---
title: The Perihelion Precession of Mercury
module: Tests of General Relativity
moduleNumber: 7
lessonNumber: 1
order: 701
summary: >
  A single extra term in the Schwarzschild orbit equation, cubic in the inverse
  radius, keeps a bound orbit from closing. The perturbation advances the
  perihelion by 6πGM/(c²a(1−e²)) per revolution, which for Mercury is 43 arcseconds
  per century — exactly the anomaly left after Newtonian planetary perturbations
  are subtracted. A note on frame dragging closes the lesson.
topics: [Tests of General Relativity]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 10 — Solar-System Tests of General Relativity; §10.2 The Perihelion Precession of Mercury"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §7 — The Schwarzschild Solution and Black Holes (perihelion precession), arXiv:gr-qc/9712019"
---

A Newtonian orbit under an inverse-square force is a closed ellipse: after one
revolution in azimuth the radius returns to its starting value and the orbit
retraces itself exactly. This closure is a special property of the $1/r$
potential, not a generic feature of central forces, and general relativity breaks
it. The Schwarzschild geometry adds a single term to the orbit equation, cubic in
the inverse radius, and the ellipse no longer closes. Its long axis rotates
slowly in the orbital plane, tracing a rosette. For Mercury the rotation is
$43''$ per century, the residual that nineteenth-century celestial mechanics
could not explain and that Einstein computed to be an automatic consequence of
the field equations.

The orbit equation used here comes from the [Schwarzschild
geodesics](/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild);
this lesson takes it as the starting point and extracts the precession. Geometrized
constants are restored so every number carries units. The signature is $(-,+,+,+)$.

## The orbit equation and its relativistic term

A massive particle on a bound geodesic in the equatorial plane of the
Schwarzschild geometry has two conserved quantities per unit mass, from the time
and azimuthal Killing vectors:

$$
\tilde E = \left(1 - \frac{2GM}{c^2 r}\right)c^2\,\frac{\d t}{\d\tau},
\qquad
\tilde L = r^2\,\frac{\d\phi}{\d\tau}.
$$

The normalization $g_{\mu\nu}\,\dot x^\mu\dot x^\nu = -c^2$ turns into a radial
energy equation, and substituting the inverse radius $u = 1/r$ with $\d r/\d\tau =
-\tilde L\,\d u/\d\phi$ converts it into an equation for the shape $u(\phi)$ of
the orbit. Differentiating once in $\phi$ removes the constant term and leaves

$$
\frac{\d^2 u}{\d\phi^2} + u = \frac{GM}{\tilde L^2} + \frac{3GM}{c^2}\,u^2 .
$$

Every piece except the last reproduces the Newtonian orbit. Dropping the final
term gives $u'' + u = GM/\tilde L^2$, a harmonic oscillator in $\phi$ driven by a
constant, whose solution is the conic

$$
u_0(\phi) = \frac{GM}{\tilde L^2}\bigl(1 + e\cos\phi\bigr),
\qquad
r(\phi) = \frac{\tilde L^2/GM}{1 + e\cos\phi}.
$$

The perihelion (minimum $r$, maximum $u$) sits at $\phi = 0$, and because the
solution has period $2\pi$ in $\phi$, the next perihelion is at $\phi = 2\pi$: the
ellipse closes.

> **Definition (Semi-latus rectum).** The constant $\ell = \tilde L^2/GM$ is the
> semi-latus rectum of the Newtonian ellipse, related to the semi-major axis $a$
> and eccentricity $e$ by $\ell = a(1-e^2)$. Equivalently $\tilde L^2 = GM\,a(1-e^2)$.

