Съдържание

IOAA 2014, Теория, предварителен вариант — Задача 12. 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, 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

Inside a photo camera a plane mirror is placed along the optical axis of the lens of the objective (as seen in figure 13). The length of the mirror is half of the focal distance of the lens of the objective. The photo camera is oriented as on the photographic plate situated in the focal plane of the photo camera are captured two images with different illuminations of the Vega star. Find out the difference between the apparent photographical magnitudes of the two images of the Vega stars.

Photo camera: light from the VEGA star arrives along the dashed optical axis to the lens objective of radius r; inside the camera box a plane mirror of length f/2 lies along the optical axis ending at the photographical plate at distance f from the lens; on the plate two star images Σ₂ (above the axis) and Σ₁ (below, at r/2). — Figure 13

Отговори

Решение

Problem 12. 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.

Ray diagram of the camera: parallel rays from the left (letters V, E, G, A) strike the lens PQ (centre O, radius r, point C); a grey shaded beam converges from CQ to the image Σ₁ on the photographic plate below the axis; rays through the upper part PC reflect on the plane mirror MN (length f/2, on the dashed optical axis) and converge to the image Σ₂ above the axis; the distances f (lens to plate) and r/2 (N to Σ₁) are marked. — Fig. 12

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 .

Circular lens of radius r with centre O and vertical dashed diameter PQ; the chord AB crosses PC at C; the upper white segment is S₂ and the grey remainder S₁; dashed radii OA and OB, with the angle 60° at O between OC and OA and the angle 30° at B. — Fig.13 b

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}}.

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