Mercurys Perihelion

Chris Pollock

March 31, 2003

Introduction In 1846, the French Astronomer Le Verriere, doing calcu-lations based on Newtons theory of gravitation, pin-pointed the position of amass that was perturbing Uranus orbit. When fellow astronomers aimed theirtelescopes at his location, they recognized, for the first time, the eighth planet.Newtons theory had reached its zenith. Shortly after, however, it became clearto Le Verriere that additional mass, nearer to the sun than Mercury, was neededto explain the strange advance of Mercurys orbit. When no such mass was ob-served, astronomers began to doubt Newtons theory. Then, along came AlbertEinstein, whose theory nearly perfectly explained Mercurys erstwhile mysteri-ous motion. This essay is a history of Newtons theory of gravity, the enigmaof Mercury, and Einsteins convincing solution. It will blend together mathe-matical and physical theories with a narrative about the brilliant scientists whochose to tackle the problem of gravitation. In particular, this paper will showthe following computations:

1. A derivation of Keplers First Law concerning elliptical orbits from New-tons Law of Gravitation and Newtons Second Law.

2. An outline of a method to approximate non-relativistic perturbations onMercurys orbit by assuming external planets are heliocentric circles ofuniform linear mass density.

3. A calculation of relativistic perihelion shift using Einsteins theory of rel-ativity and the Schwarzschild solution.

Pre-Newtonian Theories and Ideas The problem I set out to explain,the advance of Mercurys perihelion, was of tantamount importance to the as-tronomical community. This seemingly miniscule enigma stood glaringly in theway of humanitys understanding of the solar system. Lets begin by reviewingthe early evolution of celestial theories.

In the second century CE, the Greek astronomer Claudius Ptolemy hypoth-esized that the sun, moon, and planets orbited the earth along circles calleddeferents. On a smaller scale, they travelled along smaller circles, called epicy-cles, whose centres moved along the deferents. This theory was sufficientlyaccurate to explain observations of the time. It dominated until around 1843,when Copernicus published his book, de Revolutionibus Orbium Caelestium.His solar system was heliocentric but retained the orbital deferents and epicy-cles of Ptolemys theory. It is interesting to note that Copernicus had difficultyexplaining Mercurys motion and once commented that this planet has . .. influenced many perplexities and labours on us in our investigation of itswanderings. (from Baum, 11)

More exact naked-eye planetary observations were taken by Tyco Brahe,who, I discovered while writing this paper, did not cut off his own nose. I hadalways believed, courtesy of my eleventh grade science teacher, that the greatDanish observational astronomer removed his nose to allow his face a betterseal with his telescope. My teacher, it turns out, was doubly dishonest since

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Brahe lost part of his nose in a duel with a student and telescopes had not beeninvented in 1600, when Brahe made his observations.

A young mathematician who worked with Brahe examined the elders notesand tried to calculate the orbit of Mars. Although an apparently simple task,this was disconcertingly complex since planets appear to change speed and di-rection based on the motion of not only the planet but also Earth.

When Kepler finished his calculations, he determined that Mars moved in anellipse with the sun at one focus. Kepler became quite proficient at predictingMercurys passes between the Sun and Earth, called transits. By hand, hecalculated Mercurys 7 November 1631 transit time, accurate to within five hours(Baum, 15), although he died before witnessing the event he had predicted. Asubsequent transit, on 23 October 1651, was predicted, using corrections ofKeplers calculations, with an accuracy of a few minutes.

Based on his planetary observations, Kepler made the following statements,known collectively as Keplers Laws (Stewart, 897):

1. A planet revolves around the Sun in an elliptical orbit with the Sun atone focus

2. The line joining the Sun to a planet sweeps out equal areas in equal times.

3. The square of the period of revolution of a planet is proportional to thecube of the length of the major axis of its orbit.

And so, by the middle of the 17th century, kinematic data on planetary pathswere fairly accurately known. Hence, all of the pieces were on the table for abrilliant theorist (read: Newton) to assemble, explaining the dynamics behindplanetar motion.

