What is the observed precession of Mercury?

[utesfan100]

The precession of Mercury is presented as one of the threshold moments for general relativity. Are there any publicly available summaries of the classical anomaly? Most refer to a 1947 paper that uses an outdated reference frame.

Clemence, G. M. (1947). "The Relativity Effect in Planetary Motions". Reviews of Modern Physics 19 (4): 361–364.

This source gives a precession of the equinoxes of 5025.64, an observed precession of 5599.74 and a tug from the planets of 531.63. This produces an actual precession of 574.1.

NASA gives a value of 5028.83 for the precession of the equinox, using a modern standard frame. Is there a publicly reference for the actual precession of Mercury, or the precession in J2000 used by NASA?

http://ssd.jpl.nasa.gov/?constants

This source from 2008 gives an observed precession of 5600.73, but is unclear what frame is used.

http://books.google.com/books?id=fp9wrkMYHvMC&pg=PA70#v=onepage&q&f=false

Is it inaccurate to conclude the current values suggest a classical anomaly as follows:

The precession of Mercury is $5600.73-5028.83-531.63=40.27$, a 2.71 arcsec/century variance from the 42.98 prediction of GR.

 

[D H]

Is it inaccurate to conclude the current values suggest a classical anomaly as follows:

The precession of Mercury is $5600.73-5028.83-531.63=40.27$, a 2.71 arcsec/century variance from the 42.98 prediction of GR.

Yes, it is inaccurate to conclude that.

 

[utesfan100]

Ok. I was assuming it was inaccurate to conclude that.

I was wanting to know where the error in the calculation came from, and how that error can be expressed using data no older than 1990, except for the perturbative effects like the tug from the planets and the oblateness of the sun.

I have found this link, giving a precession of the equinoxes of 5028.7955$\pm$0.0003.

http://iopscience.iop.org/1538-3881/126/1/494/fulltext

The effect from the oblateness of the sun is given as 0.0245

When I use the error measures listed at the sources I provided get that the uncertainty in the excess precession of $2.70\pm0.80$ after the GR correction of 42.98 is made.

Where is the current data showing this calculation to be in error?

 

[atyy]

Maybe try the discussion in section 3.5 of http://relativity.livingreviews.org/Articles/lrr-2006-3/ ? Will gives a reference [238] published in 1990 which is Shapiro, I.I., “Solar system tests of general relativity: Recent results and present plans” in Proceedings of the 12th International Conference on General Relativity and Gravitation, unfortunately probably not freely available.

 

[D H]

When I use the error measures listed at the sources I provided get that the uncertainty in the excess precession of $2.70\pm0.80$ after the GR correction of 42.98 is made.

I just told you you cannot do that. So why do you persist?

Where is the current data showing this calculation to be in error?

There is no current data. It is a solved problem. The most recent publication was R.L.Duncombe, Astronomical Journal, 61:174 (1956). That (not Clemence) is the source of the numbers in the text you cited.

Since then, advances in computation have enabled a switch in focus from the orbital element approach of the 1800s to numerical techniques. Astronomers, along with mission planners at various space agencies, do not need and do not precession to predict the position of the planets.

 

[DaveC426913]

I just told you you cannot do that. So why do you persist?

All you acceded to was that his conclusion was inaccurate. You were silent about what step(s) leading up to it you were referring to, forcing him to zero in on the flaw in his logic.

 

[D H]

All you acceded to was that his conclusion was inaccurate. You were silent about what step(s) leading up to it you were referring to, forcing him to zero in on the flaw in his logic.

The problem is he is using the precession of the equinoxes as expressed in ICRF but is keeping the observed precession of Mercury in whatever frame was used by Clemence. That is downright invalid.

Modern tests of general relativity will almost invariably be in the form of a parameterized post Newtonian formalism. There is plenty of recent data on these kinds of tests.

 

[Passionflower]

Modern tests of general relativity will almost invariably be in the form of a parameterized post Newtonian formalism. There is plenty of recent data on these kinds of tests.

Hmm, why is that actually necessary, any powerful computer can numerically solve integrals right?

 

[D H]

Hmm, why is that actually necessary, any powerful computer can numerically solve integrals right?

Most, if not all, of these PPN tests would not be possible without the use of modern computers.

