Orbital mechanics can feel strange because a spacecraft is always falling around Earth while also moving sideways fast enough to miss the ground. In low Earth orbit, firing the engine prograde means thrusting in the direction of motion, which adds orbital energy. The surprising result is that the spacecraft does not simply move faster everywhere.
Instead, the opposite side of the orbit rises, and the spacecraft later moves more slowly there because it is farther from Earth.
Key Facts
- Prograde thrust increases orbital energy and raises the far side of the orbit.
- Retrograde thrust decreases orbital energy and lowers the far side of the orbit.
- For a circular orbit, v = sqrt(mu / r), where mu = GM.
- Vis-viva equation: v^2 = mu(2 / r - 1 / a).
- Specific orbital energy: epsilon = v^2 / 2 - mu / r = -mu / (2a).
- Kepler's third law for Earth orbits: T = 2 pi sqrt(a^3 / mu).
Vocabulary
- Prograde burn
- A prograde burn is an engine firing in the same direction as the spacecraft's motion, increasing its orbital energy.
- Apoapsis
- Apoapsis is the point in an orbit where the spacecraft is farthest from the central body, such as Earth.
- Periapsis
- Periapsis is the point in an orbit where the spacecraft is closest to the central body.
- Semi-major axis
- The semi-major axis is half the longest width of an elliptical orbit and controls the orbit's period and energy.
- Orbital energy
- Orbital energy is the sum of a spacecraft's kinetic energy and gravitational potential energy per unit mass.
Common Mistakes to Avoid
- Thinking prograde thrust makes the spacecraft stay faster all the way around the orbit. It raises the orbit's energy, but at the higher apoapsis the spacecraft moves more slowly because it has climbed out of Earth's gravity well.
- Assuming the burn raises the spacecraft immediately at the burn point. A short prograde burn mostly raises the opposite side of the orbit, while the burn point becomes the new periapsis.
- Using circular-orbit speed for every point on an elliptical orbit. The formula v = sqrt(mu / r) only applies to circular orbits, while elliptical orbits require the vis-viva equation.
- Confusing altitude with orbital radius. Orbital formulas use distance from Earth's center, so r = Earth's radius plus altitude.
Practice Questions
- 1 A spacecraft is in a circular orbit at radius r = 6.78 x 10^6 m around Earth. Using mu = 3.986 x 10^14 m^3/s^2, calculate its circular orbital speed with v = sqrt(mu / r).
- 2 An elliptical transfer orbit has periapsis radius rp = 6.78 x 10^6 m and apoapsis radius ra = 7.78 x 10^6 m. Find the semi-major axis a, then use v^2 = mu(2 / rp - 1 / a) to calculate the speed at periapsis.
- 3 A spacecraft in low Earth orbit fires prograde for a short time. Explain which part of the orbit rises, which point becomes periapsis, and why the spacecraft can be slower after it has gained energy.