Chapter 2 Gravity in the Solar System

  The Solar System, including the Earth, is our most immediate laboratory for observing the consequences of gravity. The gravitational field in the Solar System is dominated by the Sun, which is far more massive than any of the planets.

  In orbits that are relatively close to the Sun are the four smallest planets: Mercury, Venus, Earth, and Mars. Further out, there are four much larger planets: Jupiter, Saturn, Uranus, and Neptune. The Solar System contains a number of other objects, such as comets, asteroids, moons, and man-made spacecraft. By observing the motion of these objects, and in some cases by interacting with them, we can learn a great deal about the behaviour of gravity.

  The study of experimental and observational gravity in the Solar System took of in earnest during the latter half of the 20th century. While astronomers had been tracking the motion of the planets for centuries, the development of new technologies and methods in the 20th century allowed observations and experiments to be carried out in ways that had never previously been possible. In order to try and present the results of this work in some kind of sensible order, I will classify them into experiments that probe the foundational assumptions of gravity theories; experiments that probe Newton’s law; and experiments that probe the subtle efects that result from Einstein’s theory.

  Testing foundational assumptions

  There are a number of foundational assumptions that go into the modern theory of gravity. These include the fact that the rest mass of an object should be independent of its position, and independent of its motion with respect to other bodies. Other assumptions are the idea that the speed of light should be the same in every direction, and that all objects should fall at the same rate (in the absence of non-gravitational forces). Over the course of the past century all these ideas have been tested to extremely high accuracy. I』ll outline some of the best of these tests here, before we move on to thinking about experiments that test the particulars of Newton’s and Einstein’s theories of gravity.

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  Let’s start of by thinking about mass. Recall that mass is the quantity that tells us how much force we need to apply to an object in order to make it accelerate by a fixed amount. It’s thought to be a property of the object itself. This is diferent to weight, which is the name of the downwards force that an object exerts on your hand when you hold it, and which would be diferent if you held the same object while standing on a diferent planet. It is mass that appears in Newton’s law of gravity, and it is mass that was shown to be equivalent to energy in Einstein’s famous formula (E = mc2). Because these two equations are so central to gravity, we need to have some idea of whether or not the mass of an object really is independent of its position and motion in a gravitational field. This can only be done experimentally.

  Oddly enough, the best test of whether mass depends on position is to look at how light changes colour as it travels through a gravitational field. The basic idea behind this test is that photons (particles of light) should lose energy as they escape from the gravitational fields of massive objects such as stars or planets. This is because it takes energy to pull something upwards through a gravitational field. So, just as you would have to use energy to run up a flight of stairs, it takes a photon some amount of energy to travel upwards from the surface of the Earth, or away from the surface of the Sun. A change in energy of a photon results in a change in its colour (its wavelength), and so a beam of light travelling through a gravitational field should be expected to have a diferent colour depending on how far it is above the source of that field. This is the reason why light detected far from the surface of a star is found to be slightly redder, or longer in wavelength, than it was when it was emitted. The efect is known as the gravitational redshifting of light.

  So, how can the gravitational redshifting of light be expected to tell us anything about whether the mass of an object depends on its position in a gravitational field？ To answer this, let’s consider the most direct way of measuring the mass of an object in a gravitational field: lifting it upwards with a winch and recording the amount of energy it took to do so. This energy should be directly related to the mass of the object, so recording the energy required to lift the object between two diferent heights should tell us something very direct about its mass at, and between, those two positions. Unfortunately, it’s dicult to accurately measure the amount of energy used by a winch, as they tend to be quite ineicient (they give out energy in noise and heat, and by stretching the rope and their own components). This is where the gravitational redshifting of light comes in. The frequency of light can be measured to very high precision, and the energy the photons lose by climbing out of the gravitational field is expected to be exactly the same as that which would be used to lift an object with a mass that corresponds to the same energy (as calculated using E = mc2). If we can measure the redshifting of light, we therefore have a highly accurate proxy experiment, which is expected to convey exactly the same information as the winch experiment.

