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. When the relative spins change from parallel to antiparallel, a photon is emitted.

One $\lambda = 21$ cm photon is
emitted when the spins flip from parallel to antiparallel.
Image
credit

The line center frequency is

$$\bbox[border:3px blue solid,7pt]{\nu_{10} = {8 \over 3} g_{\rm I}
\biggl( {m_{\rm e} \over m_{\rm p} }\biggr) \alpha^2 (R_{\rm M} c)
\approx
1420.405751{\rm ~MHz}}\rlap{\quad \rm {(7E1)}}$$
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 $R_{\rm M}c$ is the hydrogen Rydberg frequency (Eq. 7A2).

$$\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~}

\biggl(1 + {1 \over 1836.12}\biggr)^{-1}$$ $$\nu_{10} \approx 1420.4
{\rm ~MHz}$$

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 this
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
($n = 1$). The magnitude $\vert \mu_{\rm 10}\vert$ equals the Bohr
magneton, the intrinsic dipole moment of an electron.
Electrons have spin
angular momentum $L = \hbar /2$, classical radius $r_{\rm e} = e^2 /
(m_{\rm e}c^2)$, and charge $e$, so

$$\vert \mu_{10}^* \vert = {\hbar \over 2}{e \over m_{\rm e} c}
\approx {6.63 \times 10^{-27} {\rm ~erg~s}/(2
\pi) \over 2} \cdot { 4.8
\times
10^{-10} {\rm ~statcoul~} \over 9.11 \times 10^{-28} {\rm ~g~}
\cdot 3 \times
10^{10} {\rm ~cm~s}^{-1} }$$ $$\vert \mu_{10}^* \vert \approx 9.27
\times 10^{-21} {\rm
~erg~Gauss}^{-1} $$ 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

$$\tau_{1/2} = A_{10}^{-1} \approx
3.5 \times 10^{14} {\rm ~s~} \approx 11 {\rm ~million~years}$$ Such a
low emission coefficient implies an extremely low critical density
(defined by Eq. 7D10) $n^* \ll 1 {\rm ~cm}^{-3}$, so collisions can
easily maintain this transition in LTE throughout the diffuse
interstellar medium of a normal galaxy.

Regardless
of whether the HI is in LTE or not, we can define the HI
spin temperature $T_{\rm s}$ (the HI analog of the molecular
excitation temperature $T_{\rm x}$ defined by Equation 7B8) 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. 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 for gas in LTE at $T \approx T_{\rm s}
\approx 150$ K, so in the ISM

$$ {N_1 \over N_0} \approx
{g_1 \over g_0} = 3 \qquad {\rm and} \qquad N_{\rm H} = N_0 + N_1
\approx 4 N_0$$

Inserting these weights into Equation 7B7 for the opacity coefficient of the $\lambda = 21 {\rm ~cm}$ line gives $$ \kappa_\nu = {c^2 \over 8 \pi \nu_{10}^2 } {g_1 \over g_0} N_0 A_{10} \biggl[ 1 - \exp \biggl( - { h \nu_{10} \over k T_{\rm s}} \biggr) \biggr] \phi(\nu)$$ $$\kappa_\nu \approx { c^2 \over 8 \pi \nu_{10}^2 } \cdot 3 \cdot {N_{\rm H} \over 4} A_{10} \biggl( { h \nu_{10} \over k T_{\rm s}} \biggr) \phi(\nu)$$

$$\bbox[border:3px blue
solid,7pt]{\kappa_\nu \approx { 3 c^2 \over 32
\pi} {A_{10} N_{\rm H} \over
\nu_{10} } {h \over k T_{\rm s}} \phi (\nu) ~}\rlap{\quad \rm
{(7E4)}}$$ where $N_{\rm H}$ is the number
of
neutral hydrogen atoms per
cm$^{3}$. The neutral hydrogen
column density
along any line-of-sight is defined as

$$\bbox[border:3px blue solid,7pt]{ \eta_{\rm H} \equiv \int_{\rm los}
N_{\rm H} (s) d s}\rlap{\quad \rm {(7E5)}}$$
The total opacity $\tau$ of isothermal HI is proportional to the column
density. If $\tau \ll 1$, then the integrated HI
emission-line brightness $T_{\rm b}$ is
proportional to the column density of HI
and is independent of the spin temperature $T_{\rm s}$ because $T_{\rm
b} \approx T_{\rm s} \tau$ and $\tau \propto T_{\rm s}^{-1}$ in the
radio
limit $h \nu_{10} / (k T_{\rm s}) \ll 1$. Thus $\eta_{\rm H}$ can
be determined directly from the integrated line brightness when $\tau
\ll 1$. In
astronomically
convenient units it can be written as

