Relativity on the rocks?: Mercury perihelion precession anomaly explained

Posted: August 6, 2011 by tallbloke in Astronomy, Astrophysics, solar system dynamics

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. eusa_naughty


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.…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:…n/node115.html

  1. adolfogiurfa says:

    Troublesome “pebbles universe”. Mercury has a big EM field:

    [Moderation note] Sorry Adolfo, I’m keeping this thread focussed on the gravitation of the pebbles.

  2. Why is Mercury’s orbit eccentric? Why does it have a magnetic field, which stretches all bounds of dynamo theory? Why are the charged particles within the ionosphere of Mercury so much denser than expected? Why is the efield on Mercury so much stronger than that of earth? Are there auroras on Mercury? Why are there counless small craters on the rims of large craters? Why are so many of the craters hexagonal? Why are the craters often terraced on the sides?

    If you ask me, everything is on the rocks when it comes to that planet. ha ha

  3. tallbloke says:

    All good questions, but the only one I’ll address on this thread is the eccentricity of the orbit.

    I would think that Mercury’s orbit is eccentric because it passes between the Sun and Jupiter four times a year. And because its plane of rotation is offset to the plane of rotation of an oblate body (the Sun) which introduces a quadrupole moment. The figures for the effect of these factors on the precession of Mercury’s orbit are given in the table in the main post.

    If you want a thread on planetary electrodynamics, feel free to ask for one. Or better still, take me up on the offer I just made on the Juno thread. 🙂

  4. tallbloke says:

    Post updated with my plain English description of what I think is going on here.

  5. tallbloke says:

    Useful paper translating part of le Verrier’s 1859 paper:

    Click to access rydin2.pdf

  6. tallbloke says:

    Mercury’s rising
    Too soon, it’s surprising
    Needs must fulfill
    This moment of ill

    Set Mercury straight
    The moment won’t wait
    Put caulk in the crack
    That stems from the lack
    In le Verrierfication,
    Of Newton’s summation

    The bending of light
    Will make day out of night
    And warp the wide space
    That was so full of grace

    It’s locked us all in
    I can’t hear for the din
    Or find solid ground
    On this merry-go-round

    So widen the horizon
    We built truth and lies on
    For then we’ll all see
    The cosmos is free
    To be what it is
    Let the stars pop and fizz

  7. “So widen the horizon
    We built truth and lies on
    For then we’ll all see
    The cosmos is free
    To be what it is…” ~tb

    The way it appears in retrospect is that there was a generous application of the “principle of exclusion” to Mercury’s eccentricities, where astronomers of the time confidently convinced every one that they had explored every other variable that could possibly have caused this perturbation. With Einstein’s mathematical work and theory of gravity, a neat solution presented itself and the problem was “solved.” No more questions. I appreciate this work with applying a different numerical value for Jupiter’s effect, and wish it good success in the math.

    Other possibilities for the chalkboard could be planetary migration. That was unheard of at the time, but is now a regular occurrence for extrasolar planets (and perhaps our own solar system), in case you haven’t heard!

    Another potential interesting question would be whether Mercury’s magnetotail as it brushes Venus’ plasmasphere could further alter the planet’s orbit. Nice graphic a little way down.

  8. tallbloke says:

    It’s looking like Urbain le Verrier missed the dynamically rotating frame of the solar acceleration with respect to the centre of mass of the solar system. When this is taken into account, perfectly normal Newtonian mechanics can take care of the missing ~43 arc seconds per century in the precession of the perihelion of Mercury. At the same time, the true forces on the Sun are revealed, which leads to a natural Newtonian explanation for all the ‘coincidences’ between the planetary orbital elements and the Sun’s motion, activity, surface flows and rotation rates.

    Occam may well be on Newton’s side in this.

    Although I’ll cheerfully admit to not being a whizz at maths, it looks like ‘Alsor’ has provided an elegant solution to anomalous Lunar motion as well as resolving Mercury’s anomalous precession. I’m going to engage the help of an expert to see if it pans out.

