Many satellites orbit Earth at maximum altitudes above Earth's surface of 1000 km or less. Geosynchronous satellites, however, orbit at an altitude of 35790 km above Earth's surface. How much more energy is required to launch a 410 kg satellite into a geosynchronous orbit than into an orbit 1000 km above the surface of Earth?

6.26 times more

Explanation:

Given:

- most satellite orbit at height r_1 = 1000 km

- Geosynchronous satellites orbit at height r_2 = 35,790 km

- mass of Geosynchronous satellite m = 410 kg

- The radius of the earth r_e = 6371 km

Find:

- Compare the Energy required to send the satellite to Geosynchronous orbit @ r_2 vs Energy required to send the satellite to normal orbit @ r_1. How much more. (U_1 / U_2).

Solution:

- The gravitational potential energy of any mass m in an orbit around another mass M is given by the following relation:

U_g = - G*m*M / r

Where,

G : Gravitational constant

- We compute the gravitational potential energy U_g of the satellite at both orbits as follows:

-Normal orbit U_1 = - G*m*M / r_e + G*m*M / (r_e+r_1)

U_2 = - G*m*M / r_e + G*m*M / (r_2+r_e)

Now: Take a ratio of the two energies U_1 and U_2 as follows:

U_2 / U_1 = ( - G*m*M / r_e + G*m*M / r_2+r_e) / ( - G*m*M / r_e + G*m*M / r_1+r_e)

U_2 / U_1 = (1 / (r_2+r_e) - 1 / r_e) / (1 / (r_1 + r_e) - 1 / r_e)

- Plug values:

U_2 / U_1 = (1 / (35790+6371) - 1 / 6371) / (1 / (1000+6371) - 1 / 6371)

- Evaluate:

U_2 / U_1 = (-1.33242681 * 10^-4) / (2.12944*10^-5)

U_2 / U_1 = 6.26

- Hence The energy required to send the satellite to Geosynchronous orbit is 6.26 times more than that required for normal orbit.