Some Notes About the Solar System  

I began this page in July 2001 while doing research related to orrery development. It would be years until I actually began to build orrerys.

The fundamental question I was trying to answer is to get the most accurate available figures for the length of time it takes each planet to orbit the Sun. As I looked more deeply into the question, I discovered many complex variables that make the question more and more difficult to answer as the desired accuracy increases.

Resonances

Exact Resonances

Mercury solar day to Mercury solar year: 2:3. There are exactly two points on Mercury's surface where the Sun is directly overhead at perihelion.

Venus day, Venus year and Earth year: Every time Venus comes into inferior conjunction (closest approach) to Earth, the same side of Venus is facing Earth.

Io sidereal month to Europa sidereal month: 1:2. Opposition of Io and Europa always occurs at the same point in their orbits (relative to the stars)

Europa sidereal month to Ganymede sidereal month: 1:2. Opposition of Europa and Ganymede always occurs at the same point in their orbits (relative to the stars)

Mimas sidereal month to Tethys sidereal month: 1:2. Opposition of Mimas and Tethys always occurs at the same point in their orbits (relative to the stars)

Enceladus sidereal month to Dione sidereal month: 1:2. Opposition of Enceladus and Dione always occurs at the same point in their orbits (relative to the stars)

Titan sidereal month to Hyperion sidereal month: 3:4. Titan completes 4 orbits in precisely the same time it takes Hyperion to complete 3.

Neptune year to Pluto year: 2:3. Pluto completes 2 orbits in precisely the same time it takes Neptune to complete 3 orbits. When Pluto is at perihelion, Neptune's position is always either 1/4 orbit further ahead or 1/4 orbit behind Pluto's position.

Pluto argument of perihelion precession rate to Pluto ascending node precession rate: Pluto's perihelion is locked at the point in its orbit where it is furthest "below" the plane of the solar system.

(Most of this info is also on this page. This page has some interesting facts about resonances.)

Chaotic Motion

Except when there are resonances, the motion of the planets is generally chaotic, in the sense that any tiny perturbation today can and will cause a great change in position at some time in the future. For example, the positions of the inner planets cannot be accurately predicted more than about 20,000 years into the future, nor can they be accurately estimated for more than 20,000 into the past.

Sidereal Periods

The sidereal period of a planet is the amount of time it takes for the planet to complete one orbit, when viewed in relation to the stars. This is the most common definition of the planet orbits, and the most useful if you are looking at the whole solar system and don't want to treat any one planet as "special". There are other definitions, including:

Since I wanted to build solar system models (orreries) it seemed logical to choose a year definition that would be the same for all the planets, and that makes the sidereal year the most obvious choice. To make a precise definition, you can define the sidereal year in terms of a particular star (presumably one with little proper motion), draw a line from the center of the solar system to that star, and define the year according to when the planet crosses that line.

When meauring a sidereal period, you can either use the sun as the center of the orbit, or the barycenter of the Solar System. Both are useful, although the sun is the one more commonly used.

Even though the Sun is over 300,000 times as massive as the Earth, the distances involved are so great that the Sun's position varies by 450 km (almost 300 miles) in either direction due to its gravitational interaction with the orbiting earth.

The position of the Sun can very from the barycenter of the Solar System by as much as twice the Sun's radius in any direction, and most of that movement is due to Jupiter. To balance Jupiter, the Sun moves 750,000 km in either direction, a total wobble (1.5 Gm) that is greater than the Sun's diameter. The other large planets have similar effects and they all add together. As the sun moves in its lissajous-like "orbit", the inner planets, with their comparatively shorter orbital periods, move with it. Thus, due to Jupiter's influence the Earth's orbit shifts over a range of 1,500,000 km, and more if you count Saturn, Uranus and Neptune.

As a result, if you're looking at an inner planet it makes more sense to define the sidereal year in terms of the sun's location, but if you're looking at the planets beyond Jupiter it makes more sense to use the Solar System's barycenter.

The sidereal periods of each planet vary from one orbit to the next, due to the gravitational influence of the other planets. I recently spent a while trying to find very accurate (8 significant figures or better) estimates of the sidereal periods of the planets and discovered that they vary so much from one orbit to the next that it is hard to get a figure with more than 4 digits of accuracy!

JPL (Jet Propulsion Laboratory in Pasadena CA) has made their ephemeris and tracking system available to the public. It is probably the most accurate model available anywhere (it is used by JPL for tracking of their spacecraft). Using this system I found the time (Julian Date) of the beginning of each orbit for the 8 major planets (Mercury through Neptune) according to the following definitions:

Two tables were generated, one for solar system barycenter and one for the sun as the center.

Sidereal Year Relative To Solar System Barycenter

planet during
the years
orbits average
sidereal
period
range range / orbits notes
Mercury 1800-2199 1660 87.9691205 0.054 0.0000162 main oscillation has period 11.9 years, amplitude 0.04 days
Venus 1800-2200 650 224.700295 0.199 0.000153 main oscillation has period 11.9 years, amplitude 0.15 days
Earth 1800-2199 399 365.25519 0.38 0.00047 main oscillation has period 11.9 years, amplitude 0.3 days
Mars 1800-2199 211 686.97882 0.78 0.00184 main oscillation has period 11.9 years, amplitude 0.6 days
Ceres - - - - - -
Jupiter 1500-2497 84 4332.6045 3.1 0.0186 main oscillation has period 59 years, amplitude currently near maximum of 3 days (varies over an 880-year period)
Saturn 1525-2497 34 10759.17 19.2 0.28 currently alternating back and forth between "odd" and "even" orbits, "even" orbit is 14 days longer; "odd" orbit gradually getting longer until the 25th century when they will be of equal length and the "odd" orbit will stop getting longer and the "even" orbit will begin getting shorter (total cycle lasts 880 years)
Uranus 1002-2935 23 30684.45 22.4 0.49 currently shortening by an average of 2 days per orbit; complete cycle takes 4300 years
Neptune 1037-2849 11 60189.7 47.8 2.2 currently lengthening by an average of 5 days per orbit; complete cycle takes 4300 years
Pluto - - 90284. - - I belong to the "Pluto is not a major planet" camp; however it has been shown that Pluto's orbital period is precisely 3/2 that of Neptune due to orbital resonance.