 

[Passionflower]

Most, if not all, of these PPN tests would not be possible without the use of modern computers.

You are not answering my question.

 

[D H]

You are not answering my question.

What integrals are you talking about, and what does your question have to do with this thread?

 

[Passionflower]

I am not sure what is so difficult in understanding what I ask.

You claim that GR tests will almost invariably be in the form of a parameterized post Newtonian formalism.

I ask why? Is that perhaps not allowed here?

Or more specific why do you believe that the Perihelion Advance must be solved using PPN?

In GR we can describe Mercury's orbit using a simple differential equation or an integral as we can approximate it as a test particle in a Schwarzschild solution with the Sun as the central mass.

So why not use that? Frankly I do not see any reason why we would need PPN for this.
So that's why I ask do I perhaps miss something?

 

[atyy]

The PPN parameters predicted by GR are known. The PPN parameters are used for convenience so that everyone uses the same language - the current PPN parameters evolved from several earlier slightly different PPN formalisms. GR can be placed in PPN form only for weak gravity.

The Will review linked to above says "Of course, some systems cannot be properly described by any post-Newtonian approximation because their behavior is fundamentally controlled by strong gravity. These include the imploding cores of supernovae, the final merger of two compact objects, the quasinormal-mode vibrations of neutron stars and black holes, the structure of rapidly rotating neutron stars, and so on. Phenomena such as these must be analyzed using different techniques. Chief among these is the full solution of Einstein’s equations via numerical methods. This field of “numerical relativity” is a rapidly growing and maturing branch of gravitational physics ..."

 

[D H]

You claim that GR tests will almost invariably be in the form of a parameterized post Newtonian formalism.

Several reasons. Just a few are that it provides

- A lingua franca by which different tests of GR can be compared. It turns out that there are far stronger tests of GR than the precession of Mercury.

- A way to combine disparate tests. A given model of gravitation might pass a slew of individual tests but could fail when those tests are jointly combined in the PPN parameter space. So far GR has passed all such meta-analyses.

- The ability to test not only GR but also to test (and compare) alternative formulations of gravitation.

Or more specific why do you believe that the Perihelion Advance must be solved using PPN?

I never said that; so please stop putting words in my mouth. That is however exactly how perihelion advance is oftentimes calculated.

In GR we can describe Mercury's orbit using a simple differential equation or an integral as we can approximate it as a test particle in a Schwarzschild solution with the Sun as the central mass.

So why not use that? Frankly I do not see any reason why we would need PPN for this.
So that's why I ask do I perhaps miss something?

You are missing

- The influence of the other planets, which are an order of magnitude larger than that of general relativity.

- That orbital elements provide a very nice way to visualize orbits but not so nice way to calculate orbits. The tweaks to Keplerian orbits needed to make them accurate get downright ugly.

- That the precession of Mercury is essentially a solved problem. It is a comparatively weak test of relativity. Modern physics relies on much stricter tests of GR.

- That the PPN formalism generalizes to encompass alternative formulations of gravitation other than GR.

 

[Passionflower]

You are missing

- The influence of the other planets, which are an order of magnitude larger than that of general relativity.

I am not missing that at all.

So are you claiming that the influence of other planets is actually modeled by using GR as opposed to only the perihelion shift? I think the 'other planets' part it is copied verbatim from Newtonian theory with the argument (which I do not claim is wrong) that the difference is negligible. Do you think I am wrong about that? I so, please provide a reference to a calculation where the planetary influences are taken into account using GR.

- That orbital elements provide a very nice way to visualize orbits but not so nice way to calculate orbits. The tweaks to Keplerian orbits needed to make them accurate get downright ugly.

I guess then we have to argue what the word ugly means scientifically.

I think we can calculate Schwarzschild orbits very easily with a decent size desktop and a good math program. If we take some valid initial conditions where the perihelion is at a given coordinate location we can then solve the equation to get the next location after one orbit. Then we get the GR part and leave the extra planetary influences as 'similar to Newton'.

You think I am wrong about that?

 

[D H]

I am not missing that at all.

So are you claiming that the influence of other planets is actually modeled by using GR as opposed to only the perihelion shift? I think the 'other planets' part it is copied verbatim from Newtonian theory with the argument (which I do not claim is wrong) that the difference is negligible. Do you think I am wrong about that? I so, please provide a reference to a calculation where the planetary influences are taken into account using GR.

