Съдържание

APhO 2003, Теория — Задача 1. Satellite’s orbit transfer

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от APhO 2003

4th Asian Physics Olympiad — Theoretical Competition — 23 April 2003

Условие

In the near future we ourselves may take part in launching of a satellite which, in point of view of physics, requires only the use of simple mechanics.

A small circular orbit of radius R0 around the Earth M, with the satellite m moving on it and the velocity u0 drawn tangent to it. An elliptical transfer orbit touches the circle at point Q, where the velocity u1 is drawn, and reaches the far point P, where the velocity u2 is drawn perpendicular to the dotted line of length R1 from M to P.

a) A satellite of mass $m$ is presently circling the Earth of mass $M$ in a circular orbit of radius $R_0$ . What is the speed $(u_0)$ of mass $m$ in terms of $M, R_0$ and the universal gravitation constant $G$ ? [1 т.]

b) We are to put this satellite into a trajectory that will take it to point P at distance $R_1$ from the centre of the Earth by increasing (almost instantaneously) its velocity at point Q from $u_0$ to $u_1$ . What is the value of $u_1$ in terms of $u_0, R_0, R_1$ ? [2 т.]

c) Deduce the minimum value of $u_1$ in term of $u_0$ that will allow the satellite to leave the Earth’s influence completely. [1 т.]

d) (Referring to part b.) What is the velocity $(u_2)$ of the satellite at point P in terms of $u_0, R_0, R_1$ ? [1 т.]

e) Now, we want to change the orbit of the satellite at point P into a circular orbit of radius $R_1$ by raising the value of $u_2$ (almost instantaneously) to $u_3$ .

What is the magnitude of $u_3$ in terms of $u_2, R_0, R_1$ ? [1 т.]

Axes X and Y with the Earth M at the origin; a dotted circle labelled with an arrow “mean orbit of radius R1”; the satellite m at distance r from M, at angle θ from the X axis, on a solid perturbed trajectory that crosses the dotted circle.

f) If the satellite is slightly and instantaneously perturbed in the radial direction so that it deviates from its previously perfectly circular orbit of radius $R_1$ , derive the period of its oscillation $T$ of $r$ about the mean distance $R_1$ .

Hint: Students may make use (if necessary) of the equation of motion of a satellite in orbit:

m\left[\frac{d^2}{dt^2}r - \left(\frac{d}{dt}\theta\right)^2 r\right] = -G\frac{Mm}{r^2} \qquad \ldots\ldots\ldots\ldots\ldots\ (1)

and the conservation of angular momentum:

mr^2\frac{d}{dt}\theta = \text{constant} \qquad \ldots\ldots\ldots\ldots\ldots\ (2)

[3 т.]

g) Give a rough sketch of the whole perturbed orbit together with the unperturbed one. [1 т.]

Отговори

Решение

\frac{mu_0^{\,2}}{R_0} = \frac{GMm}{R_0^{\,2}}, \qquad u_0 = \sqrt{\frac{GM}{R_0}}

(1 point)

b) conservation of angular momentum: $mu_1R_0 = mu_2R_1$

conservation of energy: $\tfrac{1}{2}mu_2^{\,2} - \dfrac{GMm}{R_1} = \tfrac{1}{2}mu_1^{\,2} - \dfrac{GMm}{R_0}$

\left[\left(\frac{R_0}{R_1}\right)^2 - 1\right]u_1^{\,2} = 2GM\left[\frac{1}{R_1} - \frac{1}{R_0}\right]

\frac{(R_0 - R_1)(R_0 + R_1)}{R_1^{\,2}}u_1^{\,2} = (2GM)\frac{(R_0 - R_1)}{R_0R_1}

u_1 = \sqrt{\frac{GM}{R_0}}\sqrt{\frac{2R_1}{R_1 + R_0}} = u_0\sqrt{\frac{2R_1}{R_1 + R_0}}

(2 points)

\lim_{R_1 \to \infty} u_1 = \sqrt{2}\,u_0

(1 point)

u_2 = u_1\frac{R_0}{R_1} = u_0\frac{\sqrt{2}\,R_0}{\sqrt{R_1(R_1 + R_0)}}

(1 point)

u_3 = \sqrt{\frac{GM}{R_1}} = \sqrt{\frac{GM}{R_0}}\sqrt{\frac{R_0}{R_1}} = u_0\sqrt{\frac{R_0}{R_1}}

= \sqrt{\frac{R_0}{R_1}}\sqrt{\frac{R_1(R_1 + R_0)}{\sqrt{2}R_0}}\,u_2

u_3 = u_2\sqrt{\frac{R_1 + R_0}{2R_0}}

(1 point)

f) (3 points) combining equations (1) and (2) :

\frac{d^2}{dt^2}r - \frac{C/m}{r^3} = -\frac{GM}{r^2}

and for the circular orbit of radius $R_1$ we have $\dfrac{C}{m} = GMR_1$

hence

\frac{d^2}{dt^2}r - \frac{GMR_1}{r^3} = -\frac{GM}{r^2}

putting $r = R_1 + \eta$ , where $\eta \ll R_1$

\therefore\ \frac{d^2}{dt^2}\eta - \frac{GMR_1}{R_1^{\,3}\left(1 + \frac{\eta}{R_1}\right)^3} = -\frac{GM}{R_1^{\,2}\left(1 + \frac{\eta}{R_1}\right)^2}

\frac{d^2}{dt^2}\eta - \frac{GM}{R_1^{\,2}}\left(1 - 3\frac{\eta}{R_1}\right) \approx -\frac{GM}{R_1^{\,2}}\left(1 - 2\frac{\eta}{R_1}\right)

\frac{d^2}{dt^2}\eta \approx -\frac{GM}{R_1^{\,3}}\eta

the frequency of oscillation about mean distance is $f = \dfrac{1}{2\pi}\sqrt{\dfrac{GM}{R_1^{\,3}}}$

the period $T = \dfrac{1}{f} = 2\pi\sqrt{\dfrac{R_1^{\,3}}{GM}}$

Note that this period is the same as the orbital period

h) (1 point)

Axes X and Y with M at the origin; a dotted circle labelled with an arrow “unperturbed orbit” and a radius arrow R1; a dash-dotted closed curve, shifted relative to the dotted circle, labelled with an arrow “perturbed orbit”.

Оригинал в Архива: 1.pdf · официални решения: 1_sol.pdf