2020年GeCAA理论第5题-电离氢区

来自astro-init

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 $$n_p$$, the concentration of electrons $$n_e$$ 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.

中文翻译

电离氢区

亮蓝巨星 (LBV) 是大质量的、不稳定的超巨星。 它们的大气很不稳定,因此可能会在一段时间之内经历较大的质量损失。这一事件后,恒星周围会形成致密的星云。 LBV的温度也非常高,可以制造大量的高能光子。这些高能光子可以电离氢原子 ($$E_ph > hν_0 = 13.6 eV$$),制造出一个近球形的电离氢区 (HII region)。


在这道题中,我们考虑一个稳定的、均匀的、仅由氢原子构成的星云。 星云中氢原子的数密度为$$n_H = 10^8 m^{−3}$$ ,温度为 $$T_{HII} = 10^4 K$$。其由单个LBV发出的光子电离,发出光子的速率稳定,为 $$Q = 10^{49} ph/s$$。 假设单个光子只能电离一个氢原子。 在电离氢区某个特定的位置, 单位体积内光致电离的速率与再复合的速率相等。这确定了完全电离的范围。这一范围叫做斯特龙根球,半径为 $$R_S$$.


单位体积内再复合的量与质子密度、电子密度和再复合系数成正比。对于氢原子,再复合系数为 $$α(T_{HII}) = 10^{−19} m^3 s^{−1}$$。为了简化计算,忽略再复合过程中放出的光子。

(a) (5 points)推导出斯特龙根球半径的代数表达式,并用现有数据计算其值。答案的单位为秒差距。


(b) (3 points) 对于频率为$$ν_0$$的光子,基态氢原子的光致电离碰撞截面为

$$σ ≈ 10^{−21} m^2$$

计算单个光子的平均自由程 $$l_{ν0}$$。 比较 $$l_{ν0}$$ 和 $$R_S$$ 来确定该电离云的边界是否锐利。 (回答是或否)

(c) (4 points)你认为斯特龙根球会在多长的时标下形成?

(d) (4 points) 电离氢云发出的辐射常被称为自由-自由发射。其由自由电子被离子不捕获地散射而形成:电子在相互作用之前是自由的,之后也是自由的。在这个过程中,电子保有散射前的大部分能量。一个电子在经过一个孤立的氢离子时候产生了一个 $$ν = 10 GHz$$的射电光子。根据题目给定的斯特龙根球的温度计算该电离氢区里电子的平均动能。这是一个自由-自由发射的例子么

(e) (4 points) 既然电离氢云是处在热力学平衡状态下, 我们可以计算其吸收参数。它与光深成正比,表达式为$$τ_ν ∝ ν^{−2.1}$$。 这表明,在足够高的频率上,热等离子体是几乎透明的,即$$τ_ν ≪ 1$$. 其光子流量密度符合幂律谱形式$$S_ν ∝ ν^β$$。 请找出射电波段的$$β$$。


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, $$l_{ν0} = \dfrac{1} {σn_H} = \dfrac{1} {10^{−21} × 10^8}$$


$$l_{ν0} = 10^{13} m$$

(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,

$$B_ν =\dfrac{ 2k_BT_eν^2} {c^2}$$

(1.0 points)


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

$$Sν ∝ ν^2 · ν^{−2.1}$$

$$\therefore S_ν ∝ ν^{−0.1} $$

(3.0 points)