Original papers on mercury's perihelion drift

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Dear All,

I would be interrested by reading some original paper on the classical evaluation of the advance of the perihelion.

I found a few simplified models (see http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=AJPIAS&ONLINE=YES&smode=strresults&sort=chron&maxdisp=25&origquery=%28perihelion%29+&disporigquery=%28perihelion%29+&threshold=0&pjournals=&pyears=&possible1=perihelion&possible1zone=article&possible3=mercury&possible3zone=multi&bool3=and&OUTLOG=NO&viewabs=AJPIAS&key=DISPLAY&docID=1&page=1&chapter=0&aqs=" [Broken] e.g.). However, the relativistic correction represents about 10% of the total drift, and therefore these simplified models (5% precision) are not relevant in the context of the history of GR. It is necessary to:
  • either go back to the original calculations
  • or find modern references on the exact calculations

Of course, I would like to find this material for free on the web.
Did some of you find such a gold mine?


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Answers and Replies

Does earth's orbit suffer precession too?
I assume earth obeys general relativity.
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Does earth's orbit suffer precession too?
I assume earth obeys general relativity.
Of course it does. For both sentences.
But how much is it?
And has anyone measured it yet?
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The observation of the relativistic perihelion precession is a residual left after the total observed precession is corrected for the many known Newtonian perturbations that contribute and which are much larger, such as that caused by the other planets, plus any effect due to the Sun's oblateness. Measurements of the former are well determined and the latter is thought to be negligible.

But see Dicke and Goldberg's paper Phys. Rev. Letters 18, 313 (1967) in which they claim a polar diameter (5.0 +/- 0.7) parts per 105 shorter than the equatorial diameter. This would produce an extra precession in Mercury's perihelion of 3.4" per century 8% of the observed total non-Newtonian precession. This result has not been generally accepted.

Weinberg Gravitation and Cosmology pg 198 gives the following results:

GR prediction _ Observed Residual ("arc/century)

Mercury 43.03 _ 43.11± 0.45
Venus 8.6 _ 8.4 ± 4.8
Earth 3.8 _ 5.0 ± 1.2
Icarus 10.3 _ 9.8 ± 0.8

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The earth one does not seem too accurate. Why? Earth violates Relativity?
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Dear All,

Some details about my interrest for original papers:
  • I am mostly interrested in the classical part of the effect (90%)
  • I would like to check the source of errors
  • I would like to know how the calculations were done with the highest precision
  • I would like to redo that by myself and evaluate the impact of each detail on the result
  • these details maybe: eccentricities of the perturbing planets, planes of their orbits, ...
  • in additions, all models I have seen replace the perturbing planets by circles or ellipses, I would like to see how it can be proven formaly that secular averages lead indeed to this model, it is intuitively obvious but I am sure this can be demonstrated with some additional benefit
  • if that was possible, I would like to read the calculations by Le Verrier


Chris Hillman
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Computing the Newtonian part of the precession of perihelia of Mercury?

Hi all,

I don't think Michel stated his goal very clearly; as I understand it, he recognizes that he already knows (from modern gtr textbooks) how to compute the extra-Newtonian precession of the perihelia of Mercury, according to weak-field gtr, and appreciates that to model as well the perturbing effect of the presence of Jupiter (which provides a much larger contribution to the precession of the perihelia of Mercury than the tiny contribution from the de Sitter precession effect), he needs to compute this using the classical theory of perturbations in Newtonian gravitation, as independently developed by Le Verrier and Adams (following previous work by Lagrange), and then simply add the Newtonian and extra-Newtonian terms. (Modulo a tricky point discussed by Weinberg; note that in modern gtr textbooks one finds an approximate solution of the equations for timelike geodesic in the exact Schwarzschild vacuum solution, which evades the tricky point; Einstein's original computation used the weak-field solution and was not quite justified as he stated it in 1915.)

