Съдържание

IAO 2014, Теория, Група β — Задача 5. Климат

Оригинал на условието (изображение)Съобщи за грешка

Всички материали от IAO 2014

Задачи за теоретичния тур · 12 октомври 2014 г. · 8 т.

Обща забележка

Обща забележка. Възможно е не във всички задачи въпросите да са поставени коректно. Някои въпроси (възможно е да е главният въпрос на задачата или да е някой подвъпрос) може да нямат смисъл. В такъв случай в решението трябва да се напише на руски или на английски – „ситуация невозможна“ или „impossible situation“. Разбира се, решението трябва да се подкрепи с пресмятания или логически разсъждения.

Данните от таблиците (Слънчева система, звезди, константи) могат да се използват във всяка задача.

Отговорите „Да – Yes“ или „Нет – No“ трябва да се напишат на руски или на английски.

Условие

XXXII век от новата ера. По неизвестни причини Слънцето е започнало да изстива, а ъгълът на наклона на еклиптиката към равнината на небесния екватор е намалял до нула. Учените са пресметнали, че проблемът със запазването на климата на Земята и ежегодната смяна на сезоните може да се реши чрез създаване на изкуствена звезда с пренебрежимо малка маса, но аналогична по светимост на звезда от главната последователност, която да се пусне в кръгова орбита около Слънцето. Звездата ще се движи около Слънцето като вътрешна планета, в резултат на което на Земята ще се установи температурен режим, аналогичен на сегашния на Иссик-Кул: средна зимна температура −3°, средна лятна +17°.

5.1.

Оценете до каква температура е изстинало Слънцето.

5.2.

Оценете на какъв спектрален клас ще съответства звездата, която трябва да се постави на нужната орбита.

Решение

Theoretical round. Sketches for solutions

In order to get an inner orbit of the star that would provide a seasonal cycle with a periodicity of 1 year, it is necessary that its close position with the Earth would take place once a year. That is, the synodic period of its revolution should be one year. It is not difficult to find that the sidereal period of revolution of the celestial body must be equal to

T=1\ \mathrm{year}\times 1\ \mathrm{year}/(1\ \mathrm{year}+1\ \mathrm{year})=0.5\ \mathrm{years},

and the radius of its orbit, from the III Kepler’s law:

R=R_0(T_1/T_0)^{2/3}=0.63R_0.

Thus, the new celestial body will once a year become close to the Earth at a distance of $R_0-R=0.37$ au, providing summer on the planet, and once a year move away to a distance of $R_0+R=1.63$ au, leaving the Earth in winter.

The ratio of energies received by the Earth from the new star in summer and winter will be

\ddot E_S/\ddot E_W=(R_0+R)^2/(R_0-R)^2=19.4.

The ratio of these energies to the energy $\ddot E_0$ , received from the new star at a distance of 1 au is

\ddot E_S/\ddot E_0=R_0^2/(R_0-R)^2=7.3,

\ddot E_W/\ddot E_0=R_0^2/(R_0-R)^2=0.38.

At the same time, the Earth will always receive additional energy $E_1$ from the cooler Sun, that is, $E_S=E_1+\ddot E_S$ in the summer, and $E_W=E_1+\ddot E_W$ . To keep the heat balance, all the incoming energy should be radiated.

The ratio of the energy radiated by the Earth in summer and winter will be

\mathcal E_S/\mathcal E_W=T_S^4/T_W^4=(290\ \mathrm K)^4/(270\ \mathrm K)^4=1.33.

Neglecting the effects of delays of the seasons compared to the maxima and minima of the energy income, let us assume that $\mathcal E_S=E_S$ , $\mathcal E_W=E_W$ . Then we can write:

E_1+\ddot E_S=1.33(E_1+\ddot E_W),

E_1+19.4\ddot E_W=1.33(E_1+\ddot E_W),

(19.4-1.33)\ddot E_W=(1.33-1)E_1,

\ddot E_W/E_1=0.0183,\qquad\ddot E_S/E_1=0.355.

The sum of $E_1$ and the average $\ddot E$ for the period must be the same as the Earth receives nowadays, $E_0$ . To estimate the average value of $\ddot E$ , let us take $\ddot E_0$ , and so

\ddot E_0/E_1=0.0486,\qquad\ddot E_0/E_0=0.0464,\qquad E_1/E_0=0.954.