Sidereal Year Relative To the Sun

planet during
the years
orbits average
sidereal
period
range range / orbits notes
Mercury 1800-2199 1660 87.96927077 0.00150 0.00000045 main oscillation has period 1.1 years, amplitude 0.001 days (2 minutes)
Venus 1800-2200 650 224.7007826 0.0091 0.0000070 irregular
Earth 1800-2199 399 365.2562953 0.0113 0.0000141 irregular
Mars 1800-2199 211 686.98228 0.104 0.00024 irregular
Ceres - - - - -
Jupiter 1500-2497 84 4332.608 3.8 0.022 same pattern as in barycenter table above
Saturn 1525-2497 34 10759.01 16.8 0.25 same pattern as in barycenter table above
Uranus 1002-2935 23 30684.53 22.4 0.49 same pattern as in barycenter table above
Neptune 1037-2849 11 60189.8 48.0 2.2 same pattern as in barycenter table above
Pluto - - 90284. - - same note as in barycenter table above

The variations in year length are real, but depend on your definition of a year. In particular, they depend a lot on when the year starts.

For example, consider the Earth's orbit with the Sun (not the barycenter) as the center, and look at the perturbation that would be caused by Jupiter. Let's divide Earth's orbit into four parts:

part description
1 Earth-Jupiter distance is less than Sun-Jupiter distance and Earth is moving towards Jupiter
2 Earth-Jupiter distance is less than Sun-Jupiter distance and Earth is moving away from Jupiter
3 Earth-Jupiter distance is more than Sun-Jupiter distance and Earth is moving away from Jupiter
4 Earth-Jupiter distance is more than Sun-Jupiter distance and Earth is moving towards Jupiter

Jupiter is pulling on both Earth and the Sun. When Earth is in sections 1 and 2, the acceleration of the Earth by Jupiter is greater than the acceleration of the Sun due to Jupiter. Similarly, in sections 3 and 4 acceleration of the Earth by Jupiter is less than the acceleration of the Sun due to Jupiter. This means that the Earth will be effectively speeding up in its orbit (relative to the Sun) during sections 1 and 3 and will be slowing down during sections 2 and 4.

Since Jupiter moves, an Earth year does not consist of 3 months in each section 1,2,3 and 4. Instead, an Earth year will include partial sections, and sometimes the acceleration during sections 1 and 3 does not completely cancel out the deceleration during sections 2 and 4. The same effect happens with all other planets, and the total of all these perturbations makes the length of the year vary by about 16 minutes from one year to another.

The Jupiter orbit oscillation period of 59.6 years corresponds to 5.02 orbits of Jupiter, which is very nearly equal to 2.02 orbits of Saturn. During this period Jupiter and Saturn are in opposition three times. This same 59.6-year period is the reason for the every-other-orbit pattern in Saturn's sidereal year. Because it's not exactly 5-to-2, the points of opposition move slowly around the orbits and take about 880 years to return to the point where they were.

There is a weaker 1-to-3 ratio between Saturn and Uranus. Saturn completes 3.08 orbits in the same time (about 91 years) it takes Uranus to complete 1.08 orbits. During this time Saturn is in opposition with Uranus twice. After about 6.5 repetitions of this period (about 570 years) the opposition happens in the same place again.

Uranus and Neptune have a close to 2:1 ratio. Uranus completes 2.04 orbits in the time it takes Neptune to complete 1.04. Thus, each opposition (point of nearest approach) between the two planets occurs 1/25 of an orbit further along in the orbit. It takes about 4300 years for this point of nearest approach to make one full cycle around the Sun.

Leap Seconds

During the 20th century, thanks to the quartz clock (and later the atomic clock), it became possible to measure time so accurately that the fluctuations in the rate of the Earth's rotation could be measured. There fluctuations are caused by such things as convection currents in the mantle and seasonal movements of ice and water (polar ice caps and tide/current resonances). It became clear that the period of rotation of the Earth had a significant unpredictable component — it was generally slowing from one year to the next, but not at a steady rate and with no pattern.

In 1967 the second was defined in terms of the Cesium atom used in atomic clocks. The length of a second has not been redefined to accommodate the slowing of the Earth. However, since 1967 the Earth's rate of spin slowed down so much that, if no correction had been made, the atomic clocks would now be reading 12:00:00 noon an average of 25 seconds before the point when the sun is overhead (after adjusting for the known astronomical effects such as the eccentricity of the Earth's orbit, etc.) Therefore, in order to prevent the atomic clocks (from which all other clocks are set) from drifting too far away from astronomical reality, "leap seconds" have been added about once every 500 days (starting in 1972). The leap second always comes at midnight on either June 30th or Dec 31st.

For more about this go to the NIST Time and Frequency website.

References

JPL HORIZONS ephemeris data

Sidereal and tropical year, solstices and equinoxes: webexhibits calendar page


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