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IAO 2018, Теория, α+β — β-1. Distance to galaxy

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Всички материали от IAO 2018

XXIII International Astronomy Olympiad — Theoretical round, α and β · 6 октомври 2018 г. · 8 т.

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General note. Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may make no real sense. In this case you have to write in your answer (in English): «impossible situation». Of course, this answer has to be explained numerically or logically.

Data from the tables (Planetary data, stars, constants, etc.) may be used for solving every problem.

The answers «Yes» or «No» have to be written in English.

Do not forget that in clean copy you should write solutions only on even (left) pages. The odd (right) pages will be used by your team leader to translate solutions into English and also to emphasize your thoughts in solutions.

Parameters of orbits and physical characteristics of Sun, planets, some dwarf planets and Moon

Body, planet	Average distance to central body (astr. units)	Distance (mln. km)	Sidereal period (tropical years)	Period (days)	Eccentricity e	Equatorial diameter (km)	Mass (10²⁴ kg)	Average density (g/cm³)	Gravity at surface (m/s²)	Max. magnitude from Earth	Albedo
Sun	1.6·10⁹	2.5·10¹¹	2.2·10⁸	8·10¹⁰		1392000	1989000	1.409		−26.8ᵐ
Mercury	0.387	57.9	0.241	87.969	0.206	4879	0.3302	5.43	3.70	−2.2ᵐ	0.06
Venus	0.723	108.2	0.615	224.701	0.007	12104	4.8690	5.24	8.87	−4.7ᵐ	0.78
Earth	1.000	149.6	1.000	365.256	0.017	12756	5.9742	5.515	9.81		0.36
Moon	0.00257	0.38440	0.0748	27.3217	0.055	3475	0.0735	3.34	1.62	−12.7ᵐ	0.07
Mars	1.524	227.9	1.880	686.980	0.093	6794	0.6419	3.94	3.71	−2.0ᵐ	0.15
Ceres	2.77	414	4.60	1681	0.077	963	0.0009	2.16	0.27	6.7ᵐ	0.09
Jupiter	5.204	778.6	11.862	4332.59	0.048	142984	1899.8	1.33	24.86	−2.7ᵐ	0.66
Saturn	9.584	1433.7	29.458	10759.20	0.054	120536	568.50	0.70	10.41	0.7ᵐ	0.68
Uranus	19.191	2871.0	84.015	30685.93	0.046	51118	86.625	1.30	8.44	5.5ᵐ	0.74
Neptune	30.071	4498.6	164.778	60187.64	0.008	49532	102.78	1.76	11.20	7.8ᵐ	0.58
Pluto	39.482	5906.4	248.09	90613	0.249	2374	0.0130	1.86	0.61	15.1ᵐ	0.6

For outer planets and Moon — in mean opposition.

Coordinates	Hotel	Seafront in the center
λ (E)	79°52′	79°51′
φ (N)	07°00′	06°54′
Timezone	UT+05:30	UT+05:30

Some constants and formulae

Quantity	Value
Speed of light in vacuum, c (m/s)	299792458
Constant of gravitation, G (N·m²/kg²)	6.674·10⁻¹¹
Solar constant, A (W/m²)	1367
Hubble parameter, H₀ (km/s/Mpc): mean value	68
Hubble parameter: diapason of values	50–100
Planck constant, h (J·s)	6.626·10⁻³⁴
Charge of electron, e (C)	1.602·10⁻¹⁹
Mass of electron, mₑ (kg)	9.109·10⁻³¹
Proton-to-electron mass ratio	1836.15
Faraday constant, F (C/mol)	96485
Magnetic constant, μ₀ (H/m)	1.257·10⁻⁶
Universal gas constant, R (J/mol/K)	8.314
Boltzmann constant, k (J/K)	1.381·10⁻²³
Stefan-Boltzmann constant, σ (W/m²/K⁴)	5.670·10⁻⁸
Wien’s displacement constant, b (m·K)	0.002897
Laboratory wavelength of Hα (Å)	6563
Laboratory wavelength of Hβ (Å)	4861
Tropical year length, T (days)	365.242199
Sidereal year length, T (days)	365.25636
Anomalistic year length, T (days)	365.259636
Nodal period of lunar orbit (years)	−18.6
Standard pressure (Pa)	101325
Visible light extinction by the terrestrial atmosphere in zenith (minimum)	19%, 0.23ᵐ
Height of homogeneous atmosphere (m)	7991
Refractive index of water for 20°C, n	1.334
Moment of inertia of solid ball	$I=\frac25MR^2$
Moment of inertia of sphere	$I=\frac23MR^2$
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

Theoretical round. Basic criteria. For work of Jury

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

Note. 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.

Note. Jury members should elaborate more detailed criteria, and also create criteria for other correct ways of the student’s solutions.

