I would like to know about a fairly accurate Ptolemaic model of the
solar system.  Even though we know that the Earth is not the center of
our Solar System, I have read that Ptolemy?s model was tweaked in such
a way that it became fairly accurate for years with only minor
adjustments to keep it such. I figure it would be an interesting
contrast to the traditional solar system if there is some interesting
information about it. I would also like to get some historical insight
too.  I?m willing to give a $10 tip for an answer.

PirateDan

           The answer has  parts 1) and 2),  summary 3) and references 4).

  1) First the good news:

   The Ptolemaic model can be made as accurate as desired, just by
adding a sufficient number of epicycles.
   I have put the more technical articles into the REFERENCES section
at the endof this essay.
    While in Ptolemy's time this was a mere conjecture (guess, hope),
as explained here:


 "..If the astronomers of Ptolemy?s time had been asked what they were
doing with the mathematical models, they would probably have expressed
the faith that a finite number of epicycles on top of epicycles could
approximate the path of a heavenly body within the limits of
observational error for all time (more observations would only mean
more epicycles). This faith, seen from the perspective of 20th-century
harmonic analysis, amounts to the belief that the path of a celestial
body can be pictured as an almost-periodic function of time. The
2,000-year journey from faith to clear understanding is marked by many
great events...."
 http://www.ams.org/bull/1996-33-04/S0273-0979-96-00682-9/S0273-0979-96-00682-9.pdf

    Today we can prove that it is true, using a mathematical technique
(usually part of calculus college courses) called Fourier Analysis or
Harmonic Analysis.

    An interesting little-known fact is that Ptolemy himself had some
notion of this modern method, but, of course, could not of provide the
proof we know today (see references on Chords).


  2)   Then the news, bad or good,
       depending on one's point of view (it is relative :-)
     
    Today  we do not have two (or more) models of the solar system any more.
    Galileo's (see below) seminal work

                   Dialogue on the Two Chief World Systems 

     is dealing with what is, from today's point of view,  a 'false dichotomy',
     meanong a false 'either or'.

     An historical example of 'false dichotomy' was the argument between 
     proponents of Darwin and his opponents about
      'slow evolution' vs 'catastrophism.'

     Today we know (or believe ;-) that both things happened:


there were mass extinctions and there was slow evolution modified by these.
http://www.pibburns.com/catastro/extinct.htm

  In this example the answer to the 'false dichotomy' was :
        both processes exist.


 In case of geocentric vs heliocentric argument, the modern resolution is:

      There is no 'center of universe',  and 'no absolute space' and so

      one (and anyone) can place the 'frame of reference' where one wants.
       It is the task of physicists  to formulate the laws of physics
so that they are valid in all frames of reference (that is, to make
them invariant).

      Newton made the first big step in that direction, and Einstein
completed that process.
      Contrary to popular belief, Einstein did not discover the
concept of relativity or invariance, but completed the train of though
which started with Newton (see reference below).
    
   

  3)     So, in conclusion, from point of view of modern physics,
    the study of astronomy from geocentric point of view is as EQUALLY VALID
    as that of heliocentric.

 The mathematics of the planetary orbits is simpler when the frame of
reference is fixed to sun, but the frame of reference fixed to Earth
is closer to observation (from Earth).

      Based on this fact, I have recommended to a elementary school
teacher friend that she teach 'science' using the geocentric system.

 I said, 

         "to introduce children to study of science by saying: 
It looks like the sun and other stars are rotating around the Earth,
  BUT DON'T BELIEVE YOUR EYS, IGNORE THE  THE OBSERVATIONS,
   accept and parrot what the authorities say:  the stars are stationary .."

     is stupid. 

It is much better to describe the observations, 
  (such as, both Sun and 'other stars' are rotating with same speed of
15 degrees per hour, 'I wonder why' .. ')


and only later, when  some advanced math becomes available, show how
the geocentric system can be transformed into heliocentric. etc.

    I think she accepted my reasoning but said 

    " I would be run out of the school district,
      if I taught that stars rotate around the Earth ". 

It has become dogma.

   So, it may be educational to remind ourselves that Galileo was only
allowed to publish his revolutionary heliocentric theory after he
promised he would say
    
     "Heliocentric system is 'only a theory, not a fact' "

    He promised that he would also describe the other explanations aas well:

 ..  There is evidence that Galileo was warned against promoting the
Copernican theory at this time. A Dialogue Concerning the Two Chief
World Systems first made it past the censors by purporting to be
neutral on the astronomical debate. Its failure to do so was but one
of the objections raised after publication... (see references)

   Compare this with adding the "Intelligent Design Hypothesis" to the
teaching of evolution. This shows that while science is progressing,
the forces of authority and dogmatism remain strong or  unchanged.