5.1. Thus, the Sun in new conditions will emit 4.6% less energy, and so its temperature will be lower and equal to

T_1=0.954^{1/4}T_0\approx 0.988T_0\approx 5710\ \mathrm K.

5.2. To understand what spectral type will correspond to the star, we calculate by how many magnitudes it is fainter than the Sun:

\Delta M=-\frac 52\lg(0.0464)\approx 3.3.

From the HR diagram, we find that it is about the spectral type K 3.

Theoretical round. Basic criteria

Understanding nature of effect and conclusion synodic period 1 year — 1 pt.
Sidereal period 0.5 years — 1 pt.
Orbital radius from Kepler’s III law — 1 pt.
Energy balances — 2 pt.
5.1. Temperature of Sun — 1 pt.
5.2. ΔM=−(5/2)lg(0.0464)≈3.3 and from HR diagram spectral type K 3 — 2 pt.

Официални решения и оценяване

Официалните скици на решенията и критериите са запазени на оригиналния английски език.

Note for jury and team leaders

The proposed sketches are not full; the team leaders have to give more detailed explanations for students. But the correct solutions in the students’ papers (enough for 8 pts) may be shorter.

Бележка

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.

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

Body	Distance (AU)	Distance (10⁶ km)	Period (tropical years)	Period (days)	Inclination i (°)	Eccentricity	Equatorial diameter (km)	Mass (10²⁴ kg)	Density (g/cm³)	Gravity (m/s²)	Axial tilt (°)	Max. 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	7.00	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	0.07348	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

Source footnote: for outer planets and Moon — in mean opposition.

Some constants and formulae

Constant or formula	Value
Speed of light in vacuum c (m/s)	299792458
Constant of gravitation G (N·m²/kg²)	$6.674\cdot10^{-11}$
Solar constant A (W/m²)	1367
Hubble parameter H₀ (km/s/Mpc), mean	71
Hubble parameter, range	50–100
Planck constant h (J·s)	$6.626\cdot10^{-34}$
Charge of electron e (C)	$1.602\cdot10^{-19}$
Mass of electron mₑ (kg)	$9.109\cdot10^{-31}$
Proton-to-electron ratio	1836.15
Faraday constant F (C/mol)	96485
Magnetic constant μ₀ (H/m)	$1.257\cdot10^{-6}$
Universal gas constant R (J/mol/K)	8.314
Boltzmann constant k (J/K)	$1.381\cdot10^{-23}$
Stefan–Boltzmann constant σ (W/m²/K⁴)	$5.670\cdot10^{-8}$
Wien displacement constant b (m·K)	0.002897
Laboratory wavelength Hα (Å)	6562.81
Tropical year length T (days)	365.242199
Standard atmosphere (Pa)	101325
Visible light extinction by terrestrial atmosphere in zenith (minimum)	19%, 0.23 magnitudes
Height of homogeneous atmosphere (m)	7991
Refractive index of water for 20°C n	1.334
Moment of inertia of a solid ball	$I=\frac25MR^2$
Volume of a ball	$V=\frac43\pi R^3$
Area of sphere	$S=4\pi R^2$
π	3.14159265
e	2.71828183
Golden ratio φ	1.61803399

Coordinates	Bishkek	Cholpon-Ata
Longitude E	74°34′	77°05′
Latitude N	42°52′	42°39′
Timezone	UT+6	UT+6

