8.3 Orbital Maneuvers and Transfers

Orbital Maneuvers and Transfers

Orbital maneuvers are how spacecraft change their paths through space. Every maneuver comes down to one core quantity: delta-v ($\Delta v$), the change in velocity a spacecraft needs to execute. Understanding delta-v, transfer orbits, and gravity assists tells you what missions are physically possible and how much fuel they'll cost.

Concept of Delta-v in Maneuvers

Delta-v ($\Delta v$) is the total change in velocity required to perform an orbital maneuver, measured in m/s or km/s. If a maneuver has a $\Delta v$ of 1 km/s, the spacecraft must change its velocity by 1 km/s in the appropriate direction.

Why does this matter so much? Delta-v is directly tied to how much propellant a spacecraft needs. The Tsiolkovsky rocket equation makes this relationship precise: higher $\Delta v$ demands exponentially more propellant mass. That means a spacecraft's propellant tanks and engines are sized around the total $\Delta v$ budget for the entire mission.

• High $\Delta v$ maneuvers can be infeasible or cost-prohibitive, so minimizing $\Delta v$ is a central goal in mission planning.
• Mission trajectories are carefully optimized to reduce total $\Delta v$, using techniques like Hohmann transfers and gravity assists.
• The $\Delta v$ budget also determines what launch vehicle you need. A mission with lower $\Delta v$ requirements can use a smaller rocket or carry a larger payload.

Hohmann Transfer for Orbit Changes

The Hohmann transfer is the most fuel-efficient way to move a spacecraft between two coplanar circular orbits. It uses just two engine burns connected by an elliptical transfer orbit.

Here's how it works, step by step:

1. The spacecraft starts in the initial circular orbit (say, a low Earth orbit).
2. A prograde burn (firing in the direction of travel) at one point raises the opposite side of the orbit, creating an elliptical transfer orbit. The burn point becomes the periapsis (closest point) of the ellipse, and the far side reaches out to the altitude of the target orbit.
3. The spacecraft coasts along the ellipse for half an orbit until it reaches apoapsis (the farthest point), which touches the target orbit.
4. A second prograde burn at apoapsis adds enough velocity to circularize the orbit at the new, higher altitude.

This method minimizes $\Delta v$ compared to alternatives like one-tangent burns or fast transfers. It's commonly used for LEO to GEO transfers, where satellites are first launched into a low Earth orbit parking orbit, then use a Hohmann transfer to reach geostationary orbit.

The tradeoff is time. The transfer takes half the orbital period of the ellipse, which can range from several hours to days depending on the orbits involved. For time-sensitive or crewed missions, faster (but more fuel-costly) transfers may be preferred.

Energy Requirements of Orbital Transfers

Every orbital maneuver changes the spacecraft's specific orbital energy, given by:

$\epsilon = -\frac{\mu}{2a}$

where $\mu$ is the gravitational parameter of the central body and $a$ is the semi-major axis of the orbit. A few key principles follow from this:

• Raising an orbit (increasing $a$) increases orbital energy and requires a prograde burn (positive $\Delta v$).
• Lowering an orbit (decreasing $a$) decreases energy and requires a retrograde burn (negative $\Delta v$).

Plane changes are especially expensive. Changing the inclination of an orbit requires a $\Delta v$ of:

$\Delta v = 2v\sin\left(\frac{\Delta i}{2}\right)$

where $v$ is the orbital velocity and $\Delta i$ is the inclination change. Because $v$ is large in low orbits, even small inclination changes in LEO cost a lot of propellant. Plane changes are most efficient at the nodes (where the old and new orbital planes intersect) and at higher altitudes where orbital velocity is lower.

A useful trick: combining a plane change with an altitude change into a single maneuver often costs less total $\Delta v$ than doing them separately. Bi-elliptic transfers take advantage of this by performing the plane change at a high apoapsis, where the spacecraft is moving slowly.

Gravity Assists for Interplanetary Missions

A gravity assist (or gravitational slingshot) uses a planet's gravity to change a spacecraft's speed and direction without burning any propellant. During a close flyby, the spacecraft exchanges momentum with the planet. The planet is so massive that its orbit is unaffected, but the spacecraft's trajectory changes significantly.

How the velocity changes depends on the flyby geometry:

• Flying past a planet in the direction of the planet's orbital motion (prograde flyby) increases the spacecraft's speed relative to the Sun.
• Flying past opposite to the planet's motion (retrograde flyby) decreases the spacecraft's speed.

Gravity assists make missions to the outer solar system practical. Without them, reaching distant targets would require enormous amounts of propellant or much larger launch vehicles. The New Horizons mission to Pluto, for example, used a Jupiter gravity assist to gain speed and shave years off its travel time.

Multiple gravity assists can be chained together for complex trajectories:

• Voyager 2 performed a "grand tour" of the outer solar system, using gravity assists at Jupiter, Saturn, Uranus, and Neptune.
• Cassini used a Venus-Venus-Earth-Jupiter sequence of gravity assists to reach Saturn, a path that would have been impossible with propellant alone given the spacecraft's mass.

The catch is timing. Gravity assists require precise planetary alignments, so launch windows can be narrow and mission planning must account for where the planets will be years in advance.