AP1 Class Notes Section 7A

Kepler's Laws of Planetary Motion
Newton's Law of Universal Gravitation

N.B.: This section was inserted specifically for use in semestres undergoing an LOA evaluation.  This section will be rewritten at a later date to incorporate more historical notes.  In the meantime, here are the important highlights:

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here follows an extremely boiled down version of the history of astronomy.  As usual, it's a complicated story inolving a lot of very brilliant people.
Typical ancient Greek thought is that the Earth is at the center of the universe (geocentric model).  Do not trivialize their mistake; often their conclusions are well thought out, but based on faulty data.  Aristarchus (my mistake, not Hipparchus) is given credit as the first to propose the heliocentric model (sun at the center) based on the relative sizes of the earth and sun.
Ptolemaic model assumes erath at center, but is fairly good at predicting the locations of planets.  Model uses epicycles to account for retrograde motion.
Kopernik (Copernicus) proposes again a heliocentric model; however, the accuracy of his model at predicing the position of Mars is half as good as Ptolemy's model is.
Galileo shows himselt to be a 'bad' scientist when he embraces the less accurate Copernican model.
Kepler introduces his heliocentric model in which 1) the planets orbit the sun in elliptical orbits with the sun at one focus; 2) a line connecting the sun to a given planet sweeps out the same area at any point in the orbit in any given time period (which we will eventually recognizze as a statement of the conservation of angular momentum), 3) the period of the planet's orbit and the mean radius are related through:
R³/T_² = C,
where C is a constant associated with the object around which the planet is orbiting.  C has a different value for each central object.  For the special case of objects orbiting our sun, we can make the math easy by putting R in astronomical units (1 AUu is the average distance frorm the sun to the earth) and the period T in earth years, so that the constant C is equal to 1.
Since Kepler's model is five times more accurate at predicting the position of Mars than the Ptolemaic model, it is 'good' science. Unfortunately, it does not explain why the planets move in such a manner.
Newton wonders if the force causing apples to fall from trees is the same force that keeps the moon moving in orbit( namely gravity).  He postulates that the gavitational force acting on a planet is proportional to the mass of the planet and inversely proportional to the square of the distance between the sun and the planet; such a behaviour turns out to be consistent with Kepler's laws.  Newton's also guesses that the force is proportional to the mass of the sun.  In more general terms, the force that each of two masses exerts on the other is given by the formula:
F_g = GM₁M₂/r².
Attempts were made to determine the proportionality constant G, based on the attraction between the earth and objects near its surface, but the mass of the earth must also be known; this mass can be estimated by assuming an average density based on the density of rocks and the volume of the earth, which has been known since antiquity.  An accurate measurement of G was accomplished by Cavendish; looking at the attraction between metal balls; this then led to an accurate value for the mass of the earth.

Nothing like compressing two thousand years of science into a coule of paragraphs.

Examples:

Given that the average radius of the orbit of Jupiter is about 5 AU, find the period of Jupiter's orbit.
Since the sun is the central object, we can use the special case relationship:
R_J³/T_J² = 1
T_J² = R_J³ = 5³ = 125
T_J = [125]^1/2 = 11.2 earth years

Consider a satellite that is in geostationary orbit (it stays above the same spot on earth as the earth rotates, so its orbital period is about 24 hours or 1 day).  How high up is such a satellite?
We can't use R_J³/T_J² = 1, because the earth is the central object here, not the sun.  However, we can compare the motion of this satellite to another earth satellite we know something about, namely the moon.
R_S³/T_S² = C_earth = R_moon³/T_moon²
The period of the moon's orbit is about 28 days, and the moon is about 400,000 km away  from the earth's center.  So,
R_S = [T_S² R_moon³/T_moon² ]^1/3 = [1² (400000³)/28² ]^1/3 = 43,375 km.
Now, that's from the center of the earth, so to find the altitude, subtract off the radius of the earth: 43,375 - 6400 = 36,975 km above the earth's surface.  Be careful what the question asks.

Now, let's take a look at Newton's law.  We know from previous discussion that the force that the earth's gravity exerts on an object of mass m near the earth's surface is given by gm.  From our new discussion, the force of gravity will be GM_earthm/R_E².  If these laws are to be compatible, then,
GM_earth/R_E² = g
6.67x10^-11*6x10²⁴/[6.4x10⁶]² = 9.77 N/kg (pretty close!)