Is Vulcan Still Missing?
The French astronomer, Le Verrier, predicted the existence of Neptune, but had to live with Vulcan for want of better. According to Le Verrier, "a planet, or if one prefers a group of smaller planets circling in the vicinity of Mercury's orbit, would be capable of producing the anomalous perturbation felt by the latter planet....According to this hypothesis, the mass sought should exist inside the orbit of Mercury."
The only way to reconcile the fact that every hundred years in which Mercury makes its radial journey of 36,000 degrees, the perihelion of its orbit shifts 1/10,000 th beyond the appointed destination thereby creating an error of just 0.38 arcseconds per year. That doesn't seem like so much to live with except for the fact that Newton's law of universal gravitation has been so precise in all other planetary calculations.
So Vulcan was born out of the discrepancy between prediction and observation. The amateur astronomer, Lescarbault, claimed he saw a planet revolving around the sun on a path that never exceeded 8 degrees from the sun and took 20 days to complete. This would be hard to distinguish from a mere sunspot. Yet, subsequent observations of Simon Newcomb increased the advance of the perihelion to 0.43 arcsecond per year, and calculations showed that a planet, or planets should be found some 21 million miles from the sun with an orbital period of 34 days. But, by 1878 it became apparent that there was a sufficient amount of evidence for the absence of Vulcan, asteroids, or a group of planets in an intra-Mercurian orbit.
It took over three decades for another explanation to emerge, and one from a place that was hardly expected to explain Mercury's pedalling around the sun. In the words of Levenson, in his book, The Hunt for Vulcan, "no chunk of matter was required to explain Mercury's tracks, no undiscovered planet, no asteroid bel, no dust, no bugling solar belly, nothing at all--the sun with its mass creates its dent in spacetime. Mercury, so firmly embraced by our star's gravitational field, lies deep with that solar gravity well." In so doing, "Einstein's pend destroyed Vulcan--and reimagined the cosmos."
The unexplainable gap in Mercury's precession was therefore chalked up to general relativity. But general relativity does not need mass to create a gravitational field. That is supposedly due to geometry, and even in an empty universe there are 'dents'. If Einstein had sent his calculation to a modern day physics journal, he would certainly be chided for not citing the literature. And what literature was there? Even Newton allowed for exceptions to his law of gravitation. To his inverse square law, he added a cubic law which although it did no jeopardize the closure of his elliptical orbits, had them precess so that they could make more revolutions in the time it would take the original one to make.
In fact, Newton's theorem for the apsidal angle which is the angle between the apses, or the perihelion and aphelion of the orbit was used by him to establish the inverse-square law for elliptic orbits where the center of attraction was at a focus. This and the case where the center of attraction was at the center are the only two known central forces that permit closed, bound orbits. This was proved by Bertrand in 1875. So if an inverse-cubic force is added to an inverse-square force it could cause precession of the orbit.
Without an inverse-cube term, Asap Hall in 1895 used Newton's formula to determine the deviation from an inverse-square law using the 0.43 arcseconds per year deviation that was found by Newcomb. Hall found that a term 0.00000016 had to be added to the inverse-square in order to come out with the known apsidal angle. However, a search of the literature did not turn up an attempt to apply the inverse-cubic force term that would lead to precession. Such a term would be a dipole force between two masses placed at a given distance apart. This could then be compared with the rate of precession predicted by Einstein's modified ellipse equation in the weak gravitational field limit. And when this is done, the distance of separation is precisely the Schwarzschild radius between the sun and an imaginary mass placed at 1.5 km from the sun. This is nothing compared to the 21 million miles from the sun that Vulcan was imagined to be.
And there is nothing relativistic in the nature of Mercury, or and of the solar planets, that would lead one to imagine that the speed of light would enter into the determination of the advance of the perihelion of Mercury or any other planet. Mercury's prograde rotation is slow enough that on account of its small eccentricity, 0.2056, its angular orbital velocity is greater than its angular rotational velocity near the perihelion. This causes the motion of the sun to temporarily reverse. But, by comparison to their planetary speeds, the speed of light is practically infinite. So why does it enter into the expression for the apsidal precession? Numerical it enters because it gives the correct answer of 0.00000016. From Newton's formula
where the second term on the left-hand side is the apsidal expression in degrees and the second term in the parentheses on the right-hand side is what Einstein found from his expression GM/pc^2, where M is the mass of the sun and p is the semi-latus rectum. It's the square of the speed of light that cuts the number down from several thousand to something of .00000017.
Notwithstanding, what would appear to explain the prograde of Mercury's orbit surely cannot explain the retrograde of Venus's orbit nor Uranus'. According to Newton's theory that would require a force falling off more slowly than the inverse-square law. Since general relativity doesn't deal with potential and forces, it must remain mute on the subject.