Newtons Law Issac Newton was born on Christmas Day, 1642. Sincehe was a failure at farming, his mother sent him to university. In the middleof his study at Cambridge, the plage broke out, and the school was closedto students for 1665-1666. Newton returned home, and in this marvellouslycreative period, wrote about both gravitation and calculus. On the formerquestion, he considered a rock twirling around on the end of a string. Therock, he knew, tended to launch, but the string provided a counteracting force.Newton wondered, then, what provided a counteracting force in the case ofplanetary motion. Could it be gravity, the force that held people on the Earthssurface?

In his magnum opus, 1687s Philosophiae Naturalis Principia Mathematice,Newton calculated that the path of planets would be elliptical if they weresubjected to a force of gravitation that varied with the inverse square of theseparation between the planets and the Sun. Starting from Newtons SecondLaw and the Newtons Law of Gravitation, it is possible to prove each of KeplersLaws. Since Keplers First Law deals with the elliptical shape of planetaryorbits, a derivation is included below. The discussion that follows is based onChapter 11.4 of Stewarts Calculus. This is the so-called one-body problemsince the system under consideration contains a test particle (the planet) moving

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under the attraction of a massive body (the Sun). The planets mass is assumedto be so small relative to that of the Sun that the Sun remains fixed in space.Also, the effects of all other planets are neglected.

Lets begin by stating Newtons Second Law:

F = ma (1)

where F is the force experienced by the planet, m is its mass, and a is itsacceleration.

And Newtons Law of Gravitation:

F = GMmr3

r = GMmr2

u (2)

The coordinate system is radial. F is the force experienced by the planet, Gdenotes the gravitational constant, r = r(t) is the planets position vector, u isa unit vector in the direction of r, and M and m are the masses of the Sun andplanet, respectively. Further, r = |r|, v = r, and a = r.

First, we will show that the planet moves in a plane. Equating the Fs from(1) and (2) gives

a =GMr3

r (3)

hence a and r are parallel, sor a = 0

From the properties of cross products, we know that

d

dt(r v) = r v+ r v

= v v+ r a= 0+ 0= 0

So, we can conclude that r v is a constant vector, say h. We can assumethat r and v are not parallel. This implies that h 6= 0. We can now concludethat r and h are perpendicular, so the planet always lies in a plane through theorigin perpendicular to h. Planetary motion, according to Newtons Laws, isplanar.

Now, we can prove examine the shape of the orbit within this plane. Letsstart by rewriting h:

h = r v= r r= ru (ru)= ru (ru + ru)= r2(u u) + rr(u u)= r2(u u)

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So, combining the above result with (3) gives

a h = GMr2

u r2(u u)= GMu (u u)= GM [(u u)u (u u)u]

Since u is a unit vector, u u = |u|2 = 1, so

a h = GM [(u u)u u] (4)

Now, lets consider the (u u) term in (4). Since u u is constant,d

dt(u u) = 0

u u+ u u = 0 u u = 0

Substituting into (4) we get

a h = GMu

But h is a constant vector, so

(v h) = v h= a h

(v h) = GMu

Integrating both sides yields

v h = GMu+ c (5)

where c is a constant vector.Since the planets motion is confined to an arbitrary plane, we can, without

loss of generality, call this the xy-plane. Recalling that h is perpendicular to thisplane, lets define a standard basis vector k in the direction of h. Looking at (5)and recalling that (v h) and u both lie in the xy-plane, we can conclude thatc is also in the xy-plane. Lets choose the x - and y-axes such that the standardbasis vector i is in the direction of c. There is really nothing special aboutthis choice. Any set of axes would do just fine. Lets go one step further andcall the angle between c and r. Hence (r, ) is the planets position in polarcoordinates. If, at this point, it is unclear where we are heading, recall that weset out to show that the planets path is elliptical. So, ultimately, were seeking

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an expression for r(). Lets look at an expression for r (v h), substitute in(5), and turn the crank . . .

r (v h) = r (GMu+ c)= GMr u+ r c= GMru u+ |r||c| cos = GMr + rc cos

where c = |c|.Now, lets solve for r in the previous equation.

r =r (v h)

GM + c cos

Define e = cGM , and substitute into the above equation:

r =r (v h)

GM [1 + e cos ](6)

So, we now have the desired equation for r. Lets simplify the numerator of(6).

r (v h) = (r v) h= h h= h2

where h = |h|. So, (6) becomes

r =h2

GM [1 + e cos ]

=eh2

c[1 + e cos ]

And, for one final simplification, lets rename h2

c as d. So

r =ed

1 + e cos (7)

This is the equation of a conic section in polar coordinates. The value of ein the denominator determines the type of conic section (7) represents. Sinceplanets have closed orbits, 0 < e < 1 and r represents an ellipse. Further, e isthe eccentricity of Mercurys orbit, and one of the ellipses foci is located at theorigin. So, beginning with only Newtons Second Law and Law of Gravitation,

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we have verified Keplers First Law, that planets moves in an elliptical orbitwith the sun at one of the foci.

Of course, the actual solar system is far more complex than this model.Newtons central idea, that planetary motions are governed by an inverse squareforce of gravity, gave mathematicians and astronomers the tools to explain theclockwork motion of the planets, as long as precise masses and positions wereknown for all planets. Shortly after Principias publication, scientists began touse these laws to make successively more accurate predictions.

Applying Newtons Amazing Law of Gravitation Comet Halley hadbeen predicted to return, by Halley himself, in winter 1758-59. In 1757, Alexis-Claude Clairault, of France, and two assistants rushed to find a more precisereturn date. Using Newtons Law of Gravitation and considering perturbationscaused by Jupiter and Saturn, they arrived at a date of mid-April 1759. CometHalleys perihelion, the point in its orbit nearest to the sun, was only 33 daysearlier than they had predicted. This error was due mainly to computationalshortcuts they made to ensure that they finished their calculations before CometHalley arrived.

William Hanover, who was born in Hanover in 1738 but moved to England in1757, made a shocking discovery in 1781, making use of the telescope. He founda seventh planet orbiting the Sun. Uranus became the first planet discoveredsince antiquity. As predicted by Newtons theory, Uranus followed an ellipticalpath. The astronomers of the world excitedly turned their attention to this newplanet.

There is one figure who played so important a role in verifying and employingNewtons amazing theory that it is necessary to look at his life in more detail.That man is Urbain Jean Joseph Le Verriere, born n Normandy in 1811. Heattended Ecole Polytechnique and graduated with a degree in chemistry. Hereceived an appealing job offer that required him to move away from Paris andhis girlfriend. He declined the job, chose marriage, gave up chemistry, and endedup in astronomy through a fortuitous job offer.

In 1839, he began calculating the stability of the solar system and planetaryorbits. In 1841, then, he applied his unique analytic skills and almost superhu-man endurance for calculation to master the motion of Mercury. (Baum, 71)This planet orbits nearest the sun and has a large eccentricity and short period.A detailed description of its motion had, so far, eluded theorists and stood outas a critical test for Newtons Gravitation. In 1845, Le Verriere gained reknownfor calculating Mercurys 1845 transit of the Sun to within 16 seconds (Baum,73). Unsatisfied by this error, however, he did not publish his tables of Mer-curys motion. At this point, he left Mercury (!) temporarily and turned toanother problem.

Since the discovery of Uranus, its orbit had been lagging behind predictions.Le Verriere, always sure of Newton, set to calculating the position and mass ofan object that could bring about the observed lag in Uranus orbit. He beganwith an assumption that the unknown mass must be exterior to Uranuss orbit,since a perturbation had not been observed in Saturns orbit. He then tried tocalculate the location, mass, and orbital parameters of such an object.

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In mid-1845, a young Cambridge mathematician, John Couch Adams, wasalso considering the same problem. Adams was, in fact, the first to theorize thenew planets position, but his results were not well disseminated. By 1846, thefamous Le Verriere too issued a prediction as to the new planets position, and ateam at the Berlin Observatory found it almost immediately, only 55 arc minutesfrom Le Verrieres calculation, and 2.5 degrees from Adams. The director of theobservatory exclaimed that this was the most outstanding conceivable proofof the validity of universal gravitation (Baum, 117). Indeed, Newtons theoryhad reached its peak. Its power, in the hands of a talented mathematician orastronomer like Le Verriere of Adams, seemed without bounds. With anothersuccess behind him, Le Verriere then returned to the problem of Mercury.