  An experiment designed to measure the gravitational redshifting of light was performed for the first time by the scientists Robert Pound and Glen Rebka in the early 1960s. They used a tower at the Jeferson Physical Laboratory at Harvard University, and looked at the redshifting of photons as they travelled up its height. They found that the light did indeed change colour as it travelled upwards, and by an amount that was entirely consistent with mass being independent of position. The only deviations possible would have to be smaller than the accuracy of the experiment itself, which was at the level of about 1 per cent. A similar experiment has since been carried out using the light emitted from the Sun, which also had results that were consistent with mass being independent of position, again at an accuracy of about 1 per cent.

  More recently, atomic clocks have been used to probe the same efect. The idea behind these experiments is that beams of light are themselves, in some sense, like clocks. The colour of light is determined by the wavelength of the photons that make up the light, which is itself related to the frequency at which they oscillate. If we were to take these oscillations to be the basis of a clock, by saying that each oscillation is one unit of time, then we can see that the redshifting of light is equivalent to the appearance of clocks at diferent positions running at diferent rates. In fact, we don’t even need to measure the frequency of light to perform this experiment, because if a clock based on the oscillations of a photon appears to run slow, then so must every other clock. All we have to do is put two clocks at two diferent heights and arrange for them to transmit the time they display via radio signals. The diference in the rate at which they appear to tick, according to the radio signals that are observed, is then entirely equivalent to the rate at which a light signal between them should be redshifted.

  We can therefore take two highly accurate atomic clocks and put one on a rocket while leaving the other at our feet. If the clock in the rocket transmits its time via radio signals, then we can compare this signal to the time displayed by the clock we kept beside us. They should, in general, be diferent: an efect known as gravitational time dilation. This experiment was carried out for the first time by Robert Vessot and Martin Levine in 1976. They observed the time dilation efect directly, and found it to be consistent with mass being independent of position to an accuracy of about 1 part in 10,000 (about a hundred times more accurate than the Harvard experiment). This experiment therefore provides very strong evidence that mass is indeed independent of position in a gravitational field, to very high accuracy.

  Moving on from position dependence, it is possible to construct experiments that test whether the speed of light, or the mass of an object, are dependent on direction. These experiments are historically very important, because before Einstein produced his theory of gravity it was widely believed that there was a substance called the ether that permeated the whole of space. The ether was invoked as the medium through which waves of light travelled, and was popular among physicists until as late as the 20th century. If the ether existed, then the speed of light that an observer measured should depend on their motion as they travelled through it. Einstein’s theory is incompatible with the existence of an ether, as he had constructed it so that all observers measure the same speed of light, independent of their state of motion. Performing tests for the existence of the ether was therefore an important part of verifying his theory. The most famous of these tests was the Michelson—Morley experiment, carried out in 1887. It is a test to see if the speed of light depends on its direction of propagation.

  The Michelson—Morley experiment used a device known as an interferometer, consisting of two arms, built at right-angles to each other. This is illustrated in Figure 6. Light from a laser is shone down each of the arms, and re』ected back on itself by mirrors positioned at each of their ends. When the reflected light reached the intersection of the two arms it was allowed to interact. As light is known to have the properties of a wave, it would be possible to let the two beams of light interact to form a pattern(like two sets of waves interacting on the surface of a pond). The form of the pattern that is produced in the interferometer depends on the length of each of the arms, and the time it takes the laser light to travel along them. If the speed of light was diferent in diferent directions, then Michelson and Morley should have seen the consequences of this by using their apparatus.

  6. A schematic diagram of an interferometer. The beam splitter separates the laser light into two beams, each of which is re』ected of a mirror. The light then re-traces its path, and interacts, before being directed into the detector.

  Michelson and Morley’s experiment had a null result. No diference was observed in the speed of light, as it travelled in the two diferent directions. This was unexpected by many scientists at the time, as the Earth was supposed to be in motion with respect to the ether. If light was a wave in the ether, then the speed of light should have been direction independent only in a laboratory that was stationary with respect to the ether. This wasn’t thought to be the case for the Earth, which orbits the Sun at around 30,000 metres per second. The experiment performed by Michelson and Morley is therefore taken to be strong evidence against the existence of an ether, and in favour of a speed of light that is truly the same in every direction. This was very important for Einstein’s theory.

