IOAA Jr. 2022, Теория, тест с избираем отговор — Задача 5. Andromeda Galaxy Rotation

Archived Grid 5 variant: Andromeda Constellation Rotation

Our galaxy and the constellation Andromeda can be observed with the help of a telescope whose mirror has a diameter of $D=6\ \mathrm m$.

The following are known: the Earth - Andromeda distance, $R=1.42\cdot10^{11}R_0$, where $R_0$ is the radius of the Earth's circular orbit around the Sun; the mass of our Galaxy, $M_G=2.5\cdot10^{11}M_0$, where $M_0$ is the mass of the Sun; $T_0=1$ terrestrial year, the period of rotation of the Earth around the Sun; the mass of the constellation Andromeda, $M_A=3.6\cdot10^{11}M_0$.

Knowing that:

1. the photos were taken in visible light, with a wavelength $\lambda=5\cdot10^{-7}\ \mathrm m$;

2. the angular distance between two objects for which they can be observed separately is $\varphi_0=1.22\cdot\lambda/D$.

Using the photographs thus obtained, time required to highlight the rotational motions of our galaxy and the constellation Andromeda around their common center of mass, is:

a) $10^4$ ani; b) $10^3$ ani; c) $10^5$ ani; d) $10^4$ ani.

Solution supplied for the archived Grid 5 variant

From the study of light diffraction it is known that the angular distance between two objects for which they can be observed separately is approximately. $\lambda/D$. It means that the change in the position of the constellation Andromeda can be observed if its angular displacement is:

If $T$ is the period of rotation of our Galaxy and the Andromeda Constellation, respectively, around their common center of mass, then the time required for the angular displacement $\varphi_0$ of the Andromeda constellation is:

For two binary systems, using the generalized form of Kepler's third law, we can write that:

out of which, for the galaxy - Andromeda and Sun - Earth systems, results in: