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40-501: VSOP was developed and is maintained (updated with the latest data) by the scientists at the Bureau des Longitudes in Paris. The first version, VSOP82, computed only the orbital elements at any moment. An updated version, VSOP87, computed the positions of the planets directly at any moment, as well as their orbital elements with improved accuracy. Predicting the position of the planets in

80-409: A in the preceding formula is the main amplitude, the factor q the main angular velocity, which is directly related to a 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 in unit time, so q

120-412: A body's center of mass to the barycenter can be calculated as a two-body problem . If one of the two orbiting bodies is much more massive than the other and the bodies are relatively close to one another, the barycenter will typically be located within the more massive object. In this case, rather than the two bodies appearing to orbit a point between them, the less massive body will appear to orbit about

160-476: A commission in the year 1897 to extend the metric system to the measurement of time . They planned to abolish the antiquated division of the day into hours , minutes , and seconds , and replace it by a division into tenths, thousandths, and hundred-thousandths of a day . This was a revival of a dream that was in the minds of the creators of the metric system at the time of the French Revolution

200-550: 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 ,

240-461: A hundred years earlier. Some members of the Bureau of Longitude commission introduced a compromise proposal, retaining the old-fashioned hour as the basic unit of time and dividing it into hundredths and ten-thousandths. Poincaré served as secretary of the commission and took its work very seriously, writing several of its reports. He was a fervent believer in a universal metric system . But he lost

280-489: Is 10 times better than VSOP82. Over a greater interval −4000...+8000 a comparison with an internal numerical indicates that the VSOP2010 solutions are about 5 times better than VSOP2000 for the telluric planets and 10 to 50 times better for the outer planets. The VSOP2013 files contain the series of the elliptic elements for the 8 planets Mercury, Venus, Earth-Moon barycenter, Mars, Jupiter, Saturn, Uranus, and Neptune and for

320-410: Is a factor of 10-100 better than its predecessors. The uncertainty for Mercury, Venus and the Earth is reported to be around 0.1 mas (milliarcsecond) for the interval 1900–2000, and that for the other planets a few milliarcseconds. The publication of and the data for VSOP2000 are publicly available. Bretagnon's last work was on the implementation of relativistic effects, which was supposed to improve

360-512: 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

400-471: Is an analytical solution for the (spherical and rectangular) positions (rather than orbital elements) of the four planets Jupiter, Saturn, Uranus, and Neptune and the dwarf planet Pluto. This solution is fitted to the Ephemeris DE405 over the time interval +1890...+2000. The reference system in the solution TOP2010 is defined by the dynamical equinox and ecliptic J2000.0. This solution is fitted to

440-438: 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 George William Hill expanded on the theories by taking second order terms into account. Third order terms had to wait until

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480-468: Is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where: The Sun–Jupiter system, with e Jupiter  = 0.0484, just fails to qualify: 1.05 < 1.07 > 0.954 . In classical mechanics (Newtonian gravitation), this definition simplifies calculations and introduces no known problems. In general relativity (Einsteinian gravitation), complications arise because, while it

520-567: Is possible, within reasonable approximations, to define the barycenter, we find that the associated coordinate system does not fully reflect the inequality of clock rates at different locations. Brumberg explains how to set up barycentric coordinates in general relativity. The coordinate systems involve a world-time, i.e. a global time coordinate that could be set up by telemetry . Individual clocks of similar construction will not agree with this standard, because they are subject to differing gravitational potentials or move at various velocities, so

560-557: Is that the amplitudes of the perturbations are a function of the masses 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

600-449: Is the case for Pluto and Charon , one of Pluto's natural satellites , as well as for many binary asteroids and binary stars . When the less massive object is far away, the barycenter can be located outside the more massive object. This is the case for Jupiter and the Sun ; despite the Sun being a thousandfold more massive than Jupiter, their barycenter is slightly outside the Sun due to

640-588: Is used in Celestia and Orbiter . 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 six orbital elements (gravity yields second order differential equations which result in two integration constants, and there is one such equation for each direction in three-dimensional space): Without perturbations these elements would be constant and are therefore ideal to base

680-567: The Paris Observatory , separating it from the Bureau, and focused the efforts of the Bureau on time and astronomy . The Bureau was successful at setting a universal time in Paris via air pulses sent through pneumatic tubes . It later worked to synchronize time across the French colonial empire by determining the length of time for a signal to make a round trip to and from a French colony . The French Bureau of Longitude established

720-469: The Solar System . Figures are given rounded to three significant figures . The terms "primary" and "secondary" are used to distinguish between involved participants, with the larger being the primary and the smaller being the secondary. If m 1 ≫ m 2 —which is true for the Sun and any planet—then the ratio ⁠ r 1 / R 1 ⁠ approximates to: Hence, the barycenter of

760-433: The barycenter (or barycentre ; from Ancient Greek βαρύς ( barús )  'heavy' and κέντρον ( kéntron )  'center') is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important concept in fields such as astronomy and astrophysics . The distance from

800-455: The 1970s when computers became available and the vast numbers of calculations to be performed in developing a theory finally became manageable. 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

840-543: The 19th century, it was responsible for synchronizing clocks across the world. It was headed during this time by François Arago and Henri Poincaré . The Bureau now functions as an academy and still meets monthly to discuss topics related to astronomy . The Bureau was founded by the National Convention after it heard a report drawn up jointly by the Committee of Navy, the Committee of Finances and

