Apsidal Precession

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    Seeing beautiful pictures of planets, moons, and nebulae has so accustomed us to conceiving the magnifying power of an astronomical telescope purely as a means of giving us such pictures of astronomical objects that few of us know that one of the most important uses to which astronomers put that magnifying power is the measurement of the positions of the planets on the sky (i.e. relative to certain distant stars). By 1838 astronomers had made telescopic measurement accurate enough that Friedrich Bessel could measure the first parallax of a nearby star (61 Cygni, 11.2 lightyears away), accurate enough that over the remainder of the century astronomers could track the motions of the planets on their orbits to within one second of arc (1/3600 of a degree) per century. With that kind of accuracy astronomers were able to determine, in particular, that the orbit of Mercury, the little planet closest to the sun, precesses at a rate of 5600 arcseconds per century; that is, that the point of the planet's closest approach to the sun (the perihelion) as seen from the sun shifts across the sky by 5600 seconds of arc every hundred years.

    According to the celestial mechanics based on Newton's law of gravity, the orbits of planets in a multi-planet system with a rotating sun should be subject to apsidal precession (also called the advance of the perihelion). Because of the perturbing forces that the other planets and the sun's equatorial bulge exert upon it, a planet moves in such a way that its line of apsides (the long axis of the ellipse that describes the orbit, the straight line that connects the orbit's perihelion to its aphelion) shifts in the direction in which the planet moves. Add to that the fact that Earth precesses on its axis, which shifts the reference point on the sky to which astronomers refer all other celestial motions, and the net amount by which the classical theory tells us that Mercury's orbit shifts is 5557 arcseconds per century, noticeably less than the amount inferred from observation.

    That 43 arcsecond per century discrepancy between theory and measurement vexed astronomers mightily for years. Some suggested that the extra precession was caused by another, undiscovered planet, one flying closer to the sun than Mercury does. That was actually a very reasonable hypothesis, even though astronomers found no observational evidence for such a planet: a similar discrepancy between the theoretical orbit of Uranus and the orbit inferred from observations led to the discovery of Neptune in 1846. Other astronomers suggested that Newton's law of gravity was not a perfectly accurate description of the actual phenomenon of gravity and that the exponent in the inverse-square law, instead of being equal exactly to two, would be equal to two plus some small fraction. Nobody liked that idea: it felt too much like an assertion that God had failed to sand down the rough edges on the Universe after taking it out of the mold in which it was cast.

    Fortunately, Einstein was able to show that General Relativity accounts for the discrepancy. As a consequence (largely in light of the suggestion that Newton's law be modified) physicists often call General Relativity Einstein's theory of gravity: more properly we should call it Einstein's theory of relativistic motion in a Newtonian gravitational field. In devising his theory Einstein discovered, indirectly, that we must take four effects into account when we calculate a description of the orbit of Mercury. As I will show, the sun's gravity produces analogues of time dilation and of the Lorentz-Fitzgerald contraction (in two different ways) and Mercury's motion brings the time dilation and spatial contraction effects from the Lorentz Transformation into play. When we bring the algebraic descriptions of those effects together (see Appendix: A Model of Orbital Motion for a suitable description of how planets move) and work out the required numbers, we obtain a description that tells us that if those effects alone affected Mercury's orbit, then Mercury would reach each perihelion (the point on its orbit at which it is closest to the sun) a little over one tenth of an arcsecond past the previous perihelion; thus, starting at one perihelion, Mercury goes a full 360 degrees around the sun and then goes a little over one tenth of an arcsecond further before reaching the next perihelion. Doing that 415 times in a century, Mercury accumulates the required 43 arcseconds that we must add to the other theoretical contributions to the full description of the orbit to obtain a result that matches the number obtained from observations.

    Astronomers conventionally refer to the advance of Mercury's perihelion as the precession of the orbit and that fact may cause some bewilderment. We usually learn about precession through a study of spinning bodies, such as gyroscopes, and we know that we must exert a force to make a gyroscope precess. We may then ask how the distortion of space and time reflected in the Schwarzschild Transformation can exert a force upon a planet to make it shift its orbit.

    In asking that question we have confused two different kinds of precession. The precession of a gyroscope, which involves the rotation of the gyroscope's axis through a circle, comes from a continuous change in the orientation of the gyroscope's angular momentum. But only an applied torque, which comes from the force that you exert upon the gyroscope, can produce that change in angular momentum. On the other hand, the precession of a planetary orbit, the slow drift of the planet's line of apsides around the circle of the sky, involves no change in the planet's angular momentum. It comes, rather, from imperfections that the distortion of space and time imposes upon what would otherwise be a perfectly elliptical orbit.

