Lecture 3, February 10; Christopher Chyba
Homework 1 is due on Thursday; there is a review session at 7:30 PM in the Peyton Hall auditorium tomorrow (Wednesday).
The observing session is scheduled for Thursday. Come to Peyton Hall at 8 PM.

We'll see that Newton's intellectual revolution was that the laws of
physics in the heavens are the same as the laws on the Earth, in
contradiction to the Aristotelian view.  Later, we will learn that
Cecilia Payne showed that the material out of which stuff in the
heavens are made of (especially the stars) is the *same* as that of
the Earth.

  (Interestingly, we're now realizing that much of the universe is
made of dark matter and dark energy, materials very different from
things familiar on Earth.  It is sounding rather Aristotelian!  We'll
discuss this in much more detail later in the course.)

  The small-angle formula: consider a long skinny isosceles triangle
  with small angle p of height d and base a; then:
     p = a/d (measured in radians)

  The parallax angle p (half the angle by which the apparent position of
  a star moves as the Earth moves around the Sun) is related to the
  distance d of the star as:
   d = 1/p
  where here, p is measured in arcseconds, and d is measured in 
  "parsecs" (PARallax of one ARCsecond).  1 parsec (often abbreviated
  'pc') is equal to about 3.3 light years.  


   
   (Review of Copernican, Galilean and Aristotelian view of the Universe; see Lecture 2)

   Onto Johannes Kepler.  Using precise measurements of the motions of
planets on the sky carried out by Tycho Brahe, he measured the paths
of the planets (especially Mars, which has the largest eccentricity, and 
hence most irregular motions), and came up with three
empirical laws of planetary motion around the Sun that
explained/predicted the positions of planets to much greater precision
than anyone before:

     1. Planets move in ellipses, with Sun at one focus.  
        Half of the main axis of the ellipse is called the 'semi-major
         	axis', denoted with the letter a. 
	The eccentricity of an ellipse, denoted with the letter e,
	characterizes the deviation of an ellipse from a circle.  

	Closest point to the Sun is the perihelion = a(1-e)
	Furthest point is the aphelion = a(1+e)

   Mars has the most elliptical orbit of all the easily visible
   planets, e = 0.09, and therefore most directly demonstrated this
   first law. 

      2.  Planets sweep out equal areas in their orbits in equal times.   So
      the planet is moving faster at perihelion than aphelion.  

      3.  Now comparing orbits of different planets: 
         Period^2 is proportional to semi-major axis^3
	 So P^2 = C a^3, for some constant C.  

	 For P measured in years, and a as AU, the constant C is one.  

  Francis Bacon asked the question: we accepted Aristotle at face
value for 1800 years.  How could we have been so wrong for so long?
He identified a series of "idols of self-deception", ways in which we
fool ourselves by mixing our own human prejudices with our
observations of the world.  To do better, and avoid being fooled in
the future, he suggested an extreme form of empiricism, which was a
first important step toward what we now call the scientific method.
 
Problem with extreme empiricism is that there is no guidance as to what 
to observe. We will talk about that after we discuss Newton, who shattered 
the Aristotelian dicotomy between Earth and heavens.

 Newton's book, The Principia, puts down three fundamental laws of motion:
     -An object at rest tends to stay in rest, and an object in motion
     continues at the same speed and the same direction, unless acted
     on by a net force.  Sometimes called 'The Law of Inertia'. 
       (Note the difference from the Aristotelian worldview!  But
         similar to what Galileo had said.)
     So if you move in a circle, even at constant speed, you are being
   accelerated, as your direction is constantly changing.  

     -The net force on an object  = its mass x its acceleration, F=ma.
     acceleration is the change in velocity (which includes speed
     *and* direction) per unit time.   

     Think of mass m as a proportionality constant between
     acceleration and force.  Here F is the *net*, or total force,
     acting on an object.  Note this doesn't say what the force is due
     to; see below. 

     Think of units here:
        Velocity has units of meters/second
	Acceleration has units is meters/second^2
	Force: kg m/sec^2 (sometimes also called a Newton, in honor of
  	  you-know-who).  

     -For every force of one body A on another B, there is an equal and
     opposite reactive force of B on A. 

So where do these forces come from?  Well, one source is gravity, an
attraction between any two masses M and m, separated by a distance r: 

    F = G M m/r^2. 

So suppose M is the Earth, and m is an eraser falling in the
Earth's gravity (with no other forces acting on it).  The force on the
eraser is given by the above expression, and is related to the
acceleration of the eraser via F=ma.

  Thus ma = G M m/r^2, or a = G M/r^2.  The mass of the falling object
drops out of the equations, and the acceleration does not depend on
the mass of the eraser (a feather gives you the same answer, in
the absence of air resistance, as demonstrated by the lunar
astronauts).

  (Here the r is the distance to the center of the Earth.)

  The real clincher is that this law explains not only motions here on
Earth, but also the motions of planets around the Sun, and the moons
around Jupiter.  In particular, we'll see that Kepler's laws of
planetary motion follow directly and quantitatively from Newton's
laws.  So the same laws hold in the heavens and the Earth, which was
really a dramatically non-Aristotelian development.  To demonstrate
explicitly that the material of which the distant stars are made is
the same as that here on Earth only happened in the 20th century;
we'll get there in a few lectures.

  Note that in everyday experience, friction is ubiquitous, which
  makes things more complicated; a thrown eraser doesn't follow
  exactly the path that gravity alone would predict.  (But it does
  really follow the path that we would predict if we included the
  effects of air friction.  And it does a superb job of predicting,
  say, the path of the planets where friction is a complete
  non-issue).  

  Then followed a discussion of philosophy of science; how do we
decide what is "really true" in a scientific sense?  Scientific ideas
are useful to the extent that they can be tested, i.e., that they can
be falsified.  
  Does science work via induction: taking a series of observations and
drawing broad generalizations/conclusions?  Or via
extrapolation/testing from a pre-existing theory (i.e., deduction)?
Karl Popper (philosopher of science) thought only the latter was 
"real science", but in practice, science uses both approaches.   
Popper points out that you can't be completely empirical: you always 
carry out experiments/observations of Nature in the context of some 
theoretical framework.


Theory: A thoroughly tested set of ideas, that forms the
foundation of a whole field of science.  Examples include: "Theory of
gravity", "Theory of plate tectonics", "theory of evolution".   A much
weaker word would be 'hypothesis'.   

 A theory should have a lot of explanatory power, and all else being
equal, a theory should be simple.  Given the choice between two
explanations of a physical phenomena, each of which does an equally
good job of matching the observations, choose the simpler one.

  Most importantly, as Popper said, a scientific theory must be
*testable*.  It is the testing that makes it science.  Thus while one
can never prove a theory right, it can be falsified.
 
There are difficulties with this idea.  No test of a theory is pure;
there are always additional assumptions.  For example, consider the
discovery of Neptune.  The planet Uranus showed features in its orbit
that Newton's gravity (which we're about to discuss) didn't predict.
Did Newton screw up?  Or was its orbit being perturbed by another, as
yet unseen planet?  The latter turned out to be correct, and indeed
Newton's Laws allowed a prediction of where the new planet should be:
a triumph of Newton!

  The orbit of Mercury is also not well-explained by Newton.  Perhaps
another unseen planet?  No, here this represents a true failure of
Newton's laws itself (which did get explained by a more comprehensive
theory, namely Einstein's Theory of General Relativity; we'll learn
more about this later in the course).
Notes for Lecture 4