Orbital inclination

Fig. 1: One view of inclination i (green) and other orbital parameters

Orbital inclination is the minimum angle between a reference plane and the orbital plane or axis of direction of an object in orbit around another object.

Orbits

The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit. It is the angular distance of the orbital plane from the plane of reference (usually the primary's equator or the ecliptic), normally stated in degrees. In the Solar System, orbital inclination is usually stated with respect to Earth's orbit.^[1]

In the Solar System, the inclination of the orbit of a planet is defined as the angle between the plane of the orbit of the planet and the ecliptic.^[2] Therefore Earth's inclination is, by definition, zero. Inclination could instead be measured with respect to another plane, such as the Sun's equator or even Jupiter'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. On the other hand, the dwarf planets Pluto and Eris have inclinations to the ecliptic of 17 degrees and 44 degrees respectively, and the large asteroid Pallas is inclined at 34 degrees.

Inclination
	Name	Inclination to ecliptic	Inclination to Sun's equator	Inclination to invariable plane[3]
Terrestrials	Mercury	7.01°	3.38°	6.34°
	Venus	3.39°	3.86°	2.19°
	Earth	0	7.155°	1.57°
	Mars	1.85°	5.65°	1.67°
Gas giants	Jupiter	1.31°	6.09°	0.32°
	Saturn	2.49°	5.51°	0.93°
	Uranus	0.77°	6.48°	1.02°
	Neptune	1.77°	6.43°	0.72°

Natural and artificial satellites

The inclination of orbits of natural 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° means the orbiting body orbits the planet in its equatorial plane, in the same direction as the planet rotates;
• an inclination greater than 0° and less than 90° is a prograde orbit.
• an inclination greater than 90° and less than 180° is a retrograde orbit.
• 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.

For impact-generated moons of terrestrial planets not too far from their star, with a large planet–moon distance, it is expected that the orbital planes of moons will tend to be aligned with the planet's orbit around the star due to tides from the star, but if the planet–moon distance is small it may be inclined. For gas giants, the orbits of moons will tend to be aligned with the giant planet's equator because these formed in circumplanetary disks.^[4]

The term "critical inclination" is often used when describing artificial satellites in orbit around Earth. This term refers to a satellite orbiting with an inclination of 63.4°. This inclination is described as critical as there is zero apogee drift for satellites in elliptical orbits at this inclination.^[5]

Exoplanets and multiple star systems

The inclination of exoplanets or members of multiple stars is the angle of the plane of the orbit relative to the plane perpendicular to the line-of-sight from Earth to the object.

• An inclination of 0° is a face-on orbit, meaning the plane of its orbit is parallel to the sky.
• An inclination of 90° is an edge-on orbit, meaning the plane of its orbit is perpendicular to the sky.

Since the word 'inclination' is used in exoplanet studies for this line-of-sight inclination then the angle between the planet's orbit and the star's rotation must use a different word and is termed the spin-orbit angle or spin-orbit alignment. In most cases the orientation of the star's rotational axis is unknown.

Because the radial-velocity method more easily finds planets with orbits closer to edge-on, most exoplanets found by this method have inclinations between 45° and 135°, although in most cases the inclination is not known. Consequently, most exoplanets found by radial velocity have true masses no more than 70% greater than their minimum masses. If the orbit is almost face-on, especially for superjovians detected by radial velocity, then those objects may actually be brown dwarfs or even red dwarfs. One particular example is HD 33636 B, which has true mass 142 M_J, corresponding to an M6V star, while its minimum mass was 9.28 M_J.

If the orbit is almost edge-on, then the planet can be seen transiting its star.

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

Calculation

components of the calculation of the orbital inclination from the momentum vector

In astrodynamics, the inclination $i$ can be computed from the orbital momentum vector $\mathbf{h}\,$ (or any vector perpendicular to the orbital plane) as $i=\arccos{h_\mathrm{z}\over\left|\mathbf{h}\right|}$, where $h_\mathrm{z}$ is the z-component of $\mathbf{h}$.

Mutual inclination of two orbits may be calculated from their inclinations to another plane using cosine rule for angles.

See also

• Altitude (astronomy)
• Axial tilt
• Azimuth
• Beta Angle
• Kepler orbits
• Kozai effect
• Orbital inclination change

References

1. Chobotov, Vladimir A. (2002). Orbital Mechanics (3rd ed.). AIAA. pp. 28–30;. ISBN 1-56347-537-5.
2. McBride, Neil; Bland, Philip A.; Gilmour, Iain (2004). An Introduction to the Solar System. Cambridge University Press. p. 248. ISBN 0-521-54620-6.
3. "The MeanPlane (Invariable plane) of the Solar System passing through the barycenter". 2009-04-03. Retrieved 2009-04-10. (produced with Solex 10 written by Aldo Vitagliano)
4. Moon formation and orbital evolution in extrasolar planetary systems-A literature review, K Lewis – EPJ Web of Conferences, 2011 – epj-conferences.org
5. Arctic Communications System Utilizing Satellites in Highly Elliptical Orbits, Lars Løge – Section 3.1, Page 17