Secular variations of the planetary orbits
Encyclopedia
The secular variations of the planetary orbits is a concept describing long-term changes (secular variation
) in the orbits of the planet
s Mercury
to Neptune
. If one ignores the gravitational attraction between the planets and only models the attraction between the Sun and the planets, then with some further idealisations, the resulting orbits would be Keplerian ellipses
. In this idealised model the shape and orientation of these ellipses would be constant in time. In reality, while the planets are at all times roughly in Keplerian orbits, the shape and orientation of these ellipses does change slowly over time. Over the centuries increasingly complex models have been made of the deviations from simple Keplerian orbits. In addition to the models, efficient and accurate numerical approximation methods have also been developed.
One such semi-analytic theory (French: Variations Séculaires des Orbites Planétaires, abbreviated as VSOP) was developed and is maintained (updating it with the results of the latest and most accurate measurements) by the scientists at the Bureau des Longitudes
in Paris
, France
. The first version, VSOP82, computed only the orbital elements
at any moment. An updated version, VSOP87, besides providing improved accuracy, computed the positions of the planets directly, as well as their orbital elements, at any moment.
At present the difference between computational predictions and observations is sufficiently small that the observations do not support the hypothesis that the models are missing some fundamental physics. Such hypothetical deviations are often referred to as post keplerian effects.
known as the Ptolemaic system
, which was based on an Earth
-centered system. The parameters of this theory were improved during the Middle Ages by Indian and Islamic astronomers.
The work of Tycho Brahe
, Kepler, and Isaac Newton
in early modern Europe laid a foundation for a modern heliocentric system. Future planetary positions continued to be predicted by extrapolating past observed positions as late as the 1740 tables of Jacques Cassini
.
The problem is that, for example, the Earth is not only gravitationally attracted by the Sun
, which would result in a stable and easily predicted elliptical orbit, but also in varying degrees by the Moon
, the other planets and any other object in the solar system. These forces cause perturbations
to the orbit
, which change over time and which cannot be exactly calculated. They can be approximated, but to do that in some manageable way requires advanced mathematics or very powerful computers. It is customary to develop them into periodic series which are a function of time, e.g. a+bt+ct2+...×cos(p+qt+rt2+...) and so forth one for each planetary interaction. The factor a in the preceding formula is the main amplitude, the factor q the main period, which is directly related to an harmonic of the driving force, that is a planetary position. For example: q= 3×(length of Mars) + 2×(length of Jupiter). (The term 'length' in this context refers to the ecliptic
longitude, that is the angle
over which the planet has progressed in its orbit, so q is an angle over time too. The time needed for the length to increase over 360° is equal to the revolution period.)
It was Joseph Louis Lagrange
in 1781, who carried out the first serious calculations, approximating the solution using a linearization
method. Others followed, but it was not until 1897 that Hill expanded on the theories by taking second order terms into account. Third order terms had to wait until the 1970s when computers became available and the vast amounts of calculations to be performed in developing a theory finally became manageable.
A major problem in any theory is that the amplitudes of the perturbations are a function of the mass
es of the planets (and other factors, but the masses are the bottlenecks). These masses can be determined by observing the periods of the moons of each planet or by observing the gravitational deflection of spacecraft passing near a planet. More observations produce greater accuracy. Short period perturbations (less than a few years) can be quite easily and accurately determined. But long period perturbations (periods of many years up to centuries) are much more difficult, because the timespan over which accurate measurements exist is not long enough, which may make them almost indistinguishable from constant terms. Yet it is these terms which are the most important influence over the millennia.
Notorious examples are the great Venus
term and the Jupiter-Saturn
great inequality. Looking up the revolution periods of these planets, one may notice that 8×(period of Earth) is almost equal to 13×(period of Venus) and 5×(period of Jupiter) is about 2×(period of Saturn).
A practical problem with the VSOP82 was that since it provided long series only for the orbital elements of the planets, it was not easy to figure out where to truncate the series if full accuracy was not needed. This problem was fixed in VSOP87, which provides series for the positions as well as for the orbital elements of the planets.
and Neptune over 6000 years before and after J2000.
This, together with its free availability has made VSOP87 the most popular source for planetary calculations nowadays, for example it is used in Celestia
.
Another major improvement is the use of rectangular coordinates in addition to the elliptical. In traditional perturbation theory it is customary to write the base orbits for the planets down with the following 6 orbital elements (gravity yields second order differential equations which result in 2 integration constants, and there is one such equation for each direction in 3 dimensional space):
Without perturbations these elements would be constant, and are therefore ideal to base the theories on. With perturbations they slowly change, and one takes as many perturbations in the calculations as possible or desirable. The results are the orbital element at a specific time, which can be used to compute the position in either rectangular coordinates
(X,Y,Z) or spherical coordinates
: longitude, latitude and heliocentric distance. These heliocentric coordinates can then fairly easily be changed to other viewpoints, e.g. geocentric coordinates. For coordinate transformations, rectangular coordinates (X,Y,Z) are often easier to use: translations (e.g. heliocentric to geocentric coordinates) are performed through vector addition, and rotations (e.g. ecliptic
to equatorial
coordinates) through matrix multiplication.
