Съдържание

IAO 2025, Теория, Групи α и γ — Задача 2. NGC of the Year

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2025

XXIX IAO. IV International Remote Astronomy Olympiad. Theoretical round. Problems to solve. Group α γ · 6 ноември 2025 г. · 8 т.

Внимание: Бележка към темата

The separate table of constants and Hertzsprung–Russell diagram referenced by the solutions are not present in the inspected 2025 archive inventory. No replacement reference values have been invented. The official solution to 1.3 contains a malformed area equality and an inconsistent missing factor; the printed formulas and result are retained with a transcription note.

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

NGC 2025 is a small open cluster, visible as an $11^{\mathrm{m}}$ object, located in the Mensa constellation, discovered by John Herschel on February 8, 1836. The coordinates of the center of the cluster are $05^{\mathrm{h}}\,32^{\mathrm{m}}$ , $-71^{\circ}\,43'$ , and its angular size is $2.0'$ . Estimate the minimum altitude required to ascend above Dhaka in a hot air balloon to observe the whole NGC 2025. Consider that a telescope powerful enough to observe $11^{\mathrm{m}}$ objects can be mounted on this balloon, taking into account atmospheric extinction near the horizon.

Отговори

Решение

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.

Altitude and refraction

The altitude $H$ , which must be reached to see an object located slightly below the horizon (at a small angle $\eta$ ), is determined by the formula, which is easy to obtain from the properties of a right-angled triangle by drawing the appropriate drawing:

(R+H)^2=R^2+(\eta R)^2\quad\Rightarrow\quad H\approx\eta^2R/2.

where $R$ is the radius of the Earth, and the angle $\eta$ must be taken in radians. We can substitute the value of the radius of the Earth and convert radians to degrees, then we get the formula with a coefficient 970 m, connecting $H$ (in meters) with $\eta$ (in degrees):

H(\mathrm{m})\approx970\,\mathrm{m}\cdot\eta(\mathrm{degrees})^2.

To see the whole cluster, you need to see its lowest point. The upper culmination of the lowest point of NGC 2025 in Dhaka will be at an altitude of

h_0=90^\circ-\varphi+\delta-\varrho/2=90^\circ-23^\circ48'-71^\circ43'-1'=-5^\circ32'.

that is, $5^\circ32'$ below the mathematical horizon. To compensate for this angle, you need to rise up to an altitude of about 30 km. This is an unimaginable height, exceeding the record for rising up in a thermal balloon.

But have we considered everything? Recall that there is refraction at the horizon, $\rho\approx35'$ , which “raises” objects a little. Taking this refraction into account

h_1=h_0+\rho=-4^\circ57',

In this case, you need to rise up about 24 km. Now let's note that when observing from such an altitude, refraction works almost twice: the first time on the way from the celestial object to the “point of contact of the ray of vision with the surface of the Earth”, and the second time on the way from the “point of contact” to the observer.

h_2=h_0+2\rho=-4^\circ22',

In this case, you need to reach about 18.5 km altitude. This is already an achievable height for a thermal balloon. This answer is an estimation.

You can go deeper and calculate the refractive angle more accurately. Using the formula for the dependence of atmospheric pressure on altitude, it can be calculated that at an altitude of about 18.5–20 km, atmospheric pressure is approximately 10% of the pressure at sea level. This means that 90% of the atmosphere is located below, respectively, for the second time, in the area from the “touch point” to the observer, refraction must be taken into account by a factor of 0.9.

h_3=h_0+1.9\rho\approx-4^\circ25'.

Taking this amendment into account, it is necessary to rise up about 19 km.

Official grading criteria

Item	Criterion	pts
	Formula connecting altitude with angle below horizon	1.5
	Formula for the upper culmination of the lowest point of NGC 2025 in Dhaka	1
	Refraction, understanding first $35\prime$	1
	Refraction, understanding second $35\prime$ or little less than $35\prime$	1
	Calculations the upper culmination of the lowest point of NGC 2025 in Dhaka	1
	Calculations of the altitude	1
	Final answer (depending of reality)	1.5
	Total	8

General notes for the 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.

Оригинал в Архива: problems_t_12_en.pdf · официални решения: solutions_t_en.pdf · допълнителни данни: criteria_t_en.pdf