IOAA 2014, Теория, предварителен вариант — Задача 1. Lagrange Points

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

The Lagrange points are the five positions in an orbital configuration, where a small object is stationary relative to two big bodies, only gravitationally interacting with them. For example, an artificial satellite relative to Earth and Moon, or relative to Earth and Sun. In the Figure 1 are sketched two possible orbits of Earth relative to Sun and of a small satellite relative to the Sun. Find out which of the two points L31L_3^1 and L32L_3^2 could be the real Lagrange point relative to the system Earth – Sun, and calculate its position relative to Sun. You know the following data: the Earth - Sun distance dES=15⋅107 kmd_{ES} = 15\cdot 10^7\ \mathrm{km} and the Earth – Sun mass ratio ME/MS=1/332946M_E/M_S = 1/332946

Two side-by-side sketches, each with the Sun S at the centre of two dashed concentric circles and dashed cross-hair axes. Left: the Earth E on the outer circle with a velocity arrow pointing up, and the point L₃¹ on the inner circle on the opposite side of S. Right: the Earth E on the inner circle with an upward arrow, and the point L₃² on the outer circle on the opposite side of S.
Figure 1

Отговори

Решение

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