Съдържание

IAO 2009, Теория, Група β — Задача 3. Eris.

Оригинал на условието (изображение)Съобщи за грешка

Всички материали от IAO 2009

Theoretical round. Problems to solve · 8 ноември 2009 г.

Бележка

Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may have no real sense. In this case you have to write in your answer (in English or Russian): “impossible situation – ситуация невозможна”. Of course, this answer has to be explained numerically or logically.

Data from the table of planetary data may be used for solving every problem.

The answers «Да-Yes» or «Нет-No» has to be written in English or Russian.

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.

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

Body, planet	Average distance (astr. units)	Average distance (mln. km)	Sidereal period (troph. years)	Period (days)	Eccentricity	Equat. diameter (km)	Mass ( $10^{24}$ kg)	Av. density (g/cm³)	Grav. acceler. (m/s²)	Max. magn. from Earth	Albedo
Sun	$1.6\cdot10^9$	$2.5\cdot10^{11}$	$2.2\cdot10^8$	$8\cdot10^{10}$		1392000	1989000	1.409		$-26.8^{\mathrm m}$
Mercury	0.387	57.9	0.241	87.97	0.206	4879	0.3302	5.43	3.70		0.06
Venus	0.723	108.2	0.615	224.70	0.007	12104	4.8690	5.24	8.87		0.78
Earth	1.000	149.6	1.000	365.26	0.017	12756	5.9742	5.515	9.81		0.36
Moon	0.00257	0.38440	0.0748	27.3217	0.055	3475	0.0735	3.34	1.62	$-12.7^{\mathrm m}$	0.07
Mars	1.524	227.9	1.880	686.98	0.093	6794	0.6419	3.94	3.71	$-2.0^{\mathrm m}$	0.15
Jupiter	5.204	778.6	11.862	4332.59	0.048	142984	1899.8	1.33	24.86	$-2.7^{\mathrm m}$	0.66
Saturn	9.584	1433.7	29.458	10759.20	0.054	120536	568.50	0.70	10.41	$0.7^{\mathrm m}$	0.68
Eris	68.01			204862	0.434	2600	0.0167		0.8		0.86

**) For 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 constant, $H_0$ (km/s/Mpc), mean value	70
Hubble constant, diapason of values	50–100
Plank constant, $h$ (J·s)	$6.626\cdot10^{-34}$
Charge of electron, $e$ (C)	$1.602\cdot10^{-19}$
Mass of electron, $m_e$ (kg)	$9.109\cdot10^{-31}$
Proton-to-electron ratio	1836.15
Faraday constant, $F$ (C/mol)	96485
Magnetic constant, $\mu_0$ (H/m)	$1.257\cdot10^{-6}$
Universal gas constant, $R$ (J/mol/K)	8.314
Boltzmann constant, $k$ (J/K)	$1.381\cdot10^{-23}$
Standard atmosphere (Pa)	101325
Stefan-Boltzmann constant, $\sigma$ (W/m²/K⁴)	$5.670\cdot10^{-8}$
Wien's displacement constant, $b$ (m·K)	0.002897
Laboratory wavelength of H $\alpha$ (Å)	6563
Refractive index of water for20°C, $n$	1.334
Area of sphere	$S=4\pi R^2$
$\pi$	3.14159265

Data of some stars

Star	Designation	RA	DEC	m	SC
Arcturus	$\alpha$ Boo	$14^{\mathrm h}15^{\mathrm m}40^{\mathrm s}$	$19^\circ10'57''$	-0.05	K1
Vega	$\alpha$ Lyr	$18^{\mathrm h}36^{\mathrm m}56^{\mathrm s}$	$38^\circ47'01''$	0.03	A0
Deneb	$\alpha$ Cyg	$20^{\mathrm h}41^{\mathrm m}26^{\mathrm s}$	$45^\circ16'49''$	1.25	A2
Polaris	$\alpha$ UMi	$02^{\mathrm h}31^{\mathrm m}51^{\mathrm s}$	$89^\circ15'51''$	2.02	F7
Sirius	$\alpha$ CMa	$06^{\mathrm h}45^{\mathrm m}09^{\mathrm s}$	$-16^\circ42'58''$	-1.46	A1

Бележки от източника

The original two-up official key photographs clip some right-hand line endings. Ellipses mark those unreadable endings; visible formulae and printed numerical values are retained. The reference table gives the Eris period as 204862 days, whereas the key uses 204852 days. Original inline slash notation is retained in the key. The photographed alpha masthead is cropped, but the original filename and distinct alpha tasks in the official key establish its group.

Условие

The largest discovered now in our Solar system Transneptune body is the dwarf planet Eris (Эрида). Now Eris is near its aphelion point. Find the apparent stellar magnitude of Eris as visible from Earth. Approximately when will the next “Great opposition” of Eris be? What will the magnitude of Eris be as visible from Earth at this “Great opposition”?

Решение

Внимание: Непълно решение

The original official key photograph clips several right-hand line endings. All visible text, formulae and diagrams are retained; ellipses mark unreadable source endings.

Let us at first answer for the question about the “Great opposition”. And what does it mean the wo… “Great”? The Great oppositions appear for the situation when the distance to opposition planet is small… than during usual (mean) oppositions. In our case it is an opposition when the outer planet is near perihelion. Now Eris is near its aphelion point. From the value of the semiaxis of the Eris' orbit it … evident that its period is of order of centuries and we may talk about the period of the Great oppositions … “years” (in comparison with “weeks” for the Great opposition of Mars) near after of half revolution of th… dwarf planet around the Sun.

