Съдържание

Условие Русский English Дополнительные данные об α Cen / Additional data about α Cen Справочные данные / Reference data Решение

IAO 2023, Теория — Задача 3 (αβγ). Межзвёздное путешествие / Interstellar Travel

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Всички материали от IAO 2023

XXVII International Astronomy Olympiad — Theoretical round / Задачи теоретического тура · 8 т.

Только для перевода. / For translation only.

Напоминание. Большинство задач составлены таким образом, что при написании решений от участника требуется минимальное использование какого-либо языка. Таким образом, допускается минимум текстов в решениях задач (ни на родном языке, ни на одном из официальных языков IAO). Уделяйте больше внимания формулам, рисункам и графикам. Разрешается употреблять несколько ключевых слов на английском языке (например, “Yes”, “No”, “Assume”, “Answer”, “Average”, “Minimum”, “Maximum”, “Impossible situation” и т.д.), а также термины, содержащиеся в прилагаемых таблицах, и названия объектов.

Reminding. Most of the problems are compiled in such a way that when writing solutions, minimal use of any language is required from the participant. So minimum texts is allowed in solutions of the problems (neither in the native language, nor in one of the official languages of the IAO). Pay more attention to formulas, drawings and graphs. It is permitted to write a few keywords in English (for example, “Yes”, “No”, “Assume”, “Answer”, “Average”, “Minimum”, “Maximum”, “Impossible situation”, etc.), as well as terms contained in the accompanying tables and names of objects.

Source notes / Примечания к источнику

The scan repeats the Russian second statement page. Russian and English versions are retained as sections of the same seven tasks. The note after 6.6 prints 8.3/8.5/8.6; the English scan has a handwritten correction. The solution’s final magnitude for task 2 prints +5 rather than −5, the eclipse expression prints L₁−L₂, and the redshift relation prints a(tₑ)/a(tₒ); these source expressions are preserved. The criteria for 6.5/6.6 have unrelated white-dwarf text crossed out without replacement. Reference data are also present within the question PDF and are not extra tasks.

Условие

Русский

В ближайшем будущем люди посылают огромный космический корабль к планете в системе Альфа Центавра, пытаясь посетить инопланетян, проживающих на этой планете. У инопланетян голубая кожа и размеры любых органов примерно как у людей. Наблюдая за небом, они замечают, что космический корабль движется к ним по прямой линии с постоянной скоростью $0{,}152c$ .

Для разгона и торможения корабль оснащён двумя мощными двигателями, каждый из которых вырабатывает конический пучок фотонов. Угол раствора конуса составляет 4,96 градуса. Выходная мощность одного двигателя равна $P=2{,}25\cdot10^{17}$ Вт, из которых 9,92% приходится на видимый свет.

3.1. На каком расстоянии тормозящий корабль сравнится по яркости с Солнцем для наблюдателей-центаврян?

3.2. Смогут ли при этом инопланетяне визуально различить на своём небе корабль и Солнце? Запишите ваши расчёты и ответьте (по-английски) “Yes” или “No”.

English

In the near future, humans will send a massive space ship towards a planet in the Alpha Centauri system, trying to visit the aliens residing on this planet. The aliens have blue skin and all their organs are similar in size to those of humans. When they observe the sky, they notice that the space ship travel towards them in a straight line with a constant speed of $0.152c$ .

For acceleration and deceleration, the ship is equipped with two powerful engines, each of which produces a conic beam of pure photons. The cone angle of the beam is 4.96 degrees. The power output of an engine is $P=2.25\cdot10^{17}$ W, of which 9.92% is visible light.

3.1. Observed by the Centaurians, at what distance is the decelerating ship as bright as the Sun?

3.2. Herein, will the aliens be able to visually distinguish the ship and the Sun in their sky? Write down your calculations and answer (in English) with “Yes” or “No”.

Дополнительные данные об α Cen / Additional data about α Cen

Собственное движение (R.A.) / Proper motion (R.A.)	Собственное движение (DEC) / Proper motion (DEC)	Лучевая скорость / Radial velocity
−3,614″/год / −3.614″/yr	+0,803″/год / +0.803″/yr	−22,4 км/с / −22.4 km/s

Справочные данные / Reference data

28 brightest stars, other stars and objects: complete bilingual data table

Complete bilingual constants and formulae reference sheet

Решение

3.1.

