Relativistic Perihelion Precession Calculator — Mercury’s 43″/century
Compute the general-relativistic perihelion advance of an orbit with Einstein’s weak-field formula Δϖ = 6πGM/(c²a(1−e²)). Reproduces Mercury’s 43 arcsec/century.
A clear path from inputs to results.
Compute how much an orbit’s perihelion rotates each revolution because of general relativity — the tiny, cumulative twist of the ellipse that Newtonian gravity cannot explain. Using Einstein’s weak-field formula, it reproduces the famous 43 arcseconds-per-century advance of Mercury and lets you explore the effect for any compact or close orbit.
How to use
- Enter the central mass in solar masses; the engine converts it to the gravitational parameter μ = GM.
- Set the orbit’s semi-major axis a (in AU) and eccentricity e. Together they fix the semi-latus rectum p = a(1−e²) that drives the precession.
- Choose the number of revolutions to propagate so you can watch the perihelion advance accumulate orbit by orbit.
- Read the relativistic advance per orbit, shown in radians and arcseconds, and the total rotation over the revolutions you selected.
- Compare against the Newtonian ellipse, which stays fixed in space — the difference is the pure general-relativistic contribution.
The formula
Reference: Einstein’s 1915 weak-field perihelion advance from the Schwarzschild geodesic (first post-Newtonian order). Einstein, “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie” (1915); Weinberg, Gravitation and Cosmology; Misner, Thorne & Wheeler, Gravitation.
Worked example
Mercury around the Sun (1 solar mass)
- Central mass M = 1.0 M☉ (1.98847×10³⁰ kg); semi-major axis a = 0.387098 AU (5.7909×10¹⁰ m); eccentricity e = 0.20563; G = 6.67430×10⁻¹¹, c = 299,792,458 m/s.
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The gravitational parameter is μ = GM = 6.67430×10⁻¹¹ × 1.98847×10³⁰ = 1.32716×10²⁰ m³/s². The semi-latus rectum is p = a(1−e²) = 5.7909×10¹⁰ × (1 − 0.20563²) = 5.54604×10¹⁰ m. The per-orbit advance is Δϖ = 6πμ / (c²p) = 6π × 1.32716×10²⁰ / ((299,792,458)² × 5.54604×10¹⁰) = 5.0188×10⁻⁷ rad, which is 0.10352 arcseconds per orbit. Mercury completes 100 yr / 0.24084 yr = 415.21 orbits per century, so the accumulated advance is 0.10352 × 415.21 = 42.98 arcseconds per century.
Perihelion advance = 5.0188×10⁻⁷ rad = 0.10352 arcsec per orbit, totalling 42.98 arcsec per century — matching the observed anomaly that general relativity famously explained.
Interpreting the result
The advance is the extra angle the ellipse’s long axis sweeps each orbit beyond the full 360° of a closed Newtonian orbit. It scales as 1/p, so it is largest for massive central bodies and tight, nearly head-on (low-p) orbits: Mercury shows it most clearly among the planets precisely because it is closest to the Sun and modestly eccentric. The effect is strictly cumulative — small per orbit, but it adds up over many revolutions and over a century becomes the measurable 43 arcseconds.
Assumptions & limitations
- Weak-field, first post-Newtonian approximation of the Schwarzschild solution; valid when GM/(c²a) ≪ 1, which holds for planetary orbits but not near a black-hole horizon.
- A single spherically symmetric, non-rotating central mass; frame-dragging (Kerr) and the quadrupole/oblateness of the central body are not included.
- Only the general-relativistic contribution is computed. Real planets also precess from Newtonian perturbations by other planets — the dominant part of Mercury’s total ~574 arcsec/century — which this engine deliberately isolates out.
- The two-body idealization is used: the orbiting body is treated as a test particle of negligible mass.
- Constants (G, c, AU, solar mass) are treated as exact SI values; the arcsecond conversion uses 206,264.806 arcsec per radian.
Frequently asked questions
Why is Mercury’s precession 43 arcseconds per century here, not 574?
This engine computes only the general-relativistic part, Δϖ = 6πGM/(c²a(1−e²)), which comes out to 42.98 arcsec/century. The larger ~574 arcsec/century observed total is dominated by Newtonian tugs from the other planets; GR accounts for exactly the ~43 arcsec that Newtonian mechanics could not, which is why it was the first great confirmation of the theory.
Is this the full general-relativistic orbit?
No — it is the leading weak-field term of the Schwarzschild geodesic, accurate when GM/(c²a) is tiny. For planetary orbits that approximation is excellent. Very close to a compact object the full relativistic treatment (and higher-order terms) is needed, and this formula would understate the precession.
Does the orbit’s size or period change too?
To this order, no — the semi-major axis, eccentricity, and period are unchanged; it is only the orientation of the ellipse that slowly rotates in its plane. That is why the effect is called a precession rather than a decay. (Orbital decay from gravitational-wave emission is a separate, much smaller effect for planets.)
What makes the perihelion advance larger or smaller?
The advance scales as GM / (a(1−e²)). A heavier central mass increases it; a larger orbit (bigger a) decreases it; and higher eccentricity increases it through the (1−e²) term. So compact, eccentric orbits around massive bodies precess the fastest — binary pulsars show advances of degrees per year.
What is the semi-latus rectum p and why use it?
The semi-latus rectum p = a(1−e²) is the orbit’s width at the focus and is the natural length scale in the precession formula, letting it be written compactly as Δϖ = 6πμ/(c²p). For Mercury p = 5.546×10¹⁰ m.
Can I use this for exoplanets or binary stars?
Yes, as long as the weak-field condition holds. Enter the central mass in solar masses and the orbit’s a and e; the engine returns the relativistic advance per orbit. For close-in hot Jupiters or tight stellar binaries the per-orbit advance can be far larger than Mercury’s and is, in principle, observable.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.