Question 13.7: Changing the Orbit of a Satellite A space transportation veh...
Changing the Orbit of a Satellite
A space transportation vehicle releases a 470-kg communications satellite while in a circular orbit 280 km above the surface of the Earth. A rocket engine on the satellite boosts it into a geosynchronous orbit. How much energy does the engine have to provide for this boost?
Conceptualize Notice that the height of 280 km is much lower than that for a geosynchronous satellite, 36 000 km, as mentioned in Example 13.5. Therefore, energy must be expended to raise the satellite to this much higher position.
Categorize This example is a substitution problem.
Find the initial radius of the satellite’s orbit when it is still in the vehicle’s cargo bay:
r_i=R_E+280 km=6.65 \times 10^6 mUse Equation 13.19 to find the difference in energies for the satellite-Earth system with the satellite at the initial and final radii:
E=-\frac{G M m}{2 r} (circular orbits) (13.19)
\Delta E=E_f-E_i=-\frac{G M_E m}{2 r_f}-\left(-\frac{G M_E m}{2 r_i}\right)=-\frac{G M_E m}{2}\left(\frac{1}{r_f}-\frac{1}{r_i}\right)Substitute numerical values, using r_f=4.22 \times 10^7 m from Example 13.5:
\begin{aligned} \Delta E=&-\frac{\left(6.674 \times 10^{-11} N \cdot m^2 / kg^2\right)\left(5.97 \times 10^{24} kg\right)(470 kg)}{2}\\&\times\left(\frac{1}{4.22 \times 10^7 m}-\frac{1}{6.65 \times 10^6 m}\right)=1.19 \times 10^{10} J\end{aligned}which is the energy equivalent of 89 gal of gasoline. NASA engineers must account for the changing mass of the spacecraft as it ejects burned fuel, something we have not done here. Would you expect the calculation that includes the effect of this changing mass to yield a greater or a lesser amount of energy required from the engine?