During the years when Kepler was developing his laws of planetary motion the open advocacy of a heliocentric viewpoint was considered to be heresy. In the year 1600 a minor figure in the history of science, Giordano Bruno, was burned at the stake for insisting that Copernicus was right. Fortunately for the development of science the greatest intellects of that time swallowed their pride and did not follow Bruno into martyrdom.
A case in point is Galileo Galilei (1564-1642), who turned the newly discovered telescope upon celestial objects and thus acquired more evidence tending to favor a heliocentric viewpoint. His simple telescope revealed the phases of the planets Venus and Mercury as well as the four largest satellites of Jupiter. Even more important was the observation that these four Jovian satellites seemed to obey Kepler's laws of planetary motion, as they circled about the planet Jupiter, much as the planets were supposed to circle about the Sun in the Copernicus picture.
In addition Galileo observed Sunspots by projecting a telescopic image of the Sun upon a viewing screen. The motion of the Sunspots across the disk of the Sun suggested that the Sun as well as the earth is in a state of rotational motion. Galileo, using his primitive telescope also observed that the planet Saturn had peculiar "ears," which with better focus would have been identified as the marvelous rings of Saturn.
When Galileo published his finding he was brought before the Inquisition, forced to recant, and confined for the last eight years of his life. During this period of confinement there gelled in Galileo's mind an idea which was to be of paramount importance--the idea of an inertial frame. For the first time the idea of force was tied clearly to accelerations rather than to velocities. Except when acted upon by a force an object moves at uniform speed in a straight line. This profound idea is so opposed to common experience that even today students often have difficulty grasping the significance of Galileo's law of inertia. Nevertheless, this idea focusses attention properly upon what causes planets to deviate from straight-line uniform motion rather than upon what keeps them moving.
The Dutch natural philosopher and mathematician C. Huygens (1629-1695) showed that in order to maintain an object of mass m moving at speed v in a circle of radius r it is necessary to exert a force of magnitude F = mv2/r upon the body toward the center of the circle. (Illustrate this by swinging an object around in a circle on the end of a string.)
The speed v of a planet in a nearly circular orbit can be estimated if you know both the period P and the orbital radius r, for clearly Pv = 2pr, the circumference of the orbit. However, from his study of the orbits of the known planets, Kepler had inferred that the period P of a planet encircling the Sun is proportional to the 3/2 power of the radius of the orbit. Hence the speed v must vary as the -1/2 power of r. Finally, if v2 varies as the -1 power of r, we conclude that the force toward the Sun must vary as the -2 power of r. The force must vary inversely as the square of the distance!
The big question was "could an inverse square force also account for elliptical orbits with a force center at one of the foci?" One of the few people capable of answering this question was Isaac Newton (1642-1727) of England. In order to accomplish the task he had to invent what would now be called calculus. In his monumental publication, the Principia, the affirmative response to our question was established.
If one assumes that every particle of matter exerts upon every other particle a force which varies directly as the product of their masses and inversely as the square of their separation, then it can be shown, using calculus, that in the presence of a massive body (the Sun) relatively less massive bodies (the planets) will pursue paths which are conic sections, i.e., circles, ellipses, parabolas, hyperbolas or straight lines, depending upon the initial state of their motion. Furthermore, in the case of elliptical motion all three of Kepler's laws of planetary motion may be deduced from Newton's universal law of gravitation,
Here M represents, for example, the mass of the Sun, m represents the mass of a planet, and r represents the instantaneous distance between the centers of the two bodies. G is a universal constant, whose value was not known. Of course, the values of M, m and r were not known either, so one might be tempted to throw up one's hands in dispair. However, the situation is not all that hopeless, as we shall now try to show.
If you were to tie a golf ball and a bowling ball to opposite ends of a piece of rope, and proceeded to spin the balls about one another, common experience would cause you to expect that the golf ball would encircle the bowling ball, and not vice versa. Thus, if the Sun appears to be at rest with the planets encircling it, you would tend to believe that the Sun must be much more massive than the planets.
Now, one could object that Newton had no way of knowing that the Sun is at rest. At that time, for example, no one had observed parallax for even a single star. What other evidence, therefore, could be cited to establish the great mass of the Sun? One could get a rough idea of its mass by the following line of argument:
According to Newton the force exerted upon a planet encircling the Sun is given by F = GMm/r2, while Huygens had shown that for a nearly circular orbit the force must be F = mv2/r, where, as we have seen, v is determined by Pv = 2pr. These two alternative ways of computing F lead to the following expression for GM:
Remember that according to Kepler's third law, the value of r3/P2 is the same for all of the planets. Thus, we can evaluate GM, the product of the universal constant of gravitation and the mass of the Sun, by substituting the values of r and P for the earth, i.e., r = 1 AU, and P = 1 year.
Unfortunately, the value of r, the AU, was not known very accurately at the time of Newton. The Greeks had concluded that it was at least 5 million miles, and Kepler was able to boost the lower limit to 15 million miles, but that is still far below the correct value, 93 million miles. Thus, this method will underestimate the value of GM by a factor (93/15)3 = 238. You can see what a vexing problem was the obtaining of an accurate value for the AU.
Suppose, however, we accept the erroneous value r = 15 million miles. What would we then conclude concerning the mass of the Sun? If you substitute these figures and evaluate GM you get approximately GM = 108 mi3/ sec2, where I expressed 1 year in terms of seconds. Of course, you still don't know G, so how can one deduce the mass of the Sun?
