Съдържание

IAO 2014, Теория, Група β — Задача 6. Галактиката Сомбреро

Оригинал на условието (изображение)Съобщи за грешка

Всички материали от IAO 2014

Задачи за теоретичния тур · 12 октомври 2014 г. · 8 т.

Обща забележка

Обща забележка. Възможно е не във всички задачи въпросите да са поставени коректно. Някои въпроси (възможно е да е главният въпрос на задачата или да е някой подвъпрос) може да нямат смисъл. В такъв случай в решението трябва да се напише на руски или на английски – „ситуация невозможна“ или „impossible situation“. Разбира се, решението трябва да се подкрепи с пресмятания или логически разсъждения.

Данните от таблиците (Слънчева система, звезди, константи) могат да се използват във всяка задача.

Отговорите „Да – Yes“ или „Нет – No“ трябва да се напишат на руски или на английски.

Условие

Галактиката Сомбреро, M104, е известна с това, че притежава една от свръхмасивните черни дупки. Вляво са дадени снимки на галактиката и нейния център. Размерите на снимките са $10'\times7'$ и $10{,}8''\times7{,}2''$ съответно. Вдясно са дадени три спектъра (с емисионни линии на йонизирания кислород и азот) на области около центъра на галактиката, получени с космическия телескоп Хъбъл. Долният спектър е на област около самия център, двата други спектъра са на съседни области. Отгоре около пунктираните линии са дадени стойности в ангстрьоми на съответните лабораторни дължини на вълните.

Оригинални изображения на Сомбреро, стрелки към областите и трите спектъра

6.1.

Приближава ли се към нас или се отдалечава M104? (Отговорете на въпроса като нарисувате схема).

6.2.

Оценете скоростта на M104 към нас или от нас.

6.3.

Грубо оценете масата на свръхмасивната черна дупка в центъра на M104 (отговора дайте в слънчеви маси $m_\odot$ ).

Решение

Theoretical round. Sketches for solutions

6.1. All the lines in the spectra are shifted somewhat to longer wavelengths (redshifted). Obviously, M 104 is receding from us due to the expansion of the Universe.

6.2. The overall receding velocity of M 104 may be found from the redshift of the spectrum of the central region. Taking the average redshift of the four emission lines, we find that

z_{\rm av}=(\Delta\lambda/\lambda)_{\rm average}\approx 0.0035,

and since $z\ll 1$ , the overall receding velocity of M 104 is given by:

V=cz=0.0036\times 300000\ \mathrm{km/s}=1050\ \mathrm{km/s}.

6.3. According to the Hubble law the distance to Sombrero Galaxy is

L=V/H=\frac{1050\ \mathrm{km/s}}{71\ \mathrm{km/s/Mpc}}\approx 15\ \mathrm{Mpc}.

Note: In reality the distance to Sombrero Galaxy is 9 Mpc. But we continue solution according to the calculated value,15 Mpc.

The first and second spectra have different shifts than the third one. It indicates that the gas which is generating the emission lines has a motion relative to the center of M 104. The amounts of these redshifts are not too clear but evidently less than the third one. To find the mass of the smaller region near the center, we will use the first spectrum as obtained closer to the center region. The distance from the center of the galaxy to this near-center region can be found by measuring sizes at the lower picture. The angular distance is approximately 0.05 of the width of the lower image, $\alpha=0.05\times 10.8''=0.54''$ . To get the actual distance in a.u., it is easy to use the formula

R(\mathrm{a.u.})=\alpha(\mathrm{arcsec})\times L(\mathrm{pc}),

R(\mathrm{a.u.})=0.54\times 15\cdot 10^6\approx 8\cdot 10^6.

Note: In reality $R(\mathrm{a.u.})=0.54\times 9\cdot 10^6\approx 5\cdot 10^6$ .

The average difference in redshifts of the first and third spectra may be estimated as about $\Delta z\approx 0.0013$ . Thus the velocity of the gas relative to the galactic center is about

V_1=c\Delta z=0.0013\times 300000\ \mathrm{km/s}\approx 400\ \mathrm{km/s}.

Then, let us assume that the gas in the region of the third spectrum revolves around the galactic center in a circular orbit with the period T. Its revolving velocity is

V_1=2\pi R/T=400\ \mathrm{km/s}.

By comparing with the average velocity of the Earth revolving around the Sun (A=1 au, P=1 year),

V_2=2\pi A/P=30\ \mathrm{km/s},

we find that

R/T=(V_1/V_2)(A/P)\approx 13\ \mathrm{au/year}.

Assuming that the mass of the gas is much less than that of the galactic center and applying the Kepler’s III law (in units solar mass–au–year), we obtain the mass of the galactic center

M=R^3/T^2\approx 13^2R\approx 200\times 8\cdot 10^6\approx 1.6\cdot 10^9\quad(\mathrm{in}\ m_{\odot}).

Note: In reality, we should write $M\approx 200\times 5\cdot 10^6m_{\odot}\approx 10^9m_{\odot}$ .

