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:
 The synodic period (beginning of each orbit defined as when the planet crosses an imaginary line drawn through the Sun and Earth; determines how long it will be until the planet reappears in the evening sky).
 The tropical year (time from one a planet's equinox to the next; factors in the precession of the planet's rotational axis, and tells how long it takes the given planet to go through a full cycle of its weather seasons). The ecliptic coordinate system in astronomy uses the Earth's equinox as its basis.
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 lissajouslike "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:
 The X, Y and Z axes cross at 90degree angles at the solar system's barycenter or sun's center (depending on which type of sidereal year is being measured). Thus they define a reference frame that is accelerated, but nonrotating. NOTE: When sun's center is used, the axes move along with the sun mostly in response to the Jovian planets.
 The XY plane is parallel to the plane of the Earth's orbit at Julian date 2451545.0 (20000101 12:00 UTC)
 The X axis is parallel to the intersection of the plane of the Earth's equator and the plane of the Earth's orbit at Julian date 2451545.0 (20000101 12:00 UTC), and the positive end of the X axis points in the same direction as the ascending node.
 The positive end of the Z axis is the end closest to the direction of the Earth's north pole.
 For any given planet position (X,Y,Z), a phase angle theta is defined as arctan(X/Y)
 The beginning of a planet orbit is defined to be the moment when theta=0. (For the Earth, this approximately coincides with the fall equinox, around Sep 22nd)
Two tables were generated, one for solar system barycenter and one for the sun as the center.
Sidereal Year Relative To Solar System Barycenter

Sidereal Year Relative To the Sun

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:

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 59.6year period is the reason for the everyotherorbit pattern in Saturn's sidereal year. Because it's not exactly 5to2, 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 1to3 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
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2020 Mar 26. s.11