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IAO 2025, Теория, Групи α, β и γ — Задача 4. On a remote exoplanet

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Всички материали от IAO 2025

XXIX IAO. IV International Remote Astronomy Olympiad. Dhaka, Bangladesh, 6-15. XI. 2025. Theoretical round. Problems to solve · 6 ноември 2025 г. · 8 т.

Внимание: Бележка към темата

The separate numbered exoplanet data table for problem 4, the table of constants and the Hertzsprung–Russell diagram referenced by the problems and solutions are not in the archive.

Reminding

Reminding. Most of the problems are compiled in such a way that when writing solutions, minimal use of any language is required from the participant. So minimum texts are allowed in solutions of the problems (neither in the native language, nor in one of the official languages of the IAO). Pay more attention to formulas, drawings and graphs. It is permitted to write a few keywords in English (for example, “Yes”, “No”, “Assume”, “Answer”, “Average”, “Minimum”, “Maximum”, “Impossible situation”, etc.), as well as terms contained in the accompanying tables and names of objects.

Условие

The table (see on a separate sheet) shows the characteristics of the Earth-like, presumably habitable exoplanet Antechea in the planetary system of the star Erythrelatis in the constellation Ursa Major. For comparison, similar characteristics of the Earth are given. Determine the spectral class (O, B, A, F, G, K, M) of the star Erythrelatis, if possible, also specify the subclass (for example, B8, A4, G2, etc.). The orbits of the planets can be considered circular. Explain which parameter from the table you use at each step of the solution (in square brackets, for example [#4]). At the end of the solution, specify which parameters the answer to the problem does not depend on (for example, "Not depend on [#1, 4, 5, 6]").

Отговори

Решение

To determine the spectral class of the star Erythrelatis, it is necessary to find the approximate temperature of its photosphere.

The thermal balance of the planet is determined by the incoming and outgoing energy per unit of time.

W_{in}=W_{out}.

Incoming power:

W_{in}=(1-\alpha)\cdot\sigma T_{star}^4\cdot\frac{\pi(\beta/2)^2}{4\pi}\cdot4\pi R_{planet}^2=(1-\alpha)\cdot\sigma T_{star}^4\cdot\frac{\pi\beta^2}{4}\cdot R_{planet}^2,

where $\sigma$ is the Stefan–Boltzmann constant, $\alpha$ is the albedo of the planet, and $\beta$ is the angular diameter of the central star visible from the planet.

Outgoing (radiated) power:

W_{out}=\sigma T_{planet}^4\cdot4\pi R_{planet}^2=\sigma(T_{obs}-T_{GHE})^4\cdot R_{planet}^2,

where $T_{planet}$ is the effective temperature of the planet, $T_{obs}$ is the observed mean temperature of the planet, and $T_{GHE}$ is the greenhouse effect.

From these two equations:

(1-\alpha)\cdot T_{star}^4\cdot\frac{\pi\beta^2}{4}=(T_{obs}-T_{GHE})^4,

From where:

T_{star}=(T_{obs}-T_{GHE})\cdot\sqrt[4]{\frac{4}{\pi\beta^2(1-\alpha)}}.

Coefficients are not needed to compare the Sun and the star Erythrelatis, therefore:

T_{star}\sim(T_{obs}-T_{GHE})\cdot\beta^{-1/2}(1-\alpha)^{-1/4},

T_{ery}=T_{sun}\cdot\frac{(T_{obs}-T_{GHE})_{ant}}{(T_{obs}-T_{GHE})_{earth}}\cdot\left(\frac{\beta_{ery}}{\beta_{sun}}\right)^{-1/2}\cdot\left(\frac{1-\alpha_{ant}}{1-\alpha_{earth}}\right)^{-1/4}.

The angular diameter of the star Erythrelatis, visible from the planet Antechea, can be found by analyzing the data of the length of day and the duration from rise to set of the central star given in the table. For the Earth, we see that the time from sunrise to sunset is 406 seconds longer than half a day. Converting this value to angular minutes (the length of day corresponds to 21600 angular minutes), we get $\delta_{earth}=101.5'$ . It can be verified that this addition is caused by refraction (twice $\rho_{earth}=34.8'$ at sunrise and sunset, data from the table of constants) and the angular diameter of the Sun (twice the angular radius of the Sun: at sunrise and sunset, $\beta_{sun}=31.9'$ in total).

For Antechea, the time from rise to set of Erythrelatis is 324 seconds longer than half a day. Converting this value to angular minutes (note that the conversion coefficient is different here: 21 600 angular minutes corresponds to 70 832 seconds), we get $\delta_{ant}=98.8'$ . This is also the doubled refraction at the horizon plus the apparent angular diameter of Erythrelatis.

A significant error is possible here: twice subtract $\rho_{earth}=34.8'$ from $\delta_{ant}=98.8'$ , and get $\beta_{ery}=29.2'$ . (Looking ahead, we note that in this highly erroneous case, $T_{ery}\approx6300\,\mathrm K$ and spectral class F8 are obtained.)

