IOAA 2014, Теория, предварителен вариант — Задача 4. Mass function of a visual binary stellar system

8th International Olympiad on Astronomy and Astrophysics, Suceava - Gura Humorului - August 2014 — THEORETICAL TEST, 15 short problems

Indications
  1. The problems were elaborated concerning two aspects: a) To cover merely all the subjects from the syllabus;
    b) The average time for solving the items is about 15 minutes per a short problem;
  2. In your folder you will find out the following: a) Answer sheets
    b) Draft sheets
    c) The envelope with the subjects in English and the translated version of them in your mother tongue;
  3. The solutions of the problems will be written down only on the answer sheets you receive on your desk. PLEASE WRITE ONLY ON THE PRINTED SIDE OF THE PAPER SHEET. DON’T USE THE REVERSE SIDE. The evaluator will not take into account what is written on the reverse of the answer sheet.
  4. The draft sheets is for your own use to try calculation, write some numbers etc. BEWARE: These papers are not taken into account in evaluation, at the end of the test they will be collected separately . Everything you consider as part of the solutions have to be written on the answer sheets.
  5. Each problem have to be started on a new distinct answer sheet.
  6. On each answer sheet please fill in the designated boxes as follows: a) In PROBLEM NO. box write down only the number of the problem: i.e. 1 – 15 for short problems, 16 – 19 for long problems. Each sheet containing the solutions of a certain problem, should have in the box the number of the problem;
    b) In Student ID – fill in your ID you will find on your envelope, consisted of 3 letters and 2 digits.
    c) In page no. box you will fill in the number of page, starting from 1. We advise you to fill this boxes after you finish the test
  7. We don’t understand your language, but the mathematic language is universal, so use as more relationships as you think that your solution will be better understand by the evaluator. If you want to explain in words we kindly ask you to use short English propositions.
  8. Use the pen you find out on the desk. It is advisable to use a pencil for the sketches.
  9. At the end of the test: a) Don’t forget to put in order your papers;
    b) Put the answer sheets in the folder 1. Please verify that all the pages contain your ID, correct numbering of the problems and all pages are in the right order and numbered. This is an advantage of ease of understanding your solutions.
    c) Verify with the assistant the correct number of answer sheets used fill in this number on the cover of the folder and sign.
    d) Put the draft papers in the designated folder, Put the test papers back in the envelope.
    e) Go to swim

GOOD LUCK !

Условие

For a visual binary stellar system consisted of the stars σ1\sigma_1 and σ2\sigma_2, the following relation represents the mass function of the system:

f(M1;M2)=M23sin⁡3i(M1+M2)2,f\left(M_1;M_2\right) = \frac{M_2^3\sin^3 i}{\left(M_1 + M_2\right)^2},

where M1M_1 is the mass of star σ1\sigma_1, M2M_2 is the mass of star σ2\sigma_2 and ii is the angle between the plane of the stars’ orbits and a plane perpendicular on the direction of observation.

The recorded spectrum of radiations emitted by the star σ1\sigma_1, during several months, reveals a sinusoidal variation of radiation wavelength, with the period T = 7 days and a shift factor z=(Δλ)/λ=0,001.z = \left(\Delta\lambda\right)/\lambda = 0{,}001.

a. Prove that the mass function of the system is:

f(M1;M2)=M23sin⁡3i(M1+M2)2=T2πK(v1⋅sin⁡i)3,f\left(M_1;M_2\right) = \frac{M_2^3\sin^3 i}{\left(M_1 + M_2\right)^2} = \frac{T}{2\pi K}\left(v_1\cdot\sin i\right)^3,

Where: v1⋅sin⁡iv_1\cdot\sin i is the maximum speed of star σ1\sigma_1 relatively to the observer; K−K - the gravitational constant , ii is the angle between the plane of the orbits and the plane normal to the observation direction.

Assumptions: The orbits of the stars are circular,

b. Derive an expression for the mass function of the system. The following values are known: c=3x108 m/s; K=6,67x10−11 Nm2kg−2.c = 3\mathrm{x}10^8\ \mathrm{m/s};\ K = 6{,}67\mathrm{x}10^{-11}\ \mathrm{Nm^2kg^{-2}}.

Решение

Внимание: Непълно решение

Официалният файл с решения повтаря условието на задачата, но не съдържа решение.

Оригинал в Архива: th_short_ver1.pdf · официални решения: th_short_ver1_sol.pdf