That is exactly what you will do if you follow the recommendations of the International Astronomical Union and wish to keep up with the Joneses (The Institute of Applied Astronomy, The Paris Observatory, and JPL) of the solar system modeling world.

Pitjeva, E.V., "EPM ephemerides and relativity," Proceedings of the International Astronomical Union (2009), 5 : pp 170-178
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6911132

All the modern ephemerides: DE – JPL (Folkner et al., 2008), EPM – IAA RAS (Pitjeva, 2009), INPOP – IMCCE (Fienga et al., 2008) are based upon relativistic equation of motion for celestial bodies and light rays as well as relativistic time scales. The numerical integration of the equations of celestial bodies motion has been performed in the Parameterized Post–Newtonian metric for General Relativity in the TDB time scale; the relativistic effects of the signal delay (the Shapiro effect), and path-bending of the radiosignal propagation in the gravitation field of the Sun, Jupiter, Saturn and the reduction of observations from the proper time of the observer to the coordinate time of the ephemerides are taken into account while processing observations.​

Fienga A. et al., "INPOP08, a 4-D planetary ephemeris: From asteroid and time-scale computations to ESA Mars Express and Venus Express contributions," Astronomy & Astrophysics 507:3 (2009)
http://www.mendeley.com/research/inpop08-a-4d-planetary-ephemeris-from-asteroid-and-timescale-computations-to-esa-mars-express-and-venus-express-contributions/

With INPOP08, we aim to produce planetary ephemerides as fully compatible as possible with the relativistic background recently adopted by the astronomical community and summarized by the IAU2000 and IAU2006 conventions (Soffel el al., 2003).​

I'd provide one on the DE series from JPL as well, but unfortunately, JPL appears to be suffering a massive connection failure right now.

I think we can calculate Schwarzschild orbits very easily with a decent size desktop and a good math program.

That is a trivial problem. Try using GR as-is (no post-Newtonian expansion) for a solar system comprising a central star, 8 planets (one of which is rather large), a growing bunch of dwarf planets and minor planets, dozens of sizable moons, hundreds of sizable asteroids, etc. It is utterly intractable.

 

[Passionflower]

Yes you are correct!
I did not know how advanced the calculations already are.

 

[utesfan100]

The problem is he is using the precession of the equinoxes as expressed in ICRF but is keeping the observed precession of Mercury in whatever frame was used by Clemence. That is downright invalid.

Modern tests of general relativity will almost invariably be in the form of a parameterized post Newtonian formalism. There is plenty of recent data on these kinds of tests.

Actually, I am using the precession of Mercury reported by Tai L. Chow in his 2008 book "Gravity, black holes, and the very early universe". Whatever frame he used. This differs from Clements value by almost an arcsecond per century. I did not see a reference in the summary of the book available online. It may be that Chow used R.L.Duncombe.

The secondary question, then, reduces do where can I find the observed precession of Mercury relative to the ICRF?

Observed orbital parameters for Mercury to the thousandths of a degree from 1970 to 2010 would more than suffice.

 

[D H]

It may be that Chow used R.L.Duncombe.

The number in Chow, 43.11±0.45″ is the exact same number, including the uncertainty, as my 1970 version of Marion which references Duncombe.

The secondary question, then, reduces do where can I find the observed precession of Mercury relative to the ICRF?

Sigh. You don't. The perihelion precession of Mercury is a solved problem. Physicists are not paid to re-solve problems that were solved a century ago. They are paid to solve today's problems.

 

[Passionflower]

As D.H. writes the matter is lock, stock and barreled, all influences up to moons are taken into account including lower order relativistic effects (higher order is really irrelevant as all this is weak field GR) and it all matches perfectly so it is a done deal.
End of story, the theory and experiment are the same.

 

[D H]

it all matches perfectly so it is a done deal.

Not really a done deal. That gravitation is not a done deal was the justification for Gravity Probe B.