Furthermore, as I understand it, Michel is asking for historical references (to follow the development of the classical theory of perturbations in modeling the motion of the planets according to Newtonian gravitation) and also textbook references (for learning to verify the computations in detail by himself), especially references freely available on the InterNet.


I would be interrested by reading some original paper on the classical evaluation of the advance of the perihelion.


I would like to read the calculations by Le Verrier
I think that most likely you will need to do some library research. In a previous post in a recent thread, I already gave you some printed references (one historical reference plus books containing many further references). In particular, I gave a citation to a book by Le Verrier (in French) which you can probably find in a good research library.

It sounds like your real interest is however in learning how to perform a modern version of the Newtonian computations carried out by Le Verrier and Adams. Here too I strongly recommend that you find some good books; but I do mention web resources you can study below; these might provide helpful supplements as you study some good textbooks such as the three volume work by Hagihara. Then you can try to reproduce the Digital Orrery:

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C-34, No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 -- Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

I found a few simplified models (see http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=AJPIAS&ONLINE=YES&smode=strresults&sort=chron&maxdisp=25&origquery=%28perihelion%29+&disporigquery=%28perihelion%29+&threshold=0&pjournals=&pyears=&possible1=perihelion&possible1zone=article&possible3=mercury&possible3zone=multi&bool3=and&OUTLOG=NO&viewabs=AJPIAS&key=DISPLAY&docID=1&page=1&chapter=0&aqs=" [Broken]
I'm impressed that you even know what the Runge-Lenz vector is!

By the way, Michel (and pervect, if you are reading this post!), this notion is best understood in terms of Lie's theory of symmetry of systems of differential equations; the Runge-Lenz vector naturally arises as a dynamical symmetry in the Kepler system; see Stephani, Differential Equations: Their Solution Using Symmetries for a clear discussion. The point is this: as Stephani shows, in the two body problem (in Newtonian gravitation!) the Runge-Lenz vector is invariant. But in the three body problem (Sun, Mercury, Jupiter) it is not, and this models the precession effect you seek.

It is necessary to:
  • either go back to the original calculations
  • or find modern references on the exact calculations
It is indeed ironic that these days it is probably more challenging to learn the relevant theory of perturbations for Newtonian astrodynamics than the classic computation of the de Sitter effect in the context of linearized (or weak-field) gtr!

Consider a less ambitious program than trying to reproduce something like the Digital Orrery (see http://math.ucr.edu/home/baez/week107.html for some citations, and see also www.gifford.co.uk/~principia/orrery.htm) [Broken] for an alleged implementation which I haven't checked out very thoroughly, but which you can play with on-line):

Following up on a problem in the Problem Book in Relativity and Gravitation by Lightman et al., I computed the effect of a solar bulge using weak-field gtr, and reported the following results:

(Oops, too long.. I'll quote this in a followup)

This might interest you because, as you will recall, some decades ago the question briefly arose whether a previously unrecognized solar bulge might account for some of the precession, in which case gtr would have turned out to agree with experiment "only by accident". Dicke used this to argue that the Brans-Dicke scalar-tensor theory should be used instead; see the discussion and citations in MTW.

That is, in the quoted post (see below), I describe the result of my computation of the precession, in the case of a test particle orbiting an isolated massive object possessing a nonzero quadrupole moment, and compared this with the de Sitter effect. The point was that the two effects scale very differently with radius, so that one cannot simultaneously fit models to the observed motion of Mercury, Mars, the Earth, asteroids, various pulsars, and so on, by assuming very solar bugles plus Brans-Dicke gravitation (which has an extra parameter which must be chosen once and for all).