    A purely Newtonian description of gravity leads us to deduce a perfect ellipse as the orbit of a single body about its common center of mass with another body. Inclusion of Relativity in that description then deforms the ellipse; that is, it deforms the mathematical description of the ellipse through appropriate mathematical descriptions of the effects that I listed above. Thus, in order to work out a description of the apsidal precession due to relativistic effects, we must begin with a perfectly elliptical orbit and then add in the relativistic effects to see how they deform it. Let's imagine a perfect ellipse somewhere in space and imagine that a small body moves along it at a slow speed. Let's imagine further that some non-gravitational force that does not deform space and time holds that body on the ellipse, that the force emanates from one of the ellipse's foci, and that it conforms to the inverse-square law. Thus we have the perfect Newtonian description as our starting point.

    I have employed a bit of a fiction, though. Reality has no force that does not deform space and time. All forcefields involve a potential energy, which, by Einstein's theorem, has mass, which deforms space and time. My imaginary forcefield cannot exist as I describe it, but I want to use it anyway in order to start my derivation with a naive orbit, a purely Newtonian one. I intend to begin with this fictitious force and then correct its deficiencies, analyzing the relativistic effects that each correction brings in. I want to start by bringing that little body up to the speed it will have when I replace the fictitious force with gravitational force. I'll represent that speed with the letter vee, upper-case for observers far from the orbit's primary focus (the outer observers) and lower-case for observers right next to the orbit (the inner observers).

    Let me say more specifically that I want to put my outer observers far enough from the primary focus of the orbit that they will make their observations from a point effectively outside the gravitational field of the sun that I will put at that focus. But gravity has infinite range: nothing can ever lie outside any body's gravitational field. But that fact merely points up a deficiency in the English language: we just don't have a term that we can readily adapt to a case like this one. However, for convenience we define 'effectively outside the gravitational field' to comprise those points in space whence escape velocity from the field comes arbitrarily close to zero relative to escape velocity from the region under consideration, the region occupied by our orbit. So if we look at the orbit of Mercury, then 'effective infinity' is on the order of a few tens of astronomical units.

    I have begun my description of the body's orbit by saying that the orbit conforms to Newtonian dynamics. In our relativistic Universe we can only approximate a true Newtonian orbit, doing so by making the radius very large so that any body following the orbit must move very slowly. Having thus established the orbit, we then intensify the force holding the body on it, thereby shrinking the radius of the orbit. In response, in order to conserve its angular momentum, the body will move faster, achieving a speed that obliges us to use Relativity to obtain a properly accurate description of the orbit.

    Now let's calculate the angular length of the orbit. Our inner observers take the speed of the body, the period of the orbit, and the orbit's radius and calculate

(Eq'n 1)

and our outer observers will calculate from their own measurements of those parameters

(Eq'n 2)

Our observers thus have their naive descriptions of the orbit.

    In using a single radius for the orbit I have tacitly implied that our naive orbit traces a perfect circle. Zero eccentricity gives us a very simple description, certainly, but we can't determine the apsidal precession of a perfect circle, so let's imagine that the ars in our equations represent a kind of average radius. We will replace that assumption in the step-by-step process by which we remove the naivete from our calculations. Let's start, as long as we are seeing the orbit as a circle, by recalling that we can regard an elliptical orbit as a radial oscillation superimposed upon a perfectly circular motion. That radial oscillation makes the orbiting body a natural clock, so its period of oscillation will be subject to Lorentzian time dilation. But the circular motion upon which the oscillation is superimposed won't be affected; in fact, that's the motion that brings the Lorentz Transformation into play here.

    To take that effect into account we must modify our equations for the angular length of the orbit to accommodate the dilated orbital period. Because we only want to determine how the orbit would look from a great distance out in space, we need only modify the outer observers' equation. If we take upper-case tee to be the proper period of the orbit, or the period the orbit would have if the Universe were purely Newtonian, then we would have to write Equation 2 as

(Eq'n 3)

That equation tells us that the body begins at the point that will be perihelion when we give the orbit some eccentricity, the point on the orbit closest to the primary focus of the ellipse; it moves away from the primary focus as it goes around until it reaches the aphelion, the point farthest from the primary focus; and then it moves back closer to the primary focus, passing its previous perihelion before it reaches its next one.

