Съдържание

Условие The brightest stars visible from Greece The brightest stars visible from Greece (cont.)The brightest stars of the sky with

m(V)\le1.00

Решение

IOAA 2013, Анализ на данни — Задача 1. Question 1

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2013

7th International Olympiad on Astronomy & Astrophysics — Data analysis

Table of Constants

Group	Quantity	Value
Earth	Mass	$5.98\times10^{24}\ \mathrm{kg}$
Earth	Radius	$6.38\times10^6\ \mathrm{m}$
Earth	Acceleration of gravity	$9.8\ \mathrm{m/s^2}$
Earth	Obliquity of Ecliptic	$23^\circ27^\prime$
Earth	Length of Tropical Year	365.2422 mean solar days
Earth	Length of Sidereal Year	365.2564 mean solar days
Earth	Albedo	0.39
Moon	Mass	$7.35\times10^{22}\ \mathrm{kg}$
Moon	Radius	$1.74\times10^6\ \mathrm{m}$
Moon	Mean distance from Earth	$3.84\times10^8\ \mathrm{m}$
Moon	Orbital inclination with the Ecliptic	$5.14^\circ$
Moon	Albedo	0.14
Moon	Apparent magnitude (mean full moon)	−12.74
Sun	Mass	$1.99\times10^{30}\ \mathrm{kg}$
Sun	Luminosity	$3.83\times10^{26}\ \mathrm{W}$
Sun	Absolute Magnitude	4.72 mag
Sun	Angular diameter	0.5 degrees
Sun	Effective Surface Temperature	5800 K
Jupiter	Orbit semi-major axis	5.204 AU
Jupiter	Orbital period	11.8618 yr
Distances and sizes	Diameter of human pupil	6 mm
Distances and sizes	1 AU	$1.50\times10^{11}\ \mathrm{m}$
Distances and sizes	1 pc	206,265 AU
Distances and sizes	Distance from Sun to Barnard’s Star	1.83 pc
Distances and sizes	Mars orbit semi-major axis	1.52 AU
Physical constants	Gravitational constant (G)	$6.67\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}$
Physical constants	Planck constant (h)	$6.62\times10^{-34}\ \mathrm{J\,s}$
Physical constants	Boltzmann constant	$1.38\times10^{-23}\ \mathrm{J\,K^{-2}}$
Physical constants	Stefan-Boltzmann constant	$5.67\times10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$
Physical constants	Hubble constant	$72\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$
Physical constants	Speed of light	299,792,458 m/s
Physical constants	Proton mass	$938.27\ \mathrm{MeV}\,c^{-2}$
Physical constants	Deuterium mass	$1875.60\ \mathrm{MeV}\,c^{-2}$
Physical constants	Neutron mass	$939.56\ \mathrm{MeV}\,c^{-2}$
Physical constants	Helium-3 mass	$2808.30\ \mathrm{MeV}\,c^{-2}$
Physical constants	Helium-4 mass	$3727.40\ \mathrm{MeV}\,c^{-2}$

Бележки за изданието

The archive PDF contains questions and inline official answers. The Hyades photograph is already annotated; no unannotated version or Hyades-stars.txt is present in the complete 19-entry year catalogue. All 35 input rows are printed in Question 4 and also supplied in DataHyades.xlsx. Source formula/numbering defects are retained: duplicate solution step 4 in Question 1, Figure 2/Figure 5 references, missing range dash for star 10, and incorrect right-ascension conversion in Question 4 and workbook. The workbook filters differences less than 10 rather than absolute differences. The reference star list also has printed value differences between its tables.

Условие

In Figure 1, part of the constellation of Ursa Major is shown. It was taken with a digital camera with a large CCD chip ( $17\,\mathrm{mm}\times22\,\mathrm{mm}$ ). Find the focal length, $f$ , of the optical system and give the error of your results.

