Re: Astronomy - History

[palmer@sfu.ca (Leigh Palmer)]

>   Two questions and a comment re: Planetary orbits
> 1) How does one determine the orbit of a superior planet, eg Mars, from
> visual observations? A reference will do.

It is not an easy algorithm. Carl Friedrich Gauss was the first to do it.
He demonstrated that three observations (time and celestial coordinates)
were necessary and sufficient to determine a Keplerian orbit of a body of
negligible mass about the Sun. Brian Marsden has written the code that
implements Gauss's method together with modern tweaks to produce the
Keplerian elements of comets that are published by the Central Bureau for
Astronomical Telegrams and are used widely to predict the circumstances
of comets like Hale-Bopp, now visible in the southern evening skies.

> 2) Any references for a technical description of the Ptolemaic Model, with
> numerical values?

See the *Almagest*.

I'm away from my references, but the model is considerably more complicated
than is suggested in most introductory textbooks in astronomy. There are
also some features which are left out of many textbook descriptions for the
sake of simplicity which, if left in, make the model more understandable.

"The" Ptolemaic model (deferents & epicycles) appears in many variants with
many values of the parameters. It was, after all, not a cosmology so much
as it was a computational device for predicting the motions of the planets.
Ptolemy, for example, probably didn't believe the planets actually moved
according to the simple motions implied by his model. He knew very well
that the model's predictions were discrepant with observation. Ptolemy
recorded positions of Mars and plotted some of them against his model curve.
It is interesting to note that the points he plotted cluster about four
regions of the curve, the regions which would agree well with a best
fitting ellipse when reduced to a heliocentric frame of reference. One
suspects he made other observations which he omitted from his result. After
all, the rules for investigation were not well established at that time.

> 3) One can produce an ellipse by using one epicycle, see below. Is this
> well known?

It is obvious when one writes it down algebraically. It reduces to

                 x = R cos Wt + r cos (-Wt) = (R+r) cos Wt
                 y = R sin Wt + r sin (-Wt) = (R-r) sin Wt

This is the well known equation of a central ellipse.

> Is there any relationship to the Ptolemaic Model?

No. In the Ptolemaic model both angular speeds are in the same sense.

>   Consider an epicycle (small circle) of radius r, whose center moves
> on the the larger circle (of radius R) at the angular speed W.
>   Point P moves on the epicycle a the angular speed -W.
>   If one starts with the epicycle at +R on the X-axis, and P at R+r on
> the X-axis, P will trace out an ellipse whose semi-major axis is R+r and
> semi-minor axis, R-r.
>   An interesting result is that the line from the CENTER to P will trace
> out equal areas in equal times.

Not if I've analyzed your construction correctly. I take W to be constant
as it is in Ptolemy's model. The rate of sweeping out of area is just

                  .    1                         2    2    2
                  A = --- rho W     where     rho  = x  + y
                       2

rho, the distance from the center, is not a constant.

Leigh