VSOP87 comes in 6 tables:
Secular variation
The secular variation of a time series is its long-term non-periodic variation . Whether something is perceived as a secular variation or not depends on the available timescale: a secular variation over a time scale of centuries may be part of a periodic variation over a time scale of millions of...
) in the orbits of the planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
to Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...
. If one ignores the gravitational attraction between the planets and only models the attraction between the Sun and the planets, then with some further idealisations, the resulting orbits would be Keplerian ellipses
Kepler orbit
In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...
. In this idealised model the shape and orientation of these ellipses would be constant in time. In reality, while the planets are at all times roughly in Keplerian orbits, the shape and orientation of these ellipses does change slowly over time. Over the centuries increasingly complex models have been made of the deviations from simple Keplerian orbits. In addition to the models, efficient and accurate numerical approximation methods have also been developed.
One such semi-analytic theory (French: Variations Séculaires des Orbites Planétaires, abbreviated as VSOP) was developed and is maintained (updating it with the results of the latest and most accurate measurements) by the scientists at the Bureau des Longitudes
Bureau des Longitudes
The Bureau des Longitudes is a French scientific institution, founded by decree of 25 June 1795 and charged with the improvement of nautical navigation, standardisation of time-keeping, geodesy and astronomical observation. During the 19th century, it was responsible for synchronizing clocks...
in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
. The first version, VSOP82, computed only the orbital elements
Orbital elements
Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used...
at any moment. An updated version, VSOP87, besides providing improved accuracy, computed the positions of the planets directly, as well as their orbital elements, at any moment.
At present the difference between computational predictions and observations is sufficiently small that the observations do not support the hypothesis that the models are missing some fundamental physics. Such hypothetical deviations are often referred to as post keplerian effects.
History
Predicting the position of the planets in the sky was already performed in ancient times. Careful observations and geometrical calculations produced a model of the motion of the solar systemSolar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
known as the Ptolemaic system
Geocentric model
In astronomy, the geocentric model , is the superseded theory that the Earth is the center of the universe, and that all other objects orbit around it. This geocentric model served as the predominant cosmological system in many ancient civilizations such as ancient Greece...
, which was based on an Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
-centered system. The parameters of this theory were improved during the Middle Ages by Indian and Islamic astronomers.
The work of Tycho Brahe
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...
, Kepler, and Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
in early modern Europe laid a foundation for a modern heliocentric system. Future planetary positions continued to be predicted by extrapolating past observed positions as late as the 1740 tables of Jacques Cassini
Jacques Cassini
Jacques Cassini was a French astronomer, son of the famous Italian astronomer Giovanni Domenico Cassini.Cassini was born at the Paris Observatory. Admitted at the age of seventeen to membership of the French Academy of Sciences, he was elected in 1696 a fellow of the Royal Society of London, and...
.
The problem is that, for example, the Earth is not only gravitationally attracted by the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
, which would result in a stable and easily predicted elliptical orbit, but also in varying degrees by the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
, the other planets and any other object in the solar system. These forces cause perturbations
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....
to the orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
, which change over time and which cannot be exactly calculated. They can be approximated, but to do that in some manageable way requires advanced mathematics or very powerful computers. It is customary to develop them into periodic series which are a function of time, e.g. a+bt+ct2+...×cos(p+qt+rt2+...) and so forth one for each planetary interaction. The factor a in the preceding formula is the main amplitude, the factor q the main period, which is directly related to an harmonic of the driving force, that is a planetary position. For example: q= 3×(length of Mars) + 2×(length of Jupiter). (The term 'length' in this context refers to the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
longitude, that is the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
over which the planet has progressed in its orbit, so q is an angle over time too. The time needed for the length to increase over 360° is equal to the revolution period.)
It was Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
in 1781, who carried out the first serious calculations, approximating the solution using a linearization
Linearization
In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
method. Others followed, but it was not until 1897 that Hill expanded on the theories by taking second order terms into account. Third order terms had to wait until the 1970s when computers became available and the vast amounts of calculations to be performed in developing a theory finally became manageable.
