Inclination
Overview

Inclination in general is the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

between a reference plane
Plane of reference
A term used in celestial mechanics, the plane of reference is the plane by means of which orbital elements are defined. The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longitude of the ascending node.Depending on the type of body...

and another plane or axis of direction.
The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

. It is the angular
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

distance of the orbital plane from the plane of reference
Plane of reference
A term used in celestial mechanics, the plane of reference is the plane by means of which orbital elements are defined. The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longitude of the ascending node.Depending on the type of body...

(usually the primary's equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

or the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

), normally stated in degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

s.

In the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

, the inclination of the orbit of a planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

is defined as the angle between the plane of the orbit of the planet and the ecliptic — which is the plane containing Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

's orbital path.
Encyclopedia
Inclination in general is the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

between a reference plane
Plane of reference
A term used in celestial mechanics, the plane of reference is the plane by means of which orbital elements are defined. The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longitude of the ascending node.Depending on the type of body...

and another plane or axis of direction.

## Orbits

The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

. It is the angular
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

distance of the orbital plane from the plane of reference
Plane of reference
A term used in celestial mechanics, the plane of reference is the plane by means of which orbital elements are defined. The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longitude of the ascending node.Depending on the type of body...

(usually the primary's equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

or the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

), normally stated in degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

s.

In the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

, the inclination of the orbit of a planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

is defined as the angle between the plane of the orbit of the planet and the ecliptic — which is the plane containing Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

's orbital path. It could be measured with respect to another plane, such as the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

's equator or even Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

's orbital plane, but the ecliptic is more practical for Earth-bound observers. Most planetary orbits in the Solar System have relatively small inclinations, both in relation to each other and to the Sun's equator. There are notable exceptions in the dwarf planet
Dwarf planet
A dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be spherical as a result of its own gravity but has not cleared its neighboring region of planetesimals and is not a satellite...

s Pluto
Pluto
Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...

and Eris
Eris (dwarf planet)
Eris, formal designation 136199 Eris, is the most massive known dwarf planet in the Solar System and the ninth most massive body known to orbit the Sun directly...

, which have inclinations to the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

of 17 degrees and 44 degrees respectively, and the large asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

Pallas
2 Pallas
Pallas, formally designated 2 Pallas, is the second asteroid to have been discovered , and one of the largest. It is estimated to constitute 7% of the mass of the asteroid belt, and its diameter of 530–565 km is comparable to, or slightly larger than, that of 4 Vesta. It is however 20%...

, which is inclined at 34 degrees.

### Natural and artificial satellites

The inclination of orbits of natural
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....

or artificial satellites is measured relative to the equatorial plane of the body they orbit if they do so close enough. The equatorial plane is the plane perpendicular to the axis of rotation of the central body.
• an inclination of 0 degrees means the orbiting body orbits the planet in its equatorial plane, in the same direction as the planet rotates;
• an inclination greater than -90° and less than 90° is a prograde orbit.
• an inclination greater than 90° and less than 270° is a retrograde orbit
Retrograde motion is motion in the direction opposite to the movement of something else, and is the contrary of direct or prograde motion. This motion can be the orbit of one body about another body or about some other point, or the rotation of a single body about its axis, or other phenomena such...

.
• an inclination of exactly 90° is a polar orbit, in which the spacecraft passes over the north and south poles of the planet; and
• an inclination of exactly 180° is a retrograde equatorial orbit.
• an inclination of exactly 270° is a polar orbit in the opposite direction to a 90° polar orbit.

For the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

, measuring its inclination with respect to Earth's equatorial plane leads to a rapidly varying quantity and it makes more sense to measure it with respect to the ecliptic (i.e. the plane of the orbit that Earth and Moon track together around the Sun), a fairly constant quantity.

