How do we locate objects in the sky?  On the Earth, we locate objects
via their latitude and longitude.  Consider these lines of latitude and
longitude onto the sky.  
  'Celestial Equator' is projection of Earth's equator to the
  celestial sphere.  
  'Celestial poles' are in the directions of the projections of the
  Earth's pole.  
  'zenith' is the point directly over your head.
  'meridian' is the North-South line which goes through the zenith. 
  'declination' on the sky is directly analogous to latitude on
      Earth.  
  Because Princeton is in the Northern hemisphere, objects in
sufficiently Southern declinations will always be below the horizon.  

  "Right ascension" (or RA) is the celestial equivalent of longitude.
It is labelled in hours (not degrees), as the Earth rotates through
the full circle of right ascension in 24 hours.  (So one hour of right
ascension is equivalent to 15 degrees).

So where is the zero line of RA? 

  For this, we make reference to another circle.  The ecliptic is the
  intersection of the plane of the Earth's orbit around the Sun, and
  Celestial Sphere.  The axis of Earth's rotation is tilted relative
  to the perpendicular to its orbit around the Sun, by 23.5 degrees,
  so the celestial equator and the ecliptic are two great circles,
  tilted relative to one another by 23.5 degrees.  The two circles
  cross at two points, the atumnal and vernal equinoxes, and we use
  the latter as the zeropoint of the right ascension system.  

  (Note that we see the whole sky rotate through our view over 24
  hours, as the Earth rotates on its axis.  Right ascension doesn't
  follow this; the right ascension system is fixed relative to the
  stars, and doesn't change as the Earth rotates.). 

  Incidentally, it is the tilt of the ecliptic relative to the Earth's
equator that gives rise to our seasons... 

  What about constellations?  These are apparent patterns in the
distribution of stars that various cultures have noticed.  So when
someone says "the center of the Galaxy is in Sagittarius", the
direction to the center of the Milky Way lies in the direction of the
stars which comprise the Sagittarius constellation.  There is no
physical significance to the patterns; remember that the individual
stars making up a constellation may be at vastly different distances,
and are not physically associated with one another.  

A note on history of science:

  A few decades after Popper, Thomas Kuhn wrote a book on scientific
revolutions.  Periods of "normal science" are defined by
paradigms/world views, and one has paradigm shifts from one to the
other, which can be so great that communication between those living
under two paradigms is essentially impossible.  The standards of what
is considered scientific evidence will be different, and the
vocabulary is different.  (e.g., think of what a modern scientist and
Leonardo da Vinci mean by "water").  For a different riff on this
theme, see the article by Cleland and Chyba on Blackboard.  

  Note that Newton doesn't come up with an explanation *why* the law
of gravity has the form it has.  It is justified because (as we'll
see) it makes predictions that are in quantitative agreement with what
we observe in the world.

  OK, back to Newton and Kepler.  

  If you are moving at a speed v in a circle of radius r, it takes a
  time P to go once around of 2 pi r/P

  We'll find it useful to define angular velocity w (really the greek
  letter lower-case omega) = 2 pi/P.  An angle (in radians) per unit
  time, unlike v which is a distance per unit time.  (Note added
  later: we ended up never actually using this in class.)

  The centripetal acceleration for an object in circular motion at
  speed v is -v^2/r.  We'll demonstrate this in detail in a posting to
  the web.  

  But the acceleration due to gravity is GM/r^2.  Equate this to
  v^2/r, substitute v = 2 pi r/P, and we'll find Kepler's third Law.
  See if you can do it yourself! 
 
  In our derivation of the relation between period and semi-major axis
  for objects going around the Sun, we're considering the "two-body
  problem" of a single planet and the Sun; we're ignoring the
  gravitational pull of the planets on one another.  This turns out to
  be an excellent approximation for our purposes.  

  From last time, we saw, for an object of mass m moving at speed v moving at
  radius r around a massive body of mass M:
    Acceleration = v^2/r
    F = ma = mv^2/r = GMm/r^2
    But the period is 2 pi r/v.

   Eliminating v from these equations, we find: 

   a^3 = (G M/4 pi^2) P^2

  Kepler's Third Law!  "Newton's form of Kepler's Third Law", now
  generalized for orbits around any object, not just around the Sun.  

  Newton doesn't have a real explanation of what gravity is due to;
  we'll come back to this question with Einstein later in the course,
  who phrases gravity in terms of distortions in space and time, a
  completely different picture. 

   Now let's talk about energy; this is *not* the same thing as
   force.  (It is also not the same as power, which is the *rate* at
   which energy is expended per unit time).  Energy is related to (and
   has the same units as) "work": 

   Consider a force acting on a mass, to cause it to move.   Work is
   defined as the force you applied, times the distance you applied it
   over:  W = Fx.  
    Thus work (and energy) has units of force (kg m/sec^2) times
    distance (m).  1 Joule (unit of energy) is kg m^2/sec^2.  

 Kinetic energy (the energy of motion): Suppose that an object has an initial
velocity v_i.  Then apply a force to speed it up; after a little time,
its final velocity is v_f.

   But F = m a = m delta v/ delta t. 

   But the change in velocity, delta v is v_f - v_i.

   The distance the particle went is the average velocity times the
   time interval:
     x = (v_f + v_i)/2 times delta t

     So the work is:
     W = Fx = m (v_f - v_i)(v_f + v_i)/2 (note that delta t drops out!)
       = 1/2 m v_f^2 - 1/2 m v_i^2

  We define the Kinetic Energy, the energy of motion, of a body of
  mass m moving at speed v, as 1/2 m v^2. 