The period of Eris can be taken from the table (days should be recalculated into years)

T_{Ed}=204852\ \text{days}/365.25\ \text{days/year}=560.85\ \text{years}\approx560\ \text{years}

or found from the III Kepler law

(T_{Ed}/T_E)^2=(A_{Ed}/A_E)^3,

(here and below $A$ means the major semiaxis, $T$ is period, $e$ is eccentricity, $\alpha$ is albedo, $D$ are diameters … the bodies, $R$ are distances, and indexes E, Ed, Sa and S correspond to Earth, Eris, Saturn and Sun).

T_{Ed}=T_E\cdot(A_{Ed}/A_E)^{3/2}=1\cdot(68.01)^{3/2}\ \text{years}\approx560\ \text{years}.

So the years of the “Great oppositions of Eris” will be after $T_{Ed}/2\approx280$ years.

For the questions of the stellar magnitude there may be a few ways to solve this problem. The main two … them is to compare Eris with some other planet (data can be given from the table) or to calculate it direct… using formulae of the total solar irradiation.

Let us use the first one and compare Eris and, for example, Saturn.

The following aspects should be taken into account:

$R_{S\to}$ . Distance from the Sun. The intensity of light coming to the bodies is reverse proportional to th… square of these distances.
$D$ . Size (diameter) of the body. The intensity of light reflecting by the body is proportional to the square of these sizes.
$\alpha$ . Albedo. The intensity of light reflecting by the body is proportional to the albedos.
$R_{E\to}$ . Distance from the observer (i.e. from the Earth). The intensity of light coming to the telesco… is reverse proportional to the square of these distances.

The flux to Earth from Eris

F_{Ed}\sim\alpha_{Ed}\cdot D_{Ed}^2\cdot(1/R_{E-Ed})^2\cdot(1/R_{S-Ed})^2.

The flux to Earth from Saturn

F_{Sa}\sim\alpha_{Sa}\cdot D_{Sa}^2\cdot(1/R_{E-Sa})^2\cdot(1/R_{S-Sa})^2.

The ratio of fluxes

F_{Ed}/F_{Sa}=(\alpha_{Ed}/\alpha_{Sa})\cdot(D_{Ed}^2/D_{Sa}^2)\cdot(R_{E-Sa}\cdot R_{S-Sa})^2/(R_{E-Ed}\cdot R_{S-Ed})^2.

For the current situation the following distances should be used:

Since Eris is near its aphelion, $R_{S-Ed}=(1+e)A_{Ed}$ , and $R_{E-Ed}=(1+e_{Ed})A_{Ed}\pm A_E$ , but, since we do n… know the real configuration of Earth, Sun and Eris (including we do not know the inclination of the Eri… orbit), we should use the average value of $R_{E-Ed}=(1+e_{Ed})A_{Ed}$ as well.

The stellar magnitude for Saturn is given for the mean opposition, so we should not use eccentricity he… $R_{S-Sa}=A_{Sa}$ and $R_{E-Sa}=A_{Sa}-A_E$ .

F_{Ed}/F_{Sa}=(0.86/0.68)\cdot(2600/120536)^2\cdot(8.584\cdot9.584)^2/(1.434\cdot68.01)^4.

F_{Ed}/F_{Sa}\approx4.4\cdot10^{-8}.

So the difference in magnitudes between the current visible Eris and Saturn in mean opposition is

\Delta m=-2^{\mathrm m}.5\cdot\lg(F_{Ed}/F_{Sa})\approx18^{\mathrm m}.4.

and the magnitude of Eris

m_{Ed}=m_{Sa}+\Delta m=0^{\mathrm m}.7+18^{\mathrm m}.4\approx19^{\mathrm m}.1.

The real values that can be found in reports of Eris observations are $18^{\mathrm m}.7\div18^{\mathrm m}.8$ .

For the situation of the “Great opposition” of the Eris it is near its perihelion. $R_{S-Ed}=(1-e)A_E$ … $R_{E-Ed}\approx(1-e_{Ed})A_{Ed}-A_E$ (here we write “ $\approx$ ” since we do not know the inclination of the Eris' orbit) and n… changed values for Saturn to compare.

F_{Ed}/F_{Sa}=(0.86/0.68)\cdot(2600/120536)^2\cdot(8.584\cdot9.584)^2/((0.566\cdot68.01\cdot(0.566\cdot68.01-1))^2).

F_{Ed}/F_{Sa}\approx1.91\cdot10^{-6}.

\Delta m=-2^{\mathrm m}.5\cdot\lg(F_{Ed}/F_{Sa})\approx14^{\mathrm m}.3.

and the magnitude of Eris in the “Great opposition” is

m_{Ed}=m_{Sa}+\Delta m=0^{\mathrm m}.7+14^{\mathrm m}.3\approx15^{\mathrm m}.0.

And we should repeat that this way with comparing the Eris and Saturn it is not the only possible correct way for solution.

Оригинал в Архива: betatheo.jpg · допълнителни данни: theosol1.jpg · допълнителни данни: theosol2.jpg · допълнителни данни: constants.jpg · допълнителни данни: betatheodata.jpg