Mercury's Perihelion Precession

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Thomas</i>
<br />In GR the 43" precession still exists in this case but not in reality (all orbits in a 1/r^2 potential have to be closed (see Landau and Lifshi(t)z, Mechanics)).<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Incorrect. Orbits are closed only in the Newtonian approximation. But in all serious post-Newtonian theories, as well as in observations stemming from 50 years before relativity was invented, the 43"/cy exists.

Many reasons are offered for this. But the simplest and easiest to understand is MM's answer that the elysium produces refraction effects in the light-carrying medium to supplement the effects of gravitational force, causing light-bending, redshift, Shapiro delay, and perihelion precession in the exact amounts observed.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">the problem is to calculate the perturbations exactly. Any approximation will introduce a certain error into the orbit calculation and I would guess that the approximations used are as usual of first order or at best of second order here<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Such low-order approximations are used only for student exercises. Simple tests are available to determine the error of any approximation. Depending on the application, 20th or higher order may be used. (But different methods are more exact and need fewer orders to get the same overall accuracy. A 4th order Runga-Kutta integration will get you observational accuracy for centuries.)

As I said, any serious worker can and always will ensure that the error of the resulting orbit is less than the desired accuracy, which is typically 10 or more decimal places these days.

Analytic and numerical methods have completely different kinds of errors, and so can be used to check one another. Checking how well certain "constants" of the dynamics are preserved is another check. Integrating the orbit forward, then backward, is another check. Comparison with observations is another check. We now use lasers to measure the Moon's motion to a millimeter.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">As far as I can see, the calculated orbit can only be as exact as the highest order of the perturbation theory allows.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">True, but usually nothing limits how high an order can be used in the application, so nothing limits the resulting accuracy. -|Tom|-

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by tvanflandern</i>
Such low-order approximations are used only for student exercises. Simple tests are available to determine the error of any approximation. Depending on the application, 20th or higher order may be used. (But different methods are more exact and need fewer orders to get the same overall accuracy. A 4th order Runga-Kutta integration will get you observational accuracy for centuries.)
As I said, any serious worker can and always will ensure that the error of the resulting orbit is less than the desired accuracy, which is typically 10 or more decimal places these days.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">We are not talking about the accuracy of the numerical calculations here, but of the analytical approximations associated with the representation of the orbit in terms of (perturbed) orbital elements. It is the latter which is used when 'observing' the perihelion precession of Mercury.
I quote from ads.harvard.edu/books/1989fcm..book/Chapter9.pdf :
'Utility of the development of these perturbation equations relies on the approximation made in equation (9.1.3). That is, the equations are essentially first order in the perturbing potential. Attempts to include higher order terms have generally led to disaster. The problem is basically that the equations of classical mechanics are nonlinear.... Many small errors can propagate through the procedures for finding the orbital elements and then to the position vector itself. Since the equations are nonlinear, the propagation is nonlinear. In general, perturbation theory has not been terribly successful in solving problems of celestial mechanics. So the current approach is generally to solve the Newtonian equations of motion directly using numerical techniques. Awkward as this approach is, it has had great success in solving specific problems as is evidenced by the space program. The ability to send a rocket on a complicated trajectory through the satellite system of Jupiter is ample proof of that. However, one gains little general insight into the effects of perturbing potentials from single numerical solutions'.

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<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Thomas</i>
<br />We are not talking about the accuracy of the numerical calculations here, but of the analytical approximations associated with the representation of the orbit in terms of (perturbed) orbital elements.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">We are talking about orbit determination in general. The fact that analytic and numerical methods agree to the accuracy of observations is certainly relevant.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">It is the latter which is used when 'observing' the perihelion precession of Mercury.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Not so. Most modern, high-precision work uses a purely numerical approach to get the orbit, and a semi-analytical approach to convert osculating elements into mean elements for purposes such as perihelion precession. That too works well to the full accuracy of modern observations.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">I quote from ... 'the equations are essentially first order in the perturbing potential.' Attempts to include higher order terms have generally led to disaster.'<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I wish you better understood what you read. For Earth (and typically for all solar system bodies), potential is of the same order as velocity squared. And phi/c^2 or v^2/c^2 (the relevant form of post-Newtonian perturbations) is then of order 0.0000 0001. So the "good" approximations typically affect the eighth decimal place. The problems alluded to then arise at the square of that order, or in the 16th decimal place.

