3D Satellite Orbit Transfer
Example 8.8: Energy changes in orbital transfer - 3D Visualization
Example
Question:
A 400 kg satellite is in a circular orbit of radius \(2R_E\) about the Earth. How much energy is required to transfer it to a circular orbit of radius \(4R_E\)? What are the changes in the kinetic and potential energies?
Solution:
Initially,
\[
E_i = -\frac{G M_E m}{4 R_E}
\]
While finally,
\[
E_f = -\frac{G M_E m}{8 R_E}
\]
The change in the total energy is
\[
\Delta E = E_f - E_i = \frac{G M_E m}{8 R_E}
\]
Or, using \(g = 9.81\,\mathrm{m}/\mathrm{s}^2\), \(m = 400\,\mathrm{kg}\), \(R_E = 6.37 \times 10^6\,\mathrm{m}\):
\[
\Delta E = \frac{g m R_E}{8} = \frac{9.81 \times 400 \times 6.37 \times 10^6}{8} = 3.13 \times 10^9\,\mathrm{J}
\]
The kinetic energy is reduced:
\[
\Delta K = K_f - K_i = -3.13 \times 10^9\,\mathrm{J}
\]
The change in potential energy is twice the change in the total energy:
\[
\Delta V = V_f - V_i = -6.25 \times 10^9\,\mathrm{J}
\]
Results:
Satellite mass: 400 kg
Initial orbit radius: 2RE
Final orbit radius: 4RE
Total Energy Required:
3.13 × 109 J
Change in Kinetic Energy:
-3.13 × 109 J
Change in Potential Energy:
6.25 × 109 J
Class 11 Example 2.4 Simulation: Physics Explained
What This Simulation Shows:
How to calculate the energy needed to move a satellite between different circular orbits around Earth.
Step-by-Step Energy Calculation:
1. Initial Orbit Energy (at 2RE):
Einitial = - (G × ME × m) / (4RE)
2. Final Orbit Energy (at 4RE):
Efinal = - (G × ME × m) / (8RE)
3. Total Energy Needed:
ΔE = (G × ME × m) / (8RE)
Using g = G×ME/RE2:
ΔE = (g × m × RE) / 8
= (9.81 × 400 × 6.37×106) / 8
= 3.13 × 109 Joules
What Changes During Transfer:
- 🔻 Kinetic Energy decreases by 3.13 × 109 J
- 🔺 Potential Energy increases by 6.25 × 109 J
- ⚡ Total Energy increases by 3.13 × 109 J



