IAO 2018, Теория, α+β — α-1. Mercury mirror
XXIII International Astronomy Olympiad — Theoretical round, α and β · 6 октомври 2018 г. · 8 т.
Read once more before you start your work
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 | |
| Moment of inertia of sphere | |
| Volume of a ball | |
| Area of sphere | |
| 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
As you know, if one rotates a vessel with mercury, its surface will take a parabolic form. A liquid mirror obtained in this way can be used as the main mirror of a telescope. The Polar Bear-astronomer living in the Arctic decided to use this idea and build such a telescope exactly at the North Pole, and use the daily revolution of the Earth as the rotation mechanism so that the main mirror would have a long focal length.
1.1.
Is the Polar Bear idea feasible? (Write “Yes” or “No” in English.) If yes, then find the focal length of the resulting liquid mirror, and if not, prove your conclusion by doing the necessary calculations and drawings.
1.2.
Include an artistic picture with an image of the Bear-astronomer, engaged in the implementation of his project.
Решение
Imagine rotating mercury. At each point, the normal to its surface will be directed at an angle to the vertical, where is the angular velocity of rotation of mercury, is the distance from the axis of rotation, is the gravity acceleration. If the direction of the vertical (vector ) is fixed, then this angle is related to the radius of curvature of the liquid mirror as . From these two equations we get:
Substituting the values and , we get:
The focal length of such a mirror would be:
But … The result would be true for a flat Earth. The condition “if the direction of the vertical is constant” is written above not accidentally. In the case of a real Earth, when moving away from the Pole, the direction of the vertical changes 300 times faster (the Earth's radius is 300 times smaller than 1.84 million km). That is, the rotation of the Earth only reduces the curvature of the liquid surface at the pole by 1/300. The mirror designed by the Bear will be convex, its curvature will be exactly the same as the curvature of the Arctic Ocean.
Thus, the correct answer is “No”, the idea of Polar Bear is impracticable.
Basic criteria
| Item | Points |
|---|---|
| 1.1. If answer about possibility is positive | max 3 |
| If answer about possibility is negative | max 7 |
| Formulae and calculations about mirror | 3 |
| Taking into account curvature of the Earth | 1 |
| Conclusion about impossibility with proving | 3 |
| 1.2. Artistic picture of the Bear-astronomer | 1 |
Оригинал в Архива: theo.pdf · официални решения: theosol.pdf · допълнителни данни: iao2018_alpha.pdf