The Ultimate Failure of Newtonian Gravity: The Strange Behaviour of Mercury. If the solar system contained only the Sun and one planet, and if Newton were right about gravity, that planet would move in an elliptical path which retraced its tracks forever, orbit after orbit, never changing. As you now appreciate, the presence of other planets slightly messes things up because of the small gravitational tugs they exert on one another, but such effects are completely predictable. In the case of the Earth, for instance, the net effect is that the orbit precesses, which means that it slowly changes in orientation. (Please note that this is a second example of precession, not the same as the mechanism which causes a slow change in the direction in which the tip of the Earth's spin axis is pointed. You will remember that we discussed that earlier example of precession in the context of interpreting the orientation of If you have trouble visualising what the precession of the orbit means, look at page 467 of your text, where the effect is shown, in enormously exaggerated fashion, for the planet Mercury. (The change from one year to the next, or even from one century to the next, is actually very tiny.) The Earth's orbit does something similar. As you can probably imagine, the inter-planetary perturbations are complex, but can be dealt with through sophisticated numerical techniques. (The Earth's precession is completely understood in terms of the influence of the other planets.) The behaviour of Mercury is not quite so satisfactory. As far back as a couple of centuries ago, astronomers were puzzled by the particular behaviour of that tiny planet. Even after the influence of all other planets had been accounted for, the orbit of Mercury was precessing somewhat more than it should. But why? (Just to put some numbers on this, you might like to note that the calculated effects of the planets should be to make the orbital orientation shift by about 600 arc seconds - about a sixth of a degree - per century. Mercury precesses somewhat more than that, with the mysterious excess amounting to about 43 arc seconds per century.)

The Futile Search for Vulcan.

One plausible explanation was that there might be an undetected planet even closer to the sun than Mercury itself is. (Remember that of the nine known planets, Mercury is the closest to the sun.) Understandably, this reasoning was stimulated and encouraged by the prediction and quick discovery of the planet Neptune, a success which seemed to confirm Newtonian gravity in every respect. The hypothetical planet was even given the name Vulcan, in the tradition of using names from classical mythology. Since it was believed to lie very close to the sun, where it would be hellishly hot, the name of the god of fire was particularly appropriate. If such a planet existed, how might we hope to find it? Even Mercury is a very difficult target to study, because its proximity to the sun means that we can only ever see it in early dawn or late evening twilight, low on the horizon. Any hypothetical planet even closer to the sun would be a terribly difficult object to find, much less study in detail. Still, the search was begun with real enthusiasm. There are in fact three obvious ways to imagine finding the planet. 1 You might hope to see it as a bright dot of light in the early morning or late evening sky, in just the way that we observe Mercury itself. 2 You might wait for a solar eclipse, at which time the sun's light is blocked off so that you might see Vulcan as a bright dot near the obscured sun. This would mean that you could see it high in the sky, rather than having to look towards the horizon through a thick layer of the Earth's atmosphere, but since eclipses are rare this is not a very efficient strategy. 3 Finally, you might observe the sun directly (using a dark filter for safety) to see if, from time to time, you see a small dark dot crossing its face. This would be Vulcan seen in silhouette, moving between us and the sun in an event known as a transit. (By the way, there is no real danger that a sunspot might be mistaken for the sought-after planet. A sunspot is a dark feature on the surface of the sun. Since it shares the sun's slow rotation, it takes many days to cross the full face of the sun. Vulcan, on the other hand, would be orbiting so quickly that it would cross the sun's face in an hour or so.) As I mentioned in my lecture, a book has recently been published about this interesting episode in astronomical history. The search for Vulcan was pursued for many decades, and there were many claims of success followed by disproofs or withdrawals of claims. For a time, it was believed that there might exist no solid planet but perhaps instead a ring or shell of gas and dust, rather like one of Saturn's rings, which would evade detection and yet still provide the gravitational perturbations needed to explain the orbit of Mercury. In the end, though, it was recognized that there simply was no planet Vulcan close to the sun. The main reason the chase was abandoned was that an alternative, and indeed completely persuasive, answer was provided by Einstein, and the problem simply went away.

Einstein Provides the Answer.

