Angle between a reference plane and the plane of an orbit
Fig. 1: Orbital inclination represented by
i
(dark green), along with other fundamental
orbital parameters
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
[
edit
]
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
[
edit
]
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
[
edit
]
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
[
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]
Components of the calculation of the orbital inclination from the momentum vector
In
astrodynamics
, the inclination
can be computed from the
orbital momentum vector
(or any vector perpendicular to the
orbital plane
) as
![{\displaystyle i=\arccos {\frac {h_{z}}{\left|h\right|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9da25a1e357520c04a7ddc2bf3ee7fb41ea65c32)
where
![{\displaystyle h_{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/453e3ca341aeff78130e772d48b352a68eaadfd7)
is the z-component of
![{\displaystyle h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
.
Mutual inclination of two orbits may be calculated from their inclinations to another plane using
cosine rule for angles
.
Observations and theories
[
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]
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° 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
[
edit
]
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.
See also
[
edit
]
Look up
inclination
in Wiktionary, the free dictionary.
References
[
edit
]
- ^
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
- ^
a
b
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).
Universita 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
.