28 brightest stars in the sky. Some other stars

Name	Designation	RA	Dec	Parallax (arcsec)	Magnitude	Spectral class
Altair	α Aql	19h50m47s	08°52′06″	0.195	0.77	A7
Capella	α Aur	05h16m41s	45°59′53″	0.073	0.08	G5+G0
Arcturus	α Boo	14h15m38s	19°10′57″	0.089	−0.04ᵛ	K1
Canopus	α Car	06h23m57s	−52°41′45″	0.028	−0.72	F0
Miaplacidus	β Car	09h13m12s	−69°43′02″	0.029	1.68	A2
Toliman (Rigel Kent)	α Cen	14h39m36s	−60°50′07″	0.747	−0.01; 1.33	G2; K1
Hadar	β Cen	14h03m49s	−60°22′23″	0.009	0.61	B1
Sirius	α CMa	06h45m09s	−16°42′58″	0.375	−1.46	A1
Adara	ε CMa	06h58m38s	−28°58′19″	0.008	1.50	B2
Procyon	α CMi	07h39m18s	05°13′30″	0.288	0.38	F5
Acrux	α Cru	12h26m36s	−63°05′57″	0.010	0.77	B0
Becrux	β Cru	12h47m43s	−59°41′20″	0.009	1.30	B0
Gacrux	γ Cru	12h31m10s	−57°06′48″	0.037	1.60	M4
Deneb	α Cyg	20h41m26s	45°16′49″	0.002	1.25	A2
Achernar	α Eri	01h37m43s	−57°14′12″	0.026	0.46	B3
Castor	α Gem	07h34m36s	31°53′18″	0.065	1.58	A1+A2
Pollux	β Gem	07h45m19s	28°01′35″	0.097	1.14	K0
Regulus	α Leo	10h08m22s	11°58′02″	0.041	1.35	B7
Vega	α Lyr	18h36m56s	38°47′01″	0.123	0.03	A0
Betelgeuse	α Ori	05h55m10s	07°24′25″	0.005	0.5ᵛ	M2
Rigel	β Ori	05h14m32s	−08°12′06″	0.013	0.12	B8
Bellatrix	γ Ori	05h25m08s	06°20′59″	0.013	1.64	B2
Fomalhaut	α PsA	22h57m39s	−29°37′20″	0.130	1.16	A3
Antares	α Sco	16h29m24s	−26°25′55″	0.024	0.96	M1+B4
Shaula	λ Sco	17h33m37s	−37°06′14″	0.005	1.62	B2
Aldebaran	α Tau	04h35m55s	16°30′33″	0.048	0.85ᵛ	K5
Elnath	β Tau	05h26m18s	28°36′27″	0.025	1.66	B7
Spica	α Vir	13h25m12s	−11°09′41″	0.023	0.98	B1

Some other stars

Name	Designation	RA	Dec	Parallax (arcsec)	Magnitude	Spectral class
Hamal	α Ari	02h07m10s	23°27′45″	0.050	2.01	K2
Sheratan	β Ari	01h54m39s	20°48′30″	0.055	2.64	A5
Thuban	α Dra	14h04m23s	64°22′33″	0.011	3.65	A0
Rastaban	β Dra	17h30m26s	52°18′05″	0.009	2.79	G2
Eltamin	γ Dra	17h56m36s	51°29′20″	0.021	2.24	K5
Polaris	α UMi	02h31m49s	89°15′51″	0.007	1.97ᵛ	F7
Kochab	β UMi	14h50m42s	74°09′20″	0.025	2.07	K4
Pherkad	γ UMi	15h20m44s	71°50′02″	0.007	3.06ᵛ	A3
Yildun	δ UMi	17h32m19s	86°35′11″	0.019	4.35	A1
Proxima Centauri	V645 Cen, α Cen C	14h29m43s	−62°40′46″	0.769	11.05	M5.5

Source discrepancy

the key prints 262.12° in 4.2; its printed β=48°33′ gives 262°54′ for 360°−2β. The printed angle and time are retained.

Source discrepancy

the winter ratio is printed with (R₀−R)² in its denominator although the preceding winter distance is R₀+R and the numerical0.38 corresponds to the latter. The printed formula is retained. Calligraphic ℰ here represents the separate radiated-energy symbol used in the key.

Source discrepancy

the key switches from z=0.0035 to0.0036 while retaining1050 km/s; the latter multiplication would give1080. It also changes the moving gas from the first spectrum to the third in its prose. Both inconsistencies are retained. The “in reality” values above are statements of the2014 source.

Оценяване: Proxima Centauri

The printed Proxima Centauri allocations are 1 + 3×1 + 1 + 1.5 + 1.5 = 8 points, matching the whole-problem scale.

Оригинал в Архива: bgtheo5.jpg · официални решения: theosol.pdf · допълнителни данни: bgtheo2.jpg · допълнителни данни: bgtheo3.jpg · допълнителни данни: bgtheo4.jpg · допълнителни данни: bgtheo5.jpg · допълнителни данни: bgtheo6.jpg · допълнителни данни: theocriteria.pdf · допълнителни данни: sombrero.pdf · допълнителни данни: constants.pdf · допълнителни данни: hrd.pdf