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 Отговори Решение

IAO 2012, Практически тур — Задача 8 (група α). Луна

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2012

XVII Международна астрономическа олимпиада — практически кръг, българско издание · 16 октомври 2012 г. · 10 т.

Забележка (група α, β)

Забележка: Ако откриете някъде в задачите невъзможна ситуация, напишете на английски „impossible situation“.

Забележка (група α)

Забележка: Ако намерите някъде в задачите невъзможна ситуация, напишете на английски „impossible situation“

Забележка (група β)

Забележка. Ако откриете някъде в задачите невъзможна ситуация, напишете на английски „impossible situation“

Fireball: printed solution values

The official key acknowledges that point II is plotted incorrectly. Section 7.2 gives A/B latitudes 36.47°/36.10°, while section 7.4 and the final table give 36.49°/36.11°. Both are retained.

Moon: missing plotting sheet and printed discrepancies

The blank eccentric-orbit plotting sheet (a) is absent from the full year inventory. The supplied geographic grid belongs to Fireball. The official completed orbit diagram is retained in the solution. The key prints an arcsecond mark after the distance 384400, uses Moon diameter 3476.4 km (3476 in its calculation), while the reference sheet lists 3475 km; values are retained. Its culmination difference table is reproduced as printed, including entries that do not equal the adjacent clock-time difference.

Clusters: missing plotting sheets and printed discrepancies

Blank graph papers (A) and (B) are absent from the full year inventory. Complete data and completed official plots are retained. The colour-colour heading prints NGC6282; the 8.4 explanation prints NGC2686; the actual problem is NGC2682. The Hyades absolute-magnitude equation and 0.75 answer are retained without recalculation.

Source ownership

Bulgarian questions and all three original IDs are retained. English source PDFs confirm shared αβ-7, α-8 Moon and β-8 Clusters. All six key images and full practical rubric apply to this combined Bulgarian edition. Four theory-labelled reference sheets were inspected: orbital data support Moon and the H-R diagram is explicitly cited by Clusters; constants and individual-star data remain separately attributed reference context, not additional practical tasks. No Bulgarian key was found.

Условие

Корейският институт по астрономия и космически науки (KASI) издава Корейски астрономически алманах всяка година. Таблицата, в която се дава времето на кулминацията на Луната по корейско време, е взета от Алманаха за 2012 г. (Вижте отделния лист; свободните клетки можете да попълните с необходимото съдържание.)

Дата	Кулминация на Луната		Дата	Кулминация на Луната		Дата	Кулминация на Луната
2 март			1 април			1 май

Също така ви се дава милиметрова хартия за построяване на графики.

8.1. Определете датата от април 2012 г., когато Луната е била най-близо до Земята. [2 т.]

8.2. Определете датата от март 2012 г., когато Луната е била най-далеч от Земята. [2 т.]

8.3. На листа милиметрова хартия (а) е дадена ексцентричната орбита на Луната; в центъра се намира Земята. Отбележете с кръстче (×) положенията на Луната на 19 април и 23 април (означете ги с A19 и A23. [2 т.]

8.4. Пресметнете отношението на видимите ъглови размери на Луната ( $\alpha_{Moon}$ ) и Слънцето ( $\alpha_{sun}$ ) на 1 юли. [2 т.]

8.5. На листа милиметрова хартия начертайте в дадения мащаб геостационарна орбита около Земята. [2 т.]

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

Body	Average distance (au)	Average distance ( $10^6$ km)	Sidereal period (tropical years)	Period (mean days)	Inclination $i$ (°)	Eccentricity $e$	Equatorial diameter (km)	Mass ( $10^{24}$ kg)	Density (g/cm³)	Gravity (m/s²)	Axial tilt (°)	Maximum magnitude	Albedo
Sun	$1.6\cdot10^9$	$2.5\cdot10^{11}$	$2.2\cdot10^8$	$8\cdot10^{10}$			1392000	1989000	1.409			−26.74
Mercury	0.387	57.9	0.241	87.969	7.01	0.206	4879	0.3302	5.43	3.70	0.01		0.06
Venus	0.723	108.2	0.615	224.7007	3.40	0.007	12104	4.8690	5.24	8.87	177.36		0.78
Earth	1.000	149.6	1.000	365.2564	0.00	0.017	12756	5.9742	5.515	9.81	23.44		0.36
Moon	0.00257	0.38440	0.0748	27.3217	5.15	0.055	3475	0.0735	3.34	1.62	6.7	−12.7	0.07
Mars	1.524	227.9	1.880	686.98	1.85	0.093	6794	0.6419	3.94	3.71	25.19	−2.0	0.15
Jupiter	5.204	778.6	11.862	4332.59	1.30	0.048	142984	1899.8	1.33	24.86	3.13	−2.7	0.66
Saturn	9.584	1433.7	29.458	10759.20	2.48	0.054	120536	568.50	0.70	10.41	26.73	0.7	0.68
Eris	68.05			205029	43.82	0.435	2326	0.0167	2.52	0.7			0.96

Maximum magnitude from Earth; for Moon — in mean opposition. Original blank cells retained.

Отговори

Решение

Официални скици на решенията и основни критерии за оценяване, на оригиналния английски език.

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.

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.

Оригинал в Архива: bgpract.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: alphapract.pdf · допълнителни данни: betapract.pdf · допълнителни данни: data_1.png · допълнителни данни: data_2.png · допълнителни данни: data_3.png · допълнителни данни: data_4.png