It is an approximation for the mass of the supermassive black hole in the center of M 104.

Theoretical round. Basic criteria

6.1. Spectra redshifted, receding from us — 1 pt.
6.2. z_av=(Δλ/λ)_average≈0.0035 — 1 pt; V=cz=0.0036×300000 km/s=1050 km/s — 1 pt.
6.3. L=V/H=1050 km/s /71 km/s/Mpc≈15 Mpc — 1 pt; R(au)=0.54×15·10⁶≈8·10⁶ — 1 pt; other correct calculations and obtaining mass of galactic center — 3 pt.

Официални решения и оценяване

Официалните скици на решенията и критериите са запазени на оригиналния английски език.

Note for jury and team leaders

The proposed sketches are not full; the team leaders have to give more detailed explanations for students. But the correct solutions in the students’ papers (enough for 8 pts) may be shorter.

Бележка

Jury members should evaluate the student’s solutions in essence, and not by looking on formal existence the mentioned sentences or formulae. The formal presence of the mentioned positions in the text is not necessary to give the respective points. Points should be done if the following steps de facto using these positions.

Parameters of orbits. Physical characteristics of some planets, Moon, Sun and Eris

Body	Distance (AU)	Distance (10⁶ km)	Period (tropical years)	Period (days)	Inclination i (°)	Eccentricity	Equatorial diameter (km)	Mass (10²⁴ kg)	Density (g/cm³)	Gravity (m/s²)	Axial tilt (°)	Max. magnitude	Albedo
Sun	$1.6\cdot10^9$	$2.5\cdot10^{11}$	$2.2\cdot10^8$	$8\cdot10^{10}$			1392000	1989000	1.409			−26.74
Mercury	0.387	57.9	0.241	87.969	7.00	0.206	4879	0.3302	5.43	3.70	0.01		0.06
Venus	0.723	108.2	0.615	224.7007	3.40	0.007	12104	4.8690	5.24	8.87	177.36		0.78
Earth	1.000	149.6	1.000	365.2564	0.00	0.017	12756	5.9742	5.515	9.81	23.44		0.36
Moon	0.00257	0.38440	0.0748	27.3217	5.15	0.055	3475	0.07348	3.34	1.62	6.7	−12.7	0.07
Mars	1.524	227.9	1.880	686.98	1.85	0.093	6794	0.6419	3.94	3.71	25.19	−2.0	0.15
Jupiter	5.204	778.6	11.862	4332.59	1.30	0.048	142984	1899.8	1.33	24.86	3.13	−2.7	0.66
Saturn	9.584	1433.7	29.458	10759.20	2.48	0.054	120536	568.50	0.70	10.41	26.73	0.7	0.68
Eris	68.05			205029	43.82	0.435	2326	0.0167	2.52	0.7			0.96

Source footnote: for outer planets and Moon — in mean opposition.

Some constants and formulae

Constant or formula	Value
Speed of light in vacuum c (m/s)	299792458
Constant of gravitation G (N·m²/kg²)	$6.674\cdot10^{-11}$
Solar constant A (W/m²)	1367
Hubble parameter H₀ (km/s/Mpc), mean	71
Hubble parameter, range	50–100
Planck constant h (J·s)	$6.626\cdot10^{-34}$
Charge of electron e (C)	$1.602\cdot10^{-19}$
Mass of electron mₑ (kg)	$9.109\cdot10^{-31}$
Proton-to-electron ratio	1836.15
Faraday constant F (C/mol)	96485
Magnetic constant μ₀ (H/m)	$1.257\cdot10^{-6}$
Universal gas constant R (J/mol/K)	8.314
Boltzmann constant k (J/K)	$1.381\cdot10^{-23}$
Stefan–Boltzmann constant σ (W/m²/K⁴)	$5.670\cdot10^{-8}$
Wien displacement constant b (m·K)	0.002897
Laboratory wavelength Hα (Å)	6562.81
Tropical year length T (days)	365.242199
Standard atmosphere (Pa)	101325
Visible light extinction by terrestrial atmosphere in zenith (minimum)	19%, 0.23 magnitudes
Height of homogeneous atmosphere (m)	7991
Refractive index of water for 20°C n	1.334
Moment of inertia of a solid ball	$I=\frac25MR^2$
Volume of a ball	$V=\frac43\pi R^3$
Area of sphere	$S=4\pi R^2$
π	3.14159265
e	2.71828183
Golden ratio φ	1.61803399

Coordinates	Bishkek	Cholpon-Ata
Longitude E	74°34′	77°05′
Latitude N	42°52′	42°39′
Timezone	UT+6	UT+6