Newton's law applies also to objects falling in the earth's gravitational field. It was known to Galileo that objects falling near the earth's surface experience an acceleration g = 32 ft/sec 2. According to Newton's theory, such objects should experience an acceleration given by g = Gmearth/Rearth 2, where mearth is the mass of the earth, and Rearth is its radius. Since Rearth = 4000 miles, approximately, a result known to Eratosthenes, we conclude that
where I have expressed feet in terms of miles. Comparing this result with our former result
we would infer that the Sun is about a thousand times more massive than the earth.
While the error of a factor of six in the value of 1 AU causes us to underestimate the mass of the Sun by 238, we nevertheless conclude that the Sun is much more massive than the earth, and that is all we set out to establish. Members of the French Academy of Sciences under Louis XIV toward the end of the 17th century succeeded in determining the AU much more accurately by triangulating upon Mars. Using the method which we have just described they were then able to conclude that the Sun was about 300,000 times the mass of the earth.
It is extremely fortunate that the mass of the Sun far exceeds the mass of each of the planets including the giant Jupiter, for otherwise the technical difficulty of calculating the orbits of the planets would have been extraordinary. It would have been necessary to include the gravitational forces exerted by each planet upon the others. The orbits woud not then be ellipses satisfying Kepler's laws, so it is doubtful that the law of gravitation would ever have been discovered.
The law of gravitation had profound implications for the future development of astronomy. Almost everyone is aware of the fact that a spinning top precesses. (Illustrate using bicycle wheel.) It precesses because of the torque due to the downward force of gravity and the upward force upon the pivot. A torque due to any other cause would work as well. Now, the spinning earth is a little bit oblate, and it acts very much like a giant top. It will precess if you exert a torque upon it. Actually the inverse square gravitational field of the moon can be shown to exert a small torque upon the spinning earth, so the latter should precess.
Even Hipparchus knew about the precession of the earth. The period of precession is large, 26,000 years. Nevertheless, it is quite easily perceived over a period of years. To calculate the expected precession rate one must have a fairly accurate idea of the shape of the earth, a matter of paramount concern to Newton. While Newton gave theoretical reasons for believing that the earth is oblate, the prevailing opinion at that time was that it was prolate (cigar-shaped). The matter was finally settled in Newton's favor, by sending forth scientific expeditions. Thus, Newton had provided a theoretical framework for understanding far more about the solar system than ever before, and it challenged people to develop better data with which to work.
In the late 17th century James Bradley, while trying to determine the
positions of stars more accurately, discovered the phenomenon of
aberration. Because of the earth's motion in its orbit
one must tilt a telescope in the forward direction in order for light
to travel down the axis of the tube. For a star located perpendicular
to the plane of the earth's motion the amount that the telescope must
be tilted is v/c radians, where v is the orbital speed of the earth,
and c is the speed of light. Bradley perceived an apparent motion of
such stars in tiny circles of radius 20.5" corresponding to
v/c = 10-4.
Exactly the same estimate of v/c was obtained independently by Olaf
Rømer, the Dane, in 1675. Rømer found that the occultation
of Jupiter's satellites by the planet itself were delayed by differing
amounts depending upon whether the earth was nearer or farther from
Jupiter. The delay was attributed to the time it requires for light
to travel across the earth's orbit. Once the size of 1 AU had been
determined, the speed of the earth in its orbit could be inferred, and
thus the speed of light could be determined to be 186,000 miles/sec.
Hence the accurate determination of the AU assumed paramount significance in the history of astronomy, for it seemed that all other measurements depended upon that.
Today one could call someone half-way around the world and agree to view, say Venus, simultaneously, both observers carefully noting the position of that planet relative to the background of stars. From such observations and the known diameter of the earth the distance to Venus could be inferred. This was not possible even toward the end of the 18th century, because there was no way to establish simultaneity over such large distances. The clocks which existed at that time would not keep time accurately while being transported on a storm-tossed ship.
Very rarely the planet Venus transits across the face of the Sun as viewed from the earth. Two such transits occured in 1761 and 1769, and the last two transits were in 1874 and 1882. The English astronomer Edmund Halley (1656-1742) proposed to use the transits of Venus to get the first accurate determination of the distance to that planet, and hence the absolute scale of the Solar system. The transit would be observed at a number of locations at different latitudes on the earth. No timing was involved. One simply observed the apparent path of the planet across the Sun's face.
The Sun's apparent diameter is about 1/2o.
The angle between the two chords could therefore be determined.
From this angle and the known distance between the two locations on
the earth's surface the distance to Venus could be calculated. In
this way the AU was determined to 5% accuracy by the end of the 18th
century.
Although the mass of the Sun was known to be about 300,000 times the mass of the earth, the latter mass was not accurately known until somewhat later. One could guess on the basis of the earth's size and apparent composition, and probably one would arrive at a mass within a factor ten of the correct value. However, it was Henry Cavendish (1731-1810) who performed the first experiment to determine mearth (and thus G too).
What Cavendish did was to take two masses m1
and m2, place them a short distance r apart,
and measure the force F exerted by the one mass upon the other. From
F = Gm1m2/
r2 he could then infer the value of G.
Of course, the Cavendish experiment is not particularly easy to
perform, for the force F is extremely minute and apt to be hidden
unless special precautions are taken to avoid air currents and other
influences such as friction. Nevertheless, using a torsion pendulum
Cavendish succeeded in overcoming these difficulties. A working model
of a Cavendish pendulum can be found in the Museum of Science and
Industry in Chicago.
Frederick the Great is said to have remarked that everything of real importance had already been discovered in Science, and toward the end of the 18th century this did indeed appear to be so.
