The HI 21 cm Line

Hydrogen is the most abundant element in the interstellar medium (ISM), but the symmetric H$_2$ molecule has no permanent dipole moment and hence does not emit a detectable spectral line at radio frequencies. Neutral hydrogen (HI) atoms are abundant and ubiquitous in low-density regions of the ISM. They are detectable in the $\lambda \approx 21$ cm ($\nu_{10} = 1420.405751 ...$ MHz) hyperfine line. Two energy levels result from the magnetic interaction between the quantized electron and proton spins, depending on whether they are parallel or antiparallel; a photon is emitted when the nuclear spin flips.

The line center frequency is
$$\bbox[border:3px blue solid,7pt]{\nu_{10} = {8 \over 3} g_{\rm I} \biggl( {m_{\rm e} \over m_{\rm p} }\biggr) \alpha^2 (R c) \approx 1420.405751{\rm ~MHz}}\rlap{\quad \rm {(7E1)}}$$
where $g_{\rm I} \approx 5.58569$ is the nuclear $g$-factor for a proton, $\alpha \equiv e^2 / (\hbar c) \approx 1 / 137.036$ is the dimensionless fine-structure constant, and $R$ is the Rydberg constant.
$$\nu_{10} \approx { 8 \over 3} \cdot 5.58569 \cdot \biggl( { 1 \over 1836.12} \biggr) \biggl( { 1 / 137.036 } \biggr)^2 \cdot 3.28984 \times 10^{15} {\rm ~Hz~} \approx 1.42 \times 10^9 {\rm ~Hz}$$

By analogy with the emission coefficient of radiation by an electric dipole $$A_{\rm UL} \approx { 64 \pi^4 \over 3 h c^3} \nu_{\rm UL}^3 \vert \mu_{\rm UL} \vert^2~,$$
the emission coefficient of a magnetic dipole is
$$A_{\rm UL} \approx { 64 \pi^4 \over 3 h c^3} \nu_{\rm UL}^3 \vert \mu_{\rm 10}^* \vert^2~,$$
where $\mu_{\rm 10}^*$ is the mean magnetic dipole moment for HI in the ground electronic state ($n = 1$). It is sometimes called the Bohr magneton. Its magnitude is
$$\vert \mu_{10}^* \vert = {e \hbar \over 2 m_{\rm e} c} \approx { 4.8 \times 10^{-10} {\rm ~statcoul~} \cdot 6.63 \times 10^{-27} {\rm ~erg~s}/(2 \pi) \over 2 \cdot 9.11 \times 10^{-28} {\rm ~g~} \cdot 3 \times 10^{10} {\rm ~cm~s}^{-1} }$$
$$\vert \mu_{10}^* \vert \approx 9.27 \times 10^{-21} {\rm ~erg~Gauss}^{-1} $$
Thus the emission coefficient of the 21 cm line is
$$A_{10} \approx { 64 \pi^4 (1.42 \times 10^9 {\rm ~Hz})^3 \over 3 \cdot 6.63 \times 10^{-27} {\rm ~erg~s~} (3 \times 10^{10} {\rm ~cm~s}^{-1})^3 } (9.27 \times 10^{-21} {\rm ~erg~Gauss}^{-1})^2$$
$$\bbox[border:3px blue solid,7pt]{A_{10} \approx 2.85 \times 10^{-15} {\rm ~s}^{-1}}\rlap{\quad \rm {(7E2)}}$$
That is, the radiative half-life of this transition is about
$$\tau_{1/2} = A_{10}^{-1} \approx 3.5 \times 10^{14} {\rm ~s~} \approx 11 {\rm ~million~years}$$

