# Orbital inclination change

Last updated

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).

## Contents

In general, inclination changes can take a very large amount of delta v to perform, and most mission planners try to avoid them whenever possible to conserve fuel. This is typically achieved by launching a spacecraft directly into the desired inclination, or as close to it as possible so as to minimize any inclination change required over the duration of the spacecraft life. Planetary flybys are the most efficient way to achieve large inclination changes, but they are only effective for interplanetary missions.

## Efficiency

The simplest way to perform a plane change is to perform a burn around one of the two crossing points of the initial and final planes. The delta-v required is the vector change in velocity between the two planes at that point.

However, maximum efficiency of inclination changes are achieved at apoapsis, (or apogee), where orbital velocity $v\,$ is the lowest. In some cases, it can require less total delta v to raise the satellite into a higher orbit, change the orbit plane at the higher apogee, and then lower the satellite to its original altitude. 

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 apsides.[ citation needed ]

For Hohmann transfer orbits, the initial orbit and the final orbit are 180 degrees apart. Because the transfer orbital plane has to include the central body, such as the Sun, and the initial and final nodes, this can require two 90 degree plane changes to reach and leave the transfer plane. In such cases it is often more efficient to use a broken plane maneuver where an additional burn is done so that plane change only occurs at the intersection of the initial and final orbital planes, rather than at the ends. 

## Inclination entangled with other orbital elements

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.

## Calculation

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:

$\Delta {v_{i}}={2\sin({\frac {\Delta {i}}{2}}){\sqrt {1-e^{2}}}\cos(\omega +f)na \over {(1+e\cos(f))}}$ where:

• $e\,$ is the orbital eccentricity
• $\omega \,$ is the argument of periapsis
• $f\,$ is the true anomaly
• $n\,$ is the mean motion
• $a\,$ is the semi-major axis

For more complicated manoeuvres which may involve a combination of change in inclination and orbital radius, the delta v is the vector difference between the velocity vectors of the initial orbit and the desired orbit at the transfer point. These types of combined manoeuvres are commonplace, as it is more efficient to perform multiple orbital manoeuvres at the same time if these manoeuvres have to be done at the same location.

According to the law of cosines, the minimum Delta-v ($\Delta {v}\,$ ) required for any such combined manoeuvre can be calculated with the following equation


$\Delta {v}={\sqrt {V_{1}^{2}+V_{2}^{2}-2V_{1}V_{2}cos(\Delta i)}}$ Here $V_{1}$ and $V_{2}$ are the initial and target velocities.

## Circular orbit inclination change

Where both orbits are circular (i.e. $e\,$ = 0) and have the same radius the Delta-v ($\Delta {v_{i}}\,$ ) required for an inclination change ($\Delta {i}\,$ ) can be calculated using:

$\Delta {v_{i}}={2v\,\sin \left({\frac {\Delta {i}}{2}}\right)}$ Where:

• $v\,$ is the orbital velocity and has the same units as $\Delta {v_{i}}$ ## Other ways to change inclination

Some other ways to change inclination that do not require burning propellant (or help reduce the amount of propellant required) include

• aerodynamic lift (for bodies within an atmosphere, such as the Earth)
• solar sails

Transits of other bodies such as the Moon can also be done.

None of these methods will change the delta-V required, they are simply alternate means of achieving the same end result and, ideally, will reduce propellant usage.

## Related Research Articles In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics. In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits of different radii around a central body in the same plane that is sometimes tangential to both. The Hohmann transfer often uses the lowest possible amount of propellant in traveling between these orbits, but bi-elliptic transfers can use less in some cases. A geosynchronous transfer orbit or geostationary transfer orbit (GTO) is a type of geocentric orbit. Satellites that are destined for geosynchronous (GSO) or geostationary orbit (GEO) are (almost) always put into a GTO as an intermediate step for reaching their final orbit. Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

Delta-v, symbolized as v and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such as launching from or landing on a planet or moon, or an in-space orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of the vehicle. The Tsiolkovsky rocket equation, classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum. In astrodynamics and aerospace, a delta-v budget is an estimate of the total change in velocity (delta-v) required for a space mission. It is calculated as the sum of the delta-v required to perform each propulsive maneuver needed during the mission. As input to the Tsiolkovsky rocket equation, it determines how much propellant is required for a vehicle of given empty mass and propulsion system. In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse. In celestial mechanics, the longitude of the periapsis, also called longitude of the pericenter, of an orbiting body is the longitude at which the periapsis would occur if the body's orbit inclination were zero. It is usually denoted ϖ.

In spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft. For spacecraft far from Earth an orbital maneuver is called a deep-space maneuver (DSM). Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag. In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver.

The perifocal coordinate (PQW) system is a frame of reference for an orbit. The frame is centered at the focus of the orbit, i.e. the celestial body about which the orbit is centered. The unit vectors and lie in the plane of the orbit. is directed towards the periapsis of the orbit and has a true anomaly of 90 degrees past the periapsis. The third unit vector is the angular momentum vector and is directed orthogonal to the orbital plane such that: A ground track or ground trace is the path on the surface of a planet directly below an aircraft's or satellite's trajectory. In the case of satellites, it is also known as a suborbital track, and is the vertical projection of the satellite's orbit onto the surface of the Earth.

A gravity turn or zero-lift turn is a maneuver used in launching a spacecraft into, or descending from, an orbit around a celestial body such as a planet or a moon. It is a trajectory optimization that uses gravity to steer the vehicle onto its desired trajectory. It offers two main advantages over a trajectory controlled solely through the vehicle's own thrust. First, the thrust is not used to change the spacecraft's direction, so more of it is used to accelerate the vehicle into orbit. Second, and more importantly, during the initial ascent phase the vehicle can maintain low or even zero angle of attack. This minimizes transverse aerodynamic stress on the launch vehicle, allowing for a lighter launch vehicle. In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity is greatest. In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, an Austro-Hungarian-born German physicist and a founder of modern rocketry. In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

In orbital mechanics, a frozen orbit is an orbit for an artificial satellite in which natural drifting due to the central body's shape has been minimized by careful selection of the orbital parameters. Typically, this is an orbit in which, over a long period of time, the satellite's altitude remains constant at the same point in each orbit. Changes in the inclination, position of the lowest point of the orbit, and eccentricity have been minimized by choosing initial values so that their perturbations cancel out., which results in a long-term stable orbit that minimizes the use of station-keeping propellant.

The Clohessy-Wiltshire equations describe a simplified model of orbital relative motion, in which the target is in a circular orbit, and the chaser spacecraft is in an elliptical or circular orbit. This model gives a first-order approximation of the chaser's motion in a target-centered coordinate system. It is very useful in planning rendezvous of the chaser with the target.

1. Braeunig, Robert A. "Basics of Space Flight: Orbital Mechanics". Archived from the original on 2012-02-04. Retrieved 2008-07-16.
2. Owens, Steve; Macdonald, Malcolm (2013). "Hohmann Spiral Transfer With Inclination Change Performed By Low-Thrust System" (PDF). Advances in the Astronautical Sciences. 148: 719. Retrieved 3 April 2020.