Съдържание

Условие Практический тур. Таблица к задаче 8 / Practical round. Table for Problem 8 Parameters of orbits. Physical characteristics of some planets, Moon, Sun and Eris Решение

IAO 2012, Практически тур, Група α — Задача 8. Moon

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2012

IAO2012 — Practical round (α) · 10 т.

Practical round. Problems to solve

XVII International Astronomy Olympiad. Gwangju, Korea,16–24 X 2012. Group α.

Note. If you find somewhere in the problems an impossible situation, write in English «impossible situation».

Бележки от източника

The official Fireball solution explicitly acknowledges an incorrectly plotted observing point II. It gives A/B latitudes36.47°/36.10° in7.2, then36.49°/36.11° in7.4 and its final table. These printed values are retained.

Missing plotting form

The original blank Moon eccentric-orbit plotting sheet (a) referred to in8.3/8.5 is absent from the complete same-year source inventory; the available paper.png is the Fireball geographic grid.

Printed solution details

The official Moon solution prints an arcsecond mark after384400 and gives a lunar diameter3476.4km whereas the shared reference table gives3475km. Its culmination-difference table is retained as printed, including values that differ from direct subtraction of adjacent times. These are not silently recalculated.

Shared source pack

The six official practical solution images and grading sheet cover both group editions; this edition includes only its matching problem8. Four theory-labelled reference sheets were inspected. The orbital-properties table supports Moon, and the Hertzsprung-Russell diagram is explicitly mentioned by the Clusters solution. The constants and individual-star tables supply reference context, not extra practical tasks. The geographic grid belongs to Fireball.

Условие

The Korean Astronomy and Space Science Institute (KASI) publishes the Korean Astronomical Almanac every year. You have been provided with an astronomical table extracted from the Korean Astronomical Almanac of 2012 showing the Korean local time of Moon culmination. (See separate sheet, and you may fill the empty cells by necessary content.)

Date	Culmination of Moon		Date	Culmination of Moon		Date	Culmination of Moon
Mar 2			April 1			May 1

Also you are provided with a scaled marked graph paper to plot graphs.

8.1 Find the date in April 2012 when the Moon is closest to the Earth.

8.2 Find the date in March 2012 when the Moon is remotest from the Earth.

8.3 The scaled marked graph paper (a) shows the eccentric orbit of the Moon, the Earth being located at the center. Mark the positions of the Moon by the symbol × on April 19 and April 23 (with labels A19 and A23).

8.4 Calculate the ratio of the apparent angular sizes of the Moon ( $\alpha_{\mathrm{Moon}}$ ) and the Sun ( $\alpha_{\mathrm{Sun}}$ ) on July 1.

8.5 Draw on the scaled marked graph paper the geostationary orbit around the Earth in the given scale.

Практический тур. Таблица к задаче 8 / Practical round. Table for Problem 8

Дата / Date	Кульминация Луны / Culmination of Moon	Дата / Date	Кульминация Луны / Culmination of Moon	Дата / Date	Кульминация Луны / Culmination of Moon
Mar 2	19 40	April 1	20 02	May 1	20 20
Mar 3	20 31	April 2	20 52	May 2	21 10
Mar 4	21 22	April 3	21 42	May 3	22 02
Mar 5	22 14	April 4	22 33	May 4	22 57
Mar 6	23 05	April 5	23 25	May 5	23 56
Mar 7	23 56	April 6	-	May 6	-
Mar 8	-	April 7	0 19	May 7	0 57
Mar 9	0 48	April 8	1 16	May 8	2 01
Mar 10	1 41	April 9	2 16	May 9	3 03
Mar 11	2 36	April 10	3 17	May 10	4 04
Mar 12	3 32	April 11	4 19	May 11	5 00
Mar 13	4 31	April 12	5 19	May 12	5 52
Mar 14	5 30	April 13	6 15	May 13	6 40
Mar 15	6 29	April 14	7 08	May 14	7 26
Mar 16	7 26	April 15	7 57	May 15	8 09
Mar 17	8 20	April 16	8 43	May 16	8 52
Mar 18	9 11	April 17	9 28	May 17	9 35
Mar 19	9 59	April 18	10 10	May 18	10 18
Mar 20	10 45	April 19	10 53	May 19	11 02
Mar 21	11 29	April 20	11 36	May 20	11 48
Mar 22	12 12	April 21	12 19	May 21	12 36
Mar 23	12 55	April 22	13 04	May 22	13 24
Mar 24	13 38	April 23	13 51	May 23	14 14
Mar 25	14 22	April 24	14 39	May 24	15 03
Mar 26	15 08	April 25	15 27	May 25	15 52
Mar 27	15 54	April 26	16 17	May 26	16 40
Mar 28	16 43	April 27	17 06	May 27	17 27
Mar 29	17 32	April 28	17 55	May 28	18 14
Mar 30	18 22	April 29	18 43	May 29	19 02
Mar 31	19 12	April 30	19 31	May 30	19 51

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

Complete bilingual reference table of orbital periods, distances, diameters, masses and physical properties