  Evidence for the direction independence of mass was given by the experiments that were independently performed by Vernon Hughes and Ronald Drever in the early 1960s. These investigators considered the electrons that exist within lithium atoms, and that orbit the nucleus at velocities of around a million metres per second. Now, the gravitational interaction between these electrons, and anything else in their immediate environment, is extremely small. This is because electrons have such tiny masses. Nevertheless, it is still possible to put extremely tight bounds on any possible direction dependence of their mass. This is because electrons give out photons when they change energy levels within atoms, and because these photons have a very specific set of frequencies known as transition lines. If the mass of the electrons depended on their direction of motion, then this would change the precise position of the transition lines that resulted from changes in energy levels. Careful studies, by both Hughes and Drever, found no evidence for the direction dependence of the mass of the electron, to extremely high accuracy.

  Let’s now return to the idea of the Universality of Free Fall. Remember that this is the name given to the idea that all objects fall at the same rate, as demonstrated by Galileo. But Galileo’s experiment, while ground-breaking, was probably not very accurate (by modern standards). So, given the enormous significance of the Universality of Free Fall for both Newton and Einstein’s theories, there has subsequently been a lot of efort to verify it to the highest possible level of accuracy. This has now been done in a number of diferent environments, including numerous laboratory experiments as well as space-based observations.

  So, to summarize, there now exists good evidence, from a variety of diferent experiments, that the mass of an object is independent of both its position and the direction of its motion. There is also strong evidence that the speed of light is the same in every direction, and that all objects fall under gravity at the same rate. These are the foundational assumptions under which Newton’s and Einstein’s theories were constructed. Let’s now move on to experiments that aim to investigate these theories themselves.

  Experiments that probe Newton’s law

  Later on in this chapter we will consider the predictions of Einstein’s theory, which is now universally accepted to supplant that of Newton. However, we know that Newton’s inverse square law of gravity is a good enough approximation to Einstein’s theory to describe a wide variety of both terrestrial and astronomical phenomena. It’s therefore very much worthwhile testing Newton’s law to see exactly how well it describes gravity. Here we will consider some of the most important tests that have been performed so far.

  The first laboratory test of the inverse square law was performed by Henry Cavendish, at the end of the 18th century. Cavendish’s experiment made use of a torsion balance, like the one used by E’tv’s (illustrated in Figure 7). Unlike E’tv’s, however, Cavendish positioned extra masses in order to make the torsion balance rotate. He put these masses close to the ones that were attached to the torsion bar. The gravitational force between the test objects hanging from the torsion balance, and the extra masses that Cavendish introduced, could then be inferred by the rate of rotation of the apparatus. The new masses pulled at the masses attached to the torsion bar through the gravitational force between them, and caused the bar to rotate. The results of Cavendish’s experiment thus allowed for the gravitational force to be probed over a distance scale of only 23cm. He found that gravity operated on these scales in a way that was entirely consistent with Newton’s inverse square law. Today, experimenters have tested these same ideas on much smaller distance scales.

  The challenges involved in testing Newton’s law of gravity in the laboratory arise principally due to the weakness of the gravitational force compared to the other forces of nature. This weakness means that even the smallest residual electric charges on a piece of experimental equipment can totally overwhelm the gravitational force, making it impossible to measure. All experimental equipment therefore needs to be prepared with the greatest of care, and the inevitable electric charges that sneak through have to be screened by introducing metal shields that reduce their in』uence. This makes the construction of laboratory experiments to test gravity extremely dicult, and explains why we have so far only probed gravity down to scales a little below 1mm (this can be compared to around a billionth of a billionth of a millimetre for the electric force).

  There are three groups of experimenters that have led the way in laboratory tests of gravity in recent times: one at the University of Washington, another at the University of Colorado, and a third at Stanford University. The Washington group took a pendulum with ten holes cut into it and held it over a disk that contained another set of holes. The pendulum was found to twist due to the missing gravitational force from the missing mass in the holes, and this allowed gravity to be measured down to scales as small as a twentieth of a millimetre. The groups at Colorado and Stanford, on the other hand, measured the gravitational field of a vibrating mass, and used this information to probe gravity all the way down to 1/40 of a millimetre. The results of these laboratory experiments have so far all been entirely consistent with Newton’s inverse square law of gravity, even on the very smallest of scales.

  On larger distance scales, however, there are a variety of other tests that we must consider. To order the discussion of these tests sensibly, let’s first of all consider distances from tens of metres, up to about to about a few kilometres. These scales may seem like they should be the easiest to probe as they are closest to the type of distances we consider in our everyday lives. They are, however, quite problematic.