    Just as this doesn’t detract from heroic efforts made by mathematicians such as Clairaut and d’Alambert in resolving the anomalous Lunar precession with heuristic equations, the application of curved space geometries and gravity bent light to achieve useful solutions for near Earth orbits is a great achievement in it’s own right.

    But if it turns out that Newton has greater explanatory power in conjunction with Miller’s theory of a gravity entrained transmissive medium pervading space, then I’d find that a more elegant and complete solution. Maybe that’s because I’m just a humble engineer who finds it easier to understand a universe which consists of real interacting particles and forces rather than Einsteins spooky action at a distance geodesics, dark matter, invisible string and all the rest of the caulk and spackle modern physics glues the universe together with. Mostly though, it satisfies the artist and musician in me, with the natural harmonies, harmonics, proportions and inclusive synchrony of a connected solar system and wider cosmos. I can feel the rhythm.

    Newton thought like that, just look at his work on optics, acoustics and kinematics.

    Non-dilated time will tell.

  9. tallbloke says:

    Zeke, interesting stuff. My modus operandi is to get the simple pebbles stuff calculated through properly, then see how much room there is for the more complex and subtle electrical forces.

    That way, we get a view on scale of effects.At the most basic, the gravitational equation and the electrical equation (coulomb force) take the same form. So since no-one has explained what gravity consists of, and the search for the graviton has yielded no result, the field is open.

  10. tchannon says:

    What is “dynamically rotating frame of the solar acceleration”?

  11. tallbloke says:

    Some newfangled idea Galileo came up with in about 1611

  12. Tenuc says:

    Oh dear, Alsor has been banned on the Bautforum and Stroller, who was trying to give some support, has been given a slapped wrist. Inevitable, I suppose, from previous experience of the place!

    It would be good if Alsor could be contacted to explain his maths in more detail – as presented I find it far from clear. I think he could make a very useful contribution to some of the ideas being kicked around here.

  13. tallbloke says:


    Next, (and probably last) comment going up in a few minutes.

  14. tallbloke says:

    Whoah, don’t refresh page 2 and take some screen shots. They just deleted my stuff!

  15. tallbloke says:

    Ahhh, they’ve decided to split us up. My stuff has reappeared here:

    Which means I’m now being thrown to the Rottweilers by myself. Wish me luck and drop by with some support if you get the chance.


  16. One more: “If the Sun were not a perfect sphere, but had an equatorial bulge (i.e., was oblate), that would cause Mercury’s orbit to precess. Well, surprise: the Sun isn’t a perfect sphere; it is oblate! So how good is the agreement between Mercury’s orbital precession and GR?

    Clifford Will devotes a chapter to this in his 1986 book “Was Einstein Right? Putting General Relativity to the Test”. Dicke and Goldenberg claimed to have detected a much larger bulge than solar models predicted, large enough to destroy the agreement between GR and Mercury’s orbit, but not large enough to permit a newtonian explanation. Specifically, the data looks like this:

    Mercury’s perihelion precession: 574 arcseconds/century
    Newtonian perturbations from
    other planets: 531 arcseconds/century
    GR correction: 43 arcseconds/century
    Newtonian correction from Dicke bulge: 3 arcseconds/century”

  17. tallbloke says:

    Zeke, Thankyou!

    This meshes with my realisation about differential inertial forces in the sun due to the changing planetary positions. See the latest on the Newton thread. The excess bulge is due to the less dense regions of the sun being more affected by changes int he solar trajectory than the denser core.

    I Think it will also explain the faster rotation of the solar equator which is inversely proportional to Jupiters orbital period. I’ll chew on this and write what I believe will be a groundbreaking post on my return after the weekend.

    [Edit] No. I have this wrong. Need to think while I’m away.

  18. Tenuc says:

    Still trying to understand Alsor’s maths, but finding it a bit of a struggle. It is interesting to compare Alsor’s approach to that of Miles Mathis, who uses well explained vector analysis as well as SR/GR to find a solution to the problem using easy to understand maths…

    Have you been able to contact Alsor yet, Rog?

  19. AusieDan says:

    Having attempted to study special and general relativity some years ago,
    I have an intuitive feeling that Newtonian physics is all there is, at least at macro scale.
    Keep it up.

