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.
What is perihelion precession?
Perihelion precession is the slow rotation, in space, of the long axis of a planet's elliptical orbit — the line through perihelion, the Sun and aphelion — so that the direction of perihelion changes over the centuries. It is not the same as the precession of Earth's spin axis: what turns is the orbit's long axis, not the axis of rotation, and the equinox does not shift because of it.
Why does the perihelion precess?
Mainly because of gravitational perturbations from the other planets. In the idealised two-body problem of a lone planet and the Sun, the planet retraces the same fixed ellipse forever; in the real solar system Jupiter, Saturn and the rest keep pulling from different directions, and the long axis of an inner planet's orbit turns a little at a time. The change is so slow that it takes thousands of years to become obvious in a diagram.
What does Mercury's perihelion have to do with relativity?
After nineteenth-century calculations had accounted for every other planet's perturbation, Mercury's perihelion still advanced about 43 arcseconds per century more than predicted, and Newtonian mechanics could not explain the residual. General relativity's curvature of spacetime near the Sun gives exactly 43 arcseconds per century, one of the theory's first confirmations. The model on this page computes only the Newtonian perturbations: it shows the axis turning, but cannot be used to verify those 43 arcseconds.
After Mercury completes an orbit, its ellipse points a little elsewhere
A planet's elliptical orbit does not point in one direction forever. Pulls from other planets slowly rotate the direction of perihelion.
An orbit is not a line engraved in space
In a two-body exercise, a planet repeats one fixed ellipse. The real Solar System includes continuing pulls from Jupiter, Saturn, and every other planet, so the direction of the orbit's closest point slowly rotates.
That rotation is apsidal precession. The planet still completes an ordinary revolution, but one ellipse does not perfectly overlap the previous one. It is also separate from the precession of Earth's rotation axis.
The extra 43 arcseconds per century
Nineteenth-century calculations included Newtonian perturbations from other planets, yet Mercury's perihelion still advanced about 43 arcseconds per century beyond the prediction. Long, precise observations made the tiny difference measurable.
General relativity supplied the missing advance. This animation uses modern orbital elements that already contain observational and relativistic results, so it can display total precession but cannot independently re-prove the 43-arcsecond test.
The animation shows the turning, but it cannot verify relativity
Stretch the time span to several thousand years and many ellipses stack up like a fan, so the turning of the perihelion becomes easy to see. Exaggerating the angle enlarges the rotation of the long axis, which helps you spot the change; switch it off to see something close to true proportions.
The modern orbital elements this simulation uses already include the results of observation and relativistic calculation. It can show what the total precession looks like, but it cannot independently redo the “43 arcsecond test”; for that you have to go back to the original observations and theory.
Why it turns
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.
Mercury's extra 43 arcseconds
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:
Which planets get no number
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.
The long axis turning
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.
Mercury's extra 43 arcseconds
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
FAQ
Is perihelion precession the same as Earth's axial precession?
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.
What does Mercury's extra 43 arcseconds per century mean?
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.
Can this interactive diagram prove the 43 arcseconds directly?
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.
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.