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.