Printed source discrepancies

The galaxy key prints nm for the wavelengths 6563 and 4861, while the constants sheet gives Å. The Martian key reverses the question’s subpart order, prints a lunar-radius term 0.17·10⁵ km, and reuses L₂ for the semimajor axis. The satellite key calls R a diameter and prints 10h20m after subtracting 62 minutes from 11h12m. Printed formulas and labels are retained rather than silently corrected. The satellite key refers to a drawing absent from the supplied solution packet.

Условие

By research of a supernova explosion in a distant galaxy, two scientists noticed that the hydrogen line Hβ, which was observed in its spectrum, exactly matched the laboratory line Hα. However, these scientists hold different views on the numerical value of the Hubble parameter. Using different values that differ from each other by $\Delta H_0=H_2-H_1=14\ \mathrm{km/s/Mpc}$ , they obtained different values for the maximum absolute magnitude of the observed supernova: $M_1=-19.02^{\mathrm m}$ and $M_2=-18.64^{\mathrm m}$ . Find, according to the point of view of the scientists, the redshift of this distant galaxy and the distance to it.

Решение

The redshift of this galaxy is:

z=\Delta\lambda/\lambda=(\lambda_{\mathrm{H}\alpha}-\lambda_{\mathrm{H}\beta})/\lambda_{\mathrm{H}\beta}=(6563\ \mathrm{nm}-4861\ \mathrm{nm})/4861\ \mathrm{nm}=0.35,

and equally, regardless of the point of view of scientists regarding the value of the Hubble parameter (data about the wavelengths of Hα and Hβ are found in the table of constants). With such values of $z$ , in estimating problems it is still possible to ignore relativistic effects and use the equations of classical physics. So the galaxy is moving away from us with speed

V=cz=300\,000\ \mathrm{km/s}\times0.35=105\,000\ \mathrm{km/s}.

According to the Hubble rule, the receding speed of galaxy is proportional to distance to it:

V=H_0\times L,

where $H_0$ is the Hubble parameter.

Since the two scientists used different values of the Hubble parameter ( $H_1$ and $H_2$ ), they obtained different distances to the researching galaxy:

L_1=V/H_1,\qquad L_2=V/H_2

and, accordingly, observing the same apparent magnitude of the supernova, obtained different values of its absolute magnitude:

M=m-5\lg(L/10\ \mathrm{pc}),

that is,

M_1=m-5\lg(L_1/10\ \mathrm{pc}),\qquad M_2=m-5\lg(L_2/10\ \mathrm{pc}),

therefore

M_2-M_1=-5\lg L_2+5\lg L_1=-5\lg(L_2/L_1),

L_2/L_1=10^{-\Delta M/5},

L_2/L_1=10^{-0.38/5}=10^{-0.076}=0.840.

Thus, since $H_1L_1=V=H_2L_2$ and $H_2-H_1=\Delta H_0$ , we get:

H_1L_1=H_2L_2,

H_1L_1=(H_1+\Delta H_0)L_1(L_2/L_1),

H_1=H_1(L_2/L_1)+\Delta H_0(L_2/L_1),

H_1(1-L_2/L_1)=\Delta H_0(L_2/L_1),

H_1=\Delta H_0(L_2/L_1)/(1-L_2/L_1).

Numerically:

H_1=14\ \mathrm{km/s/Mpc}\times0.84/0.16\approx73\ \mathrm{km/s/Mpc},

H_2=H_1+\Delta H_0=87\ \mathrm{km/s/Mpc}.

It remains to use the formula $V=H_0\times L$ and calculate the distance to the galaxy. However, what is the value of $H_0$ should we take? It turns out that the points of view of the scientists differ on this issue:

According to the point of view of the first scientist,

L_1=V/H_1=105\,000\ \mathrm{km/s}/73\ \mathrm{km/s/Mpc}=1.44\ \mathrm{Gpc}.

From the point of view of the second scientist,

L_2=V/H_2=105\,000\ \mathrm{km/s}/87\ \mathrm{km/s/Mpc}=1.21\ \mathrm{Gpc}.

And, if we use the average value of the Hubble parameter taken from the table ( $H_0=68\ \mathrm{km/s/Mpc}$ ), it turns out that

L_0=V/H_0=105\,000\ \mathrm{km/s}/68\ \mathrm{km/s/Mpc}=1.54\ \mathrm{Gpc}.

Basic criteria

The redshift of this galaxy is:

Item	Points
Calculation z from the Doppler effect	2
Receding speed	1
Formulae and calculation H₁ and H₂	2
Issue about different values of distances and calculation distances	2
Correct using relativistic effects	1

Оригинал в Архива: theo.pdf · официални решения: theosol.pdf · допълнителни данни: iao2018_alpha.pdf