Съдържание

IOAA 2014, Теория, предварителен вариант — Задача 6. The life –time of a star from the main sequence

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2014

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

Indications

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;
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;
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.
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.
Each problem have to be started on a new distinct answer sheet.
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
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.
Use the pen you find out on the desk. It is advisable to use a pencil for the sketches.
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 plot of the function $\log\left(L/L_S\right) = f\left(\log\left(M/M_S\right)\right)$ for data collected from a large number of stars is represented in figure 3. The symbols represents: L and M the luminosity respectively the mass and of a star and $L_S$ and respectively $M_S$ the luminosity and the respectively the mass of the Sun.

Grid plot of log(L/L_S) (vertical axis, ticks −1 to 4) against log(M/M_S) (horizontal axis, ticks −0,5, 0, 0,5, 1,0) with about twenty data points scattered along a straight line rising from about (−0,4; −1,3) to about (1,15; 3,9). — Figure 6

Find an expression for the life- time for each star in the Main Sequence from Hertzprung – Rossell diagram if the time spent by Sun in the same Main Sequence is $\tau_S$ .Consider the following assumptions: for each star the percentage of its mass which changed into energy is $\eta$ , the percent of the mass of Sun which changes into energy is $\eta_S$ , the mass of each star is $M = nM_S$ and the luminosity of each star remains constant, during its entire life time.

Отговори

Решение

Problem 6. Marking scheme The life –time of a star from the main sequence

A. The analysis of the graph — 6 points
- Obtaining the formula (1) from the linearity of the graph
- Correct use of the luminosity formula (2) for finding out the final formula — 4 points

Detailed solution

A. The analysis of the graph :

The graph is linear:

y = ax + b = ax;

The same mass–luminosity grid plot with the fitted straight line, the slope angle α and a dashed right triangle with legs Δx and Δy; axis labels log(L/L_S) and log(M/M_S).

From the graph it can be obtain the following data:

\log\frac{L}{L_S} = a\cdot\log\frac{M}{M_S};

a = \tan\alpha = \frac{\Delta y}{\Delta x} = \frac{3{,}5}{1} = 3{,}5;

L \sim M^{3{,}5}.\ \textbf{(1)}

The total energy of the star is:

E = Mc^2,

So the emitted energy due to the mass variation of the star is:

\Delta E = c^2\Delta M,

According to the text

\Delta M = \eta M;

\Delta E = c^2\eta M.

By using the definition of the luminosity :

\frac{\Delta E}{\Delta t} = L;\ \textbf{(2)}

\Delta t = \tau;

\frac{c^2\eta M}{\tau} = L;

\tau = \frac{c^2\eta M}{L},\ \textbf{(2)}

Which represents the life-time of the star.

By using the results from the graph analysis

L = \frac{L_S}{M_S^{3{,}5}}\cdot M^{3{,}5},

Thus :

\tau = \frac{c^2\eta M_S^{3{,}5}}{L_S}\cdot M^{-2{,}5}.

If use the same calculations for the Sun it can be obtain

E_S = M_Sc^2;

\tau_S = \frac{c^2\eta_SM_S}{L_S},

Which is the life-time of the Sun

\tau = \frac{\eta}{\eta_S}\cdot\tau_S\cdot M_S^{2{,}5}\cdot M^{-2{,}5};

\frac{\tau}{\tau_S} = \frac{\eta}{\eta_S}\cdot\left(\frac{M}{M_S}\right)^{-2{,}5};\ M = nM_S;

\tau = \frac{\eta}{\eta_S}\left(n\right)^{-2{,}5}\tau_S.

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