Orbital transfers and maneuvers

Changing orbits

A spacecraft in orbit is continuously falling around a central body. To change its orbit, it must change its velocity. The required change in velocity is called delta-v:

$Delta v$

Mission planning often focuses on minimizing total $Delta v$ because rocket fuel is limited and expensive to launch.

A short engine burn is often approximated as an impulsive maneuver: the spacecraft position remains nearly fixed during the burn, but its velocity changes suddenly.

Prograde and retrograde burns

A prograde burn points in the direction of motion and increases speed. This raises orbital energy. In a circular orbit, a prograde burn does not simply move the spacecraft outward immediately; it places the spacecraft on an ellipse whose opposite side reaches a higher altitude.

A retrograde burn points opposite the direction of motion and reduces speed. This lowers orbital energy and can lower the opposite side of the orbit.

Vis-viva equation

For a body orbiting mass $M$, orbital speed at distance $r$ in an orbit with semi-major axis $a$ is given by the vis-viva equation:

ight)$$ For a circular orbit, $a=r$, so this reduces to $$v_c=sqrt{rac{GM}{r}}$$ The vis-viva equation is central for calculating transfer orbits. ## Hohmann transfer A Hohmann transfer is an efficient two-burn transfer between two coplanar circular orbits. Suppose the initial circular radius is $r_1$ and the final circular radius is $r_2$. The transfer ellipse has semi-major axis $$a_t=rac{r_1+r_2}{2}$$ The first burn places the spacecraft onto the transfer ellipse. The second burn circularizes the orbit at the destination radius. For outward transfer, the first burn is prograde at periapsis, and the second burn is prograde at apoapsis. For inward transfer, the burns are retrograde. ## Transfer time The time for a Hohmann transfer is half the period of the transfer ellipse: $$t_{transfer}=rac{1}{2}T_t$$ Using Kepler's third law, $$T_t=2pisqrt{rac{a_t^3}{GM}}$$ so $$t_{transfer}=pisqrt{rac{a_t^3}{GM}}$$ ## Plane changes Changing orbital inclination requires rotating the velocity vector. Plane changes are expensive in $Delta v$, especially at high speed. A pure plane change by angle $Delta i$ at speed $v$ requires $$Delta v=2vsinleft(rac{Delta i}{2} ight)$$ This is why plane changes are often performed where orbital speed is low, such as near apoapsis. ## Gravity assists A gravity assist uses a planet's motion and gravity to change a spacecraft's heliocentric energy and direction. In the planet's frame, the spacecraft enters and leaves with the same speed far from the planet, but its direction changes. In the Sun's frame, this direction change can increase or decrease the spacecraft's speed. Gravity assists are essential for many deep-space missions. ## The big idea Orbital maneuvers are controlled changes in velocity. Delta-v measures maneuver cost, the vis-viva equation relates speed to orbital energy, and Hohmann transfers provide efficient two-burn paths between circular orbits. More advanced maneuvers use plane changes, phasing, and gravity assists to shape trajectories.