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880-517: The Committee of State education. Henri Grégoire had brought to the attention of the National Convention France's failing maritime power and the naval mastery of England , proposing that improvements in navigation would lay the foundations for a renaissance in naval strength. As a result, the Bureau was established with authority over the Paris Observatory and all other astronomical establishments throughout France. The Bureau

920-451: The Sun–Earth barycenter would still be within the Sun (just over 30,000 km from the center). To calculate the actual motion of the Sun, only the motions of the four giant planets (Jupiter, Saturn, Uranus, Neptune) need to be considered. The contributions of all other planets, dwarf planets, etc. are negligible. If the four giant planets were on a straight line on the same side of the Sun,

960-519: The Sun–planet system will lie outside the Sun only if: —that is, where the planet is massive and far from the Sun. If Jupiter had Mercury 's orbit (57,900,000 km, 0.387 AU), the Sun–Jupiter barycenter would be approximately 55,000 km from the center of the Sun ( ⁠ r 1 / R 1 ⁠ ≈ 0.08 ). But even if the Earth had Eris 's orbit (1.02 × 10  km, 68 AU),

1000-497: The accuracy with another factor of 10. This version was never finished, and still had weaknesses for Uranus and Neptune. The VSOP2010 files contain the series of the elliptic elements for the 8 planets Mercury, Venus, Earth-Moon barycenter, Mars, Jupiter, Saturn, Uranus, Neptune and for the dwarf planet Pluto. The VSOP2010 solution is fitted to the DE405 numerical integration over the time interval +1890...+2000. The numerical precision

1040-654: The battle. The rest of the world outside France gave no support to the commission's proposals, and the French government was not prepared to go it alone. After three years of hard work, the commission was dissolved in 1900. Since 1970, the board has been constituted with 13 members, 3 nominated by the Académie des Sciences . Since 1998, practical work has been carried out by the Institut de mécanique céleste et de calcul des éphémérides . Barycenter In astronomy ,

1080-430: The combined center of mass would lie at about 1.17 solar radii, or just over 810,000 km, above the Sun's surface. The calculations above are based on the mean distance between the bodies and yield the mean value r 1 . But all celestial orbits are elliptical, and the distance between the bodies varies between the apses , depending on the eccentricity , e . Hence, the position of the barycenter varies too, and it

1120-452: The dwarf planet Pluto of the solution VSOP2013. The planetary solution VSOP2013 is fitted to the numerical integration INPOP10a built at IMCCE, Paris Observatory over the time interval +1890...+2000. The precision is of a few 0.1″ for the telluric planets (1.6″ for Mars) over the time interval −4000...+8000. Masses multiplied by the gravitational constant of the Sun, the planets and the five big asteroids are used values from INPOP10a. This

1160-467: The fields of astronomy and astrophysics . In a simple two-body case, the distance from the center of the primary to the barycenter, r 1 , is given by: where : The semi-major axis of the secondary's orbit, r 2 , is given by r 2 = a − r 1 . When the barycenter is located within the more massive body, that body will appear to "wobble" rather than to follow a discernible orbit. The following table sets out some examples from

1200-481: The more massive body, while the more massive body might be observed to wobble slightly. This is the case for the Earth–Moon system , whose barycenter is located on average 4,671 km (2,902 mi) from Earth's center, which is 74% of Earth's radius of 6,378 km (3,963 mi). When the two bodies are of similar masses, the barycenter will generally be located between them and both bodies will orbit around it. This

1240-411: The numerical integration INPOP10a built at IMCCE (Paris Observatory) over the time interval +1890...+2000. The reference system in the solution TOP2013 is defined by the dynamical equinox and ecliptic of J2000.0. The TOP2013 solution is the best for the motion over the time interval −4000...+8000. Its precision is of a few 0.1″ for the four planets, i.e. a gain of a factor between 1.5 and 15, depending on

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1280-658: The orbital elements of the planets. 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, the 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 2,000 years and for Uranus and Neptune over 6,000 years before and after J2000. This, together with its free availability has resulted in VSOP87 being widely used for planetary calculations; for example, it

1320-484: 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 + ct +...)×cos( p + qt + rt +...) and so forth one for each planetary interaction. The factor

1360-448: The planet, compared to VSOP2013. The precision of the theory of Pluto remains valid up to the time span from 0 to +4000. Bureau des Longitudes The Bureau des Longitudes ( French: [byʁo de lɔ̃ʒityd] ) 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

1400-416: The relatively large distance between them. In astronomy, barycentric coordinates are non-rotating coordinates with the origin at the barycenter of two or more bodies. The International Celestial Reference System (ICRS) is a barycentric coordinate system centered on the Solar System 's barycenter. The barycenter is one of the foci of the elliptical orbit of each body. This is an important concept in

1440-547: The sky was already performed in ancient times. Careful observations and geometrical calculations produced a model of the motion of the Solar System 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 , Johannes Kepler , and Isaac Newton in early modern Europe laid

1480-889: 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 six tables: The VSOP87 tables are publicly available and can be retrieved from VizieR . VSOP2000 has an accuracy that

1520-463: 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)

1560-466: The world-time must be synchronized with some ideal clock that is assumed to be very far from the whole self-gravitating system. This time standard is called Barycentric Coordinate Time (TCB). Barycentric osculating orbital elements for some objects in the Solar System are as follows: For objects at such high eccentricity, barycentric coordinates are more stable than heliocentric coordinates for

1600-471: Was charged with taking control of the seas away from the English and improving accuracy when tracking the longitudes of ships through astronomical observations and reliable clocks. The ten original members of its founding board were: By a decree of 30 January 1854, the Bureau's mission was extended to embrace geodesy, time standardisation and astronomical measurements. This decree granted independence to

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