The brightest stars visible from Greece

Star	Designation	$\alpha$ (2000), h m	$\delta$ (2000), ° ′	m(V), mag	M(V), mag	B−V	Spectral Type
Alpheratz	α And	00 08	+29 05	2.03	‒0.9	‒0.10	A0 p
Caph	β Cas	00 09	+59 09	2.26	+1.5	+0.34	F2 IV
Schedar	α Cas	00 40	+21 26	2.22	‒1.0	+1.17	K0 II
Diphda	β Cet	00 44	‒17 59	2.04	+0.2	+1.04	K0 III
Mirach	β And	01 10	+35 37	2.06	‒0.4	+1.62	M0 III
Achernar	α Eri	01 38	‒57 15	0.48	‒1.6	‒0.18	B5 IV
Almach	γ And	02 04	+42 20	2.13	‒0.1	+1.20	K2 III
Hamal	α Ari	02 07	+23 28	2.00	+0.2	+1.15	K2 III
Mira	ο Cet	02 19	‒02 59	2.0	‒1.0	+1.42	M6 e
Polaris	α UMi	02 32	+89 16	2.02	‒4.6	+ 0.6	F8 Ib
Algol	β Per	03 08	+40 57	2.2	‒0.3	‒0.1	B8 V
Mirfak	α Per	03 24	+49 51	1.80	‒4.3	+0.48	F5 Ib
Aldebaran	α Tau	04 36	+16 30	0.85	‒0.3	+1.54	K5 III
Rigel	β Ori	05 15	‒08 12	0.11	‒7.0	‒0.03	B8 Ια
Capella	α Aur	05 17	+46 00	0.08	+0.3	+0.80	G8 III
Bellalrix	γ Ori	05 25	+06 21	1.63	‒3.3	‒0.22	B2 III
EINath	β Tau	05 26	+28 36	1.65	‒1.6	‒0.13	B7 III
Mintaka	δ Ori	05 32	‒00 18	2.19	‒6.1	‒0.21	O9.5II
Alnilam	ε Ori	05 36	‒01 12	1.70	‒6.2	‒0.19	B0 Ια
Alnitak	ζ Ori	05 41	‒01 57	1.79	‒5.9	‒0.21	O9.5 Ib
Saiph	κ Ori	05 48	‒09 40	2.05	‒6.8	‒0.18	B0.5 Ια
Betelgeuse	α Ori	05 55	+07 24	0.50	‒5.6	+1.86	M2 Ιab
Menkalinan	β Aur	06 00	+44 57	1.90	+0.6	+0.03	A2 IV
Mirzam	β CMa	06 23	‒17 57	1.98	‒4.5	‒0.24	Bl II
Canopus	α Car	06 24	‒52 42	‒0.73	‒4.7	+0.16	F0 Ib
Alhena	γ Gem	06 38	+16 24	1.93	+0.0	+0.00	A0 IV
Sirius	α CMa	06 45	‒16 43	‒1.45	+1.4	+0.00	Al V
Adhara	ε CMa	06 59	‒28 58	1.50	‒5.0	‒0.22	B2 II
Wezen	δ CMa	07 08	‒26 24	1.84	‒7.3	+0.67	F8 Ιa
Castor	α Gem	07 35	+31 53	1.58	+0.8	+0.04	Al V
Procyon	α CMi	07 39	+05 14	0.35	+2.7	+0.41	F5 IV
Pollux	β Gem	07 45	+28 01	1.15	+1.0	+1.00	K0 III
Naos	ζ Pup	08 04	‒40 00	2.25	‒7.0	‒0.27	Ο5.8
―	γ Vel	08 10	‒47 20	1.83	‒4.0	‒0.26	WC 7
Avior	ε Car	08 23	‒59 30	1.87	‒2.1	+1.30	K0 II
―	δ Vel	08 45	‒54 43	1.95	+0.1	+0.0	A0 V4
Suhail	λ Vel	09 08	‒43 26	2.26	‒4.5	+1.69	K5 Ib
Miaplacidus	β Car	09 13	‒69 43	1.68	‒0.4	+0.00	A0 III

The brightest stars visible from Greece (cont.)