   It may be of philosophical interest to note that Galileo only
converted to the new view  in  1610 after he SAW  satellites orbit a
body other than Sun in his new-fangled  instrument...
http://galileoandeinstein.physics.virginia.edu/lectures/galtel.htm

    and that Newton published his discovery 77 years AFTER THAT.

http://www2.jpl.nasa.gov/galileo/ganymede/discovery.html
http://en.wikipedia.org/wiki/Galilean_moon
http://members.tripod.com/~gravitee/toc.htm

   Newton's discovery may well have been facilitated by his effort to
decide which 'system of coordinates', i.e. which 'frame of reference'
is better. His solution:

  Let's express the  state of the bodies at time t,
        in terms of the system tied to state at time t-dt 
                                              (a small step to  the past)

may have lead tohis discovery of calculus
 (Contrary to popular belief,
  Newton did not discover the 'inverse square law' for gravity.
 That was known. Newton  used that law and calculus to derive
planetary orbits). That achievement far exceeded both phenomenological
models, the one of Ptolemy and of one Copernicus.

 Newton's dynamics provides not only description, but also the
explanation or causes. Later, Einstein completed the circle  by
putting Newton's explanation into fully invariant form (so that the
laws apply for an observer fixed to the Sun, to the Earth, or even an
arbitrary spaceship).

     We do not have two, or three, .. models any more. 
     Today, we have one invariant model,
     which is valid in all frame of reference.




4) REFERENCES


Explanation of Epicycles and description of three  systems
http://galileo.rice.edu/sci/theories/ptolemaic_system.html

Description of three  systems
http://en.wikipedia.org/wiki/Ptolemaic_system
http://en.wikipedia.org/wiki/Copernican_system
http://en.wikipedia.org/wiki/Tychonian_system

Galileo 
=======

His Life 

Galileo Galileo (1564-1642)
http://scienceworld.wolfram.com/biography/Galileo.html



His work (unfortunately, images are missing)
http://www.pd.astro.it/E-MOSTRA/NEW/A1003OPE.HTM


His Book  - Full text of the  Galileo's 1632 book:

Dialogue on the Two Chief World Systems

which synthesized his arguments for the Copernican system.

http://webexhibits.org/calendars/year-text-Galileo.html


Fourier Series  :
=================
 Any periodic function can be approximated by superposition of harmonic
 to arbitrary precision.

This site shows how adding harmonics makes Fourier series approximation
 (of a square pulses) better and better 
http://www.kettering.edu/~drussell/Demos/Fourier/Fourier.html


Further explanation and examples
http://mathworld.wolfram.com/FourierSeries.html

Applets (shows transform of several periodic functions)
 (applet assume browser with enabled Java)
http://www.falstad.com/fourier/


Ptolemy's methods of Chords, as an early concept of harmonic analysis
http://hypertextbook.com/eworld/chords.shtml


Relativity
==========



http://galileoandeinstein.physics.virginia.edu/lectures/lecturelist.html

SEARCH TERM: Relativity for beginners

e.g.
http://www.ajnpx.com/html/Relativity-for-beginners.html
http://www.ajnpx.com/html/Relativity.html

http://www.ebicom.net/~rsf1/gal-rel.htm

http://www.amazon.com/gp/product/0521640660/102-1890871-9776922v=glance&n=283155



  Please, feewl free to aske fro clarifications (RFC) and
  when all is clear, provide a rating.

Hedgie

Thank you, the information about epicycles was just what I was looking
for, and the historical information was useful as well.

Here is a link that explains the basic idea behind Ptolemy's Model.

http://alpha.lasalle.edu/~smithsc/Astronomy/retrograd.html

Hope this helps.

The heliocentric universe is not something that could be established
as a definite fact in the time of Copernicus. It had great appeal,
however, because it was much simpler than Ptolemy's clever, but
elaborate, collection of rotating spheres. Epicycles (circular orbits
around circular orbits) were necessary in his model to explain the
motions at all. Careful observation, even without telescopes, however,
showed that this was not enough. Additional epicycles were added to
explain the noncircular orbits. The worst of the classical planets is
Mars. Tycho assigned Kepler the job of figuring out the orbit of Mars
and explaining it with the appropriate collection of epicycles. Kepler
was a Copernican, but epicycles were needed whether or not you
believed in a heliocentric or geocentric model. After years of work,
Kepler was having great difficulty fitting the motion of Mars to an
epicyclic model. This is when he had his great insight about
elliptical orbits and formulated his three laws of planetary motion,
which were later shown to be simple results from Newtonian physics.

