Question 4.TC.1: Satellite's Orbit Transfer In the near future we ourselves m......
Satellite’s Orbit Transfer
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) A satellite of mass m is presently circling the Earth of mass M in a circular orbit of radius R_{0}. What is the e speed (u_{0}) of mass m in terms of M, R_{0} and the universal gravitation constant G?
(b) We are to put this satellite into a trajectory that will take it to point P at distance R, 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}?
(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.
(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} ?
(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}?
(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{\Big[}{\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}}r-{\Big(}{\frac{\mathrm{d}}{\mathrm{d}t}}\theta{\Big)}^{2}r{\Big]}=-G{\frac{M m}{r^{2}}}, (1)
and the conservation of angular momentum:
m r^{2}\;{\frac{\mathrm{d}}{\mathrm{dt}}\theta}={\mathrm{constant}}. (2)
(g) Give a rough sketch of the whole perturbed orbit together with the unperturbed one.
(b) Conservation of angular momentum: mu_{1}R_{0} = mu_{2}R_{1}.
Conservation 0f energy:{\frac{1}{2}}m u_{2}^{2}-{\frac{G M m}{R_{1}}}={\frac{1}{2}}m u_{1}^{2}-{\frac{G M m}{R_{0}}}
\left[\left(\frac{R_{\mathrm{0}}}{R_{1}}\right)^{2}-1\right]u_{1}^{2}=2G M\left(\frac{1}{R_{1}}-\frac{1}{R_{\mathrm{o}}}\right)
\frac{(R_{0}-R_{1})\,(R_{0}+R_{1})}{R_{1}^{2}}\ u_{1}^{2}=2G M\frac{R_{0}-R_{1}}{R_{0}R_{1}}
u_{1}=\sqrt{\frac{G{M}}{R_{\scriptscriptstyle0}}}\,\sqrt{\frac{2{R}_{1}}{R_{1}+R_{\scriptscriptstyle0}}}\,=\,u_{\scriptscriptstyle0}\sqrt{\frac{2{R}_{1}}{R_{1}+R_{\scriptscriptstyle0}}}.
(c)\operatorname*{lim}_{R_{1}\rightarrow\infty}u_{1}=\sqrt{2}\,u_{0}.
(\mathrm{d})\;u_{2}=u_{1}\,{\frac{R_{0}}{R_{1}}}=u_{0}\,{\frac{\sqrt{2}R_{0}}{\sqrt{R_{1}\left(R_{1}+{R}_{0}\right)}}}.
\left(\mathbf{e}\right)\,u_{3}={\sqrt{\frac{G M}{R_{\mathrm{1}}}}}={\sqrt{\frac{G M}{R_{\mathrm{0}}}}}\,{\sqrt{\frac{R_{\mathrm{0}}}{R_{\mathrm{1}}}}}=u_{0}{\sqrt{\frac{R_{\mathrm{0}}}{R_{\mathrm{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}}}\,.
(f) Combining equations (1) and (2):
{\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}}r-{\frac{C}{m r^{3}}}=-{\frac{G M}{r^{2}}},and for the circular orbit of radius R_{1} we have \frac{C}{m}=G M R_{1},
hence {\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}}r-{\frac{G M R_{1}}{r^{3}}}=-{\frac{G M}{r^{2}}},
putting r=R_{1}+\eta,{\mathrm{~where~}}\eta\ll R_{1},
∴ \frac{\mathrm{d}^{2}}{\mathrm{d}t}\eta-\frac{G M R_{1}}{R_{1}^{3}\left(1+\frac{\eta}{R_{1}}\right)^{3}}=-\frac{G M}{R_{1}^{2}\left(1+\frac{\eta}{R_{1}}\right)^{2}}
{\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}}\eta-{\frac{G M}{R_{1}^{2}}}\biggl(1-3\,{\frac{\eta}{R_{1}}}\biggr)\approx-{\frac{G M}{R_{1}^{2}}}\biggl(1-2\,{\frac{\eta}{R_{1}}}\biggr)
{\frac{\mathrm{d}^{2}}{\mathrm{d}t}}\eta\approx-{\frac{G M}{R_{1}^{3}}}\eta\quad.
The frequency of oscillation about mean distance is f=\frac{1}{2\pi}\sqrt{\frac{G M}{R_{1}^{3}}}.
The period T={\frac{1}{f}}=2\pi\sqrt{\frac{R_{1}^{3}}{G M}}.
Note that this period is the same as the orbital period.
(g)
