Orbital maneuvers are crucial for spacecraft to change their paths in space. Delta-v, the change in velocity needed for these maneuvers, is key. It affects how much fuel is needed and what missions are possible.

The Hohmann transfer is a smart way to move between circular orbits. It's fuel-efficient but takes time. Gravity assists use a planet's pull to change a spacecraft's speed and direction, making far-off missions possible without extra fuel.

Orbital Maneuvers and Transfers

Concept of delta-v in maneuvers

• Represents change in velocity needed for orbital maneuver
  • Quantifies the impulse required to perform a maneuver like changing the orbit of a satellite or spacecraft
  • Measured in m/s or km/s ($\Delta v$ of 1 km/s means changing velocity by 1 km/s)
• Directly proportional to propellant required
  • Higher $\Delta v$ maneuvers need more propellant mass (Tsiolkovsky rocket equation)
  • Limited by propellant capacity of spacecraft (Falcon 9, Space Shuttle)
• Determines practicality and affordability of maneuver
  • High $\Delta v$ maneuvers may be infeasible or cost-prohibitive
  • Minimizing $\Delta v$ is a key objective in mission planning (Hohmann transfers, gravity assists)
• Drives spacecraft design and mission planning
  • Propellant tanks and engines sized based on anticipated $\Delta v$ needs
  • Mission trajectories optimized to reduce total $\Delta v$

Hohmann transfer for orbit changes

• Two-impulse maneuver between coplanar circular orbits
  • Utilizes an elliptical transfer orbit tangent to initial and final orbits
  • Requires prograde burn at periapsis to raise apoapsis, then prograde burn at apoapsis to circularize
• Most propellant-efficient method for circular orbit transfers
  • Minimizes $\Delta v$ compared to other transfer strategies (one-tangent burn, fast transfers)
  • Enables larger payload mass for a given rocket or longer satellite lifetimes
• Commonly used for LEO to GEO transfers
  • Satellites often launched into LEO parking orbit, then use Hohmann transfer to reach GEO
  • Also used for transferring between Lagrange points (SOHO mission)
• Longer transfer durations than alternative methods
  • Half-orbit period of the elliptical transfer orbit (usually several hours to days)
  • May be undesirable for time-sensitive missions or crewed flights

Energy requirements of orbital transfers

• Orbital maneuvers change orbital energy ($\epsilon = -\mu/2a$)
  • Increasing orbital altitude increases energy and requires positive $\Delta v$ (prograde burn)
  • Decreasing altitude decreases energy and requires negative $\Delta v$ (retrograde burn)
• Plane changes require large amounts of energy
  • $\Delta v$ depends on inclination change and orbital velocity ($\Delta v = 2v\sin(\Delta i/2)$)
  • Most efficient at nodes where orbital planes intersect
  • Costly in terms of propellant (i.e. inclination changes in LEO)
• Combined plane and altitude changes can be more efficient
  • Reduces total $\Delta v$ compared to separate maneuvers
  • Bi-elliptic transfers combine plane change with apoapsis burn (Moulton transfer)

Gravity assists for interplanetary missions

• Technique using a planet's gravity to alter spacecraft trajectory and speed
  • Spacecraft exchanges momentum with planet during close flyby
  • Can significantly change velocity without using propellant
• Enables missions to distant targets with less propellant
  • Reduces launch energy and $\Delta v$ requirements
  • Allows smaller launch vehicles or larger payloads (New Horizons mission to Pluto)
• Velocity change depends on flyby geometry
  • Prograde flybys (in direction of planet's motion) increase spacecraft velocity
  • Retrograde flybys (opposite to planet's motion) decrease spacecraft velocity
• Multiple gravity assists can enable complex trajectories
  • Voyager grand tour of outer solar system (Jupiter, Saturn, Uranus, Neptune)
  • Cassini mission used Venus-Venus-Earth-Jupiter gravity assists to reach Saturn