    Special Relativity offers one other effect that we may have to take into account -- the Lorentz-Fitzgerald contraction. How does it affect the orbit? Imagine that instead of a simple body, we have occupied our orbit with a belt studded with clocks. As we make the orbit shrink, the clocks tick off increasingly dilated time and, after the inner observers resynchronize them, display the temporal offset the creates the Lorentz-Fitzgerald contraction. What we observe then as a shortening of the belt marks an actual shortening of space itself, of the warped inertial frame that the belt occupies. But shortening a circular or elliptical patch of space without altering its radius also shortens the angle subtended by that patch: that is, what observers riding the belt measure as a 360-degree traverse from perihelion to perihelion the outer observers measure as slightly less than 360 degrees in the same proportion in which the belt's motion dilates the time ticked off the belt's clocks. Multiplying Equation 3 by the necessary factor, the inverse of the Lorentz factor, simply gives us back Equation 2: the effects of Special Relativity have no net effect upon the orbit of our body.

    Having thus considered and dismissed the effects that Special Relativity imposes upon the orbit, we take the next step and imagine pouring mass along the axis of our polar coordinate system onto the primary focus. That mass quenches the non-gravitational force holding the body in its orbit as the gravitational force builds up and takes over the job. We pour the mass from both the south side and the north side of our system to ensure that we exert no forces or torques upon the system. Thus we change none of the parameters of our orbit, except those that change as a consequence of the Schwarzschild Transformation coming into play.

    We know that the gravitational field will impose an additional time dilation in accordance with the time-distortion equation of the Schwarzschild Transformation in the form

(Eq'n 4)

If the inner observers measure a time t for the body making one complete revolution around the orbit, the outer observers will measure a time T, which contains more nanoseconds than does the time measured by the inner observers. Those extra nanoseconds make the orbit, as seen by the outer observers a fraction of a degree longer than it would be without the gravitational distortion of time. In making that statement I have assumed that the lower observers will detect no effects of the Schwarzschild Transformation because they sit on the orbit itself. Thus the outer observers must detect the effects imposed on the orbit by the virtual motion of their inertial frame across the orbit. In accordance with that proposition I will multiply Equation 2 by the coefficient in Equation 4, so we have

(Eq'n 5)

That equation describes the extension of the body's orbit by temporal distortion; gravitational time dilation making the body go past its previous perihelion before it reaches its next one. Now I want to take into account the additional distortion of the orbit caused by the relativistic shrinkage of space. We have two such effects to account and both of them come from the Schwarzschild Transformation. Indeed, directly or indirectly, both effects originate in the gravitational contraction of radial distance inferred by the outer observers when they compare their measurements with those made by the inner observers.

    First we look at the direct effect. We know that the orbit will appear to the outer observers to have a shorter radius than it does for the inner observers. In order to account this effect properly we state that, even though the orbital radius differs for both teams, the velocity of the body does not. We have

(Eq'n 6)

Because we established our body's orbit in what we might call the system's equatorial plane we also have

(Eq'n 7)

in which the cosines don't appear because they both equal one. Combining those equations gives us

(Eq'n 8)

But q = 360 degrees, so we must multiply Equation 5 by the coefficient in Equation 8 to obtain

(Eq'n 9)

    Indirectly gravitational contraction of space causes an additional distortion of the orbit through the law pertaining to conservation of angular momentum. When the radius of the orbit shrinks the velocity of the orbiting body must increase in the same proportion in order the keep the body's angular momentum from changing gratuitously. That increased velocity takes the body through an angle slightly greater than 360 degrees in the time of one orbital period, again in the proportion displayed in Equation 8. So we must multiply Equation 9 by that proportion and obtain

(Eq'n 10)

    That equation represents an exhaustive consideration of the relativistic effects that distort the length of our body's orbit. We can now calculate the amount by which the orbit precesses on each revolution by subtracting 360 degrees from that formula. But before I carry out that calculation I want to simplify my formula a little bit. We define the Schwarzschild radius of a body of mass M as

(Eq'n 11)

If we substitute the sun's mass into that equation, we find that our sun has a Schwarzshild radius of 2.956 kilometers. But the radii of the orbits of the sun's planets exceed fifty million kilometers, so the ratio RS/R will be very small, so small that we can replace Equation 10 with the first two terms of its Taylor series expansion with effectively no loss of accuracy. We thus have

(Eq'n 12)

As seen from a distance, then, the orbit's line of apsides advances on each revolution by an amount

(Eq'n 13)

    We can accept that equation as accurate enough for an orbit that has an eccentricity close to zero, but if we want to consider an orbit that has a high eccentricity, we must replace the factor 1/R by an average inverse radial distance. We have the perihelion distance Rp = a(1-e) and the aphelion distance Ra = a(1+e), in which equations a represents the length of the orbit's semi-major axis and e represents the orbit's eccentricity. We thus calculate the average of their inverses as

