Съдържание

IOAA 2014, Теория, предварителен вариант — Задача 7. The effective temperature on the surface of a star

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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

A star emits radiation with wavelength values in a narrow range $\Delta\lambda \ll \lambda$ , i.e. the wavelength have values between $\lambda$ and $\lambda$ and $\lambda + \Delta\lambda$ . According to Planck's relationship (for an absolute black body), the following relation define, the energy emitted by star in the unit of time, through the unit of area of its surface, per length-unit of the wavelength range:

r = \frac{2\pi hc^2}{\lambda^5\left(e^{hc/k\lambda c} - 1\right)}.

The spectral intensities of two radiations with wavelengths $\lambda_1$ and respectively $\lambda_2$ , both in the range $\Delta\lambda$ measured on Earth are $I_1\left(\lambda_1\right)$ and, respectively $I_2\left(\lambda_2\right)$ .

a. Establish the equation which, in a general case, allows determining the effective temperature on the surface of the star using only spectral measurements.

b. Find out the approximate value of the effective temperature on the star surface if $hc \gg \lambda kT$ .

c. Find out the relation between wavelength $\lambda_1$ and $\lambda_2$ , if $I_1\left(\lambda_1\right) = 2I_2\left(\lambda_2\right)$ , when $hc \ll \lambda kT$ .

You know: $h -$ Planck’s constant; $k -$ Boltzmann’s constant; $c -$ speed of light in vacuum.

Отговори

Решение

Problem 7. Marking scheme The effective temperature on the surface of a star

a. Identifying the expression of spectral intensity and correct use of the given relation for obtaining the relation (1) — 4 points
b. correct use of the assumption $hc \gg \lambda kT$ and find out relation (2) — 3 points
c. correct use of the assumption $hc \ll \lambda kT$ . and find out relation (3) — 3 points

Detailed solution

a. We start from the definition of r:

r = \frac{\Delta E}{\Delta t\cdot S_{stea}\cdot\Delta\lambda} = \frac{\Delta E}{\Delta t\cdot 4\pi R^2\cdot\Delta\lambda}

where R is the radius of the star

r = \frac{2\pi hc^2}{\lambda^5\left(e^{hc/k\lambda T} - 1\right)}

Considering d as the distance from the star to the Earth, the definition- relation of the spectral intensity can be written as follows:

I\left(\lambda\right) = \frac{\frac{\Delta E}{\Delta t\cdot\Delta\lambda}}{4\pi d^2}

I\left(\lambda\right) = \frac{2\pi hc^2R^2}{d^2\lambda^5\left(e^{hc/\lambda kT} - 1\right)};

Particularly for each wavelength:

I_1\left(\lambda_1\right) = \frac{2\pi hc^2R^2}{d^2\lambda_1^5\left(e^{hc/\lambda_1kT} - 1\right)};\ I_2\left(\lambda_2\right) = \frac{2\pi hc^2R^2}{d^2\lambda_2^5\left(e^{hc/\lambda_2kT} - 1\right)};

The ratio of the 2 above relations

\frac{I_1\left(\lambda_1\right)}{I_2\left(\lambda_2\right)} = \left(\frac{\lambda_2}{\lambda_1}\right)^5\cdot\frac{e^{hc/\lambda_2kT} - 1}{e^{hc/\lambda_1kT} - 1} \qquad (1)

Represents an equation which allow to find out the temperature of star’s surface T by using spectral measurements

b. If we consider that $hc \gg \lambda kT$ , then:

e^{hc/\lambda_1kT} - 1 \approx e^{hc/\lambda_1kT} \quad \text{and} \quad e^{hc/\lambda_2kT} - 1 \approx e^{hc/\lambda_2kT}

The relation (1) becomes

\frac{I_1\left(\lambda_1\right)}{I_2\left(\lambda_2\right)} = \left(\frac{\lambda_2}{\lambda_1}\right)^5\cdot\frac{e^{hc/\lambda_2kT}}{e^{hc/\lambda_1kT}} = \left(\frac{\lambda_2}{\lambda_1}\right)^5\cdot e^{hc/\lambda_2kT - hc/\lambda_1kT}

\frac{I_1\left(\lambda_1\right)}{I_2\left(\lambda_2\right)}\cdot\left(\frac{\lambda_1}{\lambda_2}\right)^5 = e^{hc/\lambda_2kT - hc/\lambda_1kT}

T = \frac{hc\left(\lambda_1 - \lambda_2\right)}{k\lambda_1\lambda_2\cdot\ln\left[\frac{I_1\left(\lambda_1\right)}{I_2\left(\lambda_2\right)}\cdot\left(\frac{\lambda_1}{\lambda_2}\right)^5\right]}

c. If $hc \ll \lambda kT$ , then:

\frac{hc}{k\lambda_1T} \ll 1;\ e^{hc/\lambda_1kT} - 1 \approx 1 + \frac{hc}{k\lambda_1T} - 1 = \frac{hc}{k\lambda_1T}

\frac{hc}{k\lambda_2T} \ll 1;\ e^{hc/\lambda_2kT} - 1 \approx 1 + \frac{hc}{k\lambda_2T} - 1 = \frac{hc}{k\lambda_2T}

The relation (1) becomes:

\frac{I_1\left(\lambda_1\right)}{I_2\left(\lambda_2\right)} = \left(\frac{\lambda_2}{\lambda_1}\right)^5\cdot\frac{\frac{hc}{k\lambda_2T}}{\frac{hc}{k\lambda_2T}} = \left(\frac{\lambda_2}{\lambda_1}\right)^4

I_1 = 2I_2;\ \left(\frac{\lambda_2}{\lambda_1}\right)^4 = 2;\ \lambda_2 = \lambda_1\cdot\sqrt[4]{2} \approx 1{,}2\cdot\lambda_1.

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