Monday, December 13, 2010

What Everybody Knows about Newton

Isaac Newton, during his exile in the English countryside during the plague year of 1666, conceived the idea that the force that caused things to fall to the ground might be the very same force that held the Moon in orbit around Earth.  He seems to have arrived at this hypothesis without reference to apples.

Newton, being a scientist, was not content to make a qualitative generalization.  Instead, he constructed a quantitative description of his idea, an equation intended to predict how the force of gravitation depends on the mass of the attracting body and its distance.  Then Newton, being a scientist, solved the equation for the two cases of interest.  

The first case was that of the acceleration of a falling body near the ground (at a distance of one Earth radius from Earth’s center).  For this calculation he used the best available measurement of Earth’s radius.  The second case, the motion of the Moon, specifically the acceleration required to bend the Moon’s trajectory into a closed orbit around Earth, required using the best available measurement of the Moon’s distance from Earth.  Both calculations also depended on the exact mass of Earth, since both accelerations were, by his hypothesis, proportional to Earth’s mass.  Newton realized that  the ratio of these two accelerations was therefore independent of Earth’s mass, which was fortunate because Earth’s mass was not well known.  Indeed, in both calculations the predicted  acceleration was proportional to the product GM, where G was a very poorly known constant called the Universal Gravitational Constant, and M was the equally poorly known mass of Earth.  But the numerical value of the product of G times M could be calculated with good precision from measuring the acceleration of falling objects in the laboratory-- and in the ratio of the two accelerations, the product GM cancelled out perfectly.

Newton did the calculation with the best available data and found to his chagrin that there was a small but significant discrepancy.  In effect, the product GM estimated from the Moon’s motion and the value of GM deduced from laboratory measurements were not exactly the same.  Being a scientist, Newton concluded that his hypothesis was in error, and tucked it away in a drawer.  

Several years later an astronomer made new observations of the parallax of the Moon (the apparent displacement of its position seen when two observers in different places simultaneously measured the Moon’s position precisely against the background of distant stars), from which the Moon’s distance can easily be calculated.   He published his more accurate determination of the Moon’s distance.  Newton read the article and remembered his old hypothesis, gathering dust in a drawer.  He pulled it out, inserted the new measurement of the Moon’s distance, and found that it worked!  GM was the same for the Moon and for cannonballs in the laboratory!  Encouraged by this success, Newton published the Theory of Universal Gravitation, and immediately the motions of Solar System bodies became predictable!

A contemporary of Newton, Edmund Halley, was fascinated by the apparently unpredictable motions of comets.  All contemporary wisdom held that comets were not even physical objects; they were signs from God, not subject to any law understood or even understandable by mere humans.  But Halley found in ancient records reports of a series of comet appearances that very nearly fit the same orbit, and very nearly the same orbital period.  He suspected that they were all the same comet.  Halley found by laborious calculations that the comet’s orbit changed whenever it passed close to Jupiter.  Using Newton Law of Universal Gravitation, he showed that the gravitational influence of Jupiter accounted for the changes in the comet’s orbit.  Halley then predicted the next appearance of the comet.  When the comet reappeared in accordance with his prediction, years after Halley’s death, the comet was named after him.  Comets ceased to be regarded as “signs” and became Solar System bodies with predictable orbits.  This was an astonishing verification of Universal Gravitation.

Back in his day and using this law, Newton showed that a body sent from Earth with a high enough velocity, such as a projectile from a giant cannon fired from a mountain top, could enter orbit around Earth just above the atmosphere.  That critical speed was called “circular orbital velocity”  With twice as much energy (a velocity greater by a factor of the square root of 2) the projectile would coast out to infinite distance from Earth.  Any speed equal to or above this “escape velocty” would guarantee that the projectile would never return to Earth.

Everybody knows that Newton said “what goes up must come down”.  But, like almost everything that “everybody knows”, this is errant nonsense.  In the history of the human race, the first person to prove that what goes up does not have to come down was Newton.  In doing so, he paved the way for launching Earth satellites, interplanetary probes, and the Pioneer and Voyager missions into interstellar space.

For more about comets and their interaction with Earth, see Rain of Iron and Ice and Physics and Chemistry of the Solar System.

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