Orbital inclination

Last updated November 30, 2024

Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object.

For a satellite orbiting the Earth directly above the Equator, the plane of the satellite's orbit is the same as the Earth's equatorial plane, and the satellite's orbital inclination is 0°. The general case for a circular orbit is that it is tilted, spending half an orbit over the northern hemisphere and half over the southern. If the orbit swung between 20° north latitude and 20° south latitude, then its orbital inclination would be 20°.

Orbits

The inclination is one of the six orbital elements describing the shape and orientation of a celestial orbit. It is the angle between the orbital plane and the plane of reference, normally stated in degrees. For a satellite orbiting a planet, the plane of reference is usually the plane containing the planet's equator. For planets in the Solar System, the plane of reference is usually the ecliptic, the plane in which the Earth orbits the Sun.^[1]^[2] This reference plane is most practical for Earth-based observers. Therefore, Earth's inclination is, by definition, zero.

Inclination can instead be measured with respect to another plane, such as the Sun's equator or the invariable plane (the plane that represents the angular momentum of the Solar System, approximately the orbital plane of Jupiter).

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 orbit sufficiently closely. The equatorial plane is the plane perpendicular to the axis of rotation of the central body.

An inclination of 30° could also be described using an angle of 150°. The convention is that the normal orbit is prograde, an orbit in the same direction as the planet rotates. Inclinations greater than 90° describe retrograde orbits (backward). Thus:

An inclination of 0° means the orbiting body has a prograde orbit in the planet's equatorial plane.
An inclination greater than 0° and less than 90° also describes a prograde orbit.
An inclination of 63.4° is often called a critical inclination, when describing artificial satellites orbiting the Earth, because they have zero apogee drift.^[3]
An inclination of exactly 90° is a polar orbit, in which the spacecraft passes over the poles of the planet.
An inclination greater than 90° and less than 180° is a retrograde orbit.
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, the orbital planes of moons 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 tend to be aligned with the giant planet's equator, because these formed in circumplanetary disks.^[4] Strictly speaking, this applies only to regular satellites. Captured bodies on distant orbits vary widely in their inclinations, while captured bodies in relatively close orbits tend to have low inclinations owing to tidal effects and perturbations by large regular satellites.

Exoplanets and multiple star systems

The inclination of exoplanets or members of multi-star star systems is the angle of the plane of the orbit relative to the plane perpendicular to the line of sight from Earth to the object.^[5]

An inclination of 0° is a face-on orbit, meaning the plane of the exoplanet's orbit is perpendicular to the line of sight with Earth.
An inclination of 90° is an edge-on orbit, meaning the plane of the exoplanet's orbit is parallel to the line of sight with Earth.

Since the word "inclination" is used in exoplanet studies for this line-of-sight inclination, the angle between the planet's orbit and its star's rotational axis is expressed using the term the "spin-orbit angle" or "spin-orbit alignment".^[5] 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 40% greater than their minimum masses.^{[ citation needed ]} 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.

Calculation

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

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

Observations and theories

Most planetary orbits in the Solar System have relatively small inclinations, both in relation to each other and to the Sun's equator:

Body		Inclination to
Body		Ecliptic	Sun's equator	Invariable plane ^[6]
Terre- strials	Mercury	7.01°	3.38°	6.34°
	Venus	3.39°	3.86°	2.19°
	Earth	0°	7.25°^[7]	1.57°
	Mars	1.85°	5.65°	1.67°
Gas & ice 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°
Minor planets	Pluto	17.14°	11.88°	15.55°
	Ceres	10.59°		9.20°
	Pallas	34.83°		34.21°
	Vesta	5.58°		7.13°

On the other hand, the dwarf planets Pluto and Eris have inclinations to the ecliptic of 17° and 44° respectively, and the large asteroid Pallas is inclined at 34°.

In 1966, Peter Goldreich published a classic paper on the evolution of the Moon's orbit and on the orbits of other moons in the Solar System.^[8] He showed that, for each planet, there is a distance such that moons closer to the planet than that distance maintain an almost constant orbital inclination with respect to the planet's equator (with an orbital precession mostly due to the tidal influence of the planet), whereas moons farther away maintain an almost constant orbital inclination with respect to the ecliptic (with precession due mostly to the tidal influence of the sun). The moons in the first category, with the exception of Neptune's moon Triton, orbit near the equatorial plane. He concluded that these moons formed from equatorial accretion disks. But he found that the Moon, although it was once inside the critical distance from the Earth, never had an equatorial orbit as would be expected from various scenarios for its origin. This is called the lunar inclination problem, to which various solutions have since been proposed.^[9]

Other meaning

For planets and other rotating celestial bodies, the angle of the equatorial plane relative to the orbital plane – such as the tilt of the Earth's poles toward or away from the Sun – is sometimes also called inclination, but less ambiguous terms are axial tilt or obliquity.

Related Research Articles

The ecliptic or ecliptic plane is the orbital plane of Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the background of stars. The ecliptic is an important reference plane and is the basis of the ecliptic coordinate system.