The upshot of this is that you can fit all the orbits with arbitrary
accuracy if you are prepared to introduce an arbitrary number of
epicycles. For careful observations without telescopes, though,
prediciting the orbit of Mars became untenable with a reasonable
number of epicycles.

Nice answer, but it is worth talking a bit more about the heliocentric
reference frame versus the geocentric one. I agree that it is all a
matter of reference frame and I did not want to get into a long
discussion of this in my original comment. The problem, like
epicycles, is that the heliocentric theory totally fails Occam's
razor. The extent to which this is true became even more apparent with
the discovery of stellar parallax. In Ptolemy's time, one could
propose that the stars were on a fixed sphere that rotated around the
earth at a constant rate. With the discovery of parallax, one would
have had to modify this view to say that the stars fill a three
dimensional volume, which rotates as a rigid body around the earth.
This is a major break from the original proposal and, I think, as much
of a mess as the epicycles on top of epicycles. The game of science is
to reduce our explanations to the simplest ones that can accurately
predict phenomena. This is Occam's razor and the essence of modern
science. Ptolemy's theory is hopelessly complicated and arbitrary, a
collection of ad hoc additions that is necessary to preserve the
formerly cherished and now deprecated view that the earth is the
center of the universe.

Pedagogically, I think there is merit in using this as an example of
reference frames, but one has to be careful about it so as not to
mislead. Also, rotation is not relative like uniform translational
motion. The spinning of the earth makes us lighter at equator than we
would be if the earth were not spinning. We can also measure the
increased time that it takes for light to go around a ring of mirrors
in the direction the ring is rotating (Sagnac effect). Einstein was a
great fan of Ernst Mach and wanted to formulate general relativity so
that a universe that was spinning around the earth would generate the
"centrifugal force" that makes us lighter at the equator. General
relativity does not make uniform rotation relative, however, so the
heliocentric model would cause us to have to modify relativity as
well.

kottekoe-ga

 I did not noticed your insightful comment when I wrote the answer.
 Bad habit on my part - It would make my work easier. 

   The 'pedagogical suggestion' was intended for K-2 level, for children
(and adults) who are not even aware of any rotation. The first look at
the sky and shadows. Certainly should not be carried too far.

And I did  say:

Newton's dynamics provides not only description, but also the
explanation or causes..

so we are not implying these two concepts, description of orbits and
actuall dynamics, are equivalent or have the same value as a
scientific explanation.

The other is more complex:
Invariance of General Theory of Relativity to rotation and Mach
Principle  were touched upon in GA before:

http://answers.google.com/answers/threadview?id=31905
http://answers.google.com/answers/threadview?id=33716

answers to these questions ultimately depend on cosmological model,
large scale structures, 3K radiation ... which, so far, seems to point
to a special frame.
Nevertless, as a  philosophical concept, Einstein's requirement that
laws of physics should be formally covariant, seems to be more
fundamental. Do you consider that goal as abandoned? Equations of GTR
are formally invariant, are they not?

 Kepler ..had .. great difficulty fitting the motion of Mars to an
epicyclic model. This is when he had his great insight about

 Interesting. I read that Kepler only got his great result after Tycho died
 and Kepler finally got his hands on all the data. But he was tackling Mars
orbit before that. Is there a discrepancy here? Do qwe know when
excatly he got that insight?

Hedgie,

Thanks for your reply to my comment. Yes, Einstein's equations are
covariant, meaning the equations are identical in any legitimate frame
of reference. The principle of equivalence is valid, i.e. it is
impossible to distinguish between an acceleration and a gravitational
field in a small neighborhood of spacetime. Mach's principle is not a
part of General Relativity, though it did motivate Einstein's work. It
is easy to determine that the earth is rotating by any number of
experiments that do not involve looking at the movement of celestial
objects. Any accurate gyroscope can see the spinning of the earth.
Even something as rudimentary as the Foucault Pendulum can see it.
This is fully consistent with Einstein's theory, which does not say
anything about relativity of angular motion.