The author you quote is obviously partial to numerical approaches, and is somewhat unfamiliar with analytic orbit techniques. Modern analytic theories go to very high orders, indeed. But eventually, the labor to go to higher accuracies becomes so much greater than using numerical techniques that the effort is abandoned. The main "unsolved" problems lie in implementing GR field equations, for which no exact n-body solutions are known to exist, and a first-order-in-potential limitation really is a limiting factor.

But these are all highly theoretical issues. Nothing prevents either approach from achieving the full accuracy needed for definitive comparisons with observations. So the above quote does nothing to justify your claims whatever. The perihelion precession is first order in the potential, a level of accuracy achievable with any approach, at which no one questions the accuracy of the results. -|Tom|-

If observations show the precession of Mercury's perihelion is 575" per year why does the JPL/horizons generator show a greater amount of precession? Has anyone really looked into this other than from math and observations done in the nineteenth century? Noboby gets out and looks beyond the math of this detail.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />If observations show the precession of Mercury's perihelion is 575" per year why does the JPL/horizons generator show a greater amount of precession?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Are the perihelion values with or without precession? For what equinox and epoch? (And are they affected by nutation, aberration, and E-terms?) Are you looking at osculating or mean elements? Determined with what model and what masses? Is the orbit calculated numerically or analytically? To what accuracy? What is the origin of coordinates and the focus used for the ellipse?

And why didn't you specify these things before asking your question? The answer to any one of them might provide the answer to your question.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Has anyone really looked into this other than from math and observations done in the nineteenth century?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">It seems to have escaped your attention that both observations and progress with theories occurred in the 20th century. There are hundreds of astronomers who look into such matters full time, and who meet at least every third year under the auspices of the International Astronomical Union. They are continually improving the standards, reference systems, masses, techniques, and algorithms. The Division on Dynamical Astromomy of the American Astronomical Society meets annually, and has several dozen members who are experts in these areas. But if you had read any textbooks in these fields, you would already know this.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Noboby gets out and looks beyond the math of this detail.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">If you had taken my course in computer astronomy, you would have learned all the relevant background. And your "final exam" would have consisted of computing the exact time when a star disappears at the dark limb of the Moon, then observing and timing the real event and comparing the two.

But I'm not the world's only teacher. If you want to know this stuff, sign up for a course, or do what I did. I started with the Explanatory Supplement to the Astronomical Almanac, which explains all these concepts and provides the formulas needed to compute them. (I did this in high school, before I had formally learned trig and calculus. But you might also need some spot help here and there if your math is weak.) The 1961 edition is better for classical observation types, and the 1992 edition is better for modern observation types.

And if you don't care to learn all this stuff, that's okay too. But quit going around complaining that people who have learned it don't know what they are talking about. If you don't care to acquire the relevant background, then you are forced to accept the opinion of your choice of "experts" who have -- the same as when choosing a doctor, lawyer, or auto mechanic. -|Tom|-

The way I got the precession is not as professional as your methods I will admit. What I did was locate Mercury's perihelion on the generator at two times 4,000 years apart and located the sun on the two dates. Then I drew a scale model of the orbit of Earth and Mercury. Marking the location of Earth on the two dates gave the points to sight the location of Mercury on the dates using the info posted at the generator. The distance from the sun to Mercury and Earth was ploted and then the distance from Earth ot Mercury on both dates was checked. I also noted the angle of sun/Mercury/Earth on both days to be as accurate as I could be using just a few simple tools.