The Hunt for an Intermercurial Planet In 1854, Le Verriere was ap-pointed director of the Paris Observatory. He undertook, with the assistance ofmany human computers, a study of the whole solar system. His dream was toexplain every minute movement of the jewelled clockwork that seemed, for anelusive moment, to be stable, self-adjusting, and eternal (Baum, 2).

In 1859, he began examining observations of Mercurys motion in detail,relying mainly on 14 very accurate solar transit times, recorded between 1697and 1848. He concluded that the ellipse of Mercurys orbit was precessing slowly,which was expected, since the other planets of the solar system were expectedto Mercurys perihelion to precess. The observed precession amounted to aperihelion advance of approximately 565 seconds of arc per Earth century. LeVerriere then calculated an expected precession by considering the effects forceeach planet exerted on Mercury. This was an accurate, though computationallydemanding technique. The outer planets, Le Verrier predicted, should cause anadvance of 527 seconds of arc per century, leaving a residual 38 seconds thathe was not able to explain using Newtons theory (Baum, 136). In the nextsection, we will calculate an approximation of this Newtonian perturbation.

An Approximation of the Newtonian Precession of Mercurys Or-bit It is possible to model, very precisely, the Newtonian force each planet wasexpected to exert on Mercury. An elegant approximation, however, was de-scribed by Price and Rush in their paper entitled Non-relativistic contributionto Mercurys perihelion precession. They replaced each of the outer planetsby a ring of uniform linear mass density. Since Mercurys precession is slowcompared to the ortits of even Uranus, their approach yields a fairly accuratetime-averaged effect of the moving planets effects. An sketch of their calcula-tions follows.

First, lets replace each planet by a ring with linear mass density given bythe follwing equation:

i =Mi2piRi

where i is the linear mass density along the orbit of the ith planet from thesun, Mi is the mass of the ith planet, and Ri is the radius of the ith planetsorbit, which is assumed to be circular.

We can calculate the gravitational pull exerted on Mercury by each of theexterior planets by considering the radial components of the force exerted by

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differential angular elements on Mercury using Newtons Law of Gravitationand integrating over the circle. Please see page 532 of Price and Rushs paperfor further details on this integration.

This yields

Fi = [Gpiima

R2i a2]u (8)

where Fi is the radial force exerted on Mercury by the ith planet, a is Mercurysdistance from the Sun, and u is a unit vector for Mercurys position. Note thatsince Ri > a, Fi in (8) is positive, so the force exerted on Mercury by eachexternal planet is directed outward, opposite to the force exerted by the Sun.

Now, we will refer to some results from Classical Mechanics. From the re-quirements that planetary orbits are stable and closed, the following differentialequation relates gravitational force and planetary motion:

(r) = m(r 2r)where (r) is the total applied force, and represents differentiation withrespect to time.

Also, the planets angular momentum, mr2, is a constant, say h. Substi-tuting this value into the above equation yields

(r) = m(r h2

m2r3)

Now, lets consider an orbit for Mercury that oscillates about a circularorbit of radius a. Using a Taylor series approximation and solving the resultingdifferrential equation gives an equation for apsidal angle, which is defined as theangle between the perihelion and aphelion:

=pi

3 + a[(a)

(a) ](9)

where is the apsidal angle, (a) is the net central force, and a is the radiusof the circular orbit around which we are perturbing.

Now, lets use (8) to find the sum of the forces of all of the external planetaryrings on Mercury:

F (a) = Gpim9i=2

ia

R2i a2= 7.587 1015N (10)

The numerical value was obtained by substituting masses and orbital radii forall external planets into (10).