  One of the best constraints that has so far been achieved on gravity over these everyday distance scales is due to measurements of the gravitational force that an object experiences at various heights up a tall tower. Towards the end of the 1980s, a team of scientists performed this test using the 600-metre-high WTVD tower in Garner, North Carolina. The change in gravity expected from Newton’s inverse square law at diferent points up the tower is easy to calculate, and the experimenters were able to measure it to good accuracy. At around the same time, other researchers were measuring the gravitational force due to the water in a reservoir filled to diferent levels. In essence, this allowed the water to be weighed and Newton’s inverse square law of gravity to be tested. Further tests, a couple of years later, were performed by measuring the gravitational attraction of sea water at diferent depths in the ocean. These experiments all produced results that were consistent with Newton’s inverse square law, and all at a level of accuracy of around 0.1 per cent.

  On larger scales it starts to become possible to use astronomical data, which is much more accurate than trying to measure the gravitational field of the water in reservoirs and oceans. Distance scales of between a million and a few hundreds of millions of metres can be probed by studying the orbits of man-made satellites around the Earth, the orbit of the Moon, and the orbits of the planets around the Sun. The LAGEOS satellites, launched in 1976 and 1992, are particularly useful for experiments of this type』as are the orbits of the planets, which form closed ellipses (to first approximation) when the gravitational attraction is of the form given by Newton’s inverse square law. Observations of all of these bodies, and objects, have shown consistency with Newton’s law at accuracies of between one part in a million and one part in a billion.

  We therefore have very good evidence that Newton’s inverse square law is a good approximation to gravity over a wide range of distance scales. These scales range from a fraction of a millimetre, to hundreds of millions of metres. The accuracy of the experiments that probe gravity over these distances varies wildly, between about 1 part in 1,000 (for scales of tens of metres) to about one part in a billion (for distances that correspond to the orbits of the planets). This is a great success, but it’s not the end of the story. Let’s now go beyond Newton’s law and start considering the new efects that arise from Einstein’s theory.

  Experiments that probe Einstein’s theory

  The experiments discussed earlier have all involved concepts that are familiar to most people who have studied physics at school: the constancy of mass; the Universality of Free Fall; and Newton’s inverse square law of gravity. In this section, I will introduce experiments that probe the more unfamiliar ground of Einstein’s theory of gravity. The specific efects that result from this theory are often small and dicult to detect experimentally. Nevertheless, they are very important, as they provide us with a window that can be used to view and understand gravity at a deeper level.

  There are a large number of efects that result from Einstein’s theory. Here I am going to limit myself to describing four of them. These are the anomalous orbit of the planet Mercury; the bending of starlight around the Sun; the time delay of radio signals as they pass by the Sun; and the behaviour of gyroscopes in orbit around the Earth. These are four of the most prominent relativistic gravitational efects that can be observed in the Solar System. Further efects, which become apparent in more extreme astrophysical environments, will be discussed in Chapter 3.

  Let’s start with the anomalous orbit of Mercury. Earlier on in this book, I stated that Newton’s theory explained Kepler’s observation that the planets follow elliptical orbits around the Sun. This is true for a single planet, but things start to get a bit more complicated when we consider the orbits of multiple planets simultaneously. This is because there exist gravitational forces between the planets themselves and not just between each planet and the Sun individually. These interplanetary gravitational forces are smaller but measurable, and have the efect of pulling the orbits of the planets of what might otherwise be perfect ellipses.

  Physicists have known about the efects of interplanetary gravitational forces for a long time. These forces can be easily calculated within Newton’s theory, and astronomers have measured their consequences on the orbits of the planets for centuries. In fact, the existence of the planet Neptune was deduced in the mid-19th century by carefully studying the orbit of Uranus (the next nearest planet to the Sun). The orbit of Uranus was perturbed a little from where astronomers expected it to be, and the perturbation could be explained if there was a slightly larger planet a little bit further out in the Solar System. Urbain Le Verrier and John Couch Adams both predicted the existence of this planet in 1845』and by 1846 it had been observed. This was clearly a momentous achievement.