Note that in the quoted post, I stress that the inquirer had spotted an instability in his own simulation, which was due to the fact that he was treating the test particle motion via an approximate equation which is only valid for a small number of orbits; the exact solution of the geodesic equation in the Schwarzschild spacetime would not exhibit rapid inspiralling for a quasi-Keplerian orbit. Regarding the possible effects of gravitational radiation, these turn out to be neglible for the case of Mercury in comparision to the de Sitter effect.
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Chris Hillman
Science Advisor
Computing the precession due to a nonzero mass quadrupole moment of the Sun

OK, here is the quotation which wouldn't fit in my first reply to Michel in this thread:

Subject: Re: Is Mercury's perihelion shrinking?
Date: 2 Nov 2003 21:41:17 -0500
Groups: sci.physics.research

On Sun, 26 Oct 2003, Bob Day wrote:

> Joseph Weber, in his book "General Relativity and Gravitational
> Waves", Interscience Publishers Ltd., 1961, gives an equation
> for Mercury's distance, r, from the Sun vs. orbital angle, Phi
> which takes general relativistic factors into account.
> The equation (5.37 on page 66) is:
>     r = K1/(1 + e*Sin[Phi] - K2*(e*(3*Phi -
>                                        e*Cos[Phi])*Cos[Phi] + 3 + e^2))
> Where:
>     K1 == a*(1 - e^2)
>     K2 == (G*M)/(K1*c^2)
>     a == Mercury's semimajor axis
>     e == Mercury's orbital eccentricity
>     G == the gravitational constant
>     M == the combined mass of Mercury and the Sun
>     c == the speed of light

And where, I presume, phi is some angle NOT depending upon time, so that
you have the equation for a plane curve in a radial coordinate chart.

(Point being that the motion of Mercury can be regarded as tracing out
this plane curve, but we do not try to capture here the idea that Mercury
is moving faster near the perihelia than near the aphelia, etc.).

> The advance of Mercury's perihelion can be derived from this equation.

I don't have Weber's book, but because I have carried out the textbook
computation in question (you can google up a prior post by myself in this
group, at most a year back, in which I did this in excruciating detail), I
recognize this result.

What you have here is a essentially an equation r = f(phi) for a plane
curve given in radial coordinates.  This curve arises as an approximate
solution (via "perturbation expansion methods") of "the Einstein-Binet
equation", an ordinary differential equation which arises in studying the
the motion of a test particle around a nonrotating massive object in
"linearized general relativity".

Arnold Neumaier and Phil Hobbs suggested that the phenomenon you observed
might arise from pushing your numerical integrator too hard.  This could
well be part of the problem, but a more fundamental explanation (also
pointed out by Phil Hobbs) is that the approximation solution r = f(phi)
via perturbation methods of the Einstein-Binet equation will only be valid
for a certain range -c < phi < c.  IIRC I have previously discussed here
how approximate solutions of trajectories obtained via perturbation
expansion tend to break down if we apply them over too big a range of
independent variable; it turns out that spurious inspiral or outspiral is
indeed to be expected for perturbative approximations arising from the EB
equation.  This point has arisen here before in the same context, so your
question probably belongs in the FAQ.

> But the equation also predicts that Mercury's perihelion is shrinking.

"The equation predicts" is a telling phrase-- the real question is: what
does the -theory- predict?  The point is that in order to use correctly
the formula given by Weber, you must first understand its derivation, in
particular, you need to understand all the approximations involved as well
as their limits of validity!

Here is a brief outline (see standard gtr textbooks and the above
mentioned post for detail) of these approximations:

1. "Linearized gtr" is the study of solutions (Lorentzian spacetimes) to
"the linearized EFE", a simplification of the Einstein field equation, the
fundamental equation of full gtr.  Roughly speaking, if the gravitational
field is everywhere "weak", you can replace the full EFE with its
linearized version.  For example, one (static) solution of the linearized
EFE is "the linearized Schwarzschild vacuum", which agrees with a
"perturbative expansion to first order in the mass parameter m" of the
usual Schwarzschild solution, aka "the weak-field Schwarzschild
spacetime".  In general, if the gravitational field is slowly changing (or
even better, static) and sufficiently weak over some region of spacetime,
the predictions of gtr can be treated as perturbations of the
corresponding Newtonian predictions.