The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere, a primary direction towards the March equinox, and a right-handed convention.

In astronomy, the ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets and many small Solar System bodies have orbits with only slight inclinations to the ecliptic, using it as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the March equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.

Sidereal time is a system of timekeeping used especially by astronomers. Using sidereal time and the celestial coordinate system, it is easy to locate the positions of celestial objects in the night sky. Sidereal time is a "time scale that is based on Earth's rate of rotation measured relative to the fixed stars".

In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In particular, axial precession can refer to the gradual shift in the orientation of Earth's axis of rotation in a cycle of approximately 26,000 years. This is similar to the precession of a spinning top, with the axis tracing out a pair of cones joined at their apices. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—nutation and polar motion—are much smaller in magnitude.

In astronomy, an analemma is a diagram showing the position of the Sun in the sky as seen from a fixed location on Earth at the same mean solar time, as that position varies over the course of a year. The diagram resembles a figure eight. Globes of the Earth often display an analemma as a two-dimensional figure of equation of time vs. declination of the Sun.

In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orbital plane. It differs from orbital inclination.

A lunar node is either of the two orbital nodes of the Moon, that is, the two points at which the orbit of the Moon intersects the ecliptic. The ascending node is where the Moon moves into the northern ecliptic hemisphere, while the descending node is where the Moon enters the southern ecliptic hemisphere.

Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi), or 8.317 light-minutes, in a counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes 365.256 days, during which time Earth has traveled 940 million km (584 million mi). Ignoring the influence of other Solar System bodies, Earth's orbit, also called Earth's revolution, is an ellipse with the Earth–Sun barycenter as one focus with a current eccentricity of 0.0167. Since this value is close to zero, the center of the orbit is relatively close to the center of the Sun.

Cassini's laws provide a compact description of the motion of the Moon. They were established in 1693 by Giovanni Domenico Cassini, a prominent scientist of his time.

<span class="mw-page-title-main">Orbital plane</span> Concept in geometry and physics

The orbital plane of a revolving body is the geometric plane in which its orbit lies. Three non-collinear points in space suffice to determine an orbital plane. A common example would be the positions of the centers of a massive body (host) and of an orbiting celestial body at two different times/points of its orbit.

An orbital pole is either point at the ends of the orbital normal, an imaginary line segment that runs through a focus of an orbit and is perpendicular to the orbital plane. Projected onto the celestial sphere, orbital poles are similar in concept to celestial poles, but are based on the body's orbit instead of its equator.

A lunar standstill or lunistice is when the Moon reaches its furthest north or furthest south point during the course of a month. The declination at lunar standstill varies in a cycle 18.6 years long between 18.134° and 28.725°, due to lunar precession. These extremes are called the minor and major lunar standstills.

The Moon orbits Earth in the prograde direction and completes one revolution relative to the Vernal Equinox and the stars in about 27.32 days and one revolution relative to the Sun in about 29.53 days. Earth and the Moon orbit about their barycentre, which lies about 4,670 km from Earth's centre, forming a satellite system called the Earth–Moon system. On average, the distance to the Moon is about 384,400 km (238,900 mi) from Earth's centre, which corresponds to about 60 Earth radii or 1.282 light-seconds.

A satellite ground track or satellite ground trace is the path on the surface of a planet directly below a satellite's trajectory. It is also known as a suborbital track or subsatellite track, and is the vertical projection of the satellite's orbit onto the surface of the Earth . A satellite ground track may be thought of as a path along the Earth's surface that traces the movement of an imaginary line between the satellite and the center of the Earth. In other words, the ground track is the set of points at which the satellite will pass directly overhead, or cross the zenith, in the frame of reference of a ground observer.

<span class="mw-page-title-main">Retrograde and prograde motion</span> Relative directions of orbit or rotation

Retrograde motion in astronomy is, in general, orbital or rotational motion of an object in the direction opposite the rotation of its primary, that is, the central object. It may also describe other motions such as precession or nutation of an object's rotational axis. Prograde or direct motion is more normal motion in the same direction as the primary rotates. However, "retrograde" and "prograde" can also refer to an object other than the primary if so described. The direction of rotation is determined by an inertial frame of reference, such as distant fixed stars.

Nodal precession is the precession of the orbital plane of a satellite around the rotational axis of an astronomical body such as Earth. This precession is due to the non-spherical nature of a rotating body, which creates a non-uniform gravitational field. The following discussion relates to low Earth orbit of artificial satellites, which have no measurable effect on the motion of Earth. The nodal precession of more massive, natural satellites like the Moon is more complex.

This glossary of astronomy is a list of definitions of terms and concepts relevant to astronomy and cosmology, their sub-disciplines, and related fields. Astronomy is concerned with the study of celestial objects and phenomena that originate outside the atmosphere of Earth. The field of astronomy features an extensive vocabulary and a significant amount of jargon.

