Longitude of the periapsis

Last updated
p = O + o in separate planes. Orbit1.svg
ϖ = Ω + ω in separate planes.

In celestial mechanics, the longitude of the periapsis, also called longitude of the pericenter, of an orbiting body is the longitude (measured from the point of the vernal equinox) at which the periapsis (closest approach to the central body) would occur if the body's orbit inclination were zero. It is usually denoted ϖ .

Contents

For the motion of a planet around the Sun, this position is called longitude of perihelion ϖ, which is the sum of the longitude of the ascending node Ω, and the argument of perihelion ω. [1] [2] :p.672, etc.

The longitude of periapsis is a compound angle, with part of it being measured in the plane of reference and the rest being measured in the plane of the orbit. Likewise, any angle derived from the longitude of periapsis (e.g., mean longitude and true longitude) will also be compound.

Sometimes, the term longitude of periapsis is used to refer to ω, the angle between the ascending node and the periapsis. That usage of the term is especially common in discussions of binary stars and exoplanets. [3] [4] However, the angle ω is less ambiguously known as the argument of periapsis.

Calculation from state vectors

ϖ is the sum of the longitude of ascending node Ω (measured on ecliptic plane) and the argument of periapsis ω (measured on orbital plane):

which are derived from the orbital state vectors.

Derivation of ecliptic longitude and latitude of perihelion for inclined orbits

Define the following:

i, inclination
ω, argument of perihelion
Ω, longitude of ascending node
ε, obliquity of the ecliptic (for the standard equinox of 2000.0, use 23.43929111°)

Then:

A = cos ω cos Ω – sin ω sin Ω cos i
B = cos ε (cos ω sin Ω + sin ω cos Ω cos i) – sin ε sin ω sin i
C = sin ε (cos ω sin Ω + sin ω cos Ω cos i) + cos ε sin ω sin i

The right ascension α and declination δ of the direction of perihelion are:

tan α = B/A
sin δ = C

If A < 0, add 180° to α to obtain the correct quadrant.

The ecliptic longitude ϖ and latitude b of perihelion are:

tan ϖ = sin α cos ε + tan δ sin ε/cos α
sin b = sin δ cos ε – cos δ sin ε sin α

If cos(α) < 0, add 180° to ϖ to obtain the correct quadrant.

As an example, using the most up-to-date numbers from Brown (2017) [5] for the hypothetical Planet Nine with i = 30°, ω = 136.92°, and Ω = 94°, then α = 237.38°, δ = +0.41° and ϖ = 235.00°, b = +19.97° (Brown actually provides i, Ω, and ϖ, from which ω was computed).

Related Research Articles

Keplers laws of planetary motion Scientific laws describing motion of planets around the Sun

In astronomy, Kepler's laws of planet motion are three scientific laws describing the motion of planets around the Sun, published by Johannes Kepler between 1609 and 1619. These improved the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The laws state that:

Tidal acceleration Natural phenomenon due to which tidal locking occurs

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite, and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller first, and later the larger body. The Earth–Moon system is the best-studied case.

Celestial coordinate system System for specifying positions of celestial objects

In astronomy, a celestial coordinate system is a system for specifying positions of satellites, planets, stars, galaxies, and other celestial objects relative to physical reference points available to a situated observer. Coordinate systems can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.

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.

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

Equation of time apparent solar time minus mean solar time

The equation of time describes the discrepancy between two kinds of solar time. The word equation is used in the medieval sense of "reconcile a difference". The two times that differ are the apparent solar time, which directly tracks the diurnal motion of the Sun, and mean solar time, which tracks a theoretical mean Sun with uniform motion. Apparent solar time can be obtained by measurement of the current position of the Sun, as indicated by a sundial. Mean solar time, for the same place, would be the time indicated by a steady clock set so that over the year its differences from apparent solar time would have a mean of zero.

Longitude of the ascending node one of the orbital elements used to specify the orbit of an object in space

The longitude of the ascending node is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a specified reference direction, called the origin of longitude, to the direction of the ascending node, as measured in a specified reference plane. The ascending node is the point where the orbit of the object passes through the plane of reference, as seen in the adjacent image. Commonly used reference planes and origins of longitude include:

Argument of periapsis one of the orbital elements used to specify the orbit of an object in space

The argument of periapsis, symbolized as ω, is one of the orbital elements of an orbiting body. Parametrically, ω is the angle from the body's ascending node to its periapsis, measured in the direction of motion.

Mean longitude is the ecliptic longitude at which an orbiting body could be found if its orbit were circular and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle.

In celestial mechanics true longitude is the ecliptic longitude at which an orbiting body could actually be found if its inclination were zero. Together with the inclination and the ascending node, the true longitude can tell us the precise direction from the central object at which the body would be located at a particular time.

A non-inclined orbit is an orbit coplanar with a plane of reference. The orbital inclination is 0° for prograde orbits, and π (180°) for retrograde ones. If the plane of reference is a massive spheroid body's equatorial plane, these orbits are called equatorial; if the plane of reference is the ecliptic plane, they are called ecliptic.

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.

Flight dynamics (spacecraft) Application of mechanical dynamics to model the flight of space vehicles

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

In mathematics and all natural sciences, the angular distance between two point objects, as viewed from a location different from either of these objects, is the angle of length between the two directions originating from the observer and pointing toward these two objects.

Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola

In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

Beta angle angle between the orbital plane of a spacecraft and the vector to the sun

The beta angle is a measurement that is used most notably in orbital spaceflight. The beta angle determines the percentage of time that a satellite in low Earth orbit (LEO) spends in direct sunlight, absorbing solar energy. The term is defined as the angle between the orbital plane of the satellite and the vector to the Sun. The beta angle is the smaller of the two angles between the Sun vector and the plane of the object's orbit. The beta angle does not define a unique orbital plane; all satellites in orbit with a given beta angle at a given altitude have the same exposure to the Sun, even though they may be orbiting in completely different planes around Earth.

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.

Position of the Sun Apparent location of the Sun in the sky

The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on the celestial sphere, along a circular path called the ecliptic.

Astronomical nutation is a phenomenon which causes the orientation of the axis of rotation of a spinning astronomical object to vary over time. It is caused by the gravitational forces of other nearby bodies acting upon the spinning object. Although they are caused by the same effect operating over different timescales, astronomers usually make a distinction between precession, which is a steady long-term change in the axis of rotation, and nutation, which is the combined effect of similar shorter-term variations.

References

  1. Urban, Sean E.; Seidelmann, P. Kenneth (eds.). "Chapter 8: Orbital Ephemerides of the Sun, Moon, and Planets" (PDF). Explanatory Supplement to the Astronomical Almanac. University Science Books. p. 26.
  2. Simon, J. L.; et al. (1994). "Numerical expressions for precession formulae and mean elements for the Moon and the planets". Astronomy and Astrophysics. 282: 663–683. Bibcode:1994A&A...282..663S.
  3. Robert Grant Aitken (1918). The Binary Stars. Semicentennial Publications of the University of California. D.C. McMurtrie. p.  201.
  4. "Format" Archived 2009-02-25 at the Wayback Machine in Sixth Catalog of Orbits of Visual Binary Stars Archived 2009-04-12 at the Wayback Machine , William I. Hartkopf & Brian D. Mason, U.S. Naval Observatory, Washington, D.C. Accessed on 10 January 2018.
  5. Brown, Michael E. (2017) “Planet Nine: where are you? (part 1)” The Search for Planet Nine. http://www.findplanetnine.com/2017/09/planet-nine-where-are-you-part-1.html