Einstein solved this long-standing problem with the publication of his General Theory of Relativity in the year 1915. He showed that the anomalous precession is a consequence of the way gravity distorts space and controls the motions of planets when they get especially close to massive bodies, where the curvature of space is most pronounced. Newton's formulation is not quite correct in these close confines, but the General Relativistic formulation explains the observed behaviour exactly. In my discussion of Newton's Law of Universal Gravitation, I contrasted the way in which Newton and Einstein treated gravity -- Newton thinking of it as a force, and Einstein describing it in more geometrical terms. According to Newton, the Earth curves in its path because the sun pulls on it with a certain force. Following Einstein, however, we now describe gravity in geometrical terms: a lump of matter distorts space in a way which means that particles moving nearby will follow curved trajectories. I gave a simple analogy in class: imagine a big rubber sheet stretched out flat (something like a trampoline). If you roll a marble across the sheet, it will follow a straight path. But now put a heavy bowling ball gently onto the rubber sheet, so that it sits at rest in the very center. You can appreciate that the sheet will now have a pronounced `dip' in it, and the same marble will now move in a curved path when it rolls by. This is not because the bowling ball itself applies a force directly on the marble, but rather a result of the distorted geometry of the rubber sheet. (See the figure on page 455 of your text.) In like fashion, big lumps of matter deform space itself. (You have to imagine such deformations in three dimensions, rather than the two-dimensional sheet we have been discussing. Such visualisations are not easy!) Objects passing by the sun or indeed any lump of matter move along paths which are determined by these geometrical distortions. (See page 463.) So matter distorts space (and time, too), and this has an influence on how we measure intervals of time and distance. Where there are large concentrations of matter (``strong gravitational fields'') the distortion is very pronounced; far from matter, the distortions are less.

Another Prediction: The Bending of Light.

Einstein also proposed that light itself should move through space in a way which follows the distortions introduced by lumps of matter. In other words, the trajectory of light should `curve' when it passes a lump of matter. While this is true for a lump of any mass, the effects are naturally more pronounced the more massive the object is, since its gravitational effects are greater. Consequentially, you might expect the phenomenon to be important in astronomy but not in day-to-day life. Still, the suggestion came as a surprise to astronomers, who had never seen manifestations of such an effect, even when studying the stars. This may not surprise you. After all, don't we all 'know' that light travels in straight lines? Contrast that to the behaviour of ordinary matter. If I throw a piece of chalk across the room, it follows a path which is clearly curved. On the other hand, if I throw the chalk much faster, the curve it follows is less pronounced. (A baseball pitcher's fastball reaches the plate nearly on a straight line, whereas my feeble throw curves dramatically towards the dirt.) Is it indeed possible that light does curve downward ('fall' toward the ground) as it crosses the room, but that the enormous speed at which light travels hides this fact from us? If so how could we ever hope to test Einstein's bold prediction? If the drop is infinitesimally small, how will we ever measure it? As it happens, there is a way.

The Prediction Confirmed....

Einstein realised that the only way to detect the effect would be to study light passing very close to the most massive object in our vicinity - the Sun! He predicted that stars would appear to shift positions very slightly, but measureably, if their light skims past the sun's surface. (It is worth emphasising that the predicted bending is very small, because even the sun distorts space only a small amount.) The problem, of course, is to see the stars at all when the sun is in the sky! In any event, the effects can best be seen as a difference, so here is how to accomplish the task: In January (say), take a picture of a field of stars in the midnight sky. Your telescope is pointed directly away from the sun. Six months later, take another photograph with the telescope again pointed in the direction of that field of stars. The difference is that the Earth has now moved around in its orbit and is on the other side of the sun, so the sun lies directly between you and the stars! This sounds pointless, because the sun will overexpose the picture and the daytime sky will be too bright to allow us to see the background stars. But there is a clever solution: we can do the second part of the exercise at the moment of a total solar eclipse, so that the sun is masked out and the sky is dark! The testable prediction is then that the pattern of remote stars should look somewhat distorted geometrically because of the bending of the various rays of light as they pass the sun on their way to us. This famous experiment was first done in 1919. Einstein's prediction was confirmed and he was made famous. Since that time, the experiment has been repeated many times -- but not just in eclipses, because there are some wavelengths of light (such as those used by radio astronomers) for which the sun is not outrageously bright, and the experiment can be done even if the sun is seen fully visible. In every case, Einstein's theory has been shown to be precisely correct. By the way, there is an interesting socio-political sidelight to the first eclipse experiment, one which necessitated an expedition to Madagascar (where the eclipse was visible). Einstein was German, and the eclipse expedition was coordinated and run by the British astronomer Eddington in the immediate post-World-War-I years. This had a symbolic significance about the way in which pure scientific endeavour could transcend national boundaries and differences.

...and Also on Larger Scales.