28 brightest stars in the sky. Some other stars

Name	Designation	RA	Dec	Parallax (arcsec)	Magnitude	Spectral class
Altair	α Aql	19h50m47s	08°52′06″	0.195	0.77	A7
Capella	α Aur	05h16m41s	45°59′53″	0.073	0.08	G5+G0
Arcturus	α Boo	14h15m38s	19°10′57″	0.089	−0.04ᵛ	K1
Canopus	α Car	06h23m57s	−52°41′45″	0.028	−0.72	F0
Miaplacidus	β Car	09h13m12s	−69°43′02″	0.029	1.68	A2
Toliman (Rigel Kent)	α Cen	14h39m36s	−60°50′07″	0.747	−0.01; 1.33	G2; K1
Hadar	β Cen	14h03m49s	−60°22′23″	0.009	0.61	B1
Sirius	α CMa	06h45m09s	−16°42′58″	0.375	−1.46	A1
Adara	ε CMa	06h58m38s	−28°58′19″	0.008	1.50	B2
Procyon	α CMi	07h39m18s	05°13′30″	0.288	0.38	F5
Acrux	α Cru	12h26m36s	−63°05′57″	0.010	0.77	B0
Becrux	β Cru	12h47m43s	−59°41′20″	0.009	1.30	B0
Gacrux	γ Cru	12h31m10s	−57°06′48″	0.037	1.60	M4
Deneb	α Cyg	20h41m26s	45°16′49″	0.002	1.25	A2
Achernar	α Eri	01h37m43s	−57°14′12″	0.026	0.46	B3
Castor	α Gem	07h34m36s	31°53′18″	0.065	1.58	A1+A2
Pollux	β Gem	07h45m19s	28°01′35″	0.097	1.14	K0
Regulus	α Leo	10h08m22s	11°58′02″	0.041	1.35	B7
Vega	α Lyr	18h36m56s	38°47′01″	0.123	0.03	A0
Betelgeuse	α Ori	05h55m10s	07°24′25″	0.005	0.5ᵛ	M2
Rigel	β Ori	05h14m32s	−08°12′06″	0.013	0.12	B8
Bellatrix	γ Ori	05h25m08s	06°20′59″	0.013	1.64	B2
Fomalhaut	α PsA	22h57m39s	−29°37′20″	0.130	1.16	A3
Antares	α Sco	16h29m24s	−26°25′55″	0.024	0.96	M1+B4
Shaula	λ Sco	17h33m37s	−37°06′14″	0.005	1.62	B2
Aldebaran	α Tau	04h35m55s	16°30′33″	0.048	0.85ᵛ	K5
Elnath	β Tau	05h26m18s	28°36′27″	0.025	1.66	B7
Spica	α Vir	13h25m12s	−11°09′41″	0.023	0.98	B1

Some other stars

Name	Designation	RA	Dec	Parallax (arcsec)	Magnitude	Spectral class
Hamal	α Ari	02h07m10s	23°27′45″	0.050	2.01	K2
Sheratan	β Ari	01h54m39s	20°48′30″	0.055	2.64	A5
Thuban	α Dra	14h04m23s	64°22′33″	0.011	3.65	A0
Rastaban	β Dra	17h30m26s	52°18′05″	0.009	2.79	G2
Eltamin	γ Dra	17h56m36s	51°29′20″	0.021	2.24	K5
Polaris	α UMi	02h31m49s	89°15′51″	0.007	1.97ᵛ	F7
Kochab	β UMi	14h50m42s	74°09′20″	0.025	2.07	K4
Pherkad	γ UMi	15h20m44s	71°50′02″	0.007	3.06ᵛ	A3
Yildun	δ UMi	17h32m19s	86°35′11″	0.019	4.35	A1
Proxima Centauri	V645 Cen, α Cen C	14h29m43s	−62°40′46″	0.769	11.05	M5.5

Source discrepancy

the key prints 262.12° in 4.2; its printed β=48°33′ gives 262°54′ for 360°−2β. The printed angle and time are retained.

Source discrepancy

the winter ratio is printed with (R₀−R)² in its denominator although the preceding winter distance is R₀+R and the numerical0.38 corresponds to the latter. The printed formula is retained. Calligraphic ℰ here represents the separate radiated-energy symbol used in the key.

Source discrepancy

the key switches from z=0.0035 to0.0036 while retaining1050 km/s; the latter multiplication would give1080. It also changes the moving gas from the first spectrum to the third in its prose. Both inconsistencies are retained. The “in reality” values above are statements of the2014 source.

Оценяване: Proxima Centauri

The printed Proxima Centauri allocations are 1 + 3×1 + 1 + 1.5 + 1.5 = 8 points, matching the whole-problem scale.

Оригинал в Архива: bgtheo6.jpg · официални решения: theosol.pdf · допълнителни данни: bgtheo2.jpg · допълнителни данни: bgtheo3.jpg · допълнителни данни: bgtheo4.jpg · допълнителни данни: bgtheo5.jpg · допълнителни данни: bgtheo6.jpg · допълнителни данни: theocriteria.pdf · допълнителни данни: sombrero.pdf · допълнителни данни: constants.pdf · допълнителни данни: hrd.pdf