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Starts With A Bang

When did Isaac Newton finally fail?

It took hundreds of years for Einstein to dethrone him, and even then, he was off by less than 1% of a spectacular prediction.
gravity probe b
This illustration shows NASA’s Gravity Probe B in orbit around the rotating Earth, as well as the warped spacetime that causes the Lense-Thirring effect: an effect not present in Newtonian gravity.
Credit: NASA

“To explain all nature is too difficult a task for any one man or even for any one age. ’Tis much better to do a little with certainty & leave the rest for others that come after you.” —Isaac Newton

When Isaac Newton put forth his universal theory of gravitation in the 1680s, it was immediately recognized for what it was: the first incredibly successful, predictively powerful scientific theory that described the one force ruling the largest scales of all. From objects freely falling here on Earth to the planets and celestial bodies orbiting in space, Newton’s theory of gravity captured their trajectories spectacularly. When the new planet Uranus was discovered, the deviations in its orbit from Newton’s predictions allowed a spectacular leap: the prediction of the existence, mass and position of another new world beyond it: Neptune. The very night the Berlin Observatory received the theoretical prediction of Urbain Le Verrier — working 169 years after Newton’s Principia — they found our Solar System’s 8th planet within one degree of its predicted position. And yet, Newton’s laws were about to prove insufficient for what was to come.

The problem all started not at the outer reaches of the Solar System, but in the innermost regions: with the planet Mercury, orbiting closest to the Sun. Every planet orbits the Sun not in a perfect circle, but rather in an ellipse, as Kepler noticed nearly a full century before Newton. The orbits of Venus and Earth are very close to circular, but both Mercury and Mars are noticeably more elliptical, with their closest approach to the Sun differing significantly from their greatest distance.

The orbits of the inner planets, along with a comet expected to have a near-Earth encounter in 2880. Image credit: NASA / JPL.

Mercury, in particular, reaches a distance that’s 46% greater at aphelion (its farthest point from the Sun) than at perihelion (its closest approach), as compared to just a difference of 3.4% from Earth. This doesn’t have anything to do with the theory of gravity; this is merely the conditions which these planets formed under that led to these orbital properties. But the fact that these orbits aren’t perfectly circular means we can study something interesting about them. If Kepler’s laws were absolutely perfect, then a planet orbiting the Sun would return to the exact same spot with each and every orbit. When we reached perihelion one year, then if we counted out exactly one year, we’d expect to be at perihelion once again, and we’d expect the Earth to be in the same exact position in space — relative to all the other stars and the Sun — as it was the year before.

But we know Kepler’s laws can’t be perfect, because they only apply to a massless body in orbit around a massive one, with no other masses present at all. And that doesn’t describe our Solar System at all.

We have all these other massive bodies — planets, moons, asteroids, etc. — in addition to just a single planet orbiting our Sun. In addition, the planet we’re measuring itself has mass, meaning that it doesn’t orbit the center of the Sun, but rather the center-of-mass of the planet/Sun system. And finally, for any planet that we look at that isn’t Earth, we have this other confounding feature: our planet precesses on its axis, which means that there’s a difference between how we mark time (a tropical year, which refers to the seasons and the calendar) and how the Earth returns to the same position in space (a sidereal year, which refers to a single complete orbit) from year-to-year.

Image credit: Wikimedia Commons user Tauʻolunga, of the precession of Earth’s north pole.

So we have to take all of these features into account if we want to predict how much another planet’s orbit would appear to change over time. With everything we know about Earth, Mercury, and all the other masses we’ve observed and measured, what do we expect?

For starters, the difference between a sidereal year and a tropical year is slight, but important: a sidereal year is 20 minutes and 24 seconds longer. This means that as we mark the seasons, the equinoxes and the solstices, they occur on a calendar year basis, but our perihelion shifts ever so slightly relative to that. If a circle is a 360°, then going from January 1st of one year to January 1st of the next “only” gets us 359.98604° of the way there, which means — if there are 60′ (arc-minutes) in one degree and 60″ (arc-seconds) in one arc-minute — that every planet’s perihelion will appear to shift by 5025″-per-century. That shift, if you’re wondering, appears as an advance in the orbit.

But then there are also the effects of planetary masses to take into account.

The eight planets — and a little bit more — of our Solar System. Image credit: NASA.

Each planet will affect the motion of another differently depending on its relative distance, its mass and its orbital proximity, as well as whether it’s interior or exterior to the planet in question. Mercury, being the innermost planet, is arguably the easiest one to do the calculation for: all of the planets are outer to it, and hence they all cause its perihelion to also advance. Here are the effects of those planets, in order of decreasing importance:

  • Venus: 277.9″-per-century.
  • Jupiter: 153.6″-per-century.
  • Earth: 90.0″-per-century.
  • Saturn: 7.3″-per-century.
  • Mars: 2.5″-per-century.
  • Uranus: 0.14″-per-century.
  • Neptune: 0.04″-per-century.