2. Mercury can be regarded (both in linearized gtr and in Newtonian
theory) as much less massive than the Sun.  Also, we can assume that the
gravitational effects of the rotation of the Sun, solar oblateness, etc.,
can be neglected in this context.  (This assumption can be justified both
from theoretical computations and from observational results.)  Less
obvious is the reason why we can temporarily ignore the effects of the
major planets!  (But see comments below.)  In any event, the upshot that
in order to obtain the desired precession formula, we can treat the motion
of Mercury around the Sun as the motion of a -test particle- in the
linearized Schwarzschild vacuum.  This leads to the Einstein-Binet
equation (look up my above mentioned post to see what this equation looks

3. It is hard to find an exact solution of the EB equation, despite its
simplicity, so instead we look for approximate solutions using a standard
method in differential equations, called "perturbation expansion".  This
method involves defining a parameter which is regarded as "small", and
then one finds first, second and higher order expansions wrt this
parameter, and as you would expect, higher order approximations are more
accurate over a given range of variable, or have a given accuracy over a
wider range.  (A given problem may admit various different valid choices
of parameter, and the resulting approximate formulae will look different,
but not the corresponding computer pictures of the approximate
trajectories!-- at least not over a suitable range of independent
variable.)  In the case at hand, the first order expansion of the EB
equation can be solved and the "first order approximate trajectory" is an
ellipse!  But we should have expected this, since the theory of slowly
changing weak gravitational fields in gtr must closely resemble Newtonian

4. The next approximate solution of the EB equation, obtained via a second
order perturbation expansion, is still a bit messy, but we can approximate
this approximation (!) by a simpler function containing a "correction
term" to the ellipse, of form phi sin(phi), which is an -even- function of
phi (c.f. the comment of Steven Gray) but -not- periodic.  Examining this
approximation shows that over a sufficiently small range of phi, the
trajectory resembles "a rotating ellipse"; the rotation angle per orbit is
of course the precession per orbit (valid for both perihelia and aphelia).
Note that it is precisely the -cumulative- form of the correction term
which allows the perihelion/aphelion shift to accumulate from orbit to
orbit, so that over the course of a century or so it becomes observable.
A term which accumulates in this way is called a "secular term"; c.f. the
comments of Phil Hobbs.

The outline above follows Einstein 1916, and Einstein's derivation of the
EB equation was based upon the earlier derivation by Binet (in the context
of Newtonian theory) of the Binet equation giving r = f(phi) for Keplerian
elliptical orbits.

Some further comments:

A. The parameter "a" above, the semimajor axis, clearly refers tacitly to
Keplerian trajectories.  This immediately tips off the savvy reader to the
fact that we are dealing with a perturbation of an elliptical trajectory.
Indeed, it is only by thinking of the test particle motion as "nearly
elliptical" that the notion of the precession of the perihelia even makes
sense!  Almost all authors assume sufficient familarity with perturbation
methods on the part of the reader that they can neglect to mention this

B. We can also treat various phenomena which arise in -Newtonian theory-
when studying the motion of Mercury around the Sun as perturbations
(within the formalisim of Newtonian gravitation) of Keplerian orbits.
This leads to differential equations which can also be solved
approximately using perturbation expansions.  The most important Newtonian
perturbations model the gravitational effect of Jupiter and the other
major planets on the motion of Mercury; it turns out that these
perturbations also lead to a cumulative precession of the perihelia, quite
a bit -larger- than the geodetic precession.  Similarly, as suggested by
Phil Hobbs, we could investigate the possible effect of "solar
oblateness", or the slow loss of mass of the Sun due to solar wind.  It
turns out that these effects are -small- compared to the geodesic
precession.  (For solar oblateness, see my post and the survey paper by
Will cited therein.)  But the crucial point here is that it is the
-linearity- the linearized EFE which allows us to toss in the geodesic
precession as an "extra-Newtonian perturbation" of the approximate
Newtonian trajectory!

