Orbital inclination

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Fig. 1: One view of inclination i (green) and other orbital parameters Orbit1.svg
Fig. 1: One view of inclination i (green) and other 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.

Contents

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. Thus:

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 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.

Since the word "inclination" is used in exoplanet studies for this line-of-sight inclination, 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 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 MJ, corresponding to an M6V star, while its minimum mass was 9.28 MJ.

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

Calculation

Components of the calculation of the orbital inclination from the momentum vector Orbital inclination from momentum vector.gif
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

where is the z-component of .

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:

Inclination to
Body Ecliptic Sun's equator Invariable plane [5]
Terre-
strials
Mercury 7.01°3.38°6.34°
Venus 3.39°3.86°2.19°
Earth 07.155°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°
Minor
planets
Pluto 17.14°11.88°15.55°
Ceres 10.62°9.20°
Pallas 35.06°34.43°
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. [6] 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 our 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. [7]

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.

See also

Related Research Articles

Ecliptic Apparent path of the Sun on the celestial sphere

The ecliptic is the plane of Earth's orbit 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.

Longitude geographic coordinate that specifies the east-west position of a point on the Earths surface

Longitude, is a geographic coordinate that specifies the east–west position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians connect points with the same longitude. The prime meridian, which passes near the Royal Observatory, Greenwich, England, is defined as 0° longitude by convention. Positive longitudes are east of the prime meridian, and negative ones are west.

Equatorial coordinate system celestial coordinate system used to specify the positions of celestial objects

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 vernal equinox, and a right-handed convention.

Ecliptic coordinate system celestial coordinate system used for representing the positions of Solar System objects

The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions and orbits 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 vernal (March) equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.

Axial precession Gravity-induced, slow, and continuous change in the orientation of an astronomical bodys rotational axis

In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In particular, it can refer to the gradual shift in the orientation of Earth's axis of rotation in a cycle of approximately 25,772 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, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. It differs from orbital inclination.

A geocentric orbit or Earth orbit involves any object orbiting the Earth, such as the Moon or artificial satellites. In 1997 NASA estimated there were approximately 2,465 artificial satellite payloads orbiting the Earth and 6,216 pieces of space debris as tracked by the Goddard Space Flight Center. Over 16,291 previously launched objects have decayed into the Earth's atmosphere.

Moons of Neptune The natural satellites of the planet Neptune

The planet Neptune has 14 known moons, which are named for minor water deities in Greek mythology. By far the largest of them is Triton, discovered by William Lassell on October 10, 1846, 17 days after the discovery of Neptune itself; over a century passed before the discovery of the second natural satellite, Nereid. Neptune's outermost moon Neso, which has an orbital period of about 26 Julian years, orbits further from its planet than any other moon in the Solar System.

Earths orbit Trajectory of Earth around the Sun

Earth orbits the Sun at an average distance of 149.60 million km, and one complete orbit takes 365.256 days, during which time Earth has traveled 940 million km. Ignoring the influence of other solar system bodies, Earth's orbit is an ellipse with the Earth-Sun barycenter as one focus and a current eccentricity of 0.0167; since this value is close to zero, the center of the orbit is close, relative to the size of the orbit, to the center of the Sun.

Sun-synchronous orbit type of geocentric orbit

A Sun-synchronous orbit is a nearly polar orbit around a planet, in which the satellite passes over any given point of the planet's surface at the same local mean solar time. More technically, it is an orbit arranged so that it precesses through one complete revolution each year, so it always maintains the same relationship with the Sun.

Spherical astronomy branch of astronomy about the celestial sphere

Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen at a particular date, time, and location on Earth. It relies on the mathematical methods of spherical geometry and the measurements of astrometry.

Orbital plane (astronomy) geometrical plane in which an orbit lies

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.

Orbital pole direction of imaginary projection onto the remote celestial sphere of the line perpendicular to planet Earths orbital plane and passing though the orbital center

An orbital pole is either point at the ends of an imaginary line segment that runs through the center 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.

Orbit of the Moon Orbit

The Moon orbits Earth in the prograde direction and completes one revolution relative to 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 4,600 km (2,900 mi) from Earth's center. On average, the distance to the Moon is about 385,000 km (239,000 mi) from Earth's center, which corresponds to about 60 Earth radii or 1.282 light-seconds.

Irregular moon natural satellite following a distant, inclined, and often eccentric and retrograde orbit

In astronomy, an irregular moon, irregular satellite or irregular natural satellite is a natural satellite following a distant, inclined, and often eccentric and retrograde orbit. They have been captured by their parent planet, unlike regular satellites, which formed in orbit around them. Irregular moons have a stable orbit, unlike temporary satellites which often have similarly irregular orbits but will eventually depart.

Ground track path on the surface of the Earth or another body directly below an aircraft or satellite

A ground track or ground trace is the path on the surface of a planet directly below an aircraft or satellite. In the case of a satellite, it is the projection of the satellite's orbit onto the surface of the Earth.

Retrograde and prograde motion 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.

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. 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
  5. 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).
  6. Peter Goldreich (Nov 1966). "History of the Lunar Orbit". Reviews of Geophysics . 4 (4): 411. Bibcode:1966RvGSP...4..411G. doi:10.1029/RG004i004p00411. Termed "classic" by Jihad Touma & Jack Wisdom (Nov 1994). "Evolution of the Earth-Moon system". The Astronomical Journal . 108: 1943. Bibcode:1994AJ....108.1943T. doi:10.1086/117209.
  7. Kaveh Pahlevan & Alessandro Morbidelli (Nov 26, 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.