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GeCAA 2020, Теория — Задача 5. HII region

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Всички материали от GeCAA 2020

GeCAA-2020 — Theory · 20 т.

Внимание: Бележка към темата

Eight complete native whole tasks and 27 requests, with all 25 official key pages, alternatives, marking criteria, and the complete two-page GeCAA-2020 constants companion. This bilingual Russian/English question edition is transcribed from its printed English column; task titles use the matching English key. Russian-only figure captions remain native. Printed scientific and source-edition discrepancies remain with attributed local notes. Task points total 150 by addition; no aggregate paper total or overall time limit is printed, so those fields remain unspecified. Referenced answer sheets/boxes are not in the obtained packet or complete 2020 archive-folder inventory.

Native source editions and gaps

Question masthead: GeCAA-2020 [Russian], Теоретический тур / Theory, with Russian and English columns. Official English key masthead: GeCAA - Theory Solutions, October 3, 2020. That is the printed key edition date; no competition held date/place is inferred. The designated answer sheets/boxes mentioned for Light Curves and Meteor Radiant were not obtained. The primary question/official key wording variants are recorded inside their original tasks.

Table of constants

The dated GeCAA-2020 constants companion prints Russian and English labels alongside the following native values.

Fundamental constants

Quantity	Symbol	Value
Speed of light in vacuum	$c$	$2.998\times10^8\ \mathrm{m/s}$
Planck constant	$h$	$6.626\times10^{-34}\ \mathrm{J\cdot s}$
Boltzmann constant	$k_B$	$1.381\times10^{-23}\ \mathrm{J/K}$
Stefan-Boltzmann constant	$\sigma$	$5.670\times10^{-8}\ \mathrm{W/(m^2\cdot K^4)}$
Elementary charge	$e$	$1.602\times10^{-19}\ \mathrm C$
Universal gravitational constant	$G$	$6.674\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
Universal gas constant	$R$	$8.315\ \mathrm{J/(mol\cdot K)}$
Avogadro constant	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Wien’s displacement law	$\lambda_{\max}T$	$2.898\times10^{-3}\ \mathrm{m\cdot K}$
Mass of electron	$m_e$	$9.109\times10^{-31}\ \mathrm{kg}$
Mass of proton	$m_p$	$1.673\times10^{-27}\ \mathrm{kg}$
Mass of neutron	$m_n$	$1.675\times10^{-27}\ \mathrm{kg}$
Atomic mass unit (a. m. u.)		$1.661\times10^{-27}\ \mathrm{kg}$

Astronomical data

Quantity	Symbol	Value
1 parsec	1 pc	$3.086\times10^{16}\ \mathrm m=206265\ \mathrm{au}=3.262\ \mathrm{ly}$
1 astronomical unit (au)	1 au	$1.496\times10^{11}\ \mathrm m$
1 jansky	1 Jy	$10^{-26}\ \mathrm{W/(m^2\cdot Hz)}$
Hubble constant	$H_0$	$70\ \mathrm{(km/s)/Mpc}$
Solar luminosity	$L_\odot$	$3.826\times10^{26}\ \mathrm W$
Apparent angular diameter of Sun	$\theta_\odot$	$32'$
Effective temperature of Sun	$T_{\mathrm{eff}\,\odot}$	$5778\ \mathrm K$
Obliquity of the ecliptic (Earth)	$\epsilon$	$23.5^\circ$
Inclination of the lunar orbit w. r. t. ecliptic		$5^\circ08'43''$
Apparent visual magnitude of full moon		$-12.74^{\mathrm m}$
North Ecliptic Pole (J2000.0)	$(\alpha_E,\delta_E)$	$(18^{\mathrm h}00^{\mathrm m}00^{\mathrm s},+66^\circ33'39'')$
North Galactic Pole (J2000.0)	$(\alpha_G,\delta_G)$	$(12^{\mathrm h}51^{\mathrm m}26^{\mathrm s},+27^\circ07'42'')$
1 sidereal day		$23^{\mathrm h}56^{\mathrm m}04^{\mathrm s}$
1 tropical year		365.2422 solar days
1 sidereal year		365.2564 solar days

Solar magnitudes

Quantity	Value
Apparent visual magnitude	$-26.75^{\mathrm m}$
Absolute visual magnitude	$+4.82^{\mathrm m}$
Apparent bolometric magnitude	$-26.83^{\mathrm m}$
Absolute bolometric magnitude	$+4.74^{\mathrm m}$

Solar system

Object	Mean radius (km)	Mass (kg)	Semi-major axis (au)	Eccentricity
Sun	695500	$1.988\times10^{30}$	—	—
Mercury	2440	$3.301\times10^{23}$	0.387	0.206
Venus	6052	$4.867\times10^{24}$	0.723	0.007
Earth	6378	$5.972\times10^{24}$	1.000000	0.016710
Moon	1737	$7.346\times10^{22}$	0.002572	0.054900
Mars	3390	$6.417\times10^{23}$	1.524	0.093
Jupiter	69911	$1.898\times10^{27}$	5.203	0.048
Saturn	58232	$5.683\times10^{26}$	9.537	0.054
Uranus	25362	$8.681\times10^{25}$	19.189	0.047
Neptune	24622	$1.024\times10^{26}$	30.070	0.009

Gauss’s formulae

Cosine theorem:

\cos a=\cos b\cos c+\sin b\sin c\cos A.