The Gravitational force exerted by the Sun on Mercury is F = 1.318 1022N , and the net force that Mercury experiences is given by (a) = F+F (a).Lets differentiate this expression and multiply by a.

a(a) = aF + aF(a) (11)

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Lets now examine the terms on the right-hand side of (11). Differentiating(10) yields:

F (a) = Gpim9i=2

i(R2i + a2)

R2i a2(12)

And F, the magnitude of the force exerted on Mercury by the Sun, is givenby Newtons Law of Gravity

F = GMsma2

where Ms is the mass of the Sun. Differentiating,

F =2GMsa3

=2aF

aF = 2F (13)Substituting (12) and (13) into (11) gives the following for Mercurys apsidal

equation: =

pi3 +

2F+Gmpia9

i=2 iR2i+a2

(R2ia2)2

F+F (a)

(14)

We now apply a binomial expansion on both the numerator and denominatorof (14) and neglect all terms higher than first power of F (a)F :

= pi(1 F (a)F

Gmpia

9i=2 i

R2i+a2

(R2ia2)22F

) (15)

Now, compute the numerical value for by substituting the value of F (a)calculated in (10), F, and mass and orbital radius for the exterior planets into(15):

= pi(1 + 9.884 107)where is the angle between perihelion and aphelion. Please see p533 of Priceand Rushs article for the numerical valuse used in calculating .

The precession of Mercurys orbit per revolution, then, is

2 2piT

where T is the period of Mercurys orbit, 87.969 days. So, in more conventionalunits,

precession =531.9arcseccentury

(16)

Le Verriere predicted a perihelion advance of 527 arcseconds per century,and this approximation yields 531.9 arcseconds. Still, Newtons theory was notable to explain the observed perihelion precession using known masses in the

9

solar system. Le Verrier, naturally, tried to apply the same technique that hadled him to the discovery of Neptune and began looking for a missing body inthe solar system.

The Hunt for an Intermercurial Planet An increase of approximately10 percent in Venus mass would explain Mercurys perihelion advance, but itwould also affect Earths orbit in a way that had not been observed. Since themissing mass must not affect Earth, Le Verriere decided that it must be nearerto the Sun than Mercurys orbit. And so began the hunt for Le Verrieres ghostplanet, or rather planets. He quickly realized that a single planet so near theSun would have been bright and, hence, visible during solar eclipses. Since nosuch planet had been observed during past eclipses, Le Verriere hypothesized,instead, that the mass was in the form of many small bodies.

In 1860, he visited a small-town French doctor and part-time astronomerwho had recorded observations during what he believed to be a transit of anintermercurial planet. Le Verriere, convinced by the mans story, released thenews of the new planet, which was quickly dubbed Vulcan. He had, again,delighted the French scientific community. The team of Newton and Le Verrierehad again triumphed. Or so it seemed.

Based on the doctors observations, Le Verriere calculated the planets dis-tance from the sun, 0.147AU, and period, 19 days 17 hours (Baum, 156). Theastronomical community tried again and again to observe the elusive planetVulcan, but, as time passed with no further sightings, doubts as to the planetsexistence began to mount. With so much attention directed at observing thearea around the sun during subsequent eclipses, and with no, or at least veryfew, credible sightings, most astronomers lost faith in Vulcan. Until his deathin 1877, Le Verriere remained utterly convinced that the missing mass existedand would eventually be found, verifying the supremacy of Newtons Law ofGravitation.

By 1890, however, almost nobody believed a sufficient amount of matterwould be found inside Mercurys orbit to explain the perihelion advance. In1895, Newcomb corrected some inconsistencies in planetary mass and repeatedLe Verrieres calculations. He found an extra perihelion shift of 43 per century,slightly larger than Le Verrieres result. Since hidden mass was, by this time,out of the question, perhaps, he thought, the problem lay with Newtons Law ofGravitation. If, for example, the exponent in the denominator were 2.00000016(Baum, 233), then the motion of Mercury could be more accurately explained.

This represented a true paradigm shift. While formerly observations werequestioned and Newtons theory was unassailable, now credible scientists beganto question the foundations of Newton Law of Gravitation. The observationalproblem was, essentially, closed. The theory problem, in contrast, was wideopen.