Regardless of whether the HI is in LTE or not, we can define the excitation temperature or spin temperature $T_{\rm s}$ for HI by
$$\bbox[border:3px blue solid,7pt]{{N_1 \over N_0} \equiv {g_1 \over g_0} \exp \biggl( - {h \nu_{10} \over k T_{\rm s} } \biggr)}\rlap{\quad \rm {(7E3)}}$$
where the statistical weights of the upper and lower spin states are $g_1 = 3$ and $g_0 = 1$, respectively. Collisions can maintain LTE (kinetic temperature $T \approx T_{\rm s}$) only if the mean time $\langle \tau \rangle$ between collisions is much shorter than the radiative lifetime $\tau_{\rm 1/2} \approx 11 \times 10^6$ yr. The collision cross section for two HI atoms in the rest electronic state $n = 1$ is (very roughly)
$$\sigma \sim \pi (2 a_1)^2~,$$
where $a_1 \approx 0.53 \times 10^{-8}$ cm is the Bohr radius. If the HI number density is $N_{\rm H}$, the collision mean free path $l$ satisfies
$$ l \sigma \approx N_{\rm H}^{-1}$$
so
$$ l \approx ( 4 \pi a_1^2 N_{\rm H})^{-1} \approx 3 \times 10^{15} {\rm ~cm} \cdot \biggl( {N_{\rm H} \over {\rm cm}^{-3}} \biggr)^{-1} ~.$$
The corresponding mean time between collisions depends on the average speed $\langle v \rangle$ of the HI atoms:
$$\langle \tau \rangle \approx {l \over \langle v \rangle}~.$$
At kinetic temperature $T$,
$${m_{\rm H} \langle v \rangle^2 \over 2} \approx {3 k T \over 2}$$
so
$$\langle \tau \rangle \approx l \biggl( { m_{\rm H} \over 3 k} \biggr)^{1/2} T^{-1/2} \approx 3 \times 10^{15} {\rm ~cm} \cdot N_{\rm H}^{-1} \biggl( {1.67 \times 10^{-24} {\rm ~g} \over 3 \cdot 1.38 \times 10^{16} {\rm ~erg~K}^{-1} } \biggr)^{1/2} T^{-1/2}$$
$$\langle \tau \rangle \approx 10^4 {\rm ~yr} \biggl( { N_{\rm H} \over {\rm cm}^{-3}} \biggr)^{-1} \biggl( { T \over {\rm K} } \biggr)^{-1/2}~.$$
For typical values $N \sim 1 {\rm ~cm}^{-3}$ and $T \sim 150$ K in the ISM, $\langle \tau \rangle \sim 10^5 {\rm ~yr} \ll \tau_{1/2}$ as needed to maintain $T_{\rm s} \approx T$. Note that
$$ {h \nu_{10} \over k T_{\rm s}} \approx { 6.63 \times 10^{-27} {\rm ~erg~s~} \cdot 1.42 \times 10^9 {\rm ~Hz} \over 1.38 \times 10^{-16} {\rm ~erg~K}^{-1} \cdot 150 {\rm ~K} } \approx 5 \times 10^{-4} \ll 1$$ is very small, so in the ISM
$$ {N_1 \over N_0} \approx {g_1 \over g_0} = 3 \qquad {\rm and} \qquad N_{\rm H} = N_0 + N_1 \approx 4 N_0$$

Now we are ready to evaluate the opacity coefficient of the 21~cm line: $$ \kappa_\nu = {c^2 \over 8 \pi \nu_{10}^2 } {g_1 \over g_0} N_0 A_{10} \biggl[ 1 - \exp \biggl( - { h \nu_{10} \over k T_{\rm s}} \biggr) \biggr] \phi(\nu)$$ $$\kappa_\nu \approx { c^2 \over 8 \pi \nu_{10}^2 } \cdot 3 \cdot {N_{\rm H} \over 4} A_{10} \biggl( { h \nu_{10} \over k T_{\rm s}} \biggr) \phi(\nu)$$