A planetary coordinate system is a generalization of the geographic, geodetic, and the geocentric coordinate systems for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the selenographic coordinates for the Moon. The coordinate systems for almost all of the solid bodies in the Solar System were established by Merton E. Davies of the Rand Corporation, including Mercury, Venus, Mars, the four Galilean moons of Jupiter, and Triton, the largest moon of Neptune. A planetary datum is a generalization of geodetic datums for other planetary bodies, such as the Mars datum; it requires the specification of physical reference points or surfaces with fixed coordinates, such as a specific crater for the reference meridian or the best-fitting equigeopotential as zero-level surface.

References

↑ Chobotov, Vladimir A. (2002). Orbital Mechanics (3rd ed.). AIAA. pp. 28–30. ISBN 1-56347-537-5.
↑ McBride, Neil; Bland, Philip A.; Gilmour, Iain (2004). An Introduction to the Solar System. Cambridge University Press. p. 248. ISBN 0-521-54620-6.
↑ Arctic Communications System Utilizing Satellites in Highly Elliptical Orbits, Lars Løge – Section 3.1, Page 17
↑ Moon formation and orbital evolution in extrasolar planetary systems-A literature review, K Lewis – EPJ Web of Conferences, 2011 – epj-conferences.org
1 2 Tiago L. Campante (27 October 2016). "Spin-orbit alignment of exoplanet systems: Analysis of an ensemble of asteroseismic observations" (PDF). Proceedings of the International Astronomical Union. 11 (General Assembly A29B). Cambridge University Press: 636–641. Bibcode:2016IAUFM..29B.636C. doi:10.1017/S1743921316006232. S2CID 126328423 . Retrieved 27 February 2022.
↑ Heider, K.P. (3 April 2009). "The mean plane (invariable plane) of the Solar System passing through the barycenter". Archived from the original on 3 June 2013. Retrieved 10 April 2009.
produced using
Vitagliano, Aldo. "Solex 10" (computer program). Università degli Studi di Napoli Federico II. Archived from the original on 24 May 2015. Retrieved 23 November 2010.
↑ Planetary Fact Sheets, at http://nssdc.gsfc.nasa.gov
↑ Peter Goldreich (November 1966). "History of the Lunar Orbit". Reviews of Geophysics . 4 (4): 411–439. Bibcode:1966RvGSP...4..411G. doi:10.1029/RG004i004p00411. Termed "classic" by Jihad Touma & Jack Wisdom (November 1994). "Evolution of the Earth-Moon system". The Astronomical Journal . 108: 1943. Bibcode:1994AJ....108.1943T. doi: 10.1086/117209 .
↑ Kaveh Pahlevan & Alessandro Morbidelli (26 November 2015). "Collisionless encounters and the origin of the lunar inclination". Nature. 527 (7579): 492–494. arXiv: 1603.06515 . Bibcode:2015Natur.527..492P. doi:10.1038/nature16137. PMID 26607544. S2CID 4456736.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[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] Arctic Communications System Utilizing Satellites in Highly Elliptical Orbits, Lars Løge – Section 3.1, Page 17

[4] Moon formation and orbital evolution in extrasolar planetary systems-A literature review, K Lewis – EPJ Web of Conferences, 2011 – epj-conferences.org

[Campante-5] 1 2 Tiago L. Campante (27 October 2016). "Spin-orbit alignment of exoplanet systems: Analysis of an ensemble of asteroseismic observations" (PDF). Proceedings of the International Astronomical Union. 11 (General Assembly A29B). Cambridge University Press: 636–641. Bibcode:2016IAUFM..29B.636C. doi:10.1017/S1743921316006232. S2CID 126328423 . Retrieved 27 February 2022.

[meanplane-6] Heider, K.P. (3 April 2009). "The mean plane (invariable plane) of the Solar System passing through the barycenter". Archived from the original on 3 June 2013. Retrieved 10 April 2009.
produced using
Vitagliano, Aldo. "Solex 10" (computer program). Università degli Studi di Napoli Federico II. Archived from the original on 24 May 2015. Retrieved 23 November 2010.

[NASA-7] Planetary Fact Sheets, at http://nssdc.gsfc.nasa.gov

[8] Peter Goldreich (November 1966). "History of the Lunar Orbit". Reviews of Geophysics . 4 (4): 411–439. Bibcode:1966RvGSP...4..411G. doi:10.1029/RG004i004p00411. Termed "classic" by Jihad Touma & Jack Wisdom (November 1994). "Evolution of the Earth-Moon system". The Astronomical Journal . 108: 1943. Bibcode:1994AJ....108.1943T. doi: 10.1086/117209 .

[9] Kaveh Pahlevan & Alessandro Morbidelli (26 November 2015). "Collisionless encounters and the origin of the lunar inclination". Nature. 527 (7579): 492–494. arXiv: 1603.06515 . Bibcode:2015Natur.527..492P. doi:10.1038/nature16137. PMID 26607544. S2CID 4456736.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]