![Precessing Kepler orbit 280frames e0](https://upload.wikimedia.org/wikipedia/commons/8/89/Precessing_Kepler_orbit_280frames_e0.6_smaller.gif)
In
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, apsidal precession (or apsidal advance) is the
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
(gradual rotation) of the line connecting the
apsides (line of apsides) of an
astronomical body
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...
's
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
. The apsides are the orbital points closest (periapsis) and farthest (apoapsis) from its
primary body
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
. The apsidal precession is the first
time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t.
Notation
A variety of notations are used to denote th ...
of the
argument of periapsis, one of the six main
orbital elements
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 ...
of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An apsidal period is the time interval required for an orbit to precess through 360°.
History
The ancient Greek astronomer
Hipparchus
Hipparchus (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equi ...
noted the apsidal precession of the Moon's orbit (as the revolution of the Moon's apogee with a period of approximately 8.85 years); it is corrected for in the
Antikythera Mechanism
The Antikythera mechanism ( ) is an Ancient Greek hand-powered orrery, described as the oldest example of an analogue computer used to predict astronomical positions and eclipses decades in advance. It could also be used to track the four-yea ...
(circa 80 BCE) (with the supposed value of 8.88 years per full cycle, correct to within 0.34% of current measurements). The precession of the solar apsides (as a motion distinct from the precession of the equinoxes), was first quantified in the second century by
Ptolemy of Alexandria
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
. He also calculated the effect of precession on movement of the
heavenly bodies. The apsidal precessions of the Earth and other planets are the result of a plethora of phenomena, of which a part remained difficult to account for until the 20th century when the last unidentified part of Mercury's precession was precisely explained.
Calculation
A variety of factors can lead to periastron precession such as general relativity, stellar
quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure refl ...
moments, mutual star–planet tidal deformations, and perturbations from other planets.
:
For Mercury, the perihelion precession rate due to general relativistic effects is 43″ (
arcseconds) per century. By comparison, the precession due to perturbations from the other planets in the Solar System is 532″ per century, whereas the oblateness of the Sun (quadrupole moment) causes a negligible contribution of 0.025″ per century.
From classical mechanics, if stars and planets are considered to be purely spherical masses, then they will obey a simple
inverse-square law, relating force to distance and hence execute closed elliptical orbits according to
Bertrand's theorem
In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits.
The f ...
. Non-spherical mass effects are caused by the application of external potential(s): the centrifugal potential of spinning bodies like the spinning of pizza dough causes flattening between the poles and the gravity of a nearby mass raises tidal bulges. Rotational and net tidal bulges create gravitational quadrupole fields () that lead to orbital precession.
Total apsidal precession for isolated very
hot Jupiter
Hot Jupiters (sometimes called hot Saturns) are a class of gas giant exoplanets that are inferred to be physically similar to Jupiter but that have very short orbital periods (). The close proximity to their stars and high surface-atmosphere tem ...
s is, considering only lowest order effects, and broadly in order of importance
:
with planetary tidal bulge being the dominant term, exceeding the effects of general relativity and the stellar quadrupole by more than an order of magnitude. The good resulting approximation of the tidal bulge is useful for understanding the interiors of such planets. For the shortest-period planets, the planetary interior induces precession of a few degrees per year. It is up to 19.9° per year for
WASP-12b.
Newton's theorem of revolving orbits
Newton derived an early theorem which attempted to explain apsidal precession. This theorem is ''historically'' notable, but it was never widely used and it proposed forces which have been found not to exist, making the theorem invalid. This theorem of revolving orbits remained largely unknown and undeveloped for over three centuries until 1995.
[Chandrasekhar, p. 183.] Newton proposed that variations in the angular motion of a particle can be accounted for by the addition of a force that varies as the inverse cube of distance, without affecting the radial motion of a particle. Using a forerunner of the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
, Newton generalized his theorem to all force laws provided that the deviations from circular orbits are small, which is valid for most planets in the Solar System.. However, his theorem did not account for the apsidal precession of the Moon without giving up the inverse-square law of
Newton's law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
. Additionally, the rate of apsidal precession calculated via Newton's theorem of revolving orbits is not as accurate as it is for newer methods such as by
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
.
General relativity
An apsidal precession of the planet
Mercury was noted by
Urbain Le Verrier
Urbain Jean Joseph Le Verrier FRS (FOR) H FRSE (; 11 March 1811 – 23 September 1877) was a French astronomer and mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune usin ...
in the mid-19th century and accounted for by
Einstein's general theory of relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
.
Einstein showed that for a planet, the
major semi-axis of its orbit being , the
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
of the orbit and the period of revolution , then the apsidal precession due to relativistic effects, during one period of revolution in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s, is
:
where is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. In the case of Mercury, half of the greater axis is about , the eccentricity of its orbit is 0.206 and the period of revolution 87.97 days or . From these and the speed of light (which is ~), it can be calculated that the apsidal precession during one period of revolution is = radians ( degrees or 0.104″). In one hundred years, Mercury makes approximately 415 revolutions around the Sun, and thus in that time, the apsidal perihelion due to relativistic effects is approximately 43″, which corresponds almost exactly to the previously unexplained part of the measured value.
Long-term climate
Earth's apsidal precession slowly increases its
argument of periapsis; it takes about years for the ellipse to revolve once relative to the fixed stars. Earth's polar axis, and hence the solstices and equinoxes, precess with a period of about years in relation to the fixed stars. These two forms of 'precession' combine so that it takes between and years (and on average years) for the ellipse to revolve once relative to the vernal equinox, that is, for the perihelion to return to the same date (given a calendar that tracks the seasons perfectly).
This interaction between the anomalistic and tropical cycle is important in the
long-term climate variations on Earth, called the
Milankovitch cycles
Milankovitch cycles describe the collective effects of changes in the Earth's movements on its climate over thousands of years. The term was coined and named after Serbian geophysicist and astronomer Milutin Milanković. In the 1920s, he hypot ...
. Milankovitch cycles are central to understanding the effects of apsidal precession. An equivalent is also known
on Mars.
The figure to the right illustrates the effects of precession on the northern hemisphere seasons, relative to perihelion and aphelion. Notice that the areas swept during a specific season changes through time. Orbital mechanics require that the length of the seasons be proportional to the swept areas of the seasonal quadrants, so when the
orbital eccentricity
In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values betwee ...
is extreme, the seasons on the far side of the orbit may be substantially longer in duration.
See also
*
Axial precession
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 particu ...
*
Nodal precession
*
Hypotrochoid
In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle.
The parametric equations f ...
*
Rosetta (orbit)
A Rosetta orbit is a complex type of orbit.
In astronomy, a Rosetta orbit occurs when there is a periastron shift during each orbital cycle. A retrograde Newtonian shift can occur when the central mass is extended rather than a point gravitati ...
*
Spirograph
Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well-known toy version was developed by British engineer Denys Fisher and first sold ...
Notes
{{DEFAULTSORT:Apsidal Precession
Orbits
Precession