I have a tank of fish at home, and when I stand in just the right position I can see multiple images of the same fish. The reason is that light from the fish can reach my eye along several different paths. Since light changes direction abruptly when it passes from the water through the glass to the air, careful positioning can reveal several images of the same fish. (A similar phenomenon explains why a stick put into the water can appear 'bent.' The rays of light abruptly change direction, or refract, as they move from the water to the air.) Light can behave similarly as it travels through intergalactic space. This is shown schematically in a figure on page 468 of your textbook, and then beautifully exemplified in images on page 469, and again on pages 686-687. Before I describe these further, let me just reemphasise that of course the light does not pass through different regions of water, glass, and air. Rather than changing direction abruptly, which would only happen if it moved from one medium into another, it follows gently curving trajectories, moving in response to the curvature of space which has been introduced by the presence of large amounts of matter along the line of sight. In this way, two beams of light from a source may start out in slightly different directions but be brought to the same point -- your eye, or your telescope! -- at the end of a long journey. Naturally, the effect is most readily detected when the path followed by the light is very long, since that allows the tiny bending effects to accumulate to a significant amount and increases the chance that a beam of light will pass close by some very massive lump of material. The objects most likely to show up as multiple images, then, are quasars or remote galaxies. (These are objects we have not yet encountered in the course. What matters in the present context is that they are very bright, so that we can see them a long way off.) Many examples of this kind of gravitational lensing have been observed in the heavens. Sometimes we see fuzzy images; on other occasions, we may see two or more crisp, focussed images, such as the `Einstein cross', a set of four independent images of a single object (shown on page 469). By the way, perhaps you wonder how we know that these are images of the same object. One answer is that we can take the light from each object and spread it out into a spectrum. These spectra turn out to be absolutely identical, which cannot be a mere accident. A second answer is that quasars sometimes vary a little in brightness, and when this happens, the images all vary in brightness in just the same way. (Think again of my fish tank analogy. If one image of my goldfish raises its right fin, so will the other! For a more trivial example, imagine standing in a room full of mirrors so that you see many images of yourself. If you stick out your tongue, so will all the others!) Sometimes the images produced are completely `out of focus', and yield rings and arcs. Examples are shown on pages 686-687 of your text. It is a real pity that Einstein, who died in 1955, did not ever get to see these impressive and incontestible examples of the way in which light is subject to the dictates of gravity!

The Limitations of Newtonian Gravity.

In a very early lecture, I pointed out that one of the perils we face in astronomy is the need to to scales beyond which we cannot ordinarily test them. For example, Newton realised that his Universal Law of Gravitation applied throughout the Solar System (since it explained Kepler's laws so neatly), but then boldly extrapolated the law to hypothesise that it should also be applicable to the remote stars and the most far-flung galaxies (although in fact he knew nothing of the galaxies). I noted that there was always a danger that some new law, as yet unimagined, might be correct on these largest scales. Interestingly, we now see that this fear was misplaced -- or rather that our first example of the breakdown of Newtonian gravity does not occur because we are trying to apply it over too great a distance. The actual failing arises because the local disturbance in space-time -- the 'ripple' introduced by the sun -- is too big for Newton's law to be more than approximately correct. This is a foretaste of what is to come: Newton's law breaks down, and the full Einsteinian treatment must be used, whenever space is locally strongly curved, thanks to the presence of a very densely packed lump of matter. Even objects which are small in overall size, such as neutron stars, scarcely 10 km in diameter, can introduce such extreme effects -- as too can black holes. On the very largest scales, as when we are describing the structure of the universe as a whole, Newtonian gravity must again be supplemented or supplanted by the more complex Einsteinian treatment. Only in weakish gravitational fields, where there are not dramatic changes in strength over small distances, can we treat gravity in the classical Newtonian fashion. Previous chapter:Next chapter

0: Physics 015: The Course Notes, Fall 2004 1: Opening Remarks: Setting the Scene. 2: The Science of Astronomy: 3: The Importance of Scale: A First Conservation Law. 4: The Dominance of Gravity. 5: Looking Up: 6: The Seasons: 7: The Spin of the Earth: Another Conservation Law. 8: The Earth: Shape, Size, and State of Rotation. 9: The Moon: Shape, Size, Nature. 10: The Relative Distances and Sizes of the Sun and Moon: 11: Further Considerations: Planets and Stars. 12: The Moving Earth: 13: Stellar Parallax: The Astronomical Chicken 14: Greek Cosmology: 15: Stonehenge: 16: The Pyramids: 17: Copernicus Suggests a Heliocentric Cosmology: 18: Tycho Brahe, the Master Observer: 19: Kepler the Mystic. 20: Galileo Provides the Proof: 21: Light: Introductory Remarks. 22: Light as a Wave: 23: Light as Particles. 24: Full Spectrum of Light: 25: Interpreting the Emitted Light: 26: Kirchhoff's Laws and Stellar Spectra. 27: Understanding Kirchhoff's Laws. 28: The Doppler Effect: 29: Astronomical Telescopes: 30: The Great Observatories: 31: Making the Most of Optical Astronomy: 32: Adaptive Optics: Beating the Sky. 33: Radio Astronomy: 34: Observing at Other Wavelengths: 35: Isaac Newton's Physics: 36: Newtonian Gravity Explains It All: 37: Weight: 38: The Success of Newtonian Gravity: 39: The Ultimate Failure of Newtonian Gravity: 40: Tsunamis and Tides: 41: The Organization of the Solar System: 42: Solar System Formation: 43: The Age of the Solar System: 44: Planetary Structure: The Earth. 45: Solar System Leftovers: 46: The Vulnerability of the Earth: 47: Venus: 48: Mars: 49: The Search for Martian Life: 50: Physics 015 - Parallel Readings.

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