Physicists and astronomers no longer look to Mercury because there are much more stringent tests of relativity than the precession of Mercury. Look at the graph on slide 5 of this pdf presentation, http://www.fc.up.pt/great-ws-porto/Mario_Gai.pdf. The graph is a plot of PPN parameters γ vs β. The Mercury ranging tests from 1993 allow a huge band in that γ vs β plot. It adds nothing compared to how the Cassini experiment from 2003 and the Lunar Laser Ranging experiment from 2002 zero right in on a small region around (γ=1, β=1).

There's really no reason to revisit what is a now outdated test, and hence those data on the precession of Mercury will remain expressed in a now outdated frame of reference.

 

[Passionflower]

Not really a done deal. That was, after all, the justification for Gravity Probe B.

Then I am more impressed, so even the planetary rotations where taken into account in this new model. I really need to be more up to speed with the tremendous advances that are made in verifying that our planetary movements mach up to the smallest detail with general relativity.

 

[Passionflower]

There's really no reason to revisit what is a now outdated test, and hence those data on the precession of Mercury will remain expressed in a now outdated frame of reference.

I do not see how that serves science, in the least it will only encourage people like Myron Evans to express doubts.

 

[utesfan100]

The problem is he is using the precession of the equinoxes as expressed in ICRF but is keeping the observed precession of Mercury in whatever frame was used by Clemence. That is downright invalid.

Modern tests of general relativity will almost invariably be in the form of a parameterized post Newtonian formalism. There is plenty of recent data on these kinds of tests.

My brief survey of PPN gives me the impression that it measures first order perturbative effects relative to Newtonion theories. In fact the ten parameters appear to correspond to the ten independent values of a symmetric $g_{ij}$ metric.

It is often stated that one of the key aspects of the precession of Mercury is that it is a second order test. What part of the PPN corresponds to the second order effects contributing to the precession of Mercury?

 

[Passionflower]

My brief survey of PPN gives me the impression that it measures first order perturbative effects relative to Newtonion theories. In fact the ten parameters appear to correspond to the ten independent values of a symmetric $g_{ij}$ metric.

It is often stated that one of the key aspects of the precession of Mercury is that it is a second order test. What part of the PPN corresponds to the second order effects contributing to the precession of Mercury?

We just had the confirmation from DH that it is definitely not the case that we only take the GR effects of the Mercury orbit and verbatim copy the Newtonian parts to see the 43 arcseconds per century result. So it is (lower order) general relativity across the board for all Planetary influences.

 

[D H]

Exactly. A first order weak field approximation is more than enough, at least for now. The errors that result from the errors in the observations overwhelm the errors that result from ignoring higher order terms. There will eventually be a need to move to a higher order model as ever-improving measurements accumulate over time.

 

[utesfan100]

We just had the confirmation from DH that it is definitely not the case that we only take the GR effects of the Mercury orbit and verbatim copy the Newtonian parts to see the 43 arcseconds per century result. So it is (lower order) general relativity across the board for all Planetary influences.

Thank you for your help.

I have conceded that the error in my calculation above is due to a systematic error caused by a change in the reference frame.

I am not particularly interested in the difference between the Newtonian estimate of the GR correction of 42.98 and the GR estimate of this effect of 43.11. This is far less than the measurement errors in the latest data of 0.80, or the systematic error of the referenced frames used of about 3.

DH has shifted the discussion towards the PPN, apparently due to the fact that this measurement has not been explored for over 5 decades or even published in the current reference frame in the interim. In the post you cited I was trying to shift my question into this context.

In the link you provided, the precession of Mercury is discussed. The precession is described in GR as being $\frac{2\pi m}{p}(2+2\gamma-\beta)$ and some change.

http://relativity.livingreviews.org/Articles/lrr-2006-3/

This equation is a function of GR, as its derivation depends on second order effects in the radial direction. A different theory might derive a different relation between the precession, $\gamma$ and $\beta$.

Thus I ask, what parameter of PPN cooresponds to the second order effects in the radius that is the only nonlinear effect in the calculation of the precession of Mercury. Indeed, it appears to me that PPN is first order in the metric, but I hope to be shown my error.

 

[utesfan100]

The number in Chow, 43.11±0.45″ is the exact same number, including the uncertainty, as my 1970 version of Marion which references Duncombe.


Sigh. You don't. The perihelion precession of Mercury is a solved problem. Physicists are not paid to re-solve problems that were solved a century ago. They are paid to solve today's problems.