  There are other types of energy, such as:
    Potential energy: raise an object in the gravitational field of
      the Earth; it has the potential to gain kinetic energy when you
      drop it. 

  Is energy conserved?  Yes: energy can be converted between different
  forms, but you can't get rid of energy (actually, energy and mass
  can be converted into one another, via Einstein's famous E=mc^2;
  we'll learn about that later in this class).  

  The conservation of energy turns out, by a theorem due to Emmy
  Noether, to be a consequence of the statement that the laws of
  physics do not depend on time... 

  Temperature is proportional to the average kinetic energy of
  molecules in a gas.  We measure it in Kelvin: 0 Kelvin is the point
  at which all motions stop (turns out with quantum mechanics, you
  can't stop things completely, but that's a detail for now).  
   T(Kelvin) = T(Celsius) + 273.15.  

    We know about gravity.  There are three other types of forces in the
  universe:
     Electromagnetism
     Weak force (responsible for certain types of radioactive decay)
     Strong force (responsible for holding the nuclei of atoms
          together)
A *big* question in physics: are these forces all different
manifestions of the same phenomenon?  Electromagnetism and the weak
force turn out to be unified in this sense; the other two haven't yet
been worked in completely. 

  Electromagnetic force is, like gravity, also 1/r^2. 

  Consider two electrically charged objects, charges q_1 and q_2,
  separated by a distance r.  The force between them is:

    F = k q_1 q_2/r^2.  (The constant k ~ 10^10 Newton
    meter^2/coulomb^2). 

    Notice charges come in positive and negative, so the force can be
    attractive or repulsive.  

    The force of gravity is *much* weaker than that of
electromagnetism.  Consider the protons in an atomic nucleus: could
their mutual gravity hold them together? (after all, the electrostatic
force between them is pushing them apart).  You can work it out:
working out the ratio of the gravitational and the electrostatic force
gives:
     Gm_p^2/kq^2
(notice r drops out!) where m_p is the mass of the proton, and q is
the charge of a proton.  Plugging in numbers gets the astonishingly
small value of 10^{-39}.  So gravity is 10^{39} times weaker than
electromagnetism.  The so-called "strong force" is the glue that holds
the nucleus together.   It is strong only on the scales of nuclei; it
is negligible for two protons separated by a distance much larger than
the a typical atomic nucleus.  

Even though gravity is so weak, the dynamics of the universe is
dominated by gravity; this is because in most cases, the material of
the universe is electrically neutral, and has no net charge. 

  A brief overview of the periodic table.  The number of electrons in
an atom determines its chemistry.  The basic structure of atoms, with
their nuclei and surrounding electrons, was described.  A nucleus has
both protons and neutrons.  The type of the atom depends on the number
of protons (which, for neutral atoms, is the same as the number of
electrons); different *isotopes* of a given element will have
different numbers of neutrons, but the same number of protons.  

  1 atomic mass unit (AMU) is (very close to) the mass of the proton,
  which is also very close to the mass of a proton.  

  An atom is almost entirely empty space.  A proton has a radius of
about 10^{-15} meters, while the electrons are whizzing around at a
distance of 10^{-10} meters, i.e., 100,000 times further away.  
Experiments by Rutherford at Cambridge demonstrated this, by shooting
alpha particles (the nuclei of helium atoms, emitted by a radioactive
substance) at a thin foil of gold.  The vast majority of the alpha
particles go right through, but a small number will bounce back,
because they hit something very hard, namely the nucleus.

  As far as we know, an electron has no intrinsic size; they represent
  a true mathematical point. Something unfamiliar to our everyday
  experience! 

    So do the electrons also orbit in ellipses, just as we saw with
planets?  No.  There is a real problem: an accelerated electron gives
off energy in the form of light.  So it loses energy, and will have no
choice but to spiral into the nucleus (in a fraction of a second),
destroying the atom.  So how do atoms exist?  The solution to this
dilemma lay in the development of quantum mechanics.  On the very
small scales of atoms, nature works in very unfamiliar ways.

   "Radiation": any sort of emitted energy (in the form of waves or
   particles).  For example, electromagnetic radiation (light),
   emission of charged particles by atomic nuclei. 

  Radioactive decay refers to transformation of atomic nuclei from one
type to another, usually with the release of energy (in the form of
electromagnetic radiation, for example).  Indeed, the fact that the
interior of the Earth is extremely hot is largely due to the
radioactive decay of various elements, and the interior would remain
hot even if the Sun were to go out tomorrow.  

  The mix of isotopes in a given substance often gives important clues
about where the object came from.  For example, you can look at the
Deuterium/Hydrogen ratio in the water of the oceans, and compare that
with ice in comets.  They turn out not to be the same, at least suggesting that
not all the water on Earth came from comet impacts.
  (Normal hydrogen has a single proton in the nucleus, while deuterium
  has a proton and a neutron in the nucleus).  

  We can also use isotopes for radioactive dating.  Consider Carbon;
  its most common isotope is Carbon-12 (6 protons, 6 neutrons), but
  there is also Carbon-13 (with 7 neutrons).  Turns out life has a
  very slight preference to use Carbon-12 in its reactions.  So the
  C13/C12 ratio in living objects is somewhat smaller than in
  non-organic objects.  This has been used to determine whether some
  truly ancient rocks had carbon that was once incorporated into
  living objects.  

  There is also Carbon-14 (8 neutrons).  It is not stable: it decays
  with a half-life of 5700 years (i.e., after that time, half of the
  Carbon-14 will decay to some other element).  So if you knew how
  much C14 you had initially in some object, and could measure how
  much you had in hand now, you can determine the age of the object. 
  

© Copyright 2009 Christopher Chyba, Michael A. Strauss and Anatoly Spitkovsky