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

Spectral lines 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 ideal 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: $E = h \nu$. Unlike ideal particles, real particles of momentum $p$ have wave functions whose De Broglie wavelength is $\lambda = h / p$. An electron's orbit about the nucleus of an atom must allow standing waves, so its circumference must be an integer number of wavelengths. The Planck's constant $h\approx 6.62607 \times 10^{-27}$ erg s in these two equations is a quantum of action; its dimensions are (mass$\times$length$^2 \times$time$^{-1}$), the same as (energy$\times$time) or (angular momentum) or (length$\times$momentum). Spectral lines have definite frequencies resulting from transitions between discrete energy states in physical systems, and these discrete states arise from quantization of angular momentum. Another quantum effect important to spectral lines, particularly at radio wavelengths where $h \nu \ll k T$, is stimulated emission. 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. Doppler shifts of line frequencies measure radial velocities. These velocities yield the redshifts and Hubble distances of extragalactic sources, as well as the 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 constrained by spectral-line data. Some characteristics of radio spectral lines include:

(1) The "natural"
line widths are much
smaller than Doppler-broadened line widths, so very small changes in
radial
velocity can be measured.

(2) Stimulated emission is important because $h \nu \ll k T$. This
causes
line opacities to vary as $T^{-1}$ and favors the formation of natural
masers.

(3) The ability 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) In practice,
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 can be
made.

Most of the interstellar medium (ISM)
in our Galaxy is in rough
pressure equilibrium because mass motions with speeds up to the speed
of sound act to reduce pressure gradients quickly. Temperatures
equilibrate more slowly, so there are wide ranges of the
(temperature$\times$density) product consistent with a given pressure.
Consequently, there are four important phases of
the ISM having comparable pressures:

(1) cold (10's of K) dense molecular
clouds

(2) cool ($\sim 10^2$ K) neutral HI
gas

(3) warm ($\sim 10^4$ K) ionized HII
gas

(4) hot ($\sim 10^6$ K) low-density
ionized gas (in bubbles
formed by expanding supernova remnants, for example).

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

** Recombination Line Frequencies**

The semiclassical
Bohr atom 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 $m_{\rm e}$, so the nucleus is
nearly at rest in the center-of-mass frame. The wave functions of the
electrons have De
Broglie
wavelengths

$$\lambda = {h \over p} = { h \over m_{\rm e} v}~,$$ 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
$a_{\rm n}$ of the $n$th permitted electron orbit satisfies the
quantization rule

$$ 2 \pi
a_{\rm n} = n \lambda = { n h \over m_{\rm e} v}~.$$ The requirement
that

$$a_{\rm n} = { n h \over 2 \pi m_{\rm e} v}

= {n \hbar \over m_{\rm e} v}$$ implies that the orbital angular
momentum $L = m_{\rm e} v a_{\rm
n}$ be an integer multiple of $\hbar \equiv h / (2 \pi)$.

The relation between $a_{\rm n}$ and $v$ is provided by
balancing the Coulomb and centrifugal forces. For a hydrogen atom,

$$
{e^2 \over a_{\rm n}^2} = {m_{\rm e} v^2 \over a_{\rm n}}$$ so

$$v^2 =
{e^2 \over m_{\rm e} a_{\rm n}}$$ $$\bbox[border:3px blue
solid,7pt]{a_{\rm n} = {n^2 \hbar^2 \over
m_{\rm e} e^2}}\rlap{\quad \rm {(7A1)}}$$

Example: What is the Bohr radius of a
hydrogen atom whose electron
is in the $n$th electronic energy level?

$$a_{\rm n} = { \hbar^2 \over
m_{\rm e} e^2} n^2 = {[6.63/(2 \pi) \times 10^{-27} {\rm ~erg~s}]^2
\over 9.11 \times 10^{-28} {\rm ~g} \times (4.8 \times 10^{-10} {\rm
~statcoul})^2} n^2 \approx 0.53 \times 10^{-8} {\rm ~cm} \times n^2$$
The Bohr radius of a hydrogen atom in its ground
electronic state ($n = 1$)
is only $a_1
\approx 0.53 \times 10^{-8}$ cm, but since $a_{\rm n} \propto n^2$ a
highly excited ($n \approx 100$)
radio-emitting atom in the ISM can be remarkably large: $2a_{100}
\approx 10^{-4}$ cm $= 1 \mu$m, which is bigger than most viruses!