% Ptolemy / Copernicus lab exercise

\documentstyle[12pt,aaspp4,flushrt]{article}
\begin{document}


\title{Geocentric and Heliocentric World Views} 

%\author{Guy Worthey}

\begin{abstract}

We explore in some detail how the geocentric cosmology worked, and
what observations caused the adoption of the heliocentric view. For
this lab, your writeup will consist only of these sheets plus separate
sheets with the answers to the
questions posed throughout the exercise.

\end{abstract}

\section{The Motion of Mars}

Stars appear to have a simple motion in the sky: they rise in the
east, travel along a smooth arc, and set in the west. This motion was
easily explained by early geocentric cosmologies that placed the earth
at the center of the universe. However, a small group of objects,
referred to as ``wanderers'' (or planets), did not share this motion,
but drifted among the fixed stars. The rate that a wanderer moved
among the stars varied, and (except for the moon and Sun) even the direction
of their motion changed! This westward motion is referred to as {\bf
retrograde motion} (``normal'' eastward motion is termed {\bf prograde
motion}).

In order to account for these oddities, astronomers had to modify
their early Earth-centered model. 

\begin{itemize}

\item{ Table 1 gives the celestial positions of Mars at two
different times during 1992-1993, expressed in terms of right
ascension and declination. These celestial coordinates are similar to
longitude and latitude on the earth. Right ascension is analogous to
longitude and measures positions in the sky in an east-west
direction. It has units of hours, minutes, and seconds. Declination is
analogous to latitude and measures positions in the sky in a
north-south direction. It has units of degrees, arcminutes, and
arcseconds. For this exercise it is adequate to express right
ascension in units of hours and minutes and declination in units of
degrees and arcminutes. A graph whose axes use these units is shown in
Fig. 1. {\bf Plot} the Table 1 positions on Fig 1. (Use pencil.)
}

\item{ Assume that Mars moved through the sky in a {\it uniform}
manner. Sketch on Fig. 1 the positions you would expect for Dec. 1,
1992 and Feb. 1, 1993 (one-third and two-thirds of the way to April 1,
respectively.) Connect all four points with a single straight line.
}

\item{ A sample of Mars's actual positions during these months is
given in Table 2. Plot {\it these} points on Fig. 1. Connect them
smoothly with
a dotted line.
}

\item{ {\bf Q1:} In your writeup, describe the actual motion of Mars and 
compare it with that predicted from bullet 2; uniform circular motion.  }
\vskip 0.3in
\centerline{Table 1.}

\begin{center}
\begin{tabular}{lllll}
\hline
Date && Right Ascension && Declination \\ 
\hline \hline
Oct 1, 1992   && 6$^h$ 44$^m$ && 23$^\circ$ 25$'$ \\
April 1, 1993 && 7$^h$ 21$^m$ && 24$^\circ$ 41$'$ \\
\hline
\end{tabular}

\vskip 0.6in
\centerline{Table 2.}

\begin{tabular}{lllll}
\hline
Date && Right Ascension && Declination \\ 
\hline \hline
Oct. 1, 1992   && 6$^h$ 44$^m$ && 23$^\circ$ 25$'$ \\
Nov. 1, 1992   && 7$^h$ 40$^m$ && 22$^\circ$ 46$'$ \\
Dec. 1, 1992   && 8$^h$ 01$^m$ && 23$^\circ$ 09$'$ \\
Jan. 1, 1993   && 7$^h$ 31$^m$ && 25$^\circ$ 43$'$ \\
Feb. 1, 1993   && 6$^h$ 45$^m$ && 27$^\circ$ 01$'$ \\
March 1, 1993  && 6$^h$ 44$^m$ && 26$^\circ$ 21$'$ \\
April 1, 1993  && 7$^h$ 21$^m$ && 24$^\circ$ 41$'$ \\
\hline
\end{tabular}
\end{center}