(Eq'n 14)

Making the appropriate substitution of that result into Equation 13 gives us the more accurate result

(Eq'n 15)

For Mercury we have a = 57,950,000 kilometers and e = 0.2056, so we calculate that the planet advances its perihelion at a rate of 0.103539 arcseconds per revolution due solely to relativistic effects. With a period of 87.99 days, Mercury makes 415.1 revolutions about Sol in a century of Terran years, so the relativistic contribution to the precession of the perihelion accumulates 42.98 seconds of arc per century. Thus we have accounted the discrepancy between the precession inferred from observations of the planet's motions and the precession calculated from purely Newtonian considerations. In this case General Relativity passes the test of comparison with Reality.



    Before we can work out the effects of the Schwarzschild metric on the motion of a planet we must devise a model of the planet's orbital motion that enables us to separate out the spatial and temporal parts. Let's begin by describing the simplest motion in a circle.

    In order to keep a point moving in a circle of a given radius at a given speed, we must apply a centripetal acceleration to that point (or, more precisely, to the object occupying and marking that point); that is, we must make the point accelerate toward the center of the circle. Imagine drawing two radial lines from the center of the circle to the circle itself and then drawing the chord connecting the two points where the lines cross the circle. If we make the angle between those two radii small enough to approach the infinitesimal, then we can describe the velocity of a body following the circle as

(Eq'n A.1)

If we then compare two adjacent chords of small angle representing the same small increment of time dt, then we can determine that the deflection of successive chords represents the acceleration that the body must have toward the center of the circle in order to remain on the circle: we have that acceleration described as

(Eq'n A.2)

    If the body in question is a planet in our solar system, then the gravitational attraction between the sun and the planet provides the centripetal acceleration necessary to keep the planet on its orbit. We thus equate gravitational force from Newton's law of gravity to the centripetal force and obtain

(Eq'n A.3)

We can also express that criterion for a circular orbit as the net force acting upon the body; that is,

(Eq'n A.4)

    If we strike the planet a blow that acts directly toward or away from the sun, we will perturb the planet's motion on its orbit, but we will have done so without changing the planet's angular momentum. The planet will no longer revolve about the sun on a path that has an unvarying radius. Before I explore that fact I want to ensure that the only non-constant factor in my equation is the orbital radius, which means that I must somehow eliminate the velocity from the equation. We know that the planet's angular momentum is equal to mvr and that it remains constant throughout the planet's motions. We know further that the planet's mass does not change, so we can define the planet's ambition as σ = vr and use it to replace the velocity in Equation A.4

(Eq'n A.5)

(Note: the word ambition, as I have used it here, reflects the original Latin meaning of "going around". The use of the word to describe people comes from the Romans' use of it to describe politicians who would "go around" canvassing for votes.)

    Now calculate the differential of Equation A.5 by way of the definition

(Eq'n A.6)

and obtain

(Eq'n A.7)

If we multiply Equation A.5 by 2dr/r and add the result to Equation A.7, we obtain

(Eq'n A.8)

in which I have used Equation A.5 multiplied by dr/r to replace the ambition term with its equivalent gravitational term. But now that equation looks like the equation describing Hooke's law, which describes the force exerted by a stretched spring. On the basis of that resemblance we can thus describe the orbit of the planet as a perfectly circular motion upon which we have superimposed a radial pendulum-like oscillation.

    On first impression that calculation should make any proper mathematician blanch. One of the facts that we learn in our study of the calculus is that if we have a function that is equal to zero (that is, F' = 0), then the differential of that function must also be equal to zero (that is, dF' = 0). But Equation A.8 violates that expectation and that fact should trouble any mathematicians reading this: the calculation that yielded that result appears perfectly legitimate and yet its result appears equally illegitimate. What did I do wrong?

    As it turns out I didn't commit any egregious crime against mathematics: I merely neglected to note that the zero in Equation A.4 does not represent a true constant, one whose differential must equal zero, but rather represents but one possible value of the force acting on the planet, the one that represents the planet moving on a perfect circle. Rather that giving us a constant zero, the equation gives us zero as a criterion that the planet must satisfy in order to move on the orbit. Equation A.4 itself is clearly differentiable with respect to radial distance, which fact tells us that we may put the planet into a state in which the net force acting on it is not a perfect zero. When we get dF' = -kdr as a result of differentiating Equation A.4, we understand it to mean that the force acting on the planet fluctuates as the planet oscillates about some mean radius as it attempts to move on a perfect circle. We can thus separate the planet's motion into two separate motions upon which Reality plays out the effects of Relativity.


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