Week 2-- Celestial Mechanics

I. geocentric vs. heliocentric model
In Bayesian logic, if one has a model that conveys a reasonably successful understanding of the nature of things, one requires significant observational evidence of a better model in order to replace it. The geocentric model was such a successful understanding, and was held as true in Europe for over 2000 years. The reason astronomers were loathe to replace it with a heliocentric model (which was argued for by Aristarchus around 300 BC) had to do with at least five ambiguities that could not be resolved by naked-eye observations, all of which were judged one way by the geocentric model and the other by the heliocentric model (so all five would need shifting in order to shift from one model to the other). These were:
1) the absence of stellar parallax meant either the Earth did not move or the stars were mind-bogglingly far away.
2) the fact that the brightness of Venus did not vary a huge amount meant either that its distance did not change that much (so orbited in a small epicycle), or did change a lot (so orbited the Sun) but were compensated by gibbous phases on the far side.
3) Earth is extremely large and massive, so should either be very difficult to set in motion, or very difficult to cease motion once started.
4) Some planets are much brighter than others, so either they are differently self-luminous, or else they have huge differences in size that reflect sunlight.
5) Gravity makes objects fall, either always toward the center of the universe (the Earth), or always toward the most massive object in the nearby vicinity.

II. Laws of gravity and motion
The huge body of evidence needed for such a great paradigm shift in all five of the above ambiguities was provided by:
1) Galileo's experiments on gravity and motion, where he found that all objects move under gravity in the same way, and there is a principle of inertia that says not only are massive objects hard to get moving, they are also hard to stop moving.
2) Galileo's observations with a telescope showed gibbous phases of Venus, moons of Jupiter, mountains on the Moon, and sunspots. He was also able to resolve patches of the Milky way glow into individual stars that must therefore be extremely far away.
3) Kepler found mathematical patterns in Tycho's solar system data, that showed they were elliptical shapes with the Sun at one focus, not centered on the Earth.
4) Newton established the laws of gravity and motion that could explain why objects orbited the Sun, given that the Sun was very massive and hence a very strong source of gravity.

III. The virial theorem
An exceptionally powerful ramification of Newton's laws is that the average kinetic energy of an object in orbit equals half the magnitude of its average (negative) gravitational potential energy. This connects the characteristic speed of orbiting systems to their characteristic size, and to the mass responsible for the gravity. It also has the counterintuitive property that when energy is removed from such a system, its kinetic energy increases owing to the fact that the system shrinks. Since the energy removed for a cloud of particles orbiting in all directions can be regarded as heat, and the rising kinetic energy can be associated with rising kinetic energy, this property is sometimes called a negative specific heat for self-gravitating systems. This leads to an instability in temperature that is responsible for very hot gravitationally condensed objects, i.e., stars.

IV. Relativity
In a remarkably ironic twist on the geocentric vs. heliocentric dichotomy, and the "eppur si muove" comment attributed to Galileo's trial, the modern theory of relativity asserts that all motion is relative. What this means is, although the shift in the five ambiguities listed above are still maintained in modern theory, it cannot be said that the motion of the Earth around the Sun, as opposed to the motion of the Sun around the Earth, cannot be held to be some universal truth. Instead, these models are examples of different language for talking about the same phenomena, in terms of different coordinate systems. Relativity says that coordinate systems are not real, and any can be used with the same laws of physics to predict the same behaviors. It may seem more reasonable to say the Earth spins, which is why all celestial objects show the same diurnal motion, but in relativity the distinction is purely one of human language and preference, not physical truth.