The relativistic term $3GM u^2/c^2$ is small for a planetary orbit. Its size
relative to the Newtonian driving term is

$$
\frac{3GM u^2/c^2}{GM/\tilde L^2}
= \frac{3\tilde L^2 u^2}{c^2}
\approx \frac{3 v_\perp^2}{c^2},
$$

of order $(v/c)^2$. For Mercury this ratio is about $3\times10^{-8}$, so the term
cannot change the orbit's size or shape appreciably over one revolution. What it
does instead is accumulate: a tiny frequency shift each orbit, secular in the
azimuth.

$$
% caption: A Newtonian orbit closes after one revolution; the extra cubic term
% shifts the angular frequency slightly below one, so the perihelion arrives a
% little past 2 pi and the ellipse advances into a slowly rotating rosette.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % central mass
  \fill (0,0) circle (2.2pt);
  \node[below left] at (0,0) {Sun};
  % rosette: r = p/(1+e cos(k*theta)), k slightly < 1; ~3 orbits, 40 deg advance each
  \draw[acc, very thick, domain=0:1080, samples=320, variable=\t]
    plot ({ (1.5/(1+0.42*cos(0.90*\t)))*cos(\t) },
          { (1.5/(1+0.42*cos(0.90*\t)))*sin(\t) });
  % first perihelion direction (closest approach along +x)
  \draw[black, dashed] (0,0) -- (1.25,0);
  \fill[black!75] (1.056,0) circle (1.5pt);
  \node[black!70, below right] at (1.06,0) {perihelion 1};
  % advanced perihelion after one revolution (rotated by 40 deg)
  \draw[black, dashed] (0,0) -- (40:1.35);
  \fill[black!75] (40:1.056) circle (1.5pt);
  \node[black!70, above right] at (40:1.06) {perihelion 2};
  % advance arc
  \draw[->, black] (0.6,0) arc (0:40:0.6);
  \node[black] at (0.78,0.5) {advance};
\end{tikzpicture}
$$

## Perturbation and the advance per orbit

Treat the cubic term as a perturbation of the Newtonian conic. Insert $u_0$ into
the right-hand side and keep only the part that drives the oscillator on
resonance. With $u_0 = A(1 + e\cos\phi)$ and $A = GM/\tilde L^2$,

$$
\frac{3GM}{c^2}u_0^2
= \frac{3GM A^2}{c^2}\bigl(1 + 2e\cos\phi + e^2\cos^2\phi\bigr).
$$

The constant and the $\cos^2\phi$ pieces only shift and slightly distort the
orbit; the $2e\cos\phi$ term is resonant with the homogeneous solution and
produces a secular response. Keeping it,

$$
\frac{\d^2 u}{\d\phi^2} + u = A + \frac{6GM A^2 e}{c^2}\cos\phi .
$$

A driving term $\propto\cos\phi$ on an oscillator of natural frequency one grows
without bound as $\phi\cos\phi$, the signature of resonance. That secular term is
exactly the first-order expansion of a small frequency shift: to the same order,

$$
u(\phi) = A\Bigl[1 + e\cos\bigl((1-k)\phi\bigr)\Bigr],
\qquad
k = \frac{3G^2M^2}{c^2\tilde L^2},
$$

because expanding $\cos((1-k)\phi) \approx \cos\phi + k\phi\sin\phi$ reproduces the
$\phi\cos\phi$-type secular growth with the correct coefficient. The perihelion
now recurs when the argument advances by $2\pi$, i.e. at

$$
\phi_{\text{peri}} = \frac{2\pi}{1-k} \approx 2\pi(1+k).
$$

The perihelion has moved forward by $\Delta\phi = 2\pi k$ per revolution:

$$
\Delta\phi = \frac{6\pi G^2 M^2}{c^2\,\tilde L^2}
           = \frac{6\pi GM}{c^2\,a(1-e^2)} ,
$$

using $\tilde L^2 = GM\,a(1-e^2)$. The advance is per orbit, in the direction of
motion, and depends only on the mass of the central body and the orbit's size and
shape.

