Larmor's formula

$$P = {2 q^2 \dot{v}^2 \over 3 c^3}$$ states that
electromagnetic radiation with power $P$ is produced by accelerating
(or decelerating; hence the German name bremsstrahlung
meaning "braking
radiation") an electrical charge $q$. Charges can be accelerated by
electrostatic or
magnetic forces, gravitational acceleration being negligible by
comparison. We will consider electrostatic bremsstrahlung first and
later
its magnetic counterpart magnetobremsstrahlung,
or "magnetic braking
radiation," synchrotron radiation for example.

The electric force is so much
stronger than gravity that ionized interstellar clouds have almost no
net charge on
large scales; the charges of all free electrons in an ionized cloud are
balanced by the charges of positively ions. An electron (charge $-e
\approx
-4.8 \times 10^{-10}$ statcoulombs) passing by an ion (charge $+e$
for a singly ionized atom, $+Ze$ for an ionized atom with $Z$ electrons
removed) is accelerated by their Coulomb attraction

$$\dot{v} = {f \over m_{\rm
e}} = {-Ze^2 \over m_{\rm e} l^2} ,$$ where $m_{\rm e} \approx 9.1
\times
10^{-28}$ g is the electron mass and $l$ is the distance between the
electron and ion. Such radiation is called
free-free radiation
because an initially free electron is rarely captured by the ion during
the interaction. If the
ionized interstellar cloud is reasonably dense, the electrons and ions
interact
frequently enough that they come into local thermodynamic equilibrium
(LTE) at some common temperature. Thus the radiation produced by these
interactions is sometimes called
thermal bremsstrahlung. This section will cover:

(1) astronomical sources of free-free emission

(2) the radio properties (spectrum, power, opacity) of free-free emission

(3) applications: what do we learn from radio observations of free-free sources?

Interstellar gas is primarily hydrogen plus some helium and trace amounts of heavier elements such as carbon, nitrogen, oxygen, iron, ... Astronomers often lump all of the heavier elements into the category of metals, meaning elements that readily form positive ions, even though many are not metallic in the usual sense. Much of the interstellar hydrogen is in the form of neutral atoms (called HI in astronomical terminology) or diatomic molecules (H$_2$), but some is ionized. The singly ionized hydrogen atom H$^+$ is referred to as HII, doubly ionized oxygen O$^{++}$ is called OIII, etc.

In 1939 the astronomer Bengt Strömgren realized that regions of diffuse interstellar gas are either (1) mostly neutral and with nearly all of the HI atoms in their ground electronic state or (2) almost completely ionized (HII much more abundant than HI), with very thin boundaries separating distinct HI and HII regions. Sometimes the HII regions surrounding stars are called Strömgren spheres after his early theoretical models. What is the microscopic physical basis for these ideas?

The ground electronic state of a
hydrogen atom corresponds to an atom with the smallest
(and hence most tightly bound) electron orbit around the nuclear proton
that is consistent with a stationary electronic wave function, a
standing wave. [See Sections 13.3 of Rohlfs & Wilson for a brief
discussion of the Bohr orbits in Rydberg atoms.] The electronic energy
levels permitted by quantum mechanics are characterized by their
quantum numbers $n = 1,~2,~3,...$, where $n = 1$ corresponds to the
ground state. While quantum mechanics forbids an electron in the ground
state ($n = 1)$ from radiating according to the classical Larmor
formula, it does not forbid radiative decay from higher levels ($n =
2,~3,...$), and Larmor's equation fairly accurately
predicts the
radiative lifetimes of excited hydrogen atoms. The orbital radius
$a_{\rm n}$ of an electron in the $n$th energy level is $a_{\rm n} =
n^2 a_0$, where $a_0 \approx 5.29 \times 10^{-9}$ cm is called the Bohr
radius. Applying Larmor's equation, as in problem 2 of problem
set 2,
shows
that the radiative lifetime $\tau$ is proportional to $a_{\rm n}^3$ and
hence to $n^6$. Thus we can scale the [incorrect] classical result
$\tau \approx
5.5 \times 10^{-11}$ s for $n = 1$ to estimate the radiative lifetimes
of excited states. For example, the approximate radiative lifetime of
the
$n = 2$ state would be $\tau \approx 2^6 \times 5.5 \times 10^{-11}
{\rm ~s~} \approx 3 \times 10^{-9}$ s, in reasonable agreement with the
accurate quantum-mechanical result $\tau \approx 2 \times 10^{-9}$ s.
Clearly excited hydrogen atoms will spontaneously decay very quickly to
the ground state by emitting radiation. At any one time, almost all
neutral atoms are in the ground state.