2020年GeCAA理论第5题-电离氢区
英文题目
HII region
Luminous Blue Variable (LBV) are massive, unstable, supergiant stars that can undergo episodes of very strong mass loss, due to an instability in their atmospheres. After such an event, a dense nebula is formed around the star. LBV are also very hot stars and produce a large amount of high-energy photons that are able to ionise hydrogen atoms ($$E_ph > hν_0 = 13.6 eV$$) creating a roughly spherical region of ionized hydrogen (HII region).
In this problem, we consider a static, homogeneous, pure hydrogen nebula with a concentration
of $$n_H = 10^8 m^{−3}$$ and temperature $$T_{HII} = 10^4 K$, ionized by photons from a single
LBV star with a stable rate of ionizing photons $$Q = 10^{49} ph/s$$. Assume that each photon
can ionise only one hydrogen atom. At a particular location within an HII region, the
rate of photoionization is balanced by the rate of recombination per unit volume. This
sets the radius of the fully ionized region and this region is called the Stromgren sphere
with the radius $$R_S$$.
The total number of recombinations per volume is proportional to the concentration of
protons np, the concentration of electrons ne and the recombination coefficient for hydrogen
$$α(T_{HII}) = 10^{−19} m^3 s^{−1}$$. For simplification, ignore the fact that the process of
recombination can also release ionising photons.
(a) (5 points) Derive an algebraic expression for the radius of the Stromgren sphere and
calculate its value for the given parameters. Express your answer in units of parsecs
(pc).
(b) (3 points) The photoionization cross-section of H-atoms in the ground state encountering
photons with frequency $$ν_0$$ is equal to
$$σ ≈ 10^{−21} m^2$$
Calculate the mean-free path $$l_{ν0}$$ of an ionising photon. Compare $$l_{ν0}$$ to $$R_S$$ to determine
if this ionized nebulae is sharp-edged or not? (answer “YES” or “NO”)
(c) (4 points) On what timescale (in years) do you expect the Stromgren sphere to
form?
(d) (4 points) Radiation from an ionized hydrogen cloud (HII region) is often called
free-free emission because it is produced by free electrons scattering off the ions
without being captured: the electrons are free before the interaction and remain free
afterwards. In this process, the electron retains most of its pre-scattering energy.
An electron, while passing by a much more massive singly ionized hydrogen atom,
produces a radio photon of $$ν = 10 GHz$$. Calculate the mean electron thermal energy
in the HII region, for the given temperature of the Stromgren sphere. Is this an
example of free-free emission? (answer “YES” or “NO”)
(e) (4 points) Since the HII region is in local thermodynamic equilibrium, one can
calculate the absorption coefficient that is proportional to the optical depth $$τ_ν ∝ ν^{−2.1}$$
and it turns out that at the sufficiently high radio frequencies, the hot plasma is nearly
transparent and hence, $$τ_ν ≪ 1$$.
The flux density of photons has power-law spectra of the form $$S_ν ∝ ν^β$$.
Find $$β$$ for the radio frequencies.
