[QUE/ME-12005] ME-PROBLEM

• A satellite of mass 2000 kg is to be put into a circular orbit around the earth of radius 1100 km. What is the minimum energy required?
• What will the minimum energy required to transfer it to an elliptic orbit having minimum and maximum distances 1100km and 4100km?

 \(R_e=6400 \text{km},\quad GM/R_e^2=g\Rightarrow GM=gR_e^2\) Initial energy \(E_i\) of the satellite on the surface of the earth is \begin{eqnarray} E_i &=& -\frac{GMm}{R_e} = -\frac{gR_e^2m}{R_e}\\ &=& -gR_e^2m = (9.8 \text{(m/s}^2))(6400\times 1000 \text{\,m})(2000\text{\,kg})\\ &=& -9.8\times 128 \times10^8\\ &=&-12.5\times 10^{10} J \end{eqnarray} Final energy of the satellite in the orbit at height of 1100 km be \(E_f\). \begin{eqnarray} E_f &=& -\frac{GM,}{2(R+h)} = -\frac{gR_e^2 m}{2(R_e+h)}\\ &=& -\frac{9.8\times(6400\times 6400\times 10^6)\times 2000}{2\times7500\times10^3}\\ &=& \frac{9.8\times64\times 64}{2\times75}\times10^8\\ &=& -5.35\times10^{10} \end{eqnarray} Therefore the energy required to put the satellite in the circular orbit at a height 1100km \\ \(E_f-E_i=(-5.35 +!2.5 )\times 10^{10} \text{J}=7.15\times 10^{10} \text{J}\). \paragraph*{Elliptic orbit} Let \(r_1,r_2,a\) be, respectively, the perigee, apogee and the semi-major axis of the elliptic orbit. Then \(2a=r_1+ r_2\) It is give that \[r_1= 1100 km +R_e, r_2 =4100km+ R_e\] Therefore \[ 2a= 1100+4100 + 2\times 6400 = 1800\text{\,km}\] Hence a=9000km. The energy of the satellite in the elliptic orbit is \begin{eqnarray} E_\text{ell} &=& -\frac{GMm}{2a} = -\frac{gR_e^2m}{2a}\\ &=& -\frac{9.8 \times 6400\times6400\times10^6 \times 2000}{1800\times 10^3}\\ &=& -\frac{9.8\times64^2 \times 2\times 10^5}{18}\\ &=& -4.4 \times 10^{10} J. \end{eqnarray} Therefore energy requires to transfer the satellite from the circular orbit to the elliptic orbit is \((-4.4+5.35)\times 10^{10}\text{ J} = 9.3\times 10^{9}\text{J}.\)