Relativity on the rocks?: Mercury perihelion precession anomaly explained

From Wikipedia

A number of effects in our solar system cause the perihelions of planets to precess (rotate) around the sun. The principle cause is the presence of other planets which perturb each other’s orbit. Another (much more minor) effect is solar oblateness.

Mercury deviates from the precession predicted from these Newtonian effects. This anomalous rate of precession of the perihelion of Mercury’s orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier. His re-analysis of available timed observations of transits of Mercury over the Sun’s disk from 1697 to 1848 showed that the actual rate of the precession disagreed from that predicted from Newton’s theory by 38″ (arc seconds) per tropical century (later re-estimated at 43″).^[2] A number of ad hoc and ultimately unsuccessful solutions were proposed, but they tended to introduce more problems. In general relativity, this remaining precession, or change of orientation of the orbital ellipse within its orbital plane, is explained by gravitation being mediated by the curvature of spacetime. Einstein showed that general relativity^[1]agrees closely with the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general relativity.

——————————

Did le Verrier correctly calculate the amount of precession of Mercury’s perihelion due to the perturbation of its orbit by the other planets?  As you can see from the table below, Einstein’s relativity theory plugs the gap between the observed precession and that calculated from Newtonian theory.

Sources of the precession of perihelion for Mercury

Amount (arcsec/Julian century)	Cause
5028.83 ±0.04[4]	Coordinate (due to the precession of the equinoxes)
530[5]	Gravitational tugs of the other planets
0.0254	Oblateness of the Sun (quadrupole moment)
42.98 ±0.04[6][7]	General relativity
5603.24	Total
5599.7	Observed
−3.54 (−0.0632%)	Discrepancy

But what if the perturbation due to other planets wasn’t correctly calculated? Perturbation theory is a headache area for astronomers and astrophysicists, because it requires the solution of the ‘many body problem’, which is intractable to pure maths. There are lots of cunning techniques employed to work around the fundamental difficulty, but have they found all the terms which need to be accounted for in the Newtonian frame?

A recent post on the Baut Forum caught my eye. I don’t claim to fully understand all the maths here, so I’m throwing it out to the bright sparks in the talkshop readership for comment. On the face of it, it looks like le Verrier missed a significant part of the perturbation from Jupiter of around 40 arc seconds per century. This just happens to be close to the discrepancy which relativity was adopted on the strength of.

The regular rottweilers at Baut Forum are being very cagey about this one, though it looks like Alsor may have already worked out how they operate and departed. I’ve PM’d Alsor, so I’ll update this post with any reply I get. No doubt this post will enhance my reputation as a ‘relativity denier’ (John Mashey, Tim Lambert et al). My response to them is, show us where Alsor’s maths is wrong if you disagree. I’m a historian of science and I record unexplained anomalies and paradoxes without passing judgement on whether this or that theory is supported or refuted by the solutions offered.

[UPDATE – 8th Aug] 

If I understand what Alsor is saying correctly (no guarantee!), he states that le Verrier’s calculation for the gravitational effect of the other planets on Mercury’s orbit was a static calculation which doesn’t account for sun – solar system barycentre relative motion, and that this is an important omission.

My understanding of the sun – solar system barycentre relative distance is that it varies by a solar diameter or so as Jupiter and Saturn move from conjunction to opposition (the precise amount is also affected by the other planets, predominantly the other two gas giants). Mercury orbits the Sun-Mercury barycentre more closely than the solar system barycentre, due to the Sun’s proximate and high gravity. This means the Mercury – Jupiter distance will vary by a solar diameter or so as well, in addition to the normal variation due to their orbits, because Jupiter orbits the solar system barycentre more closely than the Sun – Jupiter barycentre (due to its possession of ~3/4 of the solar system mass outside the Sun, the Sun retreats from the barycentre to counteract Jupiter’s mass – retreating further when Saturn is in conjunction, when (additionally) Jupiter is pulled towards Saturn).

So, there is an additional perturbation between Jupiter and Mercury which le Verrier didn’t account for. I’m not sufficiently clued up on perturbation theory to understand Alsor’s math fully, so I’m not sure whether my description matches his calculation, or whether the additional perturbation I’m outlining would be sufficient to account for the 40″/century Alsor gets as a result.

—————————————–

Alsor says:

Revolving orbits – Mercury precession, Moon, binary stars, etc.

The correct solution of the celebrated problem of obsidial precession.

It is simplified three body problem, or two body problem in non-inertial reference frame – varying in time (rotating) force field.

Very simplified version. I told that it’s simple, and it is really simple effect, but full rigorous derivation/calculations are quite long, too long for discussion on forum.