VSOP82
Pierre Bretagnon completed a first phase of this work by 1982 and the results of it are known as VSOP82. But because of the long period variations, his results are expected not to last more than a million years (and much less, maybe 1000 years only on very high accuracy).A major problem in any theory is that the amplitudes of the perturbations are a function of the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
es of the planets (and other factors, but the masses are the bottlenecks). These masses can be determined by observing the periods of the moons of each planet or by observing the gravitational deflection of spacecraft passing near a planet. More observations produce greater accuracy. Short period perturbations (less than a few years) can be quite easily and accurately determined. But long period perturbations (periods of many years up to centuries) are much more difficult, because the timespan over which accurate measurements exist is not long enough, which may make them almost indistinguishable from constant terms. Yet it is these terms which are the most important influence over the millennia.
Notorious examples are the great Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
term and the Jupiter-Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
great inequality. Looking up the revolution periods of these planets, one may notice that 8×(period of Earth) is almost equal to 13×(period of Venus) and 5×(period of Jupiter) is about 2×(period of Saturn).
A practical problem with the VSOP82 was that since it provided long series only for the orbital elements of the planets, it was not easy to figure out where to truncate the series if full accuracy was not needed. This problem was fixed in VSOP87, which provides series for the positions as well as for the orbital elements of the planets.
VSOP87
In VSOP87 especially these long period terms were addressed, resulting in much higher accuracy, although the calculation method itself remained similar. VSOP87 guarantees for Mercury, Venus, Earth-Moon barycenter and Mars a precision of 1" for 4000 years before and after the 2000 epoch. The same precision is ensured for Jupiter and Saturn over 2000 years and for UranusUranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...
and Neptune over 6000 years before and after J2000.
This, together with its free availability has made VSOP87 the most popular source for planetary calculations nowadays, for example it is used in Celestia
Celestia
Celestia is a 3D astronomy program created by Chris Laurel. The program is based on the Hipparcos Catalogue and allows users to travel through an extensive universe, modeled after reality, at any speed, in any direction and at any time in history...
.
Another major improvement is the use of rectangular coordinates in addition to the elliptical. In traditional perturbation theory it is customary to write the base orbits for the planets down with the following 6 orbital elements (gravity yields second order differential equations which result in 2 integration constants, and there is one such equation for each direction in 3 dimensional space):
- a semi-major axisSemi-major axisThe major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
- e eccentricityOrbital eccentricityThe orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...
- i inclinationInclinationInclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...
- Ω longitude of the ascending nodeLongitude of the ascending nodeThe 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 reference direction, called the origin of longitude, to the direction of the ascending node, measured in a reference plane...
- ω argument of perihelion (or longitude of perihelion ϖ = ω + Ω)
- T time of perihelion passage (or mean anomalyMean anomalyIn celestial mechanics, the mean anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time....
M)
Without perturbations these elements would be constant, and are therefore ideal to base the theories on. With perturbations they slowly change, and one takes as many perturbations in the calculations as possible or desirable. The results are the orbital element at a specific time, which can be used to compute the position in either rectangular coordinates
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
(X,Y,Z) or spherical coordinates
Spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...
: longitude, latitude and heliocentric distance. These heliocentric coordinates can then fairly easily be changed to other viewpoints, e.g. geocentric coordinates. For coordinate transformations, rectangular coordinates (X,Y,Z) are often easier to use: translations (e.g. heliocentric to geocentric coordinates) are performed through vector addition, and rotations (e.g. ecliptic
Ecliptic coordinate system
The ecliptic coordinate system is a celestial coordinate system that uses the ecliptic for its fundamental plane. The ecliptic is the path that the sun appears to follow across the celestial sphere over the course of a year. It is also the intersection of the Earth's orbital plane and the celestial...
to equatorial
Equatorial coordinate system
The equatorial coordinate system is a widely-used method of mapping celestial objects. It functions by projecting the Earth's geographic poles and equator onto the celestial sphere. The projection of the Earth's equator onto the celestial sphere is called the celestial equator...
coordinates) through matrix multiplication.
VSOP87 comes in 6 tables:
- VSOP87 Heliocentric ecliptic orbital elements for the equinox J2000.0; the 6 orbital elements, ideal to get an idea how the orbits are changing over time
- VSOP87A Heliocentric ecliptic rectangular coordinates for the equinox J2000.0; the most useful when converting to geocentric positions and later plot the position on a star chart
- VSOP87B Heliocentric ecliptic spherical coordinates for the equinox J2000.0
- VSOP87C Heliocentric ecliptic rectangular coordinates for the equinox of the day; the most useful when converting to geocentric positions and later compute e.g. rise/set/culmination times, or the altitude and azimuth relative to your local horizon
- VSOP87D Heliocentric ecliptic spherical coordinates for the equinox of the day
- VSOP87E Barycentric ecliptic rectangular coordinates for the equinox J2000.0, relative to the barycentre of the solar system.
See also
- Shapiro delayShapiro delayThe Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar system tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than it would if the mass of the object were not...
- DE400DE400DE400 are designations for the more recent members of a series of astronomical ephemerides produced by numerical integration at the Jet Propulsion Laboratory...
- ELP-2000