rad2.0×10-3rad0.365,so  sini0.93,
r θ1/2d10-3rad12.4Mpc12.4kpc,
so
(MM) 2.3×105(100/0.93)212.4=3.3×1010.

UGC 11707 is a relatively low-mass spiral galaxy.

Figure 7.20: Hi images of UGC 11707 [104]. The hatched circle in the lower left corner of panel (a) shows the image resolution. The contours in panels (a) and (c) outline the integrated Hi brightness distribution. Panel (b) shows contours of constant velocity separated by 20 km s-1 and the darker shading indicates approaching gas. Panel (d) is a position-velocity diagram, panel (e) is the radial Hi column-density profile, and panel (f) displays the integrated Hi spectrum.

This “total” mass is really only the mass inside the radius sampled by detectable Hi. Even though Hi extends beyond most other tracers such as molecular gas or stars, it is clear from plots of Hi rotation velocities versus radius that not all of the mass is being sampled, because we don’t see the Keplerian relation vrotr-1/2 which indicates that all of the mass is enclosed within radius r. Most rotation curves, one-dimensional position-velocity diagrams along the major axis, are flat at large r, suggesting that the enclosed mass Mr as far as we can see Hi. The large total masses implied by Hi rotation curves provided some of the earliest evidence for the existence of cold dark matter in galaxies.

Because detectable Hi is so extensive, Hi is an exceptionally sensitive tracer of tidal interactions between galaxies. Long streamers and tails of Hi trace the interaction histories of pairs and groups of galaxies. See Figure 8.11 showing Hi in the M81 group of galaxies and Figure 8.12 revealing long tidal tails of Hi from the “Antennae” galaxy pair NGC 4038/9.

Another application of the Hi spectra of galaxies is determining departures from smooth Hubble expansion in the local universe via the Tully–Fisher relation. Most galaxies obey the empirical luminosity–velocity relation [108]:

Lvm4, (7.173)

where vm is the maximum rotation speed. Arguments based on the virial theorem can explain the Tully–Fisher relation if all galaxies have the same central mass density and density profile, differing only in scale length, and also have the same mass-to-light ratio. Thus a measurement of vm yields an estimate of L that is independent of the Hubble distance dH. The Tully–Fisher distance dTF can be calculated from this “standard candle” L and the apparent luminosity. Apparent luminosities in the near infrared (λ2μm) are favored because the near-infrared mass-to-light ratio of stars is nearly constant and independent of the star-formation history, and because extinction by dust is much less than at optical wavelengths. Differences between dTF and dH are ascribed to the peculiar velocities of galaxies caused by intergalactic gravitational interactions. The magnitudes and scale lengths of the peculiar velocity distributions are indications of the average density and clumpiness of mass on megaparsec scales.

7.8.3 Dark Ages and the Epoch of Reionization (EOR)

Most of the baryonic matter in the early universe was fully ionized hydrogen and helium gas, plus trace amounts of heavier elements. This smoothly distributed gas cooled as the universe expanded, and the free protons and electrons recombined to form neutral hydrogen at a redshift z1091 when the age of the universe was about 3.8×105 years. The hydrogen remained neutral during the dark ages prior to the formation of the first ionizing astronomical sources—massive (M>100M) stars, galaxies, quasars, and clusters of galaxies—by gravitational collapse of overdense regions. These astronomical sources gradually started reionizing the universe when it was several hundred million years old (z10) and completely reionized the universe by the time it was about 109 years old (z6). This era is called the epoch of reionization.

The highly redshifted Hi signal was nearly uniform during the dark ages, and it developed structure on angular scales up to several arcmin when the first astronomical sources created bubbles of ionized hydrogen around them. As the bubbles grew and merged, the Hi signal developed frequency structure corresponding to the redshifted Hi line frequency. The characteristic size of the larger bubbles reached about 10 Mpc at z6 and produced Hi signals having angular scales of several arcmin and covering frequency ranges of several MHz. These Hi signals encode unique information about the formation of the earliest astronomical sources.

The Hi signals produced by the EOR will be very difficult to detect because they are weak (tens of mK), relatively broad in frequency, redshifted to low frequencies (100 MHz) plagued by radio-frequency interference and ionospheric refraction, and lie behind a much brighter (tens of K) foreground of extragalactic continuum radio sources. Nonetheless, the potential scientific payoff is so great that several groups around the world are developing instruments to detect the Hi signature of the EOR. Two such instruments are PAPER (Precision Array to Probe the Epoch of Reionization) [79] and the Murchison Widefield Array [54] shown in Figure 8.7.