Einsteins General Theory of Relativity In November 1915, AlbertEinstein, sitting at a desk in Berlin, wrote his General Theory of Relativity. Itis ironic to note that Newtons theory had peaked in the same town, only 70years earlier, when Neptune was discovered at the Berlin Observatory.

Einstein did not set out to solve the problem of Mercurys perihelion shift.

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In fact, the answer fell out as a neat consequence of his theory. In Newtonstheory, as demonstrated previously, the motion of planets are ellipses with theSun at one focus. Here, I will show that Einsteins Theory of General Relativitychanges this conclusion, although this change is minor in most cases.

The general relativistic calculation that follows will consider the motion of atest particle in the gravitational field of a massive body. The test particles massis assumed to be so small that it has no effect on the massive body. Fortunately,the Schwarzschild solution to Einsteins field equations describes precisely thiscase.

Here, I will assume that the reader if familiar with the Schwarzschild so-lution. A good derivation and discussion of the Schwarzschild solution can befound in Hans Stephanis General Relativity. Schwarzschilds spherically sym-metric vacuum solution has a line element

ds2 = (1 2mr)dt2 1

1 2mrdr2 r2(d2 + sin2d2) (17)

where m, in our case, is the mass of the sun in relativistic units.Starting from the Schwarzschild line element, it is possible to deduce the

motion of a test mass (a planet). Please refer to section 15.3 of Ray DInvernosIntroducing Einsteins Relativity for further details. Before we begin, though, itis interesting to note that Einstein reached the same conclusion about Mercurysperihelion precession in 1915, without the use of Schwarzschilds solution.

Since the test mass (Mercury) moves along a timelike geodesic, the La-grangian is identical to kinetic energy, and gxx = 1, where , = 0, 1, 2, 3.

So, the Lagrangian, L is as follows:

L =m

2v2

=m

2g

dx

d

dx

d

=m

2gx

x

where x = dxd and is proper time.So, from (17), the Lagrangian for Mercurys force-free motion is given by

the following equation:

L =m

2[(1 2m

r)t2 1

1 2mrr2 r22 r2sin22] = m

2(18)

Let x0 = t, x1 = r, x2 = , and x3 = . Recall the Euler-Lagrange equation:

L

xa dd

(L

xa) = 0 (19)

Lets now apply the Euler-Lagrange equation to (18). For the a = 0 case:

d

d[(1 2m

r)t] = 0 (20)

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For the a = 2 case:d

d(r2) r2sincos2 = 0 (21)

And for the a = 3 case:d

d(r2sin2) = 0 (22)

It is not necessary to take the a = 1 case, which corresponds to r. SinceMercurys motion involves four equations, t = t(), r = r(), = (), and = (), equations (18), (20), (21), and (22) provide sufficient information.We have chosen to omit the a = 1 case since differentiating (18) with respect tox1 = r will clearly make the biggest mess.

As with the Newtonian case, we first must establish whether planar motionis possible. Lets consider motion in the equatorial plane and check that it isstable. So, at = 0, lets assume that = pi2 and = 0. Since =

pi2 , (21) gives

d

d(r2) = 0

r2 = constantAt = 0, r2 = 0, so r2 = 0 for all . If we assume that r 6= 0, then = 0for all . Hence = pi2 for all , and planar motion is possible. Lets considermotion in the xy-plane.

Integrating (22) givesr2 = h (23)

where h = constant. Hence, h represents angular momentum. Next, integrate(20) to give

(1 2mr)t = k (24)

where k = constant.Now, substitute (24) and = pi2 into (18):

k2

1 2mr r

2

1 2mr r22 = 1 (25)

Let u = 1r . So

r =dr

d

=d

d(1u)

= 1u2

(du

d)(d

d)

= 1u2

(du

d)hu2

12

r = h(dud

) (26)