  It therefore wasn’t too much of a shock when, in 1859, Le Verrier announced that the orbit of Mercury (the planet closest to the Sun) also seemed to deviate a little from the path that it was expected to take. No doubt buoyed by his prediction of the existence of Neptune, Le Verrier predicted that there must be another planet still closer to the Sun than Mercury. He even named it』Vulcan. This time, however, the discovery of the predicted planet never occurred. Despite much efort, no new object could be seen between Mercury and the Sun. Nevertheless, the anomalous orbit of Mercury persisted. It seemed as if its orbit was being dragged around the Sun by a gravitational force that had no obvious origin.

  The mystery of the anomalous orbit of Mercury was solved in 1915, not by the discovery of any new massive bodies in the Solar System, but by Einstein’s revolutionary new theory. You see, according to Einstein’s new theory, the Newtonian gravitational force is only a rough approximation to the true nature of gravity. As well as the inverse square law, Einstein predicted that there should be new, smaller contributions to the gravitational force law. For an astrophysical system dominated by a large mass at the centre, like the Sun at the centre of the Solar System, Einstein calculated that the largest of these new forces should vary as the inverse of the distance cubed. Therefore, the closer one is to the Sun, the larger the contribution from this new force should be, relative to Newton’s inverse square law.

  Mercury was, and still is, the closest known planet to the Sun, and so Einstein’s new gravitational force should be more in』uential on Mercury’s orbit than on the orbits of any of the other planets.

  Einstein calculated that Mercury’s orbit should be dragged around the Sun by an extra 43 arcseconds per century (1 arcsecond is 1/360 of a degree). This is a tiny amount, but it is enough to be observed by astronomers. It is also consistent with the anomalous observations made by Le Verrier. So Einstein’s theory of gravity had its first observational success as early as 1915, by explaining the anomalous orbit of the planet Mercury.

  Modern measurements of the orbit of Mercury are much better than those that existed in the 19th century. We now know the orbits of all the planets to very high accuracy, which is essential for calculating the deviation of Mercury’s orbit from an ellipse. The extra drag caused by Venus alone, for example, is more than six times that of the correction due to Einstein’s gravity. Its position therefore needs to be known to very high accuracy. This isn’t the largest source of uncertainty in modern observations though. That distinction goes to the uncertainty associated with the shape of the Sun itself. Any deviations from a perfectly spherical shape cause efects in the orbit of Mercury that could be confused with Einstein’s new gravitational force. The shape of the Sun is hard to know exactly, so the best we can currently do is to say that the anomalous orbit of Mercury is consistent with Einstein’s theory to an accuracy of about 1 part in 1,000.

  The explanation of Mercury’s orbit is impressive but probably doesn’t count as a prediction, because the anomaly itself was known about long before Einstein was even born. What does count as a genuine prediction of Einstein’s theory is the bending of starlight around the Sun by its gravitational field. It was previously uncertain whether or not light should be afected by gravity, because in Newton’s theory the gravitational force is only between objects that have mass (which light does not). This ambiguity was removed in Einstein’s theory, as light simply follows the shortest paths available in the curved space-time, just like everything else. Einstein therefore predicted that light should be bent by the gravitational fields of massive objects.

  Einstein’s calculations showed that the deflection of light would be greatest for beams that just skim the surface of massive objects. The most massive object in the Solar System is the Sun. But to see starlight that passes very close to the Sun we have to wait for a solar eclipse, otherwise the Sun’s own light overwhelms that of the much fainter stars. The first good opportunity to test the idea of bending starlight came after the end of World War I, in 1919. The expedition to measure the positions of the stars nearby the Sun, and therefore to test Einstein’s theory of gravity, was led by Sir Arthur Eddington.

  Eddington’s expedition went to the island of Príncipe, in Africa, where the solar eclipse that was going to happen that year would be total. He made careful measurements, using the best photographic plates that were available at the time. The conditions were less than ideal, but Eddington succeeded in measuring the positions of the stars during the eclipse. He found that they had indeed shifted, just as Einstein had predicted, because the trajectory of the light had been bent by the gravitational in』uence of the Sun. Eddington’s results were suicient to confirm Einstein’s theory, but only within an accuracy of around 30 per cent.

