 # Section 5: Testing the Law of Universal Gravitation

Early successes of the law of universal gravitation included an explanation for Kepler's laws of planetary orbits and the discovery of the planet Neptune. Like any physical law, however, its validity rests on its agreement with experimental observations. Although the theory of general relativity has replaced the law of universal gravitation as our best theory of gravity, the three elements of the universal law—the universal constant of gravitation, the equality of gravitational and inertial mass, and the inverse square law—are also key elements of general relativity. To test our understanding of gravity, physicists continue to examine these elements of the universal law of gravitation with ever-increasing experimental sensitivity. Figure 10: Mass of a helium atom is not equal to the sum of the individual masses of its constituent parts.

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The mass of an object does not equal the sum of the masses of its constituents. For example, the mass of a helium atom is about one part per thousand less than the sum of the masses of the two neutrons, two protons, and two electrons that comprise it. The mass of the Earth is about five parts in 1010 smaller than the sum of the masses of the atoms that make up our planet. This difference arises from the nuclear and electrostatic binding—or potential—energy that holds the helium atom together and the gravitational binding (potential) energy that holds the earth together.

The inertial mass of an object therefore has contributions from the masses of the constituents and from all forms of binding energy that act within the object. If , gravity must act equally on the constituent masses and the nuclear, electrostatic, and gravitational binding energies. Is this indeed the case? Does the Sun's gravity act on both the atoms in the Earth and the gravitational binding energy that holds the Earth together? These are questions that have to be answered by experimental measurements. Modern tests of the universality of free fall tell us that the answer to these questions is yes, at least to within the precision that the measurements have achieved to date.

## Tests of the universality of free fall

To test the universality of free fall (UFF), experimentalists compare the accelerations of different materials under the influence of the gravitational force of a third body, called the "source." Many of the most sensitive tests have come from torsion balance measurements. A recent experiment used eight barrel-shaped test bodies attached to a central frame, with four made of beryllium (Be) on one side and four of titanium (Ti) on the other. The denser titanium bodies were hollowed out to make their masses equal to those of the beryllium bodies while preserving the same outer dimensions. All surfaces on the pendulum were coated by a thin layer of gold. The vacuum vessel that surrounded and supported the torsion fiber and pendulum rotated at a slow uniform rate about the tungsten fiber axis. Any differential acceleration of the two types of test bodies toward an external source would have led to a twist about the fiber that changed in sign as the apparatus rotated through 180°. Essential to the experiment was the removal of all extraneous (nongravitational) forces acting on the test bodies. Figure 11: Torsion pendulum to test the universality of free fall.

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For source masses, experiments have used locally constructed masses within the laboratory, local topographic features such as a hillside, the Earth itself, the Sun, and the entire Milky Way galaxy. Comparing the differential acceleration of test bodies toward the galactic center is of particular interest. Theorists think that dark matter causes roughly 30 percent of our solar system's acceleration about the center of the galaxy. The same dark matter force that helps to hold the solar system in orbit about the galactic center acts on the test bodies of a torsion pendulum. A dark matter force that acts differently on different materials would then lead to an apparent breakdown of the UFF. Because physicists have observed no differential acceleration in the direction of the galactic center, they conclude that dark matter interacts with ordinary matter primarily through gravity.

No violation of the UFF has yet been observed. Physicists use tests of the UFF to search for very weak new forces that may act between objects. Such forces would lead to an apparent violation of the UFF and would be associated with length scales over which the new forces act. Different experimental techniques have been used to test the UFF (and search for new forces) at different length scales. For example, there is a region between 103 meters and 105 meters over which torsion balances fail to produce reliable constraints on new weak forces. This is because over this length scale, we do not have sufficient knowledge of the density homogeneity of the Earth to calculate reliably the direction of the new force—it might point directly parallel to the fiber axis and not produce a torque on the pendulum. In this length range, the best limits on new forces come from modern "drop tower" experiments that directly compare the accelerations of different materials in free fall at the Earth's surface.

## UFF tests in space

The future for tests of the UFF may lie in space-based measurements. In a drag-free satellite, concentric cylinders of different composition can be placed in free fall in the Earth's gravitational field. Experimentalists can monitor the relative displacement (and acceleration) of the two cylinders with exquisite accuracy for long periods of time using optical or superconducting sensors. Satellite-based measurements might achieve a factor of 1,000 times greater sensitivity to UFF violation than ground-based tests. Figure 12: Apollo mission astronauts deploy corner cube reflectors.

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One source of space-based tests of the UFF already exists. The Apollo space missions left optical corner mirror reflectors on the Moon that can reflect Earth-based laser light. Accurate measurements of the time of flight of a laser pulse to the Moon and back provide a record of the Earth-Moon separation to a precision that now approaches 1 millimeter. Because both the Earth and the Moon are falling in the gravitational field of the Sun, this lunar laser ranging (LLR) experiment provides a test of the relative accelerations of the Earth and Moon toward the Sun with precision of 2 x 10-13 of their average accelerations. Gravitational binding energy provides a larger fraction of the Earth's mass than it does for the Moon. Were the UFF to be violated because gravity acts differently on gravitational binding energy than other types of mass or binding energy, then one would expect a result about 2,000 times larger than the experimental limit from LLR.

## Validating the inverse square law

Physicists have good reason to question the validity of the inverse square law at both large and short distances. Short length scales are the domain of the quantum world, where particles become waves and we can no longer consider point particles at rest. Finding a theory that incorporates gravity within quantum mechanics has given theoretical physicists a daunting challenge for almost a century; it remains an open question. At astronomical length scales, discrepancies between observations and the expectations of ordinary gravity require dark matter and dark energy to be the dominant constituents of the universe. How sure are we that the inverse square law holds at such vast distances? Figure 13: Torsion pendulum to test the inverse square law of gravity at submillimeter distances.

Source: © Blayne Heckel. More info

The inverse square law has been tested over length scales ranging from 5 x 10-5 to 1015 meters. For the large lengths, scientists monitor the orbits of the planets, Moon, and spacecraft with high accuracy and compare them with the orbits calculated for a gravitational force that obeys the inverse square law (including small effects introduced by the theory of general relativity). Adding an additional force can lead to measurable modifications of the orbits. For example, general relativity predicts that the line connecting the perihelia and aphelia of an elliptical gravitational orbit (the points of closest and furthest approach to the Sun for planetary orbits, respectively) should precess slowly. Any violation of the inverse square law would change the precession rate of the ellipse's semi-major axis. So far, no discrepancy has been found between the observed and calculated orbits, allowing scientists to place tight limits on deviations of the inverse square law over solar system length scales. Figure 14: Experimental limits on the universality of free fall.

Source: © Blayne Heckel. More info

At the shortest distances, researchers measure the gravitational force between plates separated by about 5 x 10-5 meters, a distance smaller than the diameter of a human hair. A thin conducting foil stretched between the plates eliminates any stray electrical forces. Recent studies using a torsion pendulum have confirmed the inverse square law at submillimeter distances. To probe even shorter distances, scientists have etched miniature (micro) cantilevers and torsion oscillators from silicon wafers. These devices have measured forces between macroscopic objects as close as 10-8 meters, but not yet with enough sensitivity to isolate the gravitational force.

Does the inverse square law hold at the tiny distances of the quantum world and at the large distances where dark matter and dark energy dominate? We don't know the answer to that question. Definitive tests of gravity at very small and large length scales are difficult to perform. Scientists have made progress in recent years, but they still have much to learn.

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