It occurs to me that the data cited here can give us the answer to the observed precession relative to the ICRF, assuming that the axis of the two references are nearly aligned (within 0.25 degrees).

Mercury was observed to have a precession of 5600.73±0.41 in a frame with a precession of 5025.64±0.50. This gives a precession in an inertial frame of 575.09±0.64.

Relative to a frame precessing at 5028.83±0.04 we have an observed precession of 5603.92±0.65 relative to the ICRF.

 

[D H]

ICRF is an inertial frame.

 

[utesfan100]

ICRF is an inertial frame.

Ok. So then the precession is 575.09±0.64 relative to the ICFR, and precession of the equinoxes is irrelevant beyond the calculation from the old frame the old data was taken in.

 

[D H]

That's much closer, but it still isn't valid. That precession due to the other planets was calculated in a rotating frame in which those other planets are following orbits that deviate from Keplerian due to the frame rotation.

What, exactly, are you trying to accomplish, and why?

 

[utesfan100]

That's much closer, but it still isn't valid. That precession due to the other planets was calculated in a rotating frame in which those other planets are following orbits that deviate from Keplerian due to the frame rotation.

What, exactly, are you trying to accomplish, and why?

The value I calculated above is the observed value. The planets contribute 531.63±0.69 and GR 42.98±0.04.

Thus the remainder after these contributions are 0.48±0.94.

My goal is to determine the error in this measurement (whose meaning has been greatly clarified, thank you). Explaining my actual motivation would violate the rules of the forum :)

Instead, I will suggest that the observed precession of Mercury question has been answered and shift my focus towards understanding the PPN formalism. My basic question is now ho I get from the Schwarzschild metric to the PPN formula for the per orbit advance of $\frac{2\pi m}{p}\left(2+2\gamma-\beta\right)$?

Ok. That is more than is reasonable to expect an answer to here, but being able to do this is something I want to be able to do; and thus to be able to address my actual motives on my own.

My focus is on understanding the Eddington–Robertson–Schiff parameters. I have found where these are described as

$ds^2=\left(1-2\frac{M}{r}+2\beta\frac{M^2}{r^2}\right) dt^2-\left(1+2\gamma\frac{M}{r}\right)(dx^2+dy^2+dz^2)$

from which it is straightforward to calculate β and γ using the isotropic formulation of the metric. It is clear now that β is what I was referring to earlier as a second order effect. Am I reading this right, that Newtonian gravity about a point mass can be expressed (relative to the escape velocity [STRIKE]reference frame[/STRIKE] coordinate chart) by the metric:

$ds^2=\left(1-2\frac{M}{r}\right) dt^2-(dx^2+dy^2+dz^2)$

It seems like β might be 2 based on the precession formula above. Either way, that would be awesome insight!

Perhaps this new PPN formalism inquiry should be moved to a thread of its own.

 

[utesfan100]

I found this article, showing that the upcoming BepiColumbo mission aims to very accurately measure the orbital elements of Mercury which, when combined with observations of other planets to reduce the uncertainty of the effect of the gravitational tugs, will allow for improved estimates of γ and β, able to test the Lense-Thirring effect.

http://www.aanda.org/index.php?opti...l=/articles/aa/ref/2005/07/aa1646/aa1646.html

It would seem this test of GR is not as irrelevant as the previous discussion indicated, but we will have to wait until 2020 for this test to again become relevant relative to present methods for determining the PPN parameters. (Though shouldn't this data be attainable from Messanger?)

 

[utesfan100]

Thank you, everyone, for your help. I now understand the answers to questions I was wanting to ask, but did not even know the right terms to ask about.

I found the article below greatly helped in outlining the calculation of the precession for the PPN parameters γ and β.

http://www.math.washington.edu/~morrow/papers/Genrel.pdf

In particular, the metric does not need to be solved for because we already have the parametrized estimate. Further, the integral can be simplified by using the u=1/r substitution and finding the change in ω for the ellipse (like almost every other treatment of the Kepler problem does.)

My previous objection that this equation based on the PPN parameters depends on the theory is valid, but only if the theory adds terms to the Lagrangian to model something like gravitons carrying part of the field. For PPN this is not the case.