Pretty soon, well be at a tidy differential equation for Mercurys motion.Substitute (23) and (26) and u = 1r into (25)

k2

1 2mu h2 dud

1 2mu h2u2 = 1

Multiply through by 12muh2

k2

h2 (du

d)2 u2(1 2mu) = 1 2mu

h2

And, rearranging this expression will give a first order differential equationfor Mercurys motion:

(du

d)2 + u2 =

k2 1h2

+2mh2

u+ 2mu3 (27)

By differentiating (27) and dividing through by 2, we get a second orderdifferential equation for Mercurys motion:

d2u

d2+ u =

m

h2+ 3mu2 (28)

This term very closely resembles the differential encountered in Newtonianmotion, except there is an additional 3mu2 term in (28). This term, however,is very small. In the case of Mercury, for example, the ratio of the terms on theright-hand side of (28) is on the order of 107. We can, then, solve (28) using aperturbation method. Lets introduce a parameter = 3m

2

h2 . Also, lets expressdifferentiation with respect to with a prime and rewrite (28):

u + u =m

h2+ (

h2u2

m) (29)

All that remains is solving this differential equation. Lets assume the solu-tion has a form u = u + u1 + O(2). Now, we will differentiate this solutiontwice, substitute it into (29), and rearrange:

u + u m

h2+ (u1 + u1

h2u2m

) +O(2) = 0 (30)

For the first approximation of a solution, lets equate the coefficients of ,2, . . . to zero. Then u = mh2 (1 + ecos) is the zeroth order solution to(30). (Actually, the solution is u = mh2 (1 + ecos( )), but we can set to zero for simplicity) This solution can be verified by differentiating twice andsubstituting into (30). Remembering that u = 1r , this solution agrees with theNewtonian solution.

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Now, lets examine the coefficient of in (30):

u1 + u1 =h2u2m

=m

h2(1 + ecos)2

=m

h2(1 + 2ecos+ e2cos2)

u1 + u1 =m

h2(1 +

12e2) +

2meh2

cos+me2

2h2cos2 (31)

Lets try a general solution u1 = A + Bsin + Ccos2 and solve for itscoefficients.

u1 = Bsin+Bcos 2Csin2u1 = 2Bcos2Bsin 4Ccos2

So,u1 + u1 = (A) + (2B)cos+ (3C)cos2 (32)

Comparing (32) to (31), we arrive at the following values for A, B, and C:

A =m

h2(1 +

12e2)

B =me

h2

C = me2

6h2

Hence,

u1 =m

h2(1 +

12e2) +

me

h2sin me

2

6h2cos2

And, at last, the general solution to first order is

u u + u1

u u + mh2

[1 + esin+ e2(12 1

6cos2)] (33)

Examining the correction term in (33), we see that the esin term increasesafter each revolution, and hence becomes dominant. Lets substitute our solu-tion for u, neglect the other terms in the correction, and obtain a simplifiedversion of (33):

u mh2

(1 + ecos+ esin)

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u mh2

[1 + ecos[(1 )]] (34)It is straightforward to check that (34) satisfies (30) to first order by differ-

entiating and substituting. Examining (34) we see that Mercurys orbit is nolonger an ellipse. It is still periodic, but the period is now given by the followingequation:

period =2pi1 2pi(1 + ) (35)

And Mercurys perihelion precession per orbit, in relativistic units, is givenby subtracting 2pi from its period

precession 2pi = 6pim2r

h2r(36)

where is dimensionless, mr is the mass of the sun in relativistic units, and hris Mercurys angular momentum in relativistic units. converting mr and hr intonon-relativistic units gives a more useful form of the perihelion shift equation:

precession 6piG2m2

c2h2(37)

where G is the Gravitational constant, m is the suns mass, c is the speed oflight, and h is Mercurys angular momentum per unit mass.

Lastly, lets use Keplers second and third laws to rewrite an approximationof Mercurys perihelion precession in a more convenient form.

