Съдържание

Условие Отговори Решение

IOAA 2014, Теория, окончателен вариант — Задача 9. The Vega star in the mirror

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2014

8th International Olympiad on Astronomy and Astrophysics — Theoretical Test, 12 short problems (final edition)

Указания

In your folder you will find the following items:

a. Answer sheets

b. Rough work sheets

c. The envelope with the problems The solutions of the problems should be written only on the answer sheets you receive. 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 side of the answer sheet.
The rough worksheets are for your own use for doing calculations, write some numbers etc. BEWARE: These sheets are not taken into account for the evaluation. At the end of the test they will be collected separately. Everything you consider as part of the solutions should be written on the answer sheets.
Each problem should be started on a separate answer sheet.
On each answer sheet please fill in the designated boxes as follows:

a. In the „PROBLEM NO.” box write down only the number of the problem: i.e. 1 – 12 for each short problems, 13 – 15 for each 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 that you will find on your envelope, consisting of 3 letters and 2 digits.

c. In the „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 language of Mathematics is universal, so, please, use as many mathematical expressions as you think that may help the evaluator to better understand your solutions. If you want to explain something in words we kindly ask you to use short phrases(if possible in English).
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:

d. Don’t forget to put your papers in order.

e. Put the answer sheets in 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 will help us to understanding your solutions.

f. Verify with the assistant the correct number of answer sheets used and fill in this number on the cover of the folder and sign it.

g. Put the draft papers in the designated folder. Put the test papers back in the envelope.

h. Go to swim

GOOD LUCK !

Marking scheme

ALL THE CORRECT SOLUTIONS WHICH ARE DIFFERENT FROM THE AUTHOR’S SOLUTION WILL BE EVALUATED AND MARKED ACORDINGLY TO THE MARKING SCHEME.

Условие

Inside a camera a plane mirror is placed along the optical axis of the objective (as shown in figure). The length of the mirror is half the focal length of the objective. A photographic plate is placed at the focal plane of the camera. Two images with different brightness are captured on the photographic plate (as shown in figure). The star Vega is not on the optical axis of the lens. The distance between the optical axis and the image $\Sigma_1$ is $\frac{r}{2}$ . Find the difference between the apparent photographical magnitudes of the two images of the star Vega.

Plane mirror of length f/2 in a camera, Vega, objective lens, photographic plate and two images.

Отговори

Решение

Problem 9. Marking scheme The Vega star in the mirror

The light beam arriving from Vega Star can be considered paraxial, due to the distance from it to the observer on Earth. The explanation for the existence of two distinct images of the star is that the optical axis of the objective is not parallel with the light beam from the star.

The images on the camera plate are symmetrical placed relative to the principal optical axis.

Each of the point images of the Vega Star $\Sigma_1$ and $\Sigma_2$ didn’t concentrate the same light fluxes. In the down below figure it can be seen the sections of the lens which correspond to each image. The sector APBC is passed by the light which concentrates in the image $\Sigma_2$ and the light passing the sector ACBQ concentrates into the point image $\Sigma_1$ See the picture in figure 13 .

The ratio between the light fluxes concentrated into the two image points will directly depend on the ratio of the two sectors areas.

From the geometry of the figure 2 results :

\mathrm{MN} = \mathrm{OM};\ \mathrm{N}\Sigma_1 = \mathrm{OC} = \frac{r}{2};

\angle\left(\mathrm{CBO}\right) = 30^{\circ};\ \angle\left(\mathrm{BOC}\right) = 60^{\circ};\ \angle\left(\mathrm{AOB}\right) = 120^{\circ};

\frac{S_1}{S_2} = \frac{8\pi + 3\sqrt{3}}{4\pi - 3\sqrt{3}} \approx 4.

Using the Pogson formula :

\log\frac{E_1}{E_2} = \log\frac{\frac{\sigma T_{\mathrm{V}}^4\cdot 4\pi R_{\mathrm{V}}^2}{4\pi d_{\mathrm{PV}}^2}\cdot S_1}{\frac{\sigma T_{\mathrm{V}}^4\cdot 4\pi R_{\mathrm{V}}^2}{4\pi d_{\mathrm{PV}}^2}\cdot S_2} = -0{,}4\left(m_1 - m_2\right);

\log\frac{S_1}{S_2} = -0{,}4\left(m_1 - m_2\right);

m_2 - m_1 = 1{,}5^{\mathrm{m}}.

Fig.12. Ray paths through the lens and mirror to the two Vega images.

Circular lens split into sectors S1 and S2 with 30-degree and 60-degree angles; figure caption printed on next source page as Fig.13 b. — Fig.13 b

Оригинал в Архива: Short Theoretical.pdf · официални решения: short_final_sol.pdf · допълнителни данни: ioaa1.jpg · допълнителни данни: ioaa2.jpg · допълнителни данни: ioaa3.jpg · допълнителни данни: ioaa4.jpg · допълнителни данни: ioaa5.jpg · допълнителни данни: ioaa6.jpg · допълнителни данни: ioaa7.jpg