C. It is only because the Schwarzschild vacuum is -static- that "the time
t = 0 hyperslice" makes sense.  The trajectory r = f(phi) in question is a
curve in the equatorial plane of this hyperslice, a three dimensional
Riemannian manifold.  It only makes sense to compare Einstein-Binet
trajectories with Newtonian ones (e.g. a Keplerian ellipse) because this
hyperslice can be treated as "almost flat".  Also crucial here for
comparison with Newtonian results is the choice of coordinate chart on our
approximate solution (the "linearized Schwarzschild spacetime") to the
EFE; in my post I used a "spatially isotropic polar spherical chart",
which clarifies the levels of approximation involved, as sketched above.
But Weber probably used a different radial chart and perhaps a different
perturbation parameter, which would explain minor differences between his
formula and the one I gave.

D. As you may know, the linearized EFE yields the vitally important
prediction of gravitational radiation traveling at the speed of light.
This involves an unphysical "background metric"; small perturbations of
the Minkowski metric (where background + perturbation = "physical metric")
are treated as a second rank tensor field propagating at the speed of
light in Minkowski spacetime.  It turns out that in gtr, gravitational
waves do carry energy away from two body systems, and this leads to a
genuine inspiral in the predicted orbits.  But note that by adopting a
Schwarzschild model above (static!), we are neccessarily neglecting
radiative phenomena (dynamic!).  Standard reference books (see citations
in my above mentioned post) give the derivation of a valid formula for the
rate of inspiral due to the emission of gravitational radiation; this
turns out to be absolutely tiny for solar system phenomena.  However,
various Newtonian perturbations can also give rise to very gradual (but
much faster) inspiral/outspiral.  You can look for past posts by John Baez
in this newsgroup discussing a Newtonian computer model of the solar
system, in which this phenomenon occurs for our Moon.  This model is
supposedly accurate on a time scale of hundreds of millions of years, at
least for the larger objects.  But note that as a rule, Mercury is one of
the hardest planets to model because so many effects are relevant, and the
requisite parameters are not known to sufficient accuracy for predictions
over time scales much longer than (IIRC) tens of millions of years.

E. Phil Hobbs said "I believe that it's been proven that the orbital
elements of the planets are chaotic, which means among other things that
you can't make a closed-form expression for them."  This is a huge
oversimplification.  However, many textbooks on dynamical systems (see
voluminous references in other posts by myself) do discuss profound
results, due to Poincare and his followers, concerning chaotic phenomena
(for orbital dynamics the most relevant keyword is probably "KAM
theorem").  It does turn out that even in Newtonian dynamics, roughly
speaking, one cannot expect to find simple analytic results for n-body
systems when n > 2.  The question of what you -can- say (quite a lot!) has
fascinating answers; for details see citatations in my posts discussing
KAM, chaos, etc.  You can also look for a past post by John Baez
discussing a partial exception to the impossibility of obtaining simply
characterized results for many body systems, which came to light a few
years ago in the research of Marsden et al.

BTW, it is a good exercise to follow Einstein's model to find perturbation
expansions for more complicated linearized gtr solutions.  In the above
mentioned post I compared test particle motions in the gravitational
fields of uniform rods and disks with the field of a uniform spherical
ball of the same mass.  The first and last solutions possess additional
terms in their Einstein-Binet equations; this leads to additional terms in
the precession formulae which are a bit hard to summarize here.  However,
in the analysis of -light bending- (following Einstein 1916 again, this
prediction comes from approximate solutions via perturbation expansions of
the -null geodesic- version of the EB equation) we find the "bending
angle" of the trajectory of a radar pulse (in the plane of symmetry of a
given axially symmetric body) to be