From Keplers second law,dA

dt=

L

2where A is the area swept out by the orbit, L is angular momentum, and isreduced mass. Integrating this equation over one elliptical orbit,

piab =L

2T

hT2

For an ellipse, b2 = a2(1 e2), so

T 2 =4pi2(1 e2)a4

h2(38)

From Keplers third law,

T 2 =4pi2a3

G(m+mm)

where mm is Mercurys mass. Since mm is very small compared to m, it can beneglected:

T 2 4pi2a3

Gm(39)

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From (39), we can solve for G2m2

G2m2 =16pi4a6

T 2(1T 2

) (40)

And now, substituting (38) into (40) gives

G2m2 =4pi2a2h2

T 2(1 e2) (41)

Finally, combining (38) and (37) gives an equation for Mercurys relativisticperihelion precession per orbit:

precession =24pi3a2

cT 2(1 e2) (42)

where a is the semimajor axis of Mercurys orbit, c is the speed of light, T isthe period of Mercurys orbit, and e is the eccentricity of Mercurys orbit.

Using values from NASAs Solar System Bodies: Mercury website, this givesa perihelion shift of 42.9 seconds of arc per century. That Einsteins theoryof General Relativity so elegantly explained Mercurys perihelion shift servedas an early verification of the theory. Shortly after, during the eclipse of 21September 1922, light deflection was observed, verifying another prediction ofEinsteins theory.

A more striking example of perihelion shift than Mercurys has been ob-served. In 1975, binary pulsar PSR 1913+16 was discovered. The two massivesources, in very tight orbit, display a perihelion advance of 4.23 degrees peryear!

ConclusionsThe perihelion shift was initially observed in only Mercurys orbit since it

is nearest to the sun, and hence experiences the greatest perihelion drift. SinceMercurys orbit is farily eccentric, this perihelion shift is easily detectable (Fre-undlich, 40). If it were more circular, on the other hand, it would have beenmuch more difficult to observe.

More recently, significant perihelion drift has been observed in other planets.Einsteins calculations predict a shift for Venus of 8.6 seconds of arc per century,and 8.4 seconds has been observed. Similarly, relativity predicts a shift for Earthof 3.8 seconds of arc per century, while 5.0 seconds has been osberved (DInverno,198).

Einsteins paper elegantly and convincingly explained the observed drift inperihelion without the need for ghost planets or asteroid belts. From Einsteinsoriginal 1916 paper, entitled The Foundations of the General Theory of Relativ-ity: Calculation gives for the planet Mercury a rotation of the orbit of 43 percentury, corresponding exactly to astronomical observation (Le Verriere); forthe astronomers have discovered in the motion of the perihelion of this planet,after allowing for all disturbances by other planets, an inexplicable remainderof this magnitude. (Lorentz et al., 164)

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ReferencesBaum, Richard and Sheehan, William. In Search of Planet Vulcan: The

Ghost in Newtons Clockwork Universe. Plenum Trade, New York. 1997.Callahan, James J. The Geometry of Spacetime: An Introduction to Special

and General Relativity. Springer, New York. 1991.Davis, Phil (Web curator). Solar System Bodies: Mercury. http://solarsystem.nasa.gov/

features/planets/mercury/mercury.html. Updated 18 March 2003.DInverno, Ray. Introducing Einsteins Relativity. Oxford University Press,

Oxford. 1993.Freundlich, Erwin. The Foundations of Einsteins Theory of Gravitation.

Translated from German by Henry L. Brose. Cambridge University Press, Cam-bridge. 1920.

Price, Michael P., and Rush, William F. Non relativistic contribution toMercurys perihelion. in American Journal of Physics 47(6). 531-534. June1979.

Lorents, H. A., Einstein, A., Minkowski, H., and Weyl, H. The principleof relativity: a collection of original memoirs on the special and general theoryof relativity. contained The foundation of general relativity, by A. Einstein.Dover, New York. 1952.

Roseveare, N. T. Mercurys Perihelion from Leverriere to Einstein. CaledonPress, Oxford. 1982.

Stephani, Hans. General Relativity: An introduction to the theory of thegravitational field. Cambridge University Press, Cambridge. 1996.

Stewart, John. Calculus: Fourth Edition. Brooks/Cole, Pacific Grove. 1999.Torretti, Roberto. Relativity and Geometry. Permagon Press, Oxford. 1983.

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