$$\bbox[border:3px blue solid,7pt]{\kappa_\nu \approx { 3 c^2 \over 32 \pi} {A_{10} N_{\rm H} \over \nu_{10} } {h \over k T_{\rm s}} \phi (\nu) ~}\rlap{\quad \rm {(7E4)}}$$
where $N_{\rm H}$ is the number of neutral hydrogen atoms per cm$^{3}$. If we define the neutral hydrogen column density along the line-of-sight as
$$\bbox[border:3px blue solid,7pt]{ \eta_{\rm H} \equiv \int_{\rm los} N_{\rm H} (s) d s}\rlap{\quad \rm {(7E5)}}$$
and the optical depth $\tau \ll 1$, then in astronomically convenient units,
$$\bbox[border:3px blue solid,7pt]{\biggl( { \eta_{\rm H} \over {\rm cm}^{-2} }\biggr) \approx 1.82 \times 10^{18} \int \biggl[ { T_{\rm b} (v) \over {\rm K}} \biggr] d \biggl( { v \over {\rm km~s}^{-1} } \biggr)}\rlap{\quad \rm {(7E6)}}$$
where $T_{\rm b}$ is the observed 21 cm line brightness temperature at radial velocity $v$ and the velocity integration extends over the entire 21 cm line profile. Notice that the integrated HI emission-line brightness is proportional to the column density of HI and is independent of the spin temperature $T_{\rm s}$ because $T_{\rm b} = T_{\rm s} \tau$ and $\tau \propto T_{\rm s}^{-1}$ in the radio limit $h \nu_{10} / (k T_{\rm s}) \ll 1$. Absorption by HI in front of a continuum source with $T_{\rm b} > T_{\rm s}$, on the other hand, is weighted in favor of colder gas.

The HI absorption and emission spectra toward the source 1714-397 (Dickey, J. M. et al. 1983, ApJS, 53, 591).

The equilibrium temperature of cool interstellar HI is determined by the balance of heating and cooling. The primary heat sources are ionizing photons from stars and cosmic rays. The main coolant in the cool ISM is radiation from the fine-structure line of singly ionized carbon, CII, at $\lambda = 157.7\,\mu$m. This line is strong only when the temperature is at least
$$k T \approx h \nu = {h c \over \lambda}~,$$
so the cooling rate increases exponentially above
$$ T \approx {h c \over k \lambda} \approx {6.63 \times 10^{-27} {\rm ~erg~s} \cdot 3 \times 10^{10} {\rm ~cm~s}^{-1} \over 1.38 \times 10^{-16} {\rm ~erg~K}^{-1} \cdot 157.7 \times 10^{-4} {\rm ~cm}} \approx 91~K~.$$
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 ($\tau \gg 1$) 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.

Astronomical applications

This image shows a galactic "superbubble" in HI (green) and HII (purple) about 7 kpc distant and 3 kpc in height.  Stellar winds and supernovae in young star clusters blow these bubbles.  Images of HI away from the galactic plane are easily contaminated by sidelobe responses to the strong and widespread HI emission from the plane itself.  The low sidelobe levels of the clear-aperture GBT make such HI images possible.  Image credit

Example: Use the HI line profile of the galaxy UGC 11707 to estimate its distance
$$D \approx v_{\rm r} / H_0~,$$
where $H_0 \approx 72$ km s$^{-1}$ Mpc$^{-1}$ is the present value of the Hubble parameter, the Hubble constant. If the radial velocity $v_{\rm r} \ll c$, then we can use the nonrelativistic Doppler formula
$${v_{\rm r} \over c} \approx {\nu_0 - \nu \over \nu_0}$$
where $\nu_0$ is the rest-frame line frequency and $\nu$ is the observed frequency. This equation yields what is known as the radio velocity because radio astronomers measure frequencies, not wavelengths. Optical astronomers measure wavelengths, not frequencies, so the optical velocity is $${v_{\rm r} \over c} \approx {\lambda - \lambda_0 \over \lambda_0}~.$$ Beware of this "gotcha": the optical and radio velocities are not exactly equal. Occasionally a VLA observer mixes them up, fails to center the observing passband on the correct frequency, and ends up with only part of the HI spectrum of a galaxy.

