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Accumulated perihelion shift vs orbit number (analytic line, 1PN dots, exact dots)
Measures the advance per orbit for e = 0.05 ... 0.9 with a fresh 1PN and exact integration each.
Where Mercury's perihelion shift comes from, arcsec per century (typical values, log scale)
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How to use
Mercury's orbit is an ellipse that slowly turns: its closest point to the Sun, the perihelion, creeps forward by a tiny angle every orbit. Newton explains most of it with the pull of the other planets; the last 43 arcseconds per century were a puzzle until Einstein's general relativity in 1915. Pick a preset or drag the sliders and watch the perihelion flags and rays. Space pauses, R resets.
- Try 1: the true size. Preset Mercury, true size. The picture looks like a closed ellipse, but read the numbers: 0.1035 arcsec per orbit, 42.98 arcsec per century, matching the formula.
- Try 2: exaggerate it. Preset Mercury x100,000 turns the same orbit into a rosette. The cyan Newtonian ellipse never moves; the orange and green orbits rotate.
- Try 3: eccentricity. Drag e from 0.05 to 0.9 and watch the shift per orbit grow as 1/(1 - e^2). Then press Run eccentricity sweep and compare dots with the curve.
- Try 4: where first order fails. Preset Rosette (strong field): orange (1PN) and green (exact geodesic) separate. Look at the effective-potential graph: the extra -eps L^2/r^3 term digs a pit near the Sun.
- Try 5: other worlds. Venus, Earth, the asteroid Icarus and two binary pulsars all precess by the same law. The Hulse-Taylor pulsar gains 4.2 degrees every year.
Numerical checks run in the page; they are listed below the graphs.
Drag to pan, wheel or pinch to zoom. Perihelion zooms on the latest perihelion, Follow Mercury locks the view on the planet. The small dial at the top right magnifies the accumulated perihelion shift so that you can see it even when it is a fraction of an arcsecond.
The science
Everything here is a test particle (Mercury is much lighter than the Sun) moving in the field of a point mass GM. Lengths are measured in units of the Newtonian semi-major axis a, times in units of sqrt(a^3/GM) (one orbit = 2 pi). The only extra number is the dimensionless relativistic strength
eps = GM / (c^2 a) Mercury: 1.48 km / 5.79e7 km = 2.55e-8
Newton. r'' = -GM r / r^3 has the Kepler ellipse as its solution, which is closed, so the perihelion never moves. The sim integrates it with RK4 and a step that shrinks near the Sun.
Exact Schwarzschild orbit. For a geodesic with specific angular momentum L and proper time τ, the radial equation of motion is exact:
d²r/dτ² = -GM/r² + L²/r³ - 3 GM L² / (c² r&sup4) dφ/dτ = L/r²
The last term is the whole difference from Newton. In terms of u = 1/r the same orbit obeys u'' + u = GM/L^2 + 3GM u^2/c^2, and the effective potential is V = -GM/r + L^2/2r^2 - GM L^2/(c^2 r^3), the pit drawn in the second graph. Bound orbits need L^2 > 12 (GM/c)^2; below that the particle plunges.
First post-Newtonian (1PN) orbit. The orange orbit uses the Einstein-Infeld-Hoffmann acceleration in the test-particle limit, in Cartesian coordinates and coordinate time:
a = -GM r/r³ + (GM / c²r³) [ (4GM/r - v²) r + 4 (r·v) v ]
Both models give the same first-order result, the perihelion advance per orbit:
Δϖ = 6 π GM / ( c² a (1 - e²) ) = 6 π eps / (1 - e²) (rad per orbit)
Mercury: 5.02e-7 rad = 0.1035 arcsec per orbit; 415 orbits per century give 42.98 arcsec/century. Venus 8.6, Earth 3.8, Icarus 10.1 arcsec/century. For PSR B1913+16 (total mass 2.83 solar masses, P = 7.75 h, e = 0.617) the same formula gives 4.23 degrees per year, which is what radio timing measured.
Exaggeration. At the true eps the rosette is invisible, so the slider multiplies eps by a factor X (shown in the readout). Orbits drawn with X larger than 1 are not what the real system does; only the formula for the rate at X = 1 is quoted as the real value. For eps above about 0.01 the 1PN orbit and the exact geodesic visibly disagree, which is the expected breakdown of a first-order expansion.
Where Mercury's 5600 arcsec/century comes from (typical textbook values). About 5026 are only an artefact of measuring from the moving equinox, about 532 are Newtonian tugs from Venus, Jupiter, Earth and the others, and 42.98 are general relativity. Total 5600 observed against 5599.7 predicted. These bars are quoted values, not computed by this page; the Sun's small oblateness (J2) and the Lense-Thirring term are below 0.03 arcsec/century and are neglected.
Honest simplifications. One light body orbiting a fixed point mass; no other planets, no solar quadrupole, no gravitomagnetism, no radiation. The exact model uses proper time of the orbiting body as the clock, the 1PN model uses coordinate time, so at large eps their orbital phases drift apart. Sun, Mercury and flags are drawn much larger than to scale; the orbiter circles Mercury for decoration and does not follow the physics. Mercury's 3:2 spin-orbit resonance is drawn as a rotation rate of 1.5 times the mean motion.
Numerical checks in this page. (1) Newtonian period and energy over 20 orbits; (2) 1PN integration against 6 pi eps/(1-e^2) at eps = 1e-4; (3) the proper-time geodesic integrator against an independent integration of the orbit equation in the angle (a different equation, different variable); (4) the real Mercury integrated without exaggeration against 42.98 arcsec/century; (5) the Hulse-Taylor and Icarus formulas against published rates.