  uniform disk (mass m, radius a):  6 pi m/r + pi a^2/r^2

  uniform ball (mass m, radius a): 6 pi m/r

  uniform rod (mass m, semi-length a): 6 pi m/r - pi a^2/r^2

Note the new terms scale like 1/r; here r is the distance of closest
approach, and again we are tacitly appealing to the "nearly euclidean"
geometry of the constant time hyperslices in question.  This illustrates
how we can separate the effect of solar oblateness (note that a disk is a
maximally oblate distortion of a spherical ball!) from the geodesic
precession prediction of linearized gtr, e.g. in observations in radio
wavelengths of pulsars passing near the Sun ("near on the celestial
sphere, that is!), and also confirms that, as we should expect, we can
recover the -Newtonian prediction- entirely in the context of linearized
gtr.  Similar remarks hold for the much larger Newonian perterbations due
to the effect on the motion of Mercury of the major planets.  This
timescale is MUCH shorter than the time for inspiral due to gravitational
radiation which is predicted by linearized gtr in the case of a simple
"nearly Keplerian model" of the motion of Mercury.

(Exercise: does the sign of the terms +/- pi a^2/r^2 above make intuitive
sense physically?)

One final comment: if you are looking for fun stuff to explore with
computer experiments, you could not do better than the "evolution of
random graphs", as mentioned in my recent post on Shannon 1949. The theory
of random graphs, unlike gtr, requires very few mathematical
prerequisites.  It is tricky and subtle enough to be (very!) interesting
without providing zillions of pitfalls even for relatively well prepared
neophytes.  This theory is also very highly developed, and its
foundational geniuses include figures (like Erdos) who are no less
interesting than Einstein. It is no less universal or beautiful than gtr,
but it is much more relevant to zillions of everyday but intriguing
phenomena (e.g. "small world phenomena").  It is highly applicable to
physics (including percolation phenomena) and leads to all kinds of
intriguing phenomena, such as an unexpected connection with mathematical
logic and the work of that occasional EFE solver, Kurt Goedel.

Chris Hillman
Some of this (like the advice about random graphs) is a bit off-topic, but I'd like to repeat it anyway: random graphs are a very interesting and timely topic!

Oh yes, another thing to look for in the post is the distinction between

1. finding a parameterized timelike geodesic in the spacetime model,

2. finding the projection of this into a spatial hyperslice [itex]t=t_0[/itex] (which only makes sense in a static spacetime model!),

3. finding the unparameterized curves (trajectories) corresponding to these.

And of course, one most not get confused between "solution" in the sense of spacetime model (e.g. Schwarzschild solution, or an approximate two body solution in weak-field gtr) and a "solution" (approximate or exact) in the sense of solving the geodesic equations once the spacetime model is chosen. Note that another reasonable project would be finding an approximate solution to the motion of a test particle (modeling Mercury) in a spacetime model which is an approximate solution in weak-field gtr to the two body problem (the two massive bodies modeling the Sun and Jupiter).
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Thanks a lot Chris,

I will need some time to assimilate these informations.
At first reading, I immediately noted this passage:
By the way, Michel (and pervect, if you are reading this post!), this notion is best understood in terms of Lie's theory of symmetry of systems of differential equations; the Runge-Lenz vector naturally arises as a dynamical symmetry in the Kepler system; see Stephani, Differential Equations: Their Solution Using Symmetries for a clear discussion. The point is this: as Stephani shows, in the two body problem (in Newtonian gravitation!) the Runge-Lenz vector is invariant. But in the three body problem (Sun, Mercury, Jupiter) it is not, and this models the precession effect you seek.
These days I have some attraction to the Hamilton-Jacobi theory, but I cannot figure out how it could be used for general problem!!! And therefore I am not sure of its practical applicability.

Besides its famous link with QM, the H-J approach yields -for me- a very nice approach to the perihelion drift in GR, as I judge it from reading Landau-Lifchitz. For the 10-planets problem, it may well be the clearest approach too: the perturbations are reduced to a first order expansion of an integral. To be checked.


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