## LECTURE 10: UNIVERSAL GRAVITY, PART II

Key Questions:
• How do Newton's laws of motion and gravity explain
• ocean tides on the Earth?
• orbits of comets?
• gravitational effects of planets on each other?
• What experiment is needed to "weigh the Earth"?
• How could an "astronomer" around a distant star infer the existence of Jupiter?
• What is the relation between physics and astronomy?
• What does the "scientific revolution" teach us about science?

### APPLICATION TO TIDES

• Moon pulls more strongly on near side of Earth than on center, more strongly on center than on far side.
• Therefore tends to "stretch" the earth into a (slight) oval.
• Hard to stretch solid earth, but liquid oceans are easier.
• Get "tidal bulge" in oceans, pointing roughly towards moon.
• Lags moon slightly because of friction, running into continents.
• As Earth rotates once per day, get two high tides and two low tides.
• Tidal effect of Sun is comparable to Moon.
• Get strong ("spring") tides when Sun and Moon align, reinforce each other.
• Get weak ("neap") tides when Sun and Moon are at right angles, partially cancel.
• Many complications due to geography, weather.
Tidal locking:
• "Friction" from ocean tides is gradually slowing rotation of the earth.
• Conservation of angular momentum means Moon is slowly getting further away (1-2 centimeters per year) to compensate.
• Earth tide on Moon is stronger, because Earth is more massive.
• Tidal friction has halted Moon's rotation relative to Earth, i.e., Moon's "tidal bulge" always points towards Earth, so no more friction.
• This is why we never see the far side of the Moon from Earth.

### APPLICATION TO COMETS

• Comets moving under gravity of Sun.
• But come from far away, very elongated elliptical orbits.
• Turn rapidly when close to Sun.
• Newton used laws of motion and gravity to explain observed motions of comets, predict return of Halley's comet.

### EFFECT OF PLANETS ON EACH OTHER

• Sun 1000 times more massive than Jupiter, which is most massive planet.
• Gravitational effects of planets on each other much smaller than effect of Sun.
• But not zero.
• Newton predicts: When Jupiter is "catching up" to Saturn, Saturn should slow down. When Jupiter has passed it, Saturn should speed up.
• Effect had already been seen, never understood.
• Newton's laws explain it precisely.
• Smoking gun evidence for universal gravity.

### DISCOVERY OF NEPTUNE

• Planet Uranus discovered serendipitously in 1781, during telescopic survey of sky.
• Fifty years later, showing deviations from expected orbit.
• Two mathematicians show these deviations could be produced by gravity of a more distant planet.
• Predicted where the planet should be.
• Planet Neptune found within 1 degree of predicted position.

### WEIGHING THE EARTH

• Can get size of Earth, distances of planets and Sun from precise observations and geometry.
• Can get relative masses of Sun and Earth by comparing acceleration of Earth to acceleration of Moon.
• Similarly, can get relative mass of Sun and Jupiter.

How can we get absolute masses (in kg)?

• Sounds easy.
• From Newton's law of gravity, we know:
g = 9.8 m/sec2 = G Mearth / Rearth2.
• Rearth measured from geometry.
• Look up G in back of book.
• Solve for Mearth.

• Newton's problem: no book.
• Accelerations of planets, moons, etc. depend on GMsun, GMearth, GMjupiter, etc., not G alone.
• To measure G, need to measure gravitational force between two objects whose mass is known independently, i.e., laboratory scale objects.
• Very hard, because gravity is very weak.
• Finally done by Henry Cavendish in 1798, beautiful experiment.
• Result: G = 6.67 x 10-11 newton m2 / kg2 = 6.67 x 10-11 m3 / (kg - sec2).
Cavendish called this experiment "weighing the earth."
Why? Sounds even better than "measuring Newton's constant."
Mearth = g x Rearth2 / G = 9.8 m/sec2 x (6.4 x 106 m)2 / [6.67 x 10-11 m3 / (kg - sec2)] = 6 x 1024 kg.
Once G is known, can also get mass of Sun, mass of Jupiter, etc.

### APPLICATION TO EXTRA-SOLAR PLANET SEARCHES

• Newton's 3rd Law: If Sun pulls on Jupiter, Jupiter also pulls on the Sun.
(Other planets too, but Jupiter's effect is biggest.)
• Jupiter causes Sun to accelerate. But Sun is 1000 times more massive, accelerates 1000 times less.
• How could an astronomer around a distant star infer the existence of Jupiter?
• Too faint to see against the much brighter glow of the nearby Sun.
• But can watch for slight wobble of Sun.
• Small effect, requires very precise measurement.
• This is how we detect planets around other stars.

### EVALUATION OF NEWTON'S ACHIEVEMENT

Hard to find enough superlatives.

• Simultaneously achieves giant advance in astronomy and physics.
• Each requires the other: need laws of motion and gravity to understand motions of planets and moon, but need motions of planets and moon to figure out and prove laws of motion and gravity.
• Has to invent major new branch of mathematics, calculus, to carry this out. Calculus is the mathematics of motion and change.
• Calculations are hard, but principles are simple and few: three laws of motion plus law of gravity.
• These explain observed motions of planets, tie them physically to motion of the moon and to falling objects on earth.
• Also explain much more: motion of Jupiter's moons, tides, comets, mutual gravitational effects of planets.
• Theory makes precise predictions (e.g., orbital deviations of Saturn, position of Neptune) that are confirmed by precise observations.

### UNIVERSAL LAWS

• Perhaps most important of all, Newton gives unified explanation of terrestrial and celestial phenomena.
• Brilliant and persuasive realization of Kepler's idea: basic laws of physics apply throughout the universe.
• This idea underlies all of modern astronomy, proves astoundingly successful.
• Still the case that we use physics learned in laboratories on earth to understand planets, stars, galaxies, and the universe.
• Still the case that we use astronomical observations to learn about physics beyond the reach of terrestrial experiments.

### Scientific Revolution: A Brief Summary

• Geocentric model of Aristotle. Complex, not highly accurate.
• Heliocentric model of Aristarchus. Simple, not highly accurate.
• Geocentric model of Ptolemy. Highly complex, accurate.
• Heliocentric model of Copernicus. Highly complex, accurate.
• Observations of Tycho: Undermined Aristotelian views, provided highly precise measurements of planetary positions.
• Kepler: Simple principles explain Tycho's data, very accurately. Idea of applying physical reasoning to celestial motions.
• Galileo: Discoveries with telescope support Copernican model. Substantial progress in understanding inertia, force, acceleration.
• Newton: Laws of motion and universal gravity explain Kepler's laws. Explain and predict many other phenomena.
Newton: "If I have seen further, it is because I stood on the shoulders of giants." (Copernicus, Tycho, Kepler, Galileo)

### LESSONS

What do we learn about science from the history of the scientific revolution?

• Scientific progress usually messy. Sometimes go right direction for wrong reason. Success may be hard to recognize at the time.
• Much more than collection of facts.
• Major advances in interpretations of facts require great creative leaps.
• Advances often enabled by new experimental or theoretical technology (e.g., Tycho's instruments, telescope, calculus).
• Occam's razor (seek the simplest explanation) is a powerful guide, but not always easy to follow.
• Most important principle is testing ideas against experiment and observation.
• Successful theories explain or predict phenomena beyond (often far beyond) their original scope. Do more than they were "designed" to do.
This success makes most scientists believe that highly successful theories contain elements of "objective truth," and are not merely a compact way of summarizing a large number of empirical facts.