Star	Designation	$\alpha$ (2000), h m	$\delta$ (2000), ° ′	m(V), mag	M(V), mag	B−V	Spectral Type
Scutulum	ι Car	09 17	‒59 16	2.24	‒4.5	+0.18	F0 Ib
Alphard	α Hya	09 28	‒08 40	1.99	‒0.4	+1.43	K3 III
Regulus	α Leo	10 08	+11 58	1.35	‒0.6	‒0.11	B7 V
Algeiba	γ Leo	10 20	+19 51	2.1	‒0.5	+1.12	K0 III
Dubhe	α UMa	11 03	+61 45	1.79	‒0.7	+1.06	K0 III
Denebola	β Leo	11 49	+14 34	2.14	+1.6	+0.09	A3 V
Acrux	α Cru	12 27	‒63 06	0.9	‒3.5	‒0.26	Bl IV
Gacrux	γ Cru	12 31	‒57 07	1.64	‒2.5	+1.60	M3 III
Muhlifain	γ Cen	12 42	‒48 58	2.16	‒0.5	‒0.02	A0 III
Mimosa	β Cru	12 48	‒59 41	1.26	‒4.7	‒0.24	B0 III
Alioth	ε UMa	12 54	+55 57	1.78	‒0.2	‒0.02	A0 p
Mizar	ζ UMa	13 24	+54 56	2.09	+0.0	+0.03	A2 V
Spica	α Vir	13 25	‒11 09	0.96	‒3.4	‒0.23	Bl V
―	ε Cen	13 40	‒53 28	2.30	‒3.6	‒0.23	Bl V
Alkaid	η UMa	13 48	+49 19	1.86	‒1.9	‒0.19	B3 V
Hadar	β Cen	14 04	‒60 22	0.60	‒5.0	‒0.23	Bl II
Menkent	θ Cen	14 07	‒36 22	2.06	+1.0	+1.02	K0 III
Arcturus	α Boo	14 16	+19 11	‒0.06	‒0.2	+1.23	K2 IIIp
Rigil Kent	α Cen	14 40	‒60 50	‒0.1	+4.3	+0.7	G2 V
Kochab	β UMi	14 50	+74 09	2.07	‒0.5	+1.46	K4 III
Alphecca	α CrB	15 35	+26 43	2.23	+0.5	‒0.02	A0 V
Antares	α Sco	16 29	‒26 26	1.0	‒4.7	+1.81	Ml Ib
Atria	α TrA	16 49	‒69 02	1.93	‒0.3	+1.43	K4 III
―	ε Sco	16 5O	‒34 18	2.29	+0.7	+1.15	Κ2 ΙΙΙ
Shaula	λ Sco	17 34	‒37 06	1.62	‒3.4	‒0.22	Bl IV
Ras‒Alhague	α Oph	17 35	+12 34	2.07	+0.8	+0.15	A5 III
―	θ Sco	17 37	‒43 00	1.87	‒4.5	+0.40	FO Ι
Eltanin	γ Dra	17 57	+51 29	2.22	‒0.6	+1.52	K5 III
Kaus Australis	ε Sgr	18 24	‒34 23	1.83	‒1.5	‒0.02	B9 IV
Vega	α Lyr	18 37	+38 47	0.04	+0.5	+0.00	AO V
Nunki	σ Sgr	18 55	‒26 18	2.08	‒2.5	‒0.20	B2 V
Altair	α Aql	19 51	+08 52	0.77	+2.3	+0.22	A7 IV
Sadir	γ Cyg	20 22	+40 15	2.23	‒4.7	+0.67	F8 Ib
Peacock	α Pav	20 26	‒56 44	1.93	‒2.9	‒0.20	B3 IV
Deneb	α Cyg	20 41	+45 17	1.25	‒7.3	+0.09	A2 Ia
Al Na' ir	α Gru	22 08	‒46 58	1.74	+0.2	‒0.14	B5 V
―	β Gru	22 42	‒46 53	2.20	–1.5	+ 1.6	M3 II
Fomalhaut	α PsA	22 58	‒29 37	1.16	+1.9	+0.09	A3 V

The brightest stars of the sky with $m(V)\le1.00$

Star	Designation	$\alpha$ (2000), h m	$\delta$ (2000), ° ′	m(V), mag	M(V), mag	B−V	Spectral Type
Sirius	α CMa	06 45	‒16 43	‒1.45	+1.4	+0.00	Al V
Canopus	α Car	06 24	‒52 42	‒0.73	‒4.7	+0.16	F0 Ib
Rigil Kent	α Cen	14 40	‒60 50	‒0.1	+4.3	+0.7	G2 V
Arcturus	α Boo	14 16	+19 11	‒0.00	‒0.2	+1.23	K2 IIIp
Vega	α Lyr	18 37	+38 47	0.04	+0.5	+0.00	AO V
Capella	α Aur	05 17	+46 00	0.08	+0.3	+0.80	G8 III
Rigel	β Ori	05 15	‒08 12	0.11	‒7.0	‒0.03	B8 Ια
Procyon	α CMi	07 39	+05 14	0.35	+2.7	+0.41	F5 IV
Achernar	α Eri	01 38	‒57 15	0.48	‒1.6	‒0.18	B5 IV
Betelgeuse	α Ori	05 55	+07 24	0.50	‒5.6	+1.86	M2 Ιab
Hadar	β Cen	14 04	‒60 22	0.60	‒5.0	‒0.23	Bl II
Altair	α Aql	19 51	+08 52	0.77	+2.3	+0.22	A7 IV
Aldebaran	α Tau	04 36	+16 30	0.85	‒0.3	+1.54	K5 III
Acrux	α Cru	12 27	‒63 06	0.90	‒3.5	‒0.26	Bl IV
Spica	α Vir	13 25	‒11 09	0.96	‒3.4	‒0.23	Bl V
Antares	α Sco	16 29	‒26 26	1.00	‒4.7	+ 1.81	Ml Ib