> **Result.** A bound orbit in the Schwarzschild geometry precesses by
> $$\Delta\phi = \frac{6\pi GM}{c^2\,a(1-e^2)}$$
> radians per revolution. The factor $6\pi$ is fixed by the coefficient $3$ of the
> relativistic term in the orbit equation; the smallness comes entirely from
> $GM/(c^2 a)$, the ratio of the gravitational radius to the orbit size.

$$
% caption: The Newtonian ellipse (dashed) returns its perihelion to the same
% direction each orbit; the Schwarzschild orbit (solid) carries the perihelion
% forward by a fixed angle every revolution.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill (0,0) circle (2pt);
  % Newtonian closed ellipse, perihelion along +x, focus at origin
  \draw[black, dashed] (0,0) ++(1.4,0)
    \foreach \a in {0,10,...,360} { -- ($ (0,0) + ({ (1.9/(1+0.42*cos(\a)))*cos(\a) },{ (1.9/(1+0.42*cos(\a)))*sin(\a) }) $) };
  % Schwarzschild orbit advanced by ~30 deg (single loop, rotated conic)
  \draw[acc, very thick, domain=0:360, samples=160, variable=\a]
    plot ({ (1.9/(1+0.42*cos(\a-30)))*cos(\a) },
          { (1.9/(1+0.42*cos(\a-30)))*sin(\a) });
  \draw[black, dashed] (0,0) -- (1.35,0);
  \draw[acc, dashed] (0,0) -- (30:1.55);
  \node[black, below right] at (1.3,0) {Newtonian axis};
  \node[acc, above right] at (30:1.5) {advanced axis};
  \draw[->, black] (0.8,0) arc (0:30:0.8);
  \node[black, right] at (15:1.02) {advance};
\end{tikzpicture}
$$

## Mercury's residual

Mercury has semi-major axis $a = 5.79\times10^{10}\ \text{m}$ and eccentricity
$e = 0.206$, so $a(1-e^2) = 5.55\times10^{10}\ \text{m}$. The Sun contributes
$GM_\odot/c^2 = 1.477\times10^{3}\ \text{m}$, the gravitational radius of the Sun.
Then the advance per orbit is

$$
\Delta\phi = \frac{6\pi(1.477\times10^{3})}{5.55\times10^{10}}
           = 5.02\times10^{-7}\ \text{rad}.
$$

Mercury completes $415.2$ orbits per Julian century (orbital period $87.97$ days).
The precession per century is

$$
415.2 \times 5.02\times10^{-7}\ \text{rad}
= 2.08\times10^{-4}\ \text{rad}
= 43.0'',
$$

converting with $1\ \text{rad} = 206\,265''$. The measured value, once the two
larger classical effects are removed, is $(43.11 \pm 0.45)''$ per century.

> **Worked example.** The observed precession of Mercury's perihelion relative to
> the fixed stars is about $5600''$ per century. Two Newtonian effects dominate it:
> - **Precession of the equinoxes** — the slow rotation of the Earth-based
>   coordinate frame in which the observation is recorded, about $5025''$ per
>   century. This is a property of the measuring frame, not of Mercury.
> - **Perturbations by the other planets** — chiefly Venus, Jupiter, and Earth,
>   totaling about $532''$ per century by Newtonian celestial mechanics.
>
> Subtracting both leaves a residual near $43''$ per century that no Newtonian
> source accounts for. Le Verrier isolated this discrepancy in 1859 and attributed
> it to an unseen inner planet, "Vulcan," never found. General relativity supplies
> the missing $43''$ with no free parameters: it is $6\pi GM_\odot/(c^2 a(1-e^2))$
> summed over a century.