The value of refraction is proportional to the density of the atmosphere at the surface of the planet, which is the same as the ratio of pressure to temperature. (That this is the relationship can also be understood from the refraction formula given in the table of constants.) The temperature near the surface on both planets is the same, so the value of refraction is proportional to the atmospheric pressure:

\rho_{ant}=\rho_{earth}\cdot\frac{P_{ant}}{P_{earth}}=15.08'.

Thus, the angular diameter of the star Erythrelatis, as seen from the planet Antechea, is:

\beta_{ery}=\delta_{ant}-2\rho_{ant}\approx68.6'.

According to the formula obtained above for $T_{ery}$ :

T_{ery}=5777\,\mathrm K\cdot\frac{259\,\mathrm K}{254\,\mathrm K}\cdot\left(\frac{68.6'}{31.9'}\right)^{-1/2}\cdot\left(\frac{0.64}{0.71}\right)^{-1/4}=4123\,\mathrm K\approx4100\,\mathrm K.

Using the Hertzsprung-Russell diagram, we determine that stars of the K7 class have such a temperature.

The answer to the problem does not depend on neither the equatorial radius of the planet [#1], the mass of the planet [#2] nor the acceleration of gravity [#3].

Answer: the spectral class of the star Erythrelatis is K7. Not depending on [#1, 2, 3]

Theoretical round. Sketches for solutions. For jury job ONLY

These solutions are written for jury members, so they contain a lot of explanations in the form of text. Students are instructed to use mostly formulas, calculations, drawings and graphs, writing texts is not allowed in the solutions. It is permitted to write a few keywords in English (for example, “Yes”, “No”, “Assume”, “Answer”, “Average”, “Minimum”, “Maximum”, “Impossible situation”, etc.), as well as terms contained in the accompanying tables and names of objects.

If you find typos, errors or missing points in the solution, please inform the organizing committee. Any suggestions for modifying the solutions are welcome, write justifications to gavrilovissp@list.ru and gavrilov@issp.ac.ru (to both addresses for reliability), if the suggestions are reasonable, the proposal can be approved.

Theoretical round. Basic criteria. For work of Jury

Note. Jury members should evaluate the student’s solutions in essence, and not by looking on formal existence the mentioned sentences or formulae. The formal presence of the mentioned positions in the text is not necessary to give the respective points. Points should be done if the following steps de facto use these positions. (For example, except the final conclusion and artistic picture, all the points in solution of problem 2 may be in only one common formula.) Of course, fractions of the mentioned points should be awarded if only part of the item was done.

Note. Jury members may elaborate more detailed criteria, and also create criteria for other correct ways of the student’s solutions.

Note. Everywhere “formula” means “correct formula”, “calculation” means “correct calculation”, “picture” means “correct picture”, etc., and in all cases: based on correct facts and intermediate formulae or calculations.

Note. Everywhere final calculation means also using correct number of significant digits. Excessive number of significant digits, especially in estimational problems, will lead to subtraction of the points (column “digits”, minus 1-2 points).

Note. Solutions with absurd or almost absurd answers should be rated lower (maybe ignoring the scoring schemes below, and maybe down to zero, column “absurd”).

Any suggestions for modifying the criteria are welcome, write justifications to gavrilovissp@list.ru and gavrilov@issp.ac.ru (to both addresses for reliability), if the suggestions are reasonable, the proposal can be approved.

3. Comet Lemmon

		pts
3.1.	Actual semimajor axis of the orbit	1
	Actual orbital period	1
	Actual eccentricity of the orbit	1
3.2.	Steps to get new eccentricity of the orbit	2.5
	Actual eccentricity of the orbit	1
	Actual orbital period	1
	Final conclusion	0.5
	Total	8

4. On a remote exoplanet

		pts
	Thermal balance	0.8
	Formula for incoming power	0.5
	Formula for outgoing power	0.4
	Using albedo	0.4
	Using greenhouse effect	0.4
	Analyzing time from sunrise to sunset vs half a day, additional angular sizes	0.8
	Using dependence on the angular size of the central star	0.8
	Using any refraction	0.4
	Using refraction depending on atmospheric pressure	1.1
	Using spectral class of the Sun or temperature of the Sun	0.4
	Final formula and/or correct calculations	0.8
	Final correct result in a range K5-K9	0.8
	Postulating “Not depending on [#1, 2, 3]”	0.4
	Total	8

5. Canopus and Sirius

		pts
5.1.	Understanding about upper culmination	0.5
	Using right ascension of Canopus	1
	Using difference in days since autumnal equinox	0.5
	Using Dhaka is located about 0.5° East from the 90° meridian	0.5
	Final formula and calculations	1
5.2.	Altitude of Canopus at upper culmination	0.6
	Altitude of Sirius at the time of upper culmination of Canopus	0.6
	Extinction in atmosphere	1
	Using limit of visibility by the naked eye	0.4
	Calculation the layers of atmosphere for Canopus and Sirius	0.8
	Final formula and/or correct calculations, final answer	1.1
	Total	8

Оригинал в Архива: problems_t_345_en.pdf · официални решения: solutions_t_en.pdf · допълнителни данни: criteria_t_en.pdf