Orbital inclination change

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Orbital inclination change is an orbital maneuver aimed at changing the inclination of an orbiting body's orbit. This maneuver is also known as an orbital plane change as the plane of the orbit is tipped. This maneuver requires a change in the orbital velocity vector (delta v) at the orbital nodes (i.e. the point where the initial and desired orbits intersect, the line of orbital nodes is defined by the intersection of the two orbital planes).

Maximum efficiency of inclination change is achieved at apoapsis, (or apogee), where orbital velocity $v\,$ is the lowest. In general, inclination changes require the most delta v to perform, and most mission planners try to avoid them whenever possible to conserve fuel. This can sometimes be achieved by launching a spacecraft directly into the desired inclination, or as close to it as possible so as to minimize the inclination change required.

An important subtlety of performing an inclination change is that Keplerian orbital inclination is defined by the angle between ecliptic North and the vector normal to the orbit plane, (i.e. the angular momentum vector). This means that inclination is always positive and is entangled with other orbital elements primarily the argument of periapsis which is in turn connected to the longitude of the ascending node. This can result in two very different orbits with precisely the same inclination.

For the most efficient example mentioned above, targeting an inclination at apoapsis also changes the argument of periapsis. However, targeting in this manner limits the mission designer to changing the plane only along the line of apsis.

In a pure inclination change, only the inclination of the orbit is changed while all other orbital characteristics (radius, shape.. etc) remains the same as before. Delta-v ( $\Delta{v_i}\,$ ) required for an inclination change ( $\Delta{i}\,$ ) can be calculated as follows:

$\Delta{v_i}= {\Delta{i}\sqrt{1-e^2}\cos(w+f) \over {na(1+e\cos(f))}}$

where:

$e\,$ is the eccentricity
$w\,$ is the argument of perigee
$f\,$ is the true anomaly
$n\,$ is the mean motion
$a\,$ is the semimajor axis

For more complicated maneuvers which may involve a combination of change in inclination and orbital radius, the amount of delta v is the vector difference between the velocity vectors of the initial orbit and the desired orbit at the transfer point.

v • d • e Orbits

Orbit types by characteristics:	Box orbit \| Circular orbit \| Ecliptic orbit \| Elliptic orbit \| Highly Elliptical Orbit \| Graveyard orbit \| Hyperbolic trajectory \| Inclined orbit \| Osculating orbit \| Parabolic trajectory \| Capture orbit \| Escape orbit \| Semi-synchronous orbit \| Subsynchronous orbit \| Synchronous orbit
Earth-centered orbits:	Geosynchronous orbit \| Geostationary orbit \| Sun-synchronous orbit \| Low Earth orbit \| Medium Earth Orbit \| Molniya orbit \| Near equatorial orbit \| Orbit of the Moon \| Polar orbit \| Polar sun synchronous orbit \| Tundra orbit
Orbits about other bodies:	Areosynchronous orbit \| Areostationary orbit \| Lunar orbit \| Heliocentric orbit \| Heliosynchronous orbit

Orbital elements:	Inclination ( $i\,\!$ ) \| Longitude of the ascending node ( $\Omega\,\!$ ) \| Eccentricity ( $e\,\!$ ) \| Argument of periapsis ( $\omega\,\!$ ) \| Semimajor axis ( $a\,\!$ ) \| Mean anomaly at epoch ( $M_o\,\!$ )
Other orbital parameters:	True anomaly ( $v\,$ ) \| Semi-major axis ( $a\,$ ) \| Semi-minor axis ( $b\,$ ) \| Linear eccentricity ( $\epsilon\,$ ) \| Eccentric anomaly ( $E\,$ ) \| Mean longitude ( $L\,$ ) \| True longitude ( $l\,$ ) \| Orbital period ( $T\,$ )
Orbital maneuvers:	Bi-elliptic transfer \| Geostationary transfer orbit \| Gravitational slingshot \| Hohmann transfer orbit \| Orbital inclination change \| Orbit phasing \| Space rendezvous

Other orbital mechanics topics:	Apsis \| Celestial coordinate system \| Delta-v budget \| Epoch \| Ephemeris \| Equatorial coordinate system \| Ground track \| Interplanetary Transport Network \| Kepler's laws of planetary motion \| Lagrangian point \| List of orbits \| n-body problem \| Orbit equation \| Orbital state vectors \| Retrograde and direct motion \| Specific orbital energy \| Specific relative angular momentum