Three bodies: 1-2—–3
Earth-Moon and Sun or
Sun – Mercury and Jupiter or
S1-S2 and X3 – binary star and satelite (probably DI Herculis, ect.)

Additional radial force (averaged along orbit – path integral in force field; the field is conservative, but varying in time, simple difference of potentials is incorrect method!):
F_e(r) = Fc * r/d sin(th); (forces per unit mass == accelerations)

d – distance from local barycenter (1-2) to third body (Jacobi coordinates).
Fc = W^2 * rc – centrifugal force at local barycenter (Earth-Moon);
it’s of course, equilibrated by gravity of third body:
Fc = W^2 * rc = G(M+m3)/d^3 * d m3/(M+m3) = Gm3/d^2 = F3; M = m1 + m2;

F_e(r) = F3 r/d sin(th);

th = 2pi * W/w; angle of rotation around barycenter (of three bodies)
per one orbit (Moon about 30 degrees: T = ~1 month = 1/12 year);
W – angular velocity on second orbit (Sun – Earth+Moon).

Newton’s theorem utilisation.
Gravitational force is of course:
F(r) = GM/r^2;

Comparision of both forces:
k = F3 a/d sin(\th) / GM/a^2 = m3/M (a/d)^3 sin(th);

If force is proportional to r^1, then factor is 3/2 (quadrupole: 1/r^4 – factor is 1):
dw = 3/2 k w = 2pi * 3/2*k / T = d\phi / T;

Finally – advance in radians per orbit:
I. d\phi = 2pi 3/2 k = 3pi m3/M (a/d)^3 sin(th);

It is only additional precession… Le Verrier did not know that.

Moon’s absidal precession.
A. 1.5 degrees – Sun’s tidal forces:
Ft(r) = Gm3/d^2 * r/d (3cos(f)^2 – 1);
average: Gm3/d^2 * a/d * 1/2;
and this force is responsible for only half of observed precession (Newton has calculated).

B. Motion of whole orbit in time-varying potential:
F_e(r) = Gm3/d^2 * a/d * sin(phi);
and because phi ~ 30 degrees then it’s second half.
(hiere: rc/d = m3/(m1+m2+m3)~= 1, because m3 = mass of Sun).

Mercury precession.
Jupiter mas: m3/M = 1/1047; distance: d = 5.2 au;
Mercury m2 = 1/6mln ~= 0; a = 0.387 au;
displacement angle: th = 360/48 = 7.5 deg;

Directly from formula I, but in degrees:
dphi = 360*1.5 m3/M (a/d)^3 sin(th) =
360*1.5 /1047 (0.387/5.2)^3 * sin(7.5) = 360 7.71e-8 = 0.1” / orbit;
400 synodic orbits * 0.1” = 40” / 100 years;
Good enough. Saturn has something to say in this matter.

Comparision – tidal forces:
F_e = Gm3/d^2 a/d * sin(7.5);
1/2 : sin 7.5 = 3.8;

For different motion – hiperbolic or linear, body accumulates different gradient.
Only specjal circular patchs are free of forces (something like geodesic lines).
Especially Jupiter: sin(0) = 0.

Numerical methods – why in simulations this precession is invisible?
Because standard algorithms are inadequate for so subtle effects. 

———————————

Lunar’s solution works for Mercury too.
n’ = 1/11.86y; Jupiter
n = 1/0.24y; Mercury

m = n’/n = 1/49.4;

360 * m (3/4 m + 225/32 m^2 + … ) / orbit =~ 196521” / cy;

Too much?
Of course, because here is smaller the acceleration of baycenter!
Mass of our system (Sun + Mercury) is 1047 times greater than the third mass.

196521” / 1047 = 188” / cy

Part from tidal forces: 360 * m (3/4 m) / 1047 = 152”
Rest is consequence of rotation: 188-152 = 36” / cy

…

Le Verrier summed up only static forces, omitting the impact of planets on the Sun – rotation.
Similarly, you can calculate the precession of the Moon and you will receive only 1.5 degrees / orbit, that is what is now Newton calculated.

http://math.berkeley.edu/~mpejic/pdf…Precession.pdf

A. Moon-Earth
360 3/4 m/M * (a/d)^3 / sqrt{1-e^2} = 1.5 degrees / orbit
m / M = 330000, m – mass of the Sun, M – Earth + Moon; a/d = 384 Mm/ 150 Gm = 1/390

B. Mercury-Sun
360 * … =~ 160”/ cy
m / M = 1 / 1047, m – mass of Jupiter, M – Sun + Mercury; a/d = 0.387au / 5.2au = 1/13.4

Other similar improvisations:
http://www.mathpages.com/home/kmath280/kmath280.htm
http://farside.ph.utexas.edu/teachin…n/node115.html