    You guys may be on to something, but you’ll not be popular.
    Deniers and flat earthers will be just the start of your oponents “analysis”.

  20. The perihelion precession of Mercury has already been fully explained. Details on:
    To get full page (partly blocked by ad) click the red square with white arrow in it in the right, top corner of the page.

  21. tallbloke says:

    Andrzej Lechowski: Thank you, the page you link looks very interesting. Please tell us more about the implications for the standard model if you can in everyday language.


  22. as says:

    What is this number?
    The planetary perturbations goes over 550”.

    530 never existed.
    Relativity is not correction, but the apparent forgery: GR = 43”=> Newton = 530”.

  23. Hello,
    I have developed a new theory (model) of gravitation. It is independent of general relativity.
    Would you please like to comment on it? The (detaile) abstract is given below.Thank you.
    Raghu Singh

    A Constructive Model of Gravitation
    Raghubansh P. Singh

    Abstract for a conference

    This paper proposes a physical model in which gravitational interaction between masses is mediated by their mass-momentum fields. A mass in the mass-momentum field of another mass experiences two types of gravitational forces: repulsion due to separation; and attraction due to motions but under specific conditions.

    The model addresses: gravitational interaction between matter and matter and between matter and energy; gravity’s effect on spectral lines, clock time periods, and the length of an object; and gravitational radiation from an accelerating mass. It estimates the speed of gravitational wave; revisits pseudogravity; and makes new predictions.

    The model compares the predictions of the model with those of general relativity. It rediscovers that gravity bends a light ray, increases the wavelength of emitted light, dilates the time period of an atomic clock, and elongates a material rod. An experiment similar to the Pound-Rebka’s is suggested to test the lengthening of rods. The model rediscovers that at a black hole: time run virtually stops; light virtually turns flat in waveform but still propagates; light passing nearby might go around it and return toward its source; and rod flattens to the point where it disintegrates. It finds that time is physically meaningless in the absence of mass. The model, despite lacking observed values of its three ‘constants,’ agrees on the older predictions well within an order of magnitude.

    The model makes new predictions: gravitational interaction is also repulsive; the classical gravitational ‘constant’ is not constant; an accelerating mass emits gravitational radiation (but not when the acceleration and momentum vectors are orthogonal); gravitational waves in turn accelerate masses at 45 degrees to the direction of propagation; a gravitational wave has four degrees of polarization; and the speed of gravitational wave is about 57 percent of the speed of electromagnetic wave. It finds that orbits cannot close if the gravitational ‘constant’ is not constant.

    The model introduces for gravity mass-momentum field, which, being of the type similar to electric-magnetic field, could help in the unification of gravity with other fundamental interactions.

    For the life cycle of the universe, the model seems to favor one or more cycles of expansion-steady-contraction states over an expansion-for-ever state.

    February 16, 2012

  24. als says:

    What your model says about galaxy rotations, and other similar problems?

    And if speed of gravity is finite, then where are consequences, aberrations, ect. – what are predictions?

  25. tallbloke says:

    Hi Alsor and welcome. I was hoping you’d be able to tell us the answer, rather than ask more difficult questions… 😉

    Did you mean to post this on the ‘speed of gravity’ thread?