$$
% caption: The measured perihelion motion decomposes into a large frame-precession
% term, a planetary-perturbation term, and a small residual; the residual matches
% the general-relativistic prediction. Bar lengths use a compressed scale so the
% 43-arcsecond residual stays visible beside the 5000-arcsecond terms.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (0,4.6) node[above] {arcsec / century};
  \draw[black] (-0.1,0) -- (6.2,0);
  % compressed (sqrt-like) heights, values annotated exactly
  % equinox 5025 -> 4.0 ; planets 532 -> 2.3 ; residual 43 -> 0.85 ; GR 43 -> 0.85
  \draw[black] (0.4,0) rectangle (1.4,4.0);
  \node[black!70, above] at (0.9,4.0) {5025};
  \node[black!70, below] at (0.9,-0.05) {equinoxes};
  \draw[black] (1.9,0) rectangle (2.9,2.3);
  \node[black!70, above] at (2.4,2.3) {532};
  \node[black!70, below] at (2.4,-0.05) {planets};
  \fill[acc, draw=black] (3.4,0) rectangle (4.4,0.85);
  \node[acc, above] at (3.9,0.9) {43};
  \node[black!70, below] at (3.9,-0.05) {residual};
  \fill[acc!12, draw=acc] (4.9,0) rectangle (5.9,0.85);
  \node[acc, above] at (5.4,0.9) {43};
  \node[black!70, below] at (5.4,-0.05) {predicted};
\end{tikzpicture}
$$

$$
% caption: The residual perihelion advance and the general-relativistic
% prediction, both near 43 arcseconds per century, agree within the measurement
% uncertainty shown as an error bar.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (0,3.9) node[above] {arcsec / century};
  \draw[black] (-0.1,0) -- (5.0,0);
  % scale: 43 -> 3.0
  \draw[black] (0.8,0) rectangle (1.9,3.01);
  \node[black!70, below] at (1.35,-0.05) {observed};
  \node[above] at (1.35,3.2) {43};
  % error bar
  \draw[black] (1.35,2.98) -- (1.35,3.04);
  \draw[black] (1.28,2.98) -- (1.42,2.98);
  \draw[black] (1.28,3.04) -- (1.42,3.04);
  \fill[acc, draw=black] (2.9,0) rectangle (4.0,2.99);
  \node[black!70, below] at (3.45,-0.05) {GR prediction};
  \node[acc, above] at (3.45,3.2) {43};
\end{tikzpicture}
$$

## Frame dragging

The perihelion advance is a static effect: it uses only the mass $M$ in the
Schwarzschild metric and needs no rotation of the source. A rotating mass produces
a second, smaller relativistic effect, the dragging of inertial frames
(Lense–Thirring), which the [Kerr
geometry](/relativity/black-holes/rotating-and-charged-black-holes) describes
exactly. A gyroscope orbiting a spinning body precesses even with no torque on it,
because the local inertial frames themselves are dragged around in the direction
of the spin.

Two experiments have measured this in the weak field of the Earth.

- **Gravity Probe B** carried four gyroscopes in a $642\ \text{km}$ polar orbit and
  measured two precessions: the geodetic (de Sitter) precession from the Earth's
  mass, $6.6''$ per year, and the much smaller frame-dragging precession from the
  Earth's rotation, $0.039''$ per year. Both matched general relativity to their
  measured precision.
- **LAGEOS and LAGEOS 2**, dense laser-ranged satellites, register the
  Lense–Thirring dragging of their orbital planes at the level of tens of
  milliarcseconds per year, again consistent with the prediction.

The perihelion precession, the geodetic precession, and frame dragging probe
successively finer features of the metric: the static mass term, the same term's
effect on a transported spin vector, and the off-diagonal $g_{t\phi}$ term that
only a rotating source produces.[^hartle][^carroll]

[^hartle]: Hartle, _Gravity: An Introduction to Einstein's General Relativity_,
§10.2 (perihelion precession) and §14.5 (gyroscopes in curved spacetime, frame
dragging).
[^carroll]: Carroll, _Lecture Notes on General Relativity_, §7 (orbits in the
Schwarzschild geometry and the precession of the perihelion),
arXiv:gr-qc/9712019.