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IAO 2018, Теория, α+β — β-2. Martian observations

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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

In July this year, within one day, two unique astronomical phenomena occurred. In the morning of July 27, Mars came into a great opposition (see the attached ephemeris), and in the evening, the inhabitants of the Earth (those lucky with the weather and geographical location) observed a long lunar eclipse. But the rapprochement of the Earth and Mars is also a good time for Martian astronomers to study the Earth and The Moon. The Martian Space Agency planned two expeditions to the near-Mars piloted orbital station to observe:

Geocentric ephemerides for Mars, Moon and Sun, 2018; all printed rows, headings and events.

2.1.

transit of the Earth across the disk of the Sun (it is assumed that the entire disk of the Earth must pass at least along the edge of the Sun) to study the atmosphere of the Earth;

2.2.

transit of the Moon across the disk of the Earth (it is assumed that the entire disk of the Moon must pass over the disk of the Earth) to study the lunar atmosphere.

For each of these observations, determine in which constellation the objects will be observed from Mars, whether the orbital expedition is feasible (write “Yes” or “No” in English), and if so, find the minimum orbital period of Martian orbital station of the expedition, and if not, prove it by making the necessary calculations and drawings.

Решение

Martians will observe the Sun, the Earth and the Moon in the constellation opposite to that in which we observe the full Moon and the opposition of Mars in the end of July. That is, this is exactly the constellation in which at the end of July the Sun is located for observers at the Earth. In the constellation of Cancer.

2.1.

Using the data of the ephemeris, we can interpolate the values of the right ascension and declination of the Moon and Mars at the time of the mid-point of the lunar eclipse. For the Moon, we obtain $\alpha_L=20^{\mathrm h}28^{\mathrm m}$ , $\delta_L=-18^\circ55'$ . For Mars at the same time — $\alpha_M=20^{\mathrm h}32^{\mathrm m}$ , $\delta_M=-25^\circ33'$ . We see that the difference in right ascensions is negligible compared with the difference in declinations, therefore the angular distance between the centers of the Moon and Mars at this time with great accuracy will be $\Delta\delta=6^\circ38'$ .

The center of the moon will be at a distance

L_0=R_{EL}\times\sin\Delta\delta=4.06\cdot10^5\ \mathrm{km}\times0.116\approx4.69\cdot10^4\ \mathrm{km}

from the line connecting the centers of Mars and Earth, and the northern edge of the Moon is at a distance

L_1=L_0-R_E+R_L=4.69\cdot10^4\ \mathrm{km}-0.64\cdot10^4\ \mathrm{km}+0.17\cdot10^5\ \mathrm{km}\approx4.23\cdot10^4\ \mathrm{km}

from the line connecting the center of Mars and the northern edge of the Earth. The angle between the line connecting the northern points of the Earth and the Moon, and the line connecting the centers of Mars and Earth, will be

\varphi=\arcsin(L_1/R_{EL})\approx5^\circ59'.

Thus, to observe the transit of the entire disk of the Moon across the disk of the Earth, the Martian orbital station must be at a distance

L_2=R_{EM}\times\sin\varphi=0.386\ \mathrm{au}\times1.496\cdot10^8\ \mathrm{km/au}\times0.104\approx6.02\cdot10^6\ \mathrm{km}.

from the center of Mars. This means that the orbital station must have a semi-major axis of the orbit equal to at least

L_2=(L_2+R_M)/2\approx L_2/2\approx3\cdot10^6\ \mathrm{km}.

Orbiting of bodies around Mars in orbit with such a large semi-axis is impossible, such an orbit goes beyond Hill’s sphere.

R_H=L_{SM}\times\left(\frac{M_M}{3(M_S+M_M)}\right)^{1/3}\approx R_{SM}\times\left(\frac{M_M}{3M_S}\right)^{1/3}\approx\frac1{210}R_{SM}\approx1.1\cdot10^6\ \mathrm{km}.

2.2.

This part of the problem is solved similarly to the previous one. In order to observe the passage of the Earth along the solar disk, the Martians must hit a point that is directly opposite to the position of the center of the Sun for the earth observer (with an accuracy of the angular radius of the Sun seen from Mars). From the ephemeris of the Sun, we see that at the time of the Mars opposition, the declination of such a point is approximately $\delta_{-S}=-19^\circ13'$ . For Mars, this value is $\delta_M=-25^\circ30'$ . The angular distance between the desired point and Mars at this time with high accuracy will be $\Delta\delta=6^\circ17'$ . With a correction for the angular radius of the Sun, the minimum angle required for Martians from Mars will be $\psi=6^\circ06'$ .

Thus, in order to observe the passage of the Earth across the disk of the Sun, the Martian orbital station must be at a distance

L_2=R_{EM}\times\sin\psi=0.386\ \mathrm{au}\times1.496\cdot10^8\ \mathrm{km/au}\times0.106\approx6.14\cdot10^6\ \mathrm{km}.

from the center of Mars. This means that the orbital station must have a semi major axis of the orbit equal to at least

L_2=(L_2+R_M)/2\approx L_2/2\approx3\cdot10^6\ \mathrm{km}.

Orbiting of bodies around Mars in orbit with such a large semi-axis is impossible, such an orbit goes beyond Hill’s sphere.

R_H=L_{SM}\times\left(\frac{M_M}{3(M_S+M_M)}\right)^{1/3}\approx R_{SM}\times\left(\frac{M_M}{3M_S}\right)^{1/3}\approx\frac1{210}R_{SM}\approx1.1\cdot10^6\ \mathrm{km}.

Basic criteria

Item	Points
Correct constellation	1
Formulae, calculation distance from Mars for the first expedition	1
Formulae, calculation distance from Mars for the second expedition	1
Formulae, calculation theoretical periods of orbital stations	1
Solution with possibility to send orbital station	max. 4
Formulae, calculations and issue about impossibility to send orbital station	3
Formulae, calculation diameter of Hill sphere	1

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