Решение

α-8. Moon

Date	Culmination of Moon	$\Delta T$ (min)	Date	Culmination of Moon	$\Delta T$ (min)	Date	Culmination of Moon	$\Delta T$ (min)
Mar 2	19 40	51	April 1	20 02	50	May 1	20 20	50
Mar 3	20 31	51	April 2	20 52	50	May 2	21 10	52
Mar 4	21 22	52	April 3	21 42	51	May 3	22 02	55
Mar 5	22 14	51	April 4	22 33	52	May 4	22 57	59
Mar 6	23 05	51	April 5	23 25	54	May 5	23 56	61
Mar 7	23 56	52	April 6	-		May 6	-
Mar 8	-		April 7	0 19	57	May 7	0 57	65
Mar 9	0 48	53	April 8	1 16	57	May 8	2 01	62
Mar 10	1 41	55	April 9	2 16	60	May 9	3 03	61
Mar 11	2 36	56	April 10	3 17	61	May 10	4 04	56
Mar 12	3 32	59	April 11	4 19	62	May 11	5 00	52
Mar 13	4 31	59	April 12	5 19	60	May 12	5 52	48
Mar 14	5 30	59	April 13	6 15	56	May 13	6 40	46
Mar 15	6 29	57	April 14	7 08	53	May 14	7 26	43
Mar 16	7 26	54	April 15	7 57	49	May 15	8 09	43
Mar 17	8 20	51	April 16	8 43	46	May 16	8 52	43
Mar 18	9 11	48	April 17	9 28	45	May 17	9 35	43
Mar 19	9 59	46	April 18	10 10	43	May 18	10 18	42
Mar 20	10 45	44	April 19	10 53	46	May 19	11 02	46
Mar 21	11 29	43	April 20	11 36	43	May 20	11 48	48
Mar 22	12 12	43	April 21	12 19	45	May 21	12 36	48
Mar 23	12 55	43	April 22	13 04	47	May 22	13 24	50
Mar 24	13 38	44	April 23	13 51	48	May 23	14 14	50
Mar 25	14 22	46	April 24	14 39	48	May 24	15 03	49
Mar 26	15 08	46	April 25	15 27	50	May 25	15 52	48
Mar 27	15 54	49	April 26	16 17	50	May 26	16 40	47
Mar 28	16 43	49	April 27	17 06	49	May 27	17 27	47
Mar 29	17 32	50	April 28	17 55	48	May 28	18 14	48
Mar 30	18 22	50	April 29	18 43	48	May 29	19 02	49
Mar 31	19 12	50	April 30	19 31	49	May 30	19 51

8.1

In the table above, we compute the difference in culmination times of two neighboring days. On average it is about 48 minutes, but differs every day due to the elliptical nature of the lunar orbit. Figure 1 shows the variation of culmination time for the three months. When the moon is at the perigee, the difference in culmination is the largest. Therefore, the moon is at the perigee on Mar 13, April 11, May 7. Therefore, the answer is 11.

Official culmination-time-difference graph with red perigee arrows — Fig 1. Culmination time difference in minutes. On the dates marked by the red arrows the moon is at perigee.

8.2

14 days after Mar 13 (or 14 days before April 11) the moon is located on the opposite side of the perigee. Therefore, the answer is Mar 27.

8.3

There are 8 days between April 11 and April 19, which correspond to 8/27.3 of the lunar orbital period. The angle from the perigee is 105.5 degrees. There are 12 days corresponding to $12/27.3\times360=158.2$ degrees. The locations are indicated in Fig. 2. (See just below.)

Official orbit diagram with A19, A23 and geostationary orbit

8.4

July 1 is 54 days after May 7th, where the Moon is at perigee. The Sun is near aphelion, whereas the Moon is near perigee. Therefore the sun’s apparent angular diameter is $31^\prime59^{\prime\prime}\times0.983=31^\prime27^{\prime\prime}$ . The real diameter of the moon is $D=3476.4$ km. The distance is $384400^{\prime\prime}\times(1-0.055)=362900$ km. Hence the apparent angular size is $3476/362900\times180/3.14\times60=32^\prime54^{\prime\prime}$ . Therefore, the moon appears 1.05 larger than the Sun.

8.5

The approximate size of the geostationary orbit may be found using the Kepler’s III law, with the additional fact that geostationary orbit has 27.3 times shorter period. The ratio or the orbital sizes is $27.3^{2/3}$ , or approximately 9. The answer is shown in Fig. 2.

Practical round. Basic grading criteria

Part	Criteria
8.1	April (11 ± 1) – 2 pts, ± 2 days – 1 pt.
8.2	Mrch (27 ± 1) – 2 pts, ± 2 days – 1 pt.
8.3	Similar to the Figure in the answer sheet – 2 pts, ± 10 degrees – 1 pt.
8.4	Correct ratio ± 0.01 – 2 pts, ± 0.04 – 1 pts.
8.5	Similar to the figure in the solutions – 2 pts, radius less than 1 cm – 1 pts.

Теми и трудност

Трудност: неоценена.

Време и календари · Небесна механика · Светлина и астрономически измервания · Периоди и цикли · Орбитални елементи · Ъглов размер

Предпоставки: Разширена олимпиадна подготовка

Оригинал в Архива: alphapract.pdf · официални решения: practsol1.png · допълнителни данни: practsol2.png · допълнителни данни: practsol3.png · допълнителни данни: practsol4.png · допълнителни данни: practsol5.png · допълнителни данни: practsol6.png · допълнителни данни: pract_cr.png · допълнителни данни: paper.png · допълнителни данни: data_1.png · допълнителни данни: data_2.png · допълнителни данни: data_3.png · допълнителни данни: data_4.png