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Earth Moon Sun Jupiter Saturn Physical constants Условие Решение

IOAA 2017, Теория — Задача 8. The Star of Bethlehem

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Всички материали от IOAA 2017

IOAA 2017 — Theoretical Examination · 20 т.

Original Examination Instructions

The theoretical examination lasts for 5 hours and is worth a total of 300 marks.
Dedicated IOAA Summary Answer Sheets are provided for writing your answers. Enter the final answers into the appropriate boxes in the corresponding Summary Answer Sheet. On each Summary Answer Sheet, please fill in
- Student Code (Country Code and 1 digit)
There are Answer Sheets for carrying out detailed work/rough work. On each Answer Sheet, please fill in
- Student Code (Country Code and 1 digit)
- Question no.
- Page no. and total number of pages.
Start each problem on a separate Answer Sheet. Please write only on the printed side of the sheet. Do not use the reverse side. If you have written something on any sheet which you do not want to be evaluated, cross it out.
Use as many mathematical expressions as you think may help the evaluator to better understand your solutions. The evaluator may not understand your language. If it is necessary to explain something in words, please use short phrases (if possible in English).
You are not allowed to leave your work desk without permission. If you need any assistance (malfunctioning calculator, need to visit a restroom, need more Answer Sheets, etc.), please draw the attention of the proctor using the Help card.
The beginning and end of the examination will be indicated by a long sound signal. Additionally, there will be a buzzer sound, fifteen minutes before the end of the examination (before the final sound signal).
At the end of the examination you must stop writing immediately. Sort and put your sheets in separate stacks, a) Stack 1: Summary Answer Sheets, Answer Sheets of part 1
b) Stack 2: Summary Answer Sheets, Answer Sheets of part 2
c) Stack 3: Summary Answer Sheets, Answer Sheets of part 3
d) Stack 4: question papers and paper sheets you do not want to be graded.
Wait at your table until your envelope is collected. Once all envelopes are collected, your student guide will escort you out of the examination room.
A list of constants and a table of the mark distribution for this exam are given on the next two pages.

Original List of Constants

Earth

Quantity	Value
Mass ( $M_\oplus$ )	$5.98\times10^{24}$ kg
Radius ( $R_\oplus$ )	$6.38\times10^6$ m
Acceleration of gravity ( $g$ )	$9.8 \mathrm{m\cdot s^{-1}}$
Obliquity of Ecliptic	$23^\circ27'$
Length of Tropical Year	365.2422 mean solar days
Length of Sidereal Year	365.2564 mean solar days
Albedo	0.39

Moon

Quantity	Value
Mass (M with subscript ☾)	$7.35\times10^{22}$ kg
Radius (R with subscript ☾)	$1.74\times10^6$ m
Mean distance from Earth	$3.84\times10^8$ m
Orbital inclination with the Ecliptic	$5.14^\circ$
Albedo	0.14
Apparent magnitude (mean full moon)	−12.74

Sun

Quantity	Value
Mass ( $M_\odot$ )	$1.99\times10^{30}$ kg
Radius ( $R_\odot$ )	$6.96\times10^8$ m
Luminosity ( $L_\odot$ )	$3.83\times10^{26}$ W
Absolute Magnitude ( $\mathcal M_\odot$ )	4.80 mag
Angular diameter	0.5 degrees
Rotational velocity in the Galaxy	$220 \mathrm{km\,s^{-1}}$
Distance from Galactic centre	8.5 kpc

Jupiter

Quantity	Value
Mass	$1.89\times10^{27}$ kg
Orbital semi-major axis	5.20 au
Orbital period	11.86 year

Saturn

Quantity	Value
Mass	$5.68\times10^{26}$ kg
Orbital semi-major axis	9.58 au
Orbital period	29.45 year

Physical constants

Quantity	Value
1 au	$1.50\times10^{11}$ m
1 pc	206,265 au
Gravitational constant ( $G$ )	$6.67\times10^{-11} \mathrm{N\cdot m^2\cdot kg^{-2}}$
Planck constant ( $h$ )	$6.62\times10^{-34} \mathrm{J\cdot s}$
Boltzmann constant ( $k_B$ )	$1.38\times10^{-23} \mathrm{J\cdot K^{-1}}$
Stefan-Boltzmann constant ( $\sigma$ )	$5.67\times10^{-8} \mathrm{W\cdot m^{-2}\cdot K^{-4}}$
Hubble constant ( $H_0$ )	$67.8 \mathrm{km\,s^{-1}\,Mpc^{-1}}$
Speed of light in vacuum ( $c$ )	$299,792,458 \mathrm{m\cdot s^{-1}}$
Proton mass	$938.27 \mathrm{MeV\cdot c^{-2}}$
Deuterium mass	$1875.60 \mathrm{MeV\cdot c^{-2}}$
Neutron mass	$939.56 \mathrm{MeV\cdot c^{-2}}$
Helium-3 mass	$2808.30 \mathrm{MeV\cdot c^{-2}}$
Helium-4 mass	$3727.40 \mathrm{MeV\cdot c^{-2}}$

Original Mark Distribution

Problem number	Marks
T1	10
T2	10
T3	10
T4	10
T5	10
T6	15
T7	20
T8	20
T9	20
T10	25
T11	50
T12	40
T13	60
Total	300

Условие

A great conjunction is a conjunction of Jupiter and Saturn for observers on Earth. Assume that Jupiter and Saturn have circular orbits in the ecliptic plane.

