IOAA 2021 — Задача 1. Formation of the Venus-2

Theory — Q14-1 / Q14-2, English (Official), International Olympiad on Astronomy and Astrophysics 14 · 35 т.

Условие

A comet of mass $\alpha m$ is heading ("falls") radially towards the Sun. It is known that the total mechanical energy of the comet is zero. The comet crashes into Venus, whose mass is $m$. We further assume that the orbit of Venus, before the collision, is circular with radius $R_0$. After the crash, the comet and Venus form a single object, called "Venus-2".

14.1 Find the expression in terms of $M_{sun}$ and $R_0$ for the orbital speed, $v_0$ , of Venus before the collision. [1 т.]

14.2 Find an expression for the total mechanical energy of Venus in its orbit before colliding with the comet. [1 т.]

14.3 Find an expression for the radial velocity, $v_r$ , the angular momentum, $L$ , of "Venus-2" immediately after the collision. [10 т.]

14.4 Find an expression for the mechanical energy of the combined object "Venus-2" and express it in terms of energy before the collision, $E_i$ and $\alpha$ . [5 т.]

14.5 Show that the post-collision orbit of "Venus-2" is elliptical and determine the semi-major axis of the orbit. [5 т.]

14.6 Determine if the year for the inhabitants of "Venus-2" has been shortened or lengthened because of collision with the comet. Write the ratio between the period of Venus-2 and Venus. [3 т.]

14.7 What should be the value of $\alpha$ such that the post-collision orbit of Venus-2 would make it crash in the Sun? We will call this as $\alpha_c$ [5 т.]

14.8 A comet with $\alpha = \alpha_c$ collided with Venus. Calculate the percentage change in the magnitude of Venus' velocity ($\delta v$) and the change in the direction of the velocity vector ($\delta\theta$) immediately after the collision. [5 т.]

Решение

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Оригинал в Архива: TQ-14-Q.pdf