Sine theorem:

\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}.

Source note: The English native heading is “Gauss’s formulae”; its Russian counterpart is “Сферическая тригонометрия”. Blank source symbol cells remain blank. This is the dated same-competition constants PDF obtained from the GeCAA 2020 archive Constants folder.

Условие

Luminous Blue Variables (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 ( $E_{\mathrm{ph}}>h\nu_0=13.6\ \mathrm{eV}$ ) 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 $n_H=10^8/\mathrm{m^3}$ and temperature $T_{\mathrm{HII}}=10^4\ \mathrm K$ , ionized by photons from a single LBV star with a stable rate of ionizing photons $Q=10^{49}\ \mathrm{ph/s}$ . 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 $\alpha(T_{\mathrm{HII}})=10^{-19}\ \mathrm{m^3/s}$ . For simplification, ignore the fact that the process of recombination can also release ionising photons.

a) Derive an algebraic expression for the radius of the Stromgren sphere and calculate its value for the given parameters. Give the answer in parsecs. [5 т.]

b) The photoionization cross-section of H-atoms in the ground state encountering photons with frequency $\nu_0$ is equal to $\sigma\approx10^{-21}\ \mathrm{m^2}$ . Calculate the mean free path $l_{\nu_0}$ of an ionising photon. Compare $l_{\nu_0}$ to $R_S$ to determine if this ionized nebulae is sharp-edged or not? (answer “YES” or “NO”) [3 т.]

c) On what timescale do you expect the Stromgren sphere to form? Give the answer in years. [4 т.]

d) 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 $\nu=10\ \mathrm{GHz}$ . 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”) [4 т.]

e) Since the HII region is in local thermodynamic equilibrium, one can calculate the absorption coefficient that is proportional to the optical depth $\tau_\nu\propto\nu^{-2.1}$ and it turns out that at sufficiently high radio frequencies, the hot plasma is nearly transparent and hence $\tau_\nu\ll1$ . The flux density of photons has power-law spectra of the form $S_\nu\propto\nu^\beta$ . Find $\beta$ for the radio frequencies. Write the numeric value of beta. [4 т.]

Отговори

Решение

(a) As the number of H-atoms undergoing ionization and recombination 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_{\mathrm{recomb}}=n_{\mathrm{HII}}=Q,\qquad\text{and }n_e=n_p=n_H.\qquad[2.0]

n_{\mathrm{recomb}}=\alpha n_pn_eV_S,\qquad Q=\alpha n_H^2\frac{4\pi}3R_S^3.\qquad[1.0]

\therefore R_S=\sqrt[3]{\frac{3Q}{4\pi\alpha n_H^2}}.\qquad[1.0]

=\sqrt[3]{\frac{3\times10^{49}}{4\pi\times10^{-19}\times(10^8)^2}}=1.3\times10^{17}\ \mathrm m,\qquad\therefore R_S\approx4\ \mathrm{pc}.\qquad[1.0]

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

l_{\nu_0}=\frac1{\sigma n_H}=\frac1{\sigma10^{-21}\times10^8},\qquad l_{\nu_0}=10^{13}\ \mathrm m.\qquad[2.0]

\therefore l_{\nu_0}\approx10^{-4}R_S\ll 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]

(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=\frac{4\pi}3R_S^3n_H.\qquad[1.0]

t_S=\frac NQ=\frac{4\pi R_S^3n_H}{3Q}.\qquad[1.0]

=\frac1{\alpha n_H}=\frac1{10^{-19}\times10^8}=10^{11}\ \mathrm s\approx3000\ \mathrm{yr}.\qquad[2.0]

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 $T_e=10^4\ \mathrm K$ is

E_e\approx k_BT_e\approx1.4\times10^{-19}\ \mathrm J\approx0.9\ \mathrm{eV}.\qquad[2.0]

The energy of a photon is

E_\gamma=h\nu\approx6.6\times10^{-24}\ \mathrm J\approx4\times10^{-5}\ \mathrm{eV}.\qquad[1.0]

As the energy of the radio photon is much much smaller than that of the electron, the answer is “Yes”. [1.0]

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

B_\nu=\frac{2k_BT_e\nu^2}{c^2}.\qquad[1.0]

S_\nu\propto B_\nu\tau_\nu=\frac{2k_BT_e\nu^2}{c^2}\tau_\nu,\qquad S_\nu\propto\nu^2\cdot\nu^{-2.1},\qquad\therefore S_\nu\propto\nu^{-0.1}.\qquad[3.0]

Source note: The key’s numeric mean-free-path substitution prints an extra $\sigma$ in the denominator before $10^{-21}$ . This printed form is retained alongside its native result; it has not been silently corrected.

Теми и трудност

Оценена трудност: 3/5.

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Оригинал в Архива: GeCAA-2020_Theory.pdf · официални решения: Theory Solutions.pdf · допълнителни данни: GeCAA-2020_Constants.pdf