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

This integrated HI spectrum of UGC 11707 obtained with the 140-foot telescope (beamwidth $\approx 20$ arcmin) shows the typical two-horned profile of a spiral galaxy.

For UGC 11707, the line center frequency is $\nu \approx 1416.2$ MHz, so $$v_{\rm r} \approx c \biggl( 1 - { \nu \over \nu_0} \biggr) \approx 3 \times 10^5 {\rm ~km~s}^{-1} \biggl( 1 - {1416.2 {\rm ~MHz} \over 1420.4 {\rm ~MHz}} \biggr) \approx 890 {\rm ~km~s}^{-1}$$
$$D \approx {v_{\rm r} \over H_0} = {890 {\rm ~km~s}^{-1} \over 72 {\rm ~km~s}^{-1} {\rm ~Mpc}^{-1} } = 12.4 {\rm ~Mpc}$$

Example: What is the HI mass of UGC 11707?

Assuming $\tau \ll 1$,
$$\bbox[border:3px blue solid,7pt]{\biggl ( {M_{\rm H} \over M_\odot} \biggr) \approx 2.36 \times 10^5 \biggl( { D \over {\rm Mpc} } \biggr)^2 \int \biggl[ { S(v) \over {\rm Jy}} \biggr] \biggl( {d v \over {\rm km~s}^{-1} } \biggr)}\rlap{\quad \rm {(7E7)}}$$

The integral $\int S(v) d v$ over the line is called the line flux and is usually expressed in units of Jy km s$^{-1}$. The single-dish profile of UGC 11707 shows a line flux
$$\int S(v)\, d v \approx 0.35 {\rm ~Jy~} \times 200 {\rm ~km~s}^{-1} \approx 70 {\rm ~Jy~km~s}^{-1}$$ so $$\biggl( {M_{\rm H} \over M_\odot} \biggr) \approx 2.36 \times 10^5 \cdot (12.4)^2 \cdot 70 \approx 2.5 \times 10^9$$
Small statistical corrections for nonzero $\tau$ can be made from knowledge about the expected opacity as a function of disk inclination, galaxy mass, morphological type, etc.

An HI image of a galaxy yields the total mass $M$ if the gas flows in circular orbits about the center of mass. Such a flow yields a characteristic radial velocity field, illustrated by nearby spiral galaxy M33.

The HI radial velocity field of M33. Image credit

For a roughly spherical mass distribution,
$${ G M \over r^2} = {v^2 \over r}~,$$
where $M$ is the mass within the sphere of radius $r$, so
$$v^2 = { G M \over r}~.$$
[Ignore the virial mass derivation in Rohlfs & Wilson Section 12.8.1.] Note that the velocity $v$ is the full tangential velocity, not just the radial component $v_{\rm r}$ that contributes to the Doppler shift:
$$v = {v_{\rm r} \over \sin i}~,$$
where $i$ is the inclination angle between the galaxy disk and the line-of-sight. The inclination angle of a circular disk can be estimated from the axial ratio
$$\cos i = {\theta_{\rm m} \over \theta_{\rm M}}~,$$
where $\theta_{\rm m}$ and $\theta_{\rm M}$ are the minor- and major-axis angular diameters, respectively. Converting from cgs to astronomically convenient units,
$$\biggl[ \biggl( { v \over {\rm cm~s}^{-1} } \biggr) \biggl( { 10^5 {\rm ~cm~s}^{-1} \over {\rm km~s}^{-1} } \biggr) \biggr]^2 = $$
$$\biggl[ 6.67 \times 10^{-8} {\rm ~dyne~cm}^2 {\rm ~g}^{-2} \cdot \biggl( { M \over {\rm g} } \biggr) \biggl( { 2 \times 10^{33} {\rm ~g} \over M_\odot} \biggr) \biggr] \times$$
$$\biggl[ \biggl( { r \over {\rm cm}} \biggr) \biggl( { 3.09 \times 10^{21} {\rm ~cm} \over {\rm kpc} } \biggr) \biggr]^{-1}$$
$$10^{10} \biggl( { v \over {\rm km~s}^{-1} } \biggr)^2 = \biggl[ 6.67 \times 10^{-8} \cdot 2 \times 10^{33} \biggl( { M \over M_\odot} \biggr) \biggr] \biggl[ 3.09 \times 10^{21} \biggl( { r \over {\rm kpc}} \biggr) \biggr]^{-1}$$
we get
$$\bbox[border:3px blue solid,7pt]{ \biggl( { M \over M_\odot} \biggr) \approx 2.3 \times 10^5 \biggl( { v \over {\rm km~s}^{-1} } \biggr)^2 \biggl( { r \over {\rm kpc} } \biggr) \approx 2.3 \times 10^5 \biggl[ { (v_{\rm r} / \sin i) \over {\rm km~s}^{-1} } \biggr]^2 \biggl( { r \over {\rm kpc} } \biggr) }\rlap{\quad \rm {(7E8)}}$$

