The previous lesson argued that retrograde motion is a trick of the line of sight and that the orbit itself does not change. This one is about the way the orbit does change, only far more slowly. The long axis of the ellipse — the line from aphelion through the Sun to perihelion — rotates, so a planet never quite returns to where it started.
If the solar system held only the Sun and one planet, that planet would run round a perfectly fixed ellipse forever and the long axis would not move by a single arcsecond — that is what Kepler's laws say. It turns because the other planets pull too. Jupiter is heavy enough on its own to move the inner planets' axes by hundreds to thousands of arcseconds a century.
The rates sound tiny, but they always push the same way, so they accumulate. Mercury's 574 arcseconds per century works out at one complete turn of the axis every 226,000 years or so. It is also why this lesson has to switch element tables: across the 250 years the previous lesson's table covers, Mercury's axis turns 0.4 degrees, which is nothing you could draw.
In the mid-nineteenth century Le Verrier found that Mercury's perihelion moved slightly more than Newtonian gravity predicted. The discrepancy was small, but it was measurable and it would not go away. One proposal was an undiscovered planet inside Mercury's orbit, provisionally named Vulcan; it was never found. The books balance roughly like this, in arcseconds per century:
Neither table is wrong; the quantity is simply hard to pin down. The perihelion rate in an element table is a coefficient fitted to make positions come out right, not a directly measured physical rate, and the two tables were fitted over intervals differing by a factor of twenty-four. On top of that, when an orbit is nearly circular, "which way does perihelion point" barely means anything — Venus's eccentricity is 0.0068, so the direction of the long axis affects positions too weakly for a fit to constrain it. The honest answer for those planets is that this approximation cannot tell you, not a confident-looking number.
One planet's orbit at nine different epochs, drawn over each other. Faintest is earliest, solid is latest. The violet line is the apse line — the thing that is rotating.
The slope of this line is the precession rate. It is very nearly straight, which is what makes a single rate meaningful.
This lesson uses NASA/JPL's long-range element table (Table 2, valid 3000 BC to 3000 AD), with additional periodic terms for Jupiter through Neptune. It is a different table from the previous lesson's: the short-range one is more accurate over 1800–2050 but does not span enough time for precession to show.
| Precession rate | arcsec/century |
|---|---|
| Observed total, against the equinox of date | +5,599.74 |
| Less Earth's axial precession | −5,025.65 |
| Actual motion against fixed stars | +574.09 |
| Less the Newtonian pull of the other planets | −531.54 |
| Residual | +42.55 |
Two different things are called precession, and they are constantly confused. This lesson is about apsidal precession: the orbit turning. The other one, precession of the equinoxes, is Earth's rotation axis wobbling and dragging the equinox around the sky over about 25,800 years. That is a fact about Earth, not about planetary orbits, and it is not modelled here at all. The elements used below are referred to a fixed J2000 ecliptic and equinox, so these rates are measured against unmoving space rather than against a moving equinox.
General relativity predicts 42.98 arcseconds per century, matching the residual. Einstein published that calculation in 1915, and it was the theory's first successful test — not a new observation, but a sixty-year-old debt in the accounts that suddenly had an explanation.
To be clear: this page does not derive that 43 arcseconds. The five figures above are quoted from the literature (Clemence 1947). The orbital elements this package uses are fitted to modern observations, so they already contain the relativistic contribution and it cannot be separated out of them. Doing so needs an n-body gravitational model with relativity switched off, which is a different program.
Related tools: Solar System Orrery · The ecliptic plane · Astro Events
Orbital elements from NASA/JPL, "Keplerian Elements for Approximate Positions of the Major Planets" (Standish & Williams, 1992), valid 1800–2050. Positions are geometric — no light-time or aberration — because this is a diagram of the ecliptic plane, not a simulation of a telescope view.