Решение

Calculate the angular distance, $\varphi$ , (in degrees or minutes of arc), of two bright stars, whose coordinates are given in the List of Bright stars. Preferably these stars should be chosen to be far apart (e.g. $\alpha$ UMa [ $a_1=11^h03^m$ , $\delta_1=+61^\circ45^\prime$ ] and $\eta$ UMa [ $a_2=13^h48^m$ , $\delta_2=+49^\circ19^\prime$ ]).

1. In order to calculate the angular distance of the two stars, the coordinates should be converted to decimal degrees (e.g. $\alpha$ UMa [ $a_1=165.75^\circ$ , $\delta_1=+61.75^\circ$ ] and $\eta$ UMa [ $a_2=207.0^\circ$ , $\delta_2=+49.32^\circ$ ]). (2.5 Point)

2. Use the cosine law to calculate the angular distance between the two stars:

\varphi=\arccos(\sin\delta_1\times\sin\delta_2+\cos\delta_1\times\cos\delta_2\times\cos(a_1-a_2))\quad\to\quad\varphi=25.8448^\circ.

(10 Points)

3. Measure the distance, $d_0$ , of the two stars in mm in Figure 1. $d_0=138\,\mathrm{mm}$ . (2.5 Points)

4. The photograph in Figure 1 does not have the same dimensions as the original photograph. Measure the length, $\ell$ , of the photograph in mm. $\ell=140\,\mathrm{mm}$ . This length corresponds to 22 mm. Convert $d_0$ to the distance, $d$ , of the original photograph, $d=138\times\frac{22}{140}\,\mathrm{mm}$ . $d=21.6857\,\mathrm{mm}$ . (1.5 Points)

4. The image scale $\left(\frac{\varphi}{d}\right)$ of the original photograph is

\frac{\varphi}{d}=\frac{25.8448^\circ}{21.6857}=1.1918\ \frac{\mathrm{deg}}{\mathrm{mm}}.

(2.5 Points)

5. The focal length is given from equation (see Figure 2): $\tan\frac{\varphi}{2}=\frac{d}{2f}$ or $f=\frac{d}{2\tan(\varphi/2)}$ from which the focal length is calculated: $f=\frac{21.6857}{2\times0.2294}$ or $f=47.3\,\mathrm{mm}$ . (1.5 Points)

(If errors are included (4 Points))

Figure 1A. Calculating the distance scale.

Теми и трудност

Трудност: неоценена.

Оптика · Небесна сфера и сферична астрономия · Лещи и образи · Небесни координати

Предпоставки: Разширена олимпиадна подготовка

Оригинал в Архива: Data-Analysis.pdf · допълнителни данни: IOAA7-BrightStars-list.docx · допълнителни данни: IOAA7-Constants.doc · допълнителни данни: DataHyades.xlsx · допълнителни данни: 130501_P-24.11_B-4.17-rot.jpg · допълнителни данни: 130503_P-23.74_B-3.97_rot.jpg · допълнителни данни: 130505_P-23.34_B-3.76-rot.jpg · допълнителни данни: 130507_P-22.92_B-3.55-rot.jpg · допълнителни данни: 130509_P-22.46_B-3.33_rot.jpg · допълнителни данни: Stonyhurst.3 for Nov 09 to Nov 17 & Dec 28 to Jan 05 & June 28 to July 06 & May 07 to May 15_14cm.jpg · допълнителни данни: Stonyhurst.4 for Jan 06 to Jan 14 & Apr 28 to May 06 & July 07 to July 16 & Oct 31 to Nov 08_14cm.jpg

Съдържание

IOAA 2013, Анализ на данни — Задача 1. Question 1

Условие

The brightest stars visible from Greece

The brightest stars visible from Greece (cont.)

The brightest stars of the sky with m(V)≤1.00m(V)\le1.00m(V)≤1.00

Решение

The brightest stars of the sky with $m(V)\le1.00$