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USAAAO 2026, I кръг — Задача 12

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Всички материали от USAAAO 2026

USAAAO 2026 - First Round · 9 февруари 2026 г.

Problem writers

Problem writers: Srihari (Hari) Balaji, Vincent Bian, Ferdinand, Hagan Hensley, Elizabeth Lee, Joe McCarty, Lucas Pinheiro, Feodor Yevtushenko

Условие

An extraterrestrial form of life from a distant planetary system observed a transit of Jupiter across the Sun. The transit was central, which means that the center of Jupiter’s disk intersected with the center of the Sun’s disk throughout the transit. Neglecting the effect of limb darkening, which of the light curves below could correspond to the transit of Jupiter seen by the extraterrestrial?

The $x$ -axis on the plots corresponds to the time difference in hours from the middle of the transit, and the $y$ -axis corresponds to the apparent bolometric magnitude of the star.

Light curve (a): apparent bolometric magnitude from 9.740 to 9.752 versus time from mid-transit from about −25 to 25 h; flat at 9.740, steep drop to 9.7515 between about −16 and −13 h, flat bottom, steep rise between about 13 and 16 h. — (a)

Light curve (b): apparent bolometric magnitude from about 4.74 to 4.950 versus time from mid-transit from about −12 to 12 h; gradual V-shaped ingress from about −8 to −3 h, short flat bottom near 4.94, gradual egress from about 3 to 8 h. — (b)

Light curve (c): apparent bolometric magnitude from about 7.1255 to 7.137 versus time from mid-transit from about −35 to 35 h; flat, drop between about −23 and −19 h, flat bottom, rise between about 19 and 23 h. — (c)

Light curve (d): apparent bolometric magnitude from about 8.2348 to 8.2353 versus time from mid-transit from about −27 to 27 h; almost vertical drop at about −17 h, flat bottom at 8.2353, almost vertical rise at about 17 h. — (d)

Light curve (e): apparent bolometric magnitude from 8.225 to about 8.435 versus time from mid-transit from about −29 to 29 h; gradual ingress from about −19 to −8 h, flat bottom, gradual egress from about 8 to 19 h. — (e)

Отговори

Решение

Solution: There a few main parameters that can be used to determine which of the light curves corresponds to the transit of Jupiter. One of them is the variation in magnitude during the transit, which is caused by the planet blocking part of the star’s disk. Using Pogson’s equation:

\begin{aligned}\Delta m &= -2.5\log\left(\frac{\pi R_{\odot}^2 - \pi R_{\text{♃}}^2}{\pi R_{\odot}^2}\right)\\ &= -2.5\log\left(\frac{(6.96\times10^{8})^2 - (7.1492\times10^{7})^2}{(6.96\times10^{8})^2}\right)\\ &= 1.15\times10^{-2}\end{aligned}

Note that the exact magnitudes before and during the transit depend on the distance between the observer and the solar system, and only the variation in magnitude is relevant to determine the correct light curve.

Another important parameter is the duration of the transit (between the first and fourth contacts). In order to calculate that, it is important to first determine the orbital velocity of Jupiter, which can be calculated as follows assuming that the orbit is roughly circular:

\begin{aligned}v_{\text{♃}} &= \sqrt{\frac{GM_{\odot}}{a_{\text{♃}}}}\\ &= \sqrt{\frac{6.673\times10^{-11}\times1.989\times10^{30}}{5.2\times1.496\times10^{11}}}\\ &= 1.3\times10^{4}\ \mathrm{m/s}\end{aligned}

Therefore, the interval between the first and fourth contacts is the following:

\begin{aligned}\Delta t_{1-4} &= \frac{2(R_{\odot} + R_{\text{♃}})}{v_{\text{♃}}}\\ &= \frac{2(6.96\times10^{8} + 7.1492\times10^{7})}{1.3\times10^{4}}\\ &= 1.2\times10^{5}\ \mathrm{s}\\ &= 33\ \mathrm{h}\end{aligned}

These two parameters are already enough to determine the correct answer, but another parameter students could choose to calculate is the interval between the first and second contacts of the transit, which can be determined as follows:

\begin{aligned}\Delta t_{1-2} &= \frac{2R_{\text{♃}}}{v_{\text{♃}}}\\ &= \frac{2\times7.1492\times10^{7}}{1.3\times10^{4}}\\ &= 1.1\times10^{4}\ \mathrm{s}\\ &= 3.0\ \mathrm{h}\end{aligned}

This is the same as the interval between the third and fourth contacts. It would also be possible to use the interval between the second and third contacts as a parameter to identify the correct light curve.

Alternative A is the only light curve that has the correct variation in magnitude, transit duration, and intervals between the contacts of the transit.

Answer: A

Оригинал в Архива: usaaao_first_exam_2026.pdf · официални решения: usaaao_first_exam_solutions_2026.pdf