Съдържание

IAO 2010, Теория, Група β — Задача 3. Parallax.

Оригинал на условието (изображение)Съобщи за грешка

Всички материали от IAO 2010

Theoretical round. Problems to solve · 16 октомври 2010 г. · 8 т.

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.

Some constants and formulae

Constant	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 for 20°C, $n$	1.334
Area of sphere	$S=4\pi R^2$
$\pi$	3.14159265

Data of some stars

Star	Designation	RA	DEC	$p$	m	SC
Arcturus	$\alpha$ Boo	14h 15m 40s	19° 10′ 57″	0″.089	−0.05	K1
Vega	$\alpha$ Lyr	18h 36m 56s	38° 47′ 01″	0″.129	0.03	A0
Deneb	$\alpha$ Cyg	20h 41m 26s	45° 16′ 49″	0″.001	1.25	A2
Polaris	$\alpha$ UMi	02h 31m 51s	89° 15′ 51″	0″.007	2.02	F7
Sirius	$\alpha$ CMa	06h 45m 09s	−16° 42′ 58″	0″.379	−1.46	A1

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

Body, planet	Average distance to central body (astr. units)	Average distance (mln. km)	Sidereal (or analogous) period (troph. years)	Period (days)	Eccentricity $e$	Equat. diameter (km)	Mass ( $10^{24}$ kg)	Av. density (g/cm³)	Grav. acceler. at surf. (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		$-26.8^m$
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	$-12.7^m$	0.07
Mars	1.524	227.9	1.880	686.98	0.093	6794	0.6419	3.94	3.71	$-2.0^m$	0.15
Jupiter	5.204	778.6	11.862	4332.59	0.048	142984	1899.8	1.33	24.86	$-2.7^m$	0.66
Saturn	9.584	1433.7	29.458	10759.20	0.054	120536	568.50	0.70	10.41	$0.7^m$	0.68
Eris	68.01			204862	0.434	2600	0.0167		0.8		0.86

**) For Moon – in mean opposition.

Source discrepancies

Printed data and formulae are retained. The official key uses several ungrouped slash expressions. The Venus key initially prints divisor 1.406 before using 19.6 later. The ISS key prints 6371 + 350 = 6720 km.

Условие

Like humans, astronomers of Venus use the same method for definitions of parallaxes and of parsec but measure them in different (their own) units. For example, the distance to Sirius equals to 19.6 vpc (19.6 venusial parsec).

Describe the most evident system of angular units used by astronomers of Venus.
Calculate the venusial diurnal parallax of the Sun and write the answer in “vau” (venusial angular units – the common angular units for astronomers of Venus).

Note: citizens of Venus have two hands (as humans), and 7 fingers at each hand.

Решение

From the table “Data of some stars” we can find that the distance to Sirius is

L=1/p\;\mathrm{pc}=1/0.379\;\mathrm{pc}=2.64\;\mathrm{pc}=206\,265\times2.64\;\mathrm{a.u.}\approx5.44\cdot10^5\;\mathrm{a.u.}

This value equals to 19.6 vpc, that is

1\;\mathrm{vpc}=206\,265\;\mathrm{a.u.}/0.379/1.406=5.44\cdot10^5\;\mathrm{a.u.}/19.6\approx2.78\cdot10^4\;\mathrm{a.u.}

This distance is equal to

2.78\cdot10^4/0.723=3.84\cdot10^4

semiaxis of orbit of Venus.

By logic we should propose that Venusials use tetradecimal (base-14) numeral system, so let us find $\log_{14}(3.84\cdot10^4)$ .

\log_{14}38\,400=\ln38\,400/\ln14=3.996\approx4,

3.84\cdot10^4\approx14^4.

(In fact exactly $38\,416=14^4$ .) So it is reasonable to propose that astronomers of Venus use angle values in radians (rad), units similar to our centiradians ( $14^{-2}$ rad) and units $14^{-4}$ rad, and use the angular unit of $14^{-4}$ -radians (maybe called as microradians but “micro-” at Venus means $14^{-4}$ ) for definition of “their parsec”. So the venusial parsec is really par- $14^{-4}$ -rad.

The venusial horizontal (diurnal) parallax of the Sun is the ratio of equatorial radius of Venus and the average distance to Sun

p_V(\mathrm{sun})=r_V/a_V=d_V/2a_V=12\,104\;\mathrm{km}/2\cdot108\,200\,000\;\mathrm{km}\approx55.933\cdot10^{-6}\;\mathrm{rad}.

That is

p_V(\mathrm{sun})=55.9\cdot10^{-6}\;\mathrm{rad}/14^{-4}\;\mathrm{rad}=2.15\;\text{venusial angular units}.

Note for jury. Some students may try to write the answer using tetradecimal (base-14) numeral system, which is not necessary, and answers in both decimal and tetradecimal system should be considered as correct. Answer in tetradecimal system may be found as follows:

In decimal system $2.1487=2+1\cdot10^{-1}+4\cdot10^{-2}+8\cdot10^{-3}+7\cdot10^{-4}$ .

Hence it equals tetradecimal value $\mathrm{K.LMN}=K+L\cdot14^{-1}+M\cdot14^{-2}+N\cdot14^{-3}+\ldots$

K=[2.1487]=2,\qquad L=[(2.1487-K)/14^{-1}]=2,\qquad M=[(2.1487-K-L\cdot14^{-1})/14^{-2}]=1,

N=[(2.1487-K-L\cdot14^{-1}-M\cdot14^{-2})/14^{-3}]=2\ldots\qquad([X]\;\text{means integer part of X.})

That is, the value written in decimal numeral system as 2.1487 will be 2.212 in tetradecimal system.

Basic criteria. For work of Jury

3.1. 6 pt total, including:

$L=1/p\;\mathrm{pc}\approx5.44\cdot10^5\;\mathrm{a.u.}$ – 1 pt
$1\;\mathrm{vpc}\approx2.78\cdot10^4\;\mathrm{a.u.}$ – 1 pt
Understanding to compare 1 vpc and 0.723 a.u. – 1 pt
Understanding using tetradecimal (base-14) numeral system – 1 pt
Understanding and calculation $2.78\cdot10^4$ a.u. = $14^4$ semiaxis of orbit of Venus – 1 pt
Speculations and conclusions like “astronomers of Venus measure annual parallax in units $14^{-4}$ rad” or “the venusial parsec is really par-μ-rad” – 1 pt

3.2. 2 pt total, including:

The venusial diurnal parallax $p_V(\mathrm{sun})=r_V/a_V$ – 1 pt.
Calculation and result about 2.15 vau – 1 pt.

Оригинал в Архива: betatheo.jpg · допълнителни данни: theosol1.jpg · допълнителни данни: theosol2.jpg · допълнителни данни: theosol3.jpg · допълнителни данни: theosol4.jpg · допълнителни данни: theosol5.jpg · допълнителни данни: theocriteria1.jpg · допълнителни данни: theocriteria2.jpg · допълнителни данни: constants.jpg · допълнителни данни: orbital-elements.jpg · допълнителни данни: theodata.jpg