$$\bbox[border:3px blue solid,7pt]{\biggl( { \eta_{\rm H} \over {\rm
cm}^{-2} }\biggr) \approx
1.82 \times 10^{18} \int \biggl[ { T_{\rm b} (v) \over {\rm K}} \biggr]
d \biggl( { v \over {\rm km~s}^{-1} } \biggr)}\rlap{\quad \rm
{(7E6)}}$$ 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. Note that absorption
by HI
in front of a continuum source with continuum brightness temperagure
$> T_{\rm s}$, 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 cosmic rays and ionizing photons from hot stars. 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.

Galactic HI

Neutral hydrogen gas in the disk of
our Galaxy moves in nearly circular orbits around the galactic
center. Radial velocities $V_{\rm r}$ measured from the Doppler
shifts of HI $\lambda = 21$ 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.

The figure below shows a plan view of
the galactic disk. The Sun ($\odot$) lies in the disk and moves
in a circular orbit around the galactic center. The distance to
the galactic center $R_\odot = 8.0 \pm 0.5$ kpc and the Sun's orbital
speed $\omega_\odot R_\odot \approx 220$ km s$^{-1}$ have been measured
by a variety of means (Reid, M. J. 1993, ARAA, 31, 345). All HI
clouds at galactocentric distance $R$ are assumed to be in circular
orbits with angular velocity $\omega(R)$, where $\omega(R)$ is a
monotonically decreasing function of $R$. For cloud 1 at
galactocentric azimuth $\theta$ on the line of sight at galactic
longitude $l$, the observed radial velocity $V_{\rm r}$ relative to the
Sun is given by

$$V_{\rm r} = \omega R \cos[\pi/2 - (l + \theta)] - \omega_\odot
R_\odot \cos(\pi/2 - l)$$

Using the trigonometric identities $\cos[\pi/2 - (l + \theta)] = \sin(l
+ \theta)$ and $\sin(l+\theta) = \sin\theta \cos l + \cos\theta \sin l$
we obtain

$$V_{\rm r} = \omega R (\sin\theta \cos l + \cos\theta \sin l) -
\omega_\odot R_\odot \sin l$$ $$V_{\rm r} = R_\odot (\omega
-\omega_\odot)\sin l$$

To apply this equation, we need to determine the rotation curve
$R\omega(R)$. The maximum radial velocity on the line of sight at
longitude $l$ is called the "terminal velocity" $V_{\rm T}$.
Since
$\omega$ decreases with $R$, this velocity occurs at the minimum $R =
R_{\rm min} = R_\odot \sin l$ where the orbit is tangent to the line of
sight:

$$V_{\rm T} = R_\odot [\omega(R_{\rm min})-\omega_\odot]\sin l $$ We
can determine the rotation curve from measurements of $V_{\rm T}$
spanning a wide range of $l$ and thus of $R_{\rm min}$.

Example: At galactic longitude $l = 30^\circ$, the terminal velocity is observed to be $V_{\rm T} \approx 130 $ km s$^{-1}$. What is $R_{\rm min}$ and the orbital speed $R_{\rm min}\omega(R_{\rm min})$?

$$R_{\rm min} = R_\odot \sin l = 8.0 {\rm ~kpc} \cdot 0.5 = 4.0 {\rm ~kpc}$$ $$ V_{\rm T} = R_\odot [\omega(R_{\rm min}) - \omega_\odot]\sin l$$ $$ V_{\rm T} = R_{\rm min} \omega(R_{\rm min}) - R_\odot \omega_\odot \sin l$$ $$ R_{\rm min} \omega(R_{\rm min}) = V_{\rm T} + R_\odot \omega_\odot \sin l$$ $$R_{\rm min } \omega(R_{\rm min}) = 130 {\rm ~km~s}^{-1} + 220 {\rm ~km~s}^{-1} \times 0.5

= 240 {\rm ~km~s}^{-1}$$

Note that, for $\vert l \vert < \pi /2$, there is a distance ambiguity: clouds 1 and 2 both have the same radial velocities but different distances $d$. There is no distance ambiguity for $\vert l \vert > \pi/2$.