Example: What is the total mass of UGC 11707?

HI images of UGC 11707 (Swaters, R. A. et al. 2002, A&A, 390, 829).  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.

$$ v_{\rm r} \approx {\Delta v_{\rm r} \over 2} \approx { (1000 {\rm ~km~s}^{-1} - 800 {\rm ~km~s}^{-1}) \over 2} \approx 100 {\rm ~km~s}^{-1}$$
$$ \cos i \approx { {\rm minor~axis} \over {\rm major~axis} } \approx { 0.73 \times 10^{-3} {\rm ~rad} \over 2.0 \times 10^{-3} {\rm ~rad} } \approx 0.365 \qquad {\rm so} \qquad \sin i \approx 0.93$$
$$r \approx \theta_{1/2} D \approx 10^{-3} {\rm ~rad~} \cdot 11.8 {\rm ~Mpc} \approx 11.8 {\rm ~kpc}$$
so
$$\biggl( { M \over M_\odot} \biggr) \approx 2.3 \times 10^5 \cdot (100 / 0.93) ^2 \cdot 11.8 = 3.1 \times 10^{10}~.$$
UGC 11707 is a relatively low-mass spiral galaxy.

This "total" mass is really only the mass within the radius sampled by detectable HI, although HI extends beyond most other tracers such as molecular gas or stars. Even so, 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 $v_{\rm r} \propto r^{-1/2}$.  Most rotation curves, one-dimensional position-velocity diagrams along the major axis, are flat at large $r$, suggesting that the enclosed mass $M \propto r$ 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.

Optical (white) and HI (blue) images of the strongly interacting galaxies NGC 4038 and NGC 4039 (also known as the "antennae").  The velocity distributions of the long HI tidal tails provide strong constraints for computer models of the interaction history. Image credit

The radio continuum (red) and HI (blue) images of the post-merger pair of galaxies UGC 813 and UGC 816 indicate that the disks of these two galaxies passed through each other about 50 million years ago. Image credit

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 (Tully, R. B., & Fisher, J. R. 1977, A&A, 54, 661):

$$L \propto v_{\rm m}^4~,$$

where $v_{\rm m}$ is the maximum rotation speed. Arguments based on the virial theorem can "explain" this relation if all galaxies have 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 $v_{\rm m}$ yields an estimate of $L$ that is independent of the Hubble distance $D_{\rm H}$. The Tully-Fisher distance $D_{\rm TF}$ can be calculated from this "standard candle" $L$ and the apparent luminosity. Apparent luminosities in the near infrared ($\lambda \sim 2\,\mu$m) 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 $D_{\rm TF}$ and $D_{\rm H}$ 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.