Distance between the Sun and α Cen:

r=\frac1\pi=\frac1{0.747}=1.34\ pc=4.36\ ly=276\,000\ au

(the parallax of α Cen can be found in the table of stars and objects).

Flux of the Sun seen at α Cen:

F_\odot=1367\ W/m^2\times\left(\frac{1\ au}{r}\right)^2

(the value of the solar constant can be found in the table of constants).

Solid angle of the engine plume:

s=\pi\frac{(4.96/2)^2}{(180/\pi)^2}=5.89\cdot10^{-3}

Flux from the ship at a distance of $r_s$ :

F_s=\frac{2P\times0.0992}{sr_s^2}

When the two fluxes equals, $F_\odot=F_s$ .

1367\ W/m^2\times\left(\frac{1\ au}{r}\right)^2=\frac{2P\times0.0992}{5.89\cdot10^{-3}\cdot r_s^2}\quad\Rightarrow\quad r_s=\sqrt{\frac{4.5\cdot10^{17}\ W\times0.0992\times(276\,000\ au/1\ au)^2}{1367\ W/m^2\times5.89\cdot10^{-3}}}

r_s=2.06\cdot10^{13}\ m=137\ au.

3.2.

Considering both the ship and Alpha Centauri move at constant speed, the relative direction of them ( $A_iS_i$ ) remains the same to the departure positions (AE). So seen at Alpha Centauri, the angular distance of the ship and the sun ( $\angle EA_iS_i$ ) increases, and reaches maximum just before the arrival. At the arrival, the angular distance is equal to $\angle AER$ , which is actually the angular movement of Alpha Centauri during the transit time. Since the distance of Alpha Centauri does not change much during the years, the angular movement can easily be estimated by proper motion multiplied by duration.

There is no aberration since there is no apparent motion.

Geometry of Earth, Alpha Centauri and ship rendezvous

TLDR: $A_iS_i\parallel AE$ , $\max(\angle EAS)=\angle AER=$ proper motion × duration

Total proper motion:

\mu=\sqrt{3.614''^2+0.803''^2}=3.7''/yr

Ship transit time:

t=\frac rv=\frac{4.36\ ly}{0.152c}=28.6\ yr

Maximum angle:

\angle REA=28.6\ yr\times3.7''/yr=106''.\qquad106''>1'

It’s greater than the naked eye resolve angle of common practice.

Conclusion: “Yes”.

Basic criteria

Criterion	pts
3.1. Distance between the Sun and α Cen	0.5
Flux of the Sun seen at α Cen	0.5
Solid angle of the engine plume	0.5
Flux from the ship	0.5
General formulae and calculations	2
Final answer	1.5
3.2. Explanation and calculations	2
Final conclusion “Yes”	0.5
Total	8

Theoretical round. Sketches for solutions. For jury job ONLY

These solutions are written for jury members, so they contain a lot of explanations in the form of text. Students are instructed to minimize text, mostly use formulas, calculations, drawings and graphs, writing extra tests is not encouraged in the solutions. It is permitted to write a few keywords in English (for example, “Yes”, “No”, “Assume”, “Answer”, “Average”, “Minimum”, “Maximum”, “Impossible situation”, etc.), as well as terms contained in the accompanying tables and names of objects.

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

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.

Theoretical round. Basic criteria. For work of Jury

Note. 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 use these positions. (For example, except the final conclusion, all the points in solution of problem 2 may be in only one common formula.)

Note. Jury members should elaborate more detailed criteria, and also create criteria for other correct ways of the student’s solutions.

Note. Everywhere “formula” means “correct formula”, “calculation” means “correct calculation”, “picture” means “correct picture”, etc., and in all cases: based on correct facts and intermediate formulae or calculations.

Note. Everywhere final calculation means also using correct number of significant digits. Extra significant digits, especially in estimational problems, will lead to subtraction of the points.

Note. Solutions with absurd or almost absurd answers should be rated lower (maybe ignoring the scoring schemes below, and maybe down to zero).

Everywhere – excessive number of significant digits in the answer – minus 1–2 points.

Теми и трудност

Трудност: неоценена.

Светлина и астрономически измервания · Уреди и наблюдения · Поток и светимост · Ъглова разделителна способност

Предпоставки: Разширена олимпиадна подготовка

Оригинал в Архива: theo.pdf · официални решения: theosol.pdf · допълнителни данни: data.pdf