Mercury’s perihelion precession refers to the gradual and predictable shift of the point at which Mercury is closest to the Sun in its elliptical orbit. This point, called the perihelion, slowly moves or precesses around the Sun over time.
The perihelion precession is measured by the angular shift of the major axis of Mercury’s elliptical orbit from one perihelion passage to the next.
According to Newtonian mechanics, the predicted precession is per century. However, observations reveal an actual precession of per century, resulting in a discrepancy of per century. This discrepancy puzzled astronomers and led them to hypothesize the existence of other planets or celestial bodies near Mercury’s orbit that might be affecting its motion. Despite extensive searches, no such entity was ever found.
On November 18, 1915, shortly before finalizing the field equations of general relativity, Einstein published a derivation of Mercury’s orbital precession based on the vacuum field equations, which were carried over unchanged into the final theory. In his paper titled “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie” (“Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity“), Einstein presented the following formula for calculating the perihelion precession of a planet:
where denotes the amount of precession in a complete orbit, measured from one perihelion passage to the next. The constants and represent specific parameters:
Einstein’s formula considers only the Sun’s effects on the planet’s perihelion precession, disregarding any Newtonian gravitational influence from other entities.
For Mercury, and . With these values, yields . Since the Mercury makes orbital rounds per century (see ), its perihelion precession per century in degree is .
Finally, so Mercury’s perihelion precession per century
Adding to the Newtonian precession, we obtain the observed precession; that is,
Now the theoretical prediction matches the empirical data.
Excerpts from “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie“
Einstein’s original derivation in the context of orbital mechanics and general relativity, though groundbreaking, is notably succinct and omits many intermediate steps critical for a comprehensive understanding. His approach presumes a high level of mathematical sophistication and familiarity with the underlying physical principles, making the derivation challenging to follow for those not intimately acquainted with the subject. Therefore, additional details and explicit calculations are needed to bridge the gaps in his reasoning. This involves elaborating on the approximations and assumptions used, clarifying the nature of the governing differential equations, and providing a more thorough explanation of the physical interpretations behind the mathematical expressions. By filling in these missing steps, we can make the derivation accessible and comprehensible to a broader audience.
A: Aphelion, P: Perihelion,
Let’s start with what Einstein discovered after evaluating the Christoffel symbols to determine the geodesic equations of motion. Specifically,
where is the inverse of the planet’s radial distance from the Sun, the planet’s angular coordinate in the orbital plan and
with being the Sun’s mass in geometrical units. The symbols and are constants.
By applying the chain rule, we express (1) as
.
Considering achieves its extreme values and , it is evident that
when or .
This implies
when or .
i.e.,
Both and are roots of the cubic equation
Let and denote the first two roots, and the third root.
From (1), we deduce that
.
Integrating this from to gives:
.
Since it is generally not possible to express an elliptic integral in terms of elementary functions, we proceed as follows:
Rewrite the integral as
By Vieta’s Theorem,
As a result,
Given
we have
.
Therefore, (4) can be approximated by
.
From (see A Gem from Issac Newton), we have:
and, if
then
.
It follows that
is negligible
.
This represents the angular shift from aphelion to the subsequent perihelion, and similarly from perihelion to the following aphelion. Therefore, the total angular shift of the elliptical orbit’s major axis in one complete cycle from one perihelion to the next is
(see Natural System of Units in General Relativity)
By Kepler’s third law, (see From Dancing Planet to Kepler’s Laws)
.
See also Surpassing Einstein in Determining Mercury’s Perihelion Precession
Since
by A Worthy Indefinite Integral,
From An Ellipse in Its Polar Form,
where
We also have
and
Submitting (A-2-4) into (A-2-3),
And so,
Similarly,
Moreover,
.
Hence,
.
That is,
Let denote the number of orbital round Mercury makes per century.
Given that Mercury completes orbital round every Earth days and there are days in a year, we have:
.
Thus,
Exercise-1 Show that is applicable to Mercury.
Exercise-2 Show that
.
Exercise-3 Show that
is integratable from to .
Exercise-4 Identify an error in Einstein’s 1915 paper and explain why it is most likely a typographical mistake.
Exercise-5 Show that an alternative formula for calculating is where .