  26. From: Raghunansh Singh —
    It has been a long time since I posted the abstract. Here is what I know about GR and the three famous tests.
    1. Mercury’s orbital precession. The precession rate is 5600 arc-secs/century. After taking out observer-planet relative motion effects, a 575 as/c is left over. Classical mechanics accounts for 572 as/c. GR accounts for 572 and another left over 43 as/c. Other effects are infinitesimal and not detectable for now. There are a few recent theories which account for the residual 43 ac/s. Biswas shows that a Lorentz covariant modification of the Newtonian potential yields the 43 as/c. Barwacz shows that the second-order special-relativistic term in the energy term gives the 43 as/c. There are other — radical though — give the 43 as/c. My said model is the first in a three-part series and begins as a pure classical model, as any brand model must in order to retain physical insight.
    2. Gravitational deflection of light at the sun. GR calculates it to be 1.75 arc-secs. The 1919 expedition reported confirmation. But later expeditions found it to be from 1.5 to 3 as. I personally know for sure that there is no way that all non-gravitational effects of the solar atmosphere can be ascertained (period). Adler, Bazin, and Schiffer have determined that at least 1/5th of an observed deflection is due to non-gravitational effects. The stellar atmospheres are too violent to photons for us humans to make precise observations. My said model uses the most reliable value of 2.2 as in the classical equation to calculate other gravitational effects; the results are astonishing!
    3.Gravitational dilatation of the time period of an atomic clock , of the wavelength of light, and the length of of a material rod. GR does very good predictions here. But my said model, using classical potentials, predicts these within an order of magnitude. (Remember I did not use special-relativity.)
    4. Gravitational radiation. GR says it is quadrupole moment radiation with two degrees of polarization traveling at the speed of light. (The speed seems to be odd. Please recall that gravitational forces are 10**40 times weaker that electromagnetic forces.) My said model predicts that gravitational radiation is simply momentum pulses travelling at far less speed that the speed of light with four degrees of polarization. Physics must wait for the experimental evidences on the structure, speed, polarization, and quantization of gravitational radiation to ascertain which model/theory Nature is hiding. As you know, we have not even detected gravitational radiation.

    Finally, I want some of you to review my paper and give unbiased comments. I have given up on gravity experts. They are stuch with GR, hidden dimensions, supersymmetry, and current quantum gravity.

    I have lost my UN and PW for Tallbloke. So, please send mt e-mails.

    All the Best,
    Raghu Singh

  27. tallbloke says:

    Hi Raghunbansh and welcome. I’ve emailed you.

  28. Dick Johnson says:

    This is priceless. Apart from the precession of Mercury, Special and General Relativity are among the most precisely confirmed theories in physics – by experiment, observation, and everyday use. You might have heard of the Large Hadron Collider!

    You also may have heard of the Global Positioning System. We need to make relativistic corrections every single time anyone uses GPS – to take into account the relative velocities of the satellites, and the effect of the Earth’s gravitational field on ground and orbital clocks. We *know* relativity is correct. Anyone who says something like “I’ve got an intuitive feeling that Newtonian physics is all there is” simply doesn’t have the faintest idea what they’re talking about.

    In other words, you people are cranks.

  29. tallbloke says:

    Hi Dick,
    Yes, I’ve heard of CERN’s large hadron collider. In fact I machined several of it’s principal components when I worked as an engineer for a large Swiss multinational called Sulzer in the 80s.

    Tell me, are you aware that the thermosphere shrank by about 30% during the long solar minimum of 2007-9 and that this had a marked effect on the orbital rate of satellites due to the reduction in drag?

    If you were, I doubt you be so stupid as to tell me that solar variation didn’t have a more marked effect on satellites than the relativistic effect of gravity. The phrase ‘lost in the noise’ springs to mind.

    Forgive me for feeling a bit cranky, I haven’t had my morning coffee yet, but do me a favour and take your ignorance elsewhere. Thanks.

  30. Raghubansh P. Singh says:

    To Mr. Johnson-
    Yes, special relativity is well confirmed experimentally and in observations. No other theory seemsto duplicate SR. But – I have read other theories which arrive at the length contraction, ime dilation, and E=mc**2 equations.
    I cannot say the same about general relativity. There are gravity models which precisely reproduce GR’s predictions. People cite PSR B1913+16’s orbital shrinking as evidence of gravitational waves predicted by GR. This tells us that gravitational waves exist, but does not say anything about its physics (emission. propagation, structure, speed, and polarization). The first step to decipher gravity is to detect gravitational radiation and measure its speed. Before that, no conclusions must be made regadring GR or any other gravity theories.
    Regarding GPS, I have conflicting news for you. Some say GR is not used in the GPS calculations.
    The research communitis have changed. Many scientists have worked on GR, strings, supersymmetry, hiden diensions, and quantum gravity. They are not going to let them go; they will oppose other parallel efforts.