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.

官方解答

(a) As the number of H-atoms undergoing ionization and recomination are balanced at $$R_S$$, each photon can ionize exactly one hydrogen atom and each neutral hydrogen has exactly one proton and one electron, $$n_{recomb} = n_{HII} = Q$$

and $$n_e = n_p = n_H$$

(2.0 points)

$$n_{recomb} = αn_pn_eV_S$$

$$Q = αn_{H}^{2}\dfrac{4π}{3}R^3_S$$

(1.0 points)

$$\therefore R_S = \sqrt[3]{ \dfrac{3Q} {4παn^2_H}}$$

(1.0 points)

$$= \sqrt[3]{\dfrac{ 3 × 10^{49}}{4π × 10^{−19} × (10^8)^2}} = 1.3 × 10^{17} m$$

$$\therefore R_S ≈ 4 pc$$

(1.0 points)

(b) Per one unit length distance, a typical photon will encounter$$ σn_H$$ H-atoms. Thus, the mean free path will be,

$$l_{ν0} = dfrac{1} {σn_H} = dfrac{1} {σ10^{−21} × 10^8} l_{ν0} = 10^{13} m$$

(2.0 points)

$$\therefore l_{ν0} ≈ 10^{−4}R_S ≪ R_S$$

Thus, the boundary layer of the sphere is very thin as compared to its total size. Hence,

“YES” this ionized nebula is very sharp-edged.

(1.0 points)

(c) For Stromgren sphere to form, all H-atoms ($$N$$) inside $$R_S$$ need to be ionized. Thus, time $$t_S$$ required will be

$$N = V_Sn_H = \dfrac{4π} {3} R^3_S n_H$$

(1.0 points)

$$t_S =\dfrac {N} {Q} = \dfrac{4πR^3_S n_H} {3Q}$$

(1.0 points)

$$=\dfrac {1} {αn_H} = \dfrac{1} [10^{−19} × {10^8} = 10^{11} s ≈ 3000 yr $$

(2.0 points)

For typical nebular densities, the main-sequence lifetime of LBV stars is much longer than this ionization time, and hence our assumption of a stationary system is justified.

(d) The mean electron thermal energy in a plasma of temperature $$T_e = 10^4 K$$ is

$$E_e ≈ k_BT_e ≈ 1.4 × 10^{−19} J ≈ 0.9 eV$$

(2.0 points)

The energy of a photon is

$$E_γ = hν ≈ 6.6 × 10^{−24} J ≈ 4 × 10^{−5} eV$$

(1.0 points)

As the energy of the radio photon is much much smaller than that of the

electron, the answer is “Yes”.

(1.0 points)

(e) As the plasma is nearly transparent, using the Rayleigh-Jeans law,

$$Bν =\dfrac{ 2k_BT_eν^2} {c^2}$$

(1.0 points)

$$S_ν ∝ B_ντ_ν = \dfrac{2k_BT_eν^2} {c^2} τ_ν$$

$$Sν ∝ ν^2 · ν^{−2.1}$$

$$\therefore Sν ∝ ν^{−0.1} $$

(3.0 points)