### Exoplanets and multiple star systems

The inclination of exoplanets or members of multiple star
Star system
A star system or stellar system is a small number of stars which orbit each other, bound by gravitational attraction. A large number of stars bound by gravitation is generally called a star cluster or galaxy, although, broadly speaking, they are also star systems.-Binary star systems:A stellar...

s is the angle of the plane of the orbit relative to the plane perpendicular to the line-of-sight to the object.
• an inclination of 0° is a prograde face-on orbit, meaning the plane of its orbit is parallel to the sky;
• an inclination greater than 0° and less than 90° orbits in the same direction as the rotation of its star;
• an inclination of exactly 90° is an edge-on orbit regardless if the orbit is prograde or retrograde, meaning the plane of its orbit is perpendicular to the sky;
• an inclination greater than 90° and less than 180° orbits in the opposite direction as the rotation of its star; and
• an inclination of 180° is a retrograde face-on orbit, meaning the plane of its orbit is parallel to the sky.

Because the radial velocity method is easier to find planets with more edge-on orbits, most exoplanets would have inclinations between 45° and 135°, even though most exoplanets don't have known inclination. Correspondingly, most exoplanets would have true masses no more than 70% greater than their minimum masses. If the orbit is almost edge-on, then the planet can be seen transiting its star. If the orbit is almost face-on, especially for superjovians detected by radial velocity, then those objects may actually be brown dwarf
Brown dwarf
Brown dwarfs are sub-stellar objects which are too low in mass to sustain hydrogen-1 fusion reactions in their cores, which is characteristic of stars on the main sequence. Brown dwarfs have fully convective surfaces and interiors, with no chemical differentiation by depth...

s or even red dwarf
Red dwarf
According to the Hertzsprung-Russell diagram, a red dwarf star is a small and relatively cool star, of the main sequence, either late K or M spectral type....

s. One particular example is HD 33636
HD 33636
HD 33636 is a binary system located approximately 94 light-years away in Orion constellation. The visible member HD 33636 A is a 7th magnitude yellow main-sequence star. It is located at a distance of 93.6 light years from our home planet. It has Fe/H of −0.05 +/- 0.07.A companion was discovered in...

B, which has true mass 142 MJ, corresponding to an M6V star, while its minimum mass was 9.28 MJ. The inclinations and hence true masses for almost all the exoplanets will eventually be measured by the number of observatories in space, including the Gaia mission, Space Interferometry Mission
Space Interferometry Mission
The Space Interferometry Mission, or SIM, also known as SIM Lite , was a planned space telescope developed by the U.S. National Aeronautics and Space Administration , in conjunction with contractor Northrop Grumman...

, and James Webb Space Telescope
James Webb Space Telescope
The James Webb Space Telescope , previously known as Next Generation Space Telescope , is a planned next-generation space telescope, optimized for observations in the infrared. The main technical features are a large and very cold 6.5 meter diameter mirror, an observing position far from Earth,...

.

## Other meanings

• For planets and other rotating celestial bodies, the angle of the axis of rotation with respect to the normal to plane of the orbit is sometimes also called inclination or axial inclination, but to avoid ambiguity can be called axial tilt
Axial tilt
In astronomy, axial tilt is the angle between an object's rotational axis, and a line perpendicular to its orbital plane...

or obliquity.
• In geology, the magnetic inclination is the angle made by a compass needle with respect to the horizontal surface of the Earth at a given latitude.

## Calculation

In astrodynamics
Astrodynamics
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 Newton's law of universal gravitation. It...

, the inclination can be computed from the orbital momentum vector  (or any vector perpendicular to the orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...

) as , where is the z-component of .

• Altitude (astronomy)
• Axial tilt
Axial tilt
In astronomy, axial tilt is the angle between an object's rotational axis, and a line perpendicular to its orbital plane...

• Azimuth
• Beta Angle
Beta angle
The beta angle is a value that is used most notably in spaceflight. The beta angle determines the percentage of time an object such as a spacecraft in low Earth orbit spends in direct sunlight, absorbing solar energy. Beta angle is defined as the angle between the orbit plane and the vector from...

• Kepler orbit
Kepler orbit
In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...

s
• Kozai effect
• Orbital inclination change
Orbital inclination change
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 at the orbital nodes Orbital...