  Once again, modern observations have improved this result dramatically. This has been helped partly by the fact that there are, coincidentally, a number of very bright objects, known as quasars, which lie in just the right part of the sky to test Einstein’s prediction. As these objects pass behind the Sun, the de』ection of the light that they emit can be measured. Several million observations of these quasars have now been made using very large baseline interferometers (this is a type of telescope that mixes the signals from a number of diferent detectors to create a high-resolution image). The result of this work shows perfect consistency with Einstein’s theory, at a level of accuracy of about 1 part in 10,000.

  A more recent prediction from Einstein’s theory is the time delay of radio signals that pass by massive objects. For some reason, it took until 1964 for scientists to notice that this efect is a necessary outcome of Einstein’s gravity, but it has now been measured in a number of diferent situations. These have included the observation of radio signals that bounce of planets as they are about to pass behind the Sun, as well as by looking at the signals that are emitted from man-made probes as they do the same. The first of these methods has the benefit of the position of the planets being very well known, and their trajectories being easy to predict. This stability makes them a good target, but the imperfections in their shape can cause some problems in interpreting the re』ected signals. Man-made probes, on the other hand, emit very easily predictable signals, but their trajectories can be somewhat less certain.

  Radio targets that have been used to measure the time delay efect include the planets Mercury and Venus, and the space probes Mariners 6 and 7, Voyager 2, the Viking Mars Lander and Orbiters, and the Cassini probe. The most recent, and most accurate, of these observations was made using Cassini. The primary mission of this spacecraft was to observe Saturn, but its relevance for the study of gravity was perhaps best served in 2003, when it was announced it had confirmed the existence of the time delay efect with an accuracy of 1 part in 100,000. This was yet another spectacular confirmation of Einstein’s theory, and one which was achieved at higher accuracy than any previous experiment. This was, in part, due to the observations of the radio signals being made at multiple frequencies, which allowed interference from the Sun’s corona to be extracted.

  Now let’s move on to the final experiment I want to discuss in this section: the behaviour of gyroscopes in orbit around the Earth. A gyroscope is essentially a spinning top, whose central axis is allowed to point in any direction. According to Einstein’s theory of gravity, there should be two new efects that can be observed when a gyroscope is put in orbit around the Earth. The first of these is a change in direction of the gyroscope’s axis of rotation, as it orbits the Earth. This efect, known as geodetic precession, is due to the curvature of space-time around the Earth. The second efect is known as frame-dragging, and is due to the rotation of the Earth efectively pulling space around with it as it rotates. This is an entirely new type of gravitational interaction, and so it is of great interest to see if it can be confirmed experimentally.

  Although the prediction of the frame-dragging efect was made only a few years after Einstein published his new theory of gravity, it took until the 1960s before the efect on gyroscopes in orbit was calculated, and the experimental observation of the efects themselves did not take place until the 21st century. The LAGEOS satellite network provided an observation of the frame-dragging efect by measuring the change in the orbit of its satellites as they went around the rotating Earth. The long-awaited gyroscope experiment was performed by a mission called Gravity Probe B, in 2011. The geodetic precession and frame-dragging efects were measured by this experiment with accuracies of about 0.3 per cent and 20 per cent, respectively. The accuracy of the corresponding results from the LAGEOS satellites are estimated at between 5 and 10 per cent. All results were once again found to be consistent with Einstein’s theory.

  So the overall picture we are left with is very encouraging for Einstein’s theory of gravity. The foundational assumptions of this theory, such as the constancy of mass and the Universality of Free Fall, have been tested to extremely high accuracy. The inverse square law that formed the basis of Newton’s theory, and which is a good first approximation to Einstein’s theory, has been tested from the sub-millimetre scale all the way up to astrophysical scales. We are also now in possession of a number of accurate experimental results that probe the tiny, subtle efects that result from Einstein’s theory specifically. This data allows us direct experimental insight into the relationship between matter and the curvature of space-time, and all of it is so far in good agreement with Einstein’s predictions. This is a truly spectacular confirmation of a theory that was borne out of almost pure thought. Einstein wanted to make a theory of gravity that was compatible with a speed of light that was measured to be the same by every observer. He did so, and we have now seen the numerous consequences of his revolutionary new picture of the Universe.

  This isn’t the end of the story though: Einstein’s gravity has many more spectacular consequences, which we will discuss in the following chapters.