The time between successive conjunctions may vary slightly as viewed from the Earth. However, the average time period of the great conjunctions is the same as that of an observer at the centre of the Solar system.

a) Find the average great conjunction period (in years) and average heliocentric angle between two successive great conjunctions (in degrees). [6 т.]

b) The next great conjunction will be on 21st December 2020 with an elongation of $30.3^\circ$ East of the Sun. Suggest the constellation in which the conjunction on 21st December 2020 will occur. (Give the IAU Latin name or IAU three-letter abbreviation of the constellation, i.e. Ursa Major or UMa) [2 т.]

In 1606, Johannes Kepler determined that in some years the great conjunction can be happen three times in the same year due to the retrograde motions of the planets. He also determined that such an event happened in the year 7 BC, which could have been the event commonly known as “The Star of Bethlehem”. For the calculations below you may ignore the precession of the axis of the Earth.

c) Suggest the constellation in which the great conjunctions in 7 BC occurred. (Give the IAU Latin name or IAU three-letter abbreviation of the constellation, i.e. Ursa Major or UMa) [8 т.]

d) During the second conjunction of the series of three conjunctions in 7 BC, suggest the constellation the Sun was in as viewed by the observer on Earth. (Give the IAU Latin name or IAU three-letter abbreviation of the constellation, i.e. Ursa Major or UMa) [4 т.]

Решение

a) Solution

Assume that the average Earth’s orbit is at the Sun. Therefore, average great conjunction period equals to the synodic period between Jupiter and Saturn.

\frac1{P_{Synodic}}=\frac1{P_{Jupiter}}-\frac1{P_{Saturn}},

\frac1{P_{Synodic}}=\frac1{11.86}-\frac1{29.45}.

[3.0]

P_{Synodic}=19.86\ \mathrm{years}.

[1.0]

From average great conjunction period, in 19.86 years Jupiter and Saturn move

Heliocentric angle $=\frac{19.86}{11.86}\times360^\circ=602.7^\circ=242.7^\circ$

in an eastward direction through the zodiac, or $360^\circ-242.7^\circ=117.3^\circ$ in a westward direction through the zodiac. [2.0]

b) Solution

On 21st December 2020 (winter solstice), the Sun will be in the constellation of Sagittarius. Planets are always near the ecliptic. The current 12 zodiac constellations can be roughly divided into 12 equal parts of 30 degrees each.

The conjunction will locate elongation $30.3^\circ$ East to the Sun. Therefore, the conjunction will be in the constellation of Capricornus (Cap). [2.0]

Justification is NOT necessary for the full mark for this part.

c) Solution

The duration between 7BC and 2020 is 2026 years or 2027 years.

Each conjunction occurs on the average every 19.86 years. Therefore, the number of conjunction between 7BC and 2020 is

Integer value of $2026/19.86=102$ . [2.0]

Correct number of conjunctions 1.0 mark. Correct number of conjunctions with integer 2.0 marks.

From a conjunction to its previous successive conjunction, the conjunction moves $243^\circ$ westwards. The position of the conjunction in 7BC is

102\times242.7^\circ=24758.33^\circ=278.3^\circ

[3.0]

westward, or equivalently $81.7^\circ$ eastward from the position of the conjunction in 2020. Therefore, the conjunction was in the constellation of Aries (Ari) or Pisces (Psc). [3.0]

Justification is NOT necessary for the full mark for this part. Correct answer only.

d) Solution

At the second conjunction, Jupiter and Saturn had retrograde motions. For observers on the Earth, both of them were in opposition. Therefore, the sun was in the constellation of Libra (Lib) (if answer c) is Aries) or Virgo (Vir) (if answer c) is Pisces). [4.0]

This question is marked only if students get c) correctly.

Source arithmetic retained: the printed multiplication in c) is $102\times242.7^\circ=24758.33^\circ$ , as shown in the original key.

Бележки за изданието

The original question and key contain several differing expressions. The Galactic-arm question uses a flat curve beyond 5 kpc and its worked key uses 2 kpc. The HI survey key prints differing flux-threshold exponents and an unusual Hubble-constant unit. The Bethlehem key has differing multiplication and remainder values. The Galactic Outflow key heading says T2 while its full prompt matches T9. GOTO assigns a different pair of marks to parts a) and d) in the question and key. The Shipwreck key prints differing sidereal-time values in its main and alternative methods. Visible missing-font glyph squares and the Exomoon key’s grading dependency are preserved. The original constants table prints the acceleration unit as m·s^-1. These printed forms remain literal.

Оригинал в Архива: Theory.pdf · официални решения: Theory_Solution.pdf