The other effects, like the massiveness of the individual planet in question itself, the Sun’s motion around the Solar System’s barycenter, the contribution of the asteroids and the Kuiper belt objects, and the oblateness (non-sphericity) of the Sun and planets, all contribute 0.01″-per-century or less, and so can be safely ignored.

An illustration of the known and anticipated objects in the Solar System. Image credit: NASA / JPL-Caltech / R. Hurt.

All told, these effects add up to 532″-per-century of advance, which gives us a total of 5557″-per-century when we add in the effects of the Earth’s precession. But when we look at what nature actually gives us, we saw that there’s more: we get 5600″-per-century of perihelion advance. In fact, this was known back in the late 1800s, thanks to the incredible observations of Tycho Brahe going back to the late 1500s! When you have a 300 year baseline of observations, you can detect effects that small.

There’s more precession than Newton predicts, and the big question is why. There were a few hints, if we knew where to look.

Candidate range for the hypothetical planet Vulcan. Image credit: Wikimedia Commons user Reyk.

The first idea was that there was a planet interior to Mercury with the right properties to cause that additional advance, or that the Sun’s corona was very massive; either one of those could cause the additional gravitational effects needed. But the Sun’s corona isn’t massive, and there is no Vulcan (and we’ve looked!), so that’s out.

The second idea came from two scientists — Simon Newcomb and Asaph Hall — who determined that if you replaced Newton’s inverse square law, which says that gravity falls off as one over the distance to the power of 2, with a law that says gravity falls off as one over the distance to the power of 2.0000001612, you could get that extra precession. As we know today, that would mess up the observed orbits of the Moon, Venus and Earth, so that’s out.

And the third hint came from Henri Poincare, who noted that if you took Einstein’s special relativity into account — the fact that Mercury moves around the Sun at 48 km/s on average, or 0.016% the speed of light — you get part (but not all) of the missing precession.

The overall precession of an object orbiting a central, large mass, greatly exaggerated in magnitude. Image credit: Wikimedia Commons user Mpfiz.

It was putting those second and third ideas together that led to general relativity. The idea that there was a fabric — a spacetime — came from one of Einstein’s former teachers, Hermann Minkowski, and when Poincare applied that concept to the problem of Mercury’s orbit, there was an important step towards the missing solution. The idea by Newcomb and Hall, although incorrect, showed that if gravity were stronger than Newton’s predictions by the orbit of Mercury, additional precession could occur.

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Einstein’s big idea, of course, was that the presence of matter/energy results in a curvature of space, and that the closer you are to a more massive object, the stronger gravity behaves. Not only that, but the greater the departure is from the predictions of Newtonian gravity as well.

The effect would be most extreme close to extremely massive, compact objects, like black holes, neutron stars and white dwarfs. Image credit: ESO/L. Calçada.

When Einstein finally made enough headway on his theory to predict this additional precession, his prediction — of an extra 43″-per-century — was actually thought to be too much; the Newtonian contributions were estimated slightly incorrectly, and so only 38″-per-century were predicted at the time. This discrepancy was cited as an argument against general relativity, or that general relativity at best would be an approximation to the correct step forward.

It really took the prediction that light would be bent when passing by a massive body — like the limb of the Sun — to test whether Newton’s or Einstein’s theory was correct.

Positive development of the photographic plate from the solar eclipse of 1919. You can see the stars marked by vertical lines. Image credit: F. W. Dyson, A. S. Eddington, and C. Davidson, 1919.

Newton’s theory predicted, if we want to be literal about it, that starlight would not deflect at all when it passed by the Sun, since light is massless. But if you assigned light a mass based on Einstein’s E = mc^2 (or m = E/c^2), you could find that starlight should deflect by 0.87″ when it passed by the Sun’s extreme outer limit. For a contrast, though, Einstein’s theory gave twice that amount: 1.75″ of deflection.

These were small numbers, but a joint expedition by Arthur Eddington and Andrew Crommelin during the 1919 solar eclipse, were able to measure to the necessary accuracy. The deflection they came up with was 1.61″ ± 0.30″, which agreed (within the errors) with Einstein’s predictions, and not with Newton’s. Newtonian gravity was busted.

Images credit: New York Times, 10 November 1919 (L); Illustrated London News, 22 November 1919 (R).

And that’s the story — the real story — of not only Newton’s gravity being superseded, but in what way(s) Newton’s theory came up short. There have been many other victories for general relativity since (including the 101-year-in-the-making detection of gravitational waves), but in all the cases where Newton’s and Einstein’s theories differ, it’s Einstein, with stronger gravitational effects close to massive bodies, who emerges victorious. Science marches forward, but sometimes each new step takes avery long time!

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