\clearpage

\item{
Early geocentric models assumed everything revolved around the Earth
in a {it uniform, steady} manner and on a perfect circle. Under these
circumstances, Mars would cover the same arc distance along its orbit
in equal amounts of time. For example, the amount of time Mars would
need to move from position 1 to 2 in Fig. 2 would be the same as to
move from position 2 to 3. Suppose that Mars moved in this
fashion. On Fig. 2, draw a straight line from the Earth to each
position of Mars, and extend it to just beyond that stars. At the end
of each of these lines draw a small circle to represent Mars.
}


\item{Connect the predicted Mars positions with a line. {\bf Q2:} Does the
motion of Mars produced by the above figure agree with that predicted
by uniform circular motion (bullet 2)? Why or why not?
}

\item{{\bf Q3:} Is there any way to produce the {\it actual} motion of Mars with
the Earth-centered, uniform motion model?
}

\item{ In an attempt to explain the loop-like retrograde motion of
Mars, an ingenious fix, the {\it epicycle}, was added to the geocentric
model. This revised model is called the {\bf Ptolemaic model} in honor
of Claudius Ptolemy (c. 140 A.D.) who published big tables predicting
future planetary positions based on this model. In this model, the
orbit of Mars looks like Fig. 3.

In such a model, Mars has two motions. Mars revolves on its epicycle,
and the center of the epicycle revolves around earth. By assigning
speeds to each of these motions, the observed retrograde motion of
Mars could be produced.

On Fig. 3, draw arrows to illustrate these motions.
}

\item{Let us visualize the motions produced by a model like Fig. 3,
and draw them on Fig. 4. 

A. Assume Mars moved along its epicycle, but {\it did not} revolve
around the earth. Illustrate how Mars would appear to move against
star field A in Fig. 4. (Act it out: draw some stars on a blackboard,
Joe student stands next to the board and holds up his fist to
represent Mars; and spins around. How does Joe's fist move as seen by
Jill and Jack?)

B. Now let the epicycle move slowly around earth. Use star field B in
Fig. 4 to show the path Mars would take among the distant stars. (Act
it out: Joe walks past the board as he spins. Careful, Joe, don't get
too dizzy.)  }

\item{{\bf Q4:} Explain in words how the observed retrograde motion of Mars
could be produced in this modified geocentric model.
}


\item{{\bf Q5:} What observational data is needed to force a revision of the
early uniform-motion geocentric model?
}

\item{{\bf Q6:} Explain how the addition of the epicycle ``fixed'' the
geocentric model.
}

\item{Make sure you can define the terms: retrograde motion, right ascension and
declination, epicycle, and Ptolemaic model.
}

\end{itemize}

\section{The Ptolemaic Model}

During the time of Galileo Galilei (1564-1642), the Ptolemaic model,
shown in Fig. 5, was widely accepted. It placed each planet on its own
epicycle. Planets possessed two motions, one along its epicycle and
the other around the Earth. This model agreed with the observational
data of Ptolemy's era, and was able to predict the future positions of
the planets (to within a few degrees, but this was considered ``good
enough.'')


One of the Ptolemaic model's important properties was that the
epicycles of Mercury and Venus were attached to a bar-like structure
connecting the Earth and the Sun. 

\begin{itemize}

\item{In Fig. 5, locate Mercury and Venus, their respective
epicycles, and the bar-like structure.}

\item{{\bf Q7:} How did the epicycle allow this model to explain retrograde
motion?}

\item{{\bf Q8:} Why was this bar-like structure needed in the Ptolemaic model?
}

\item{{\bf Q9:} Why was the epicycle of Mercury smaller than that of Venus?
}

\item{{\bf Q10:} Why, in this model, are Mars, Jupiter, and Saturn {\it not} connected
to the bar-like structure? That is, what observation requires these
planets to be free?
}

\item{ Figure 6 shows an enlarged view of the Ptolemaic model showing
the Earth and Venus. On Fig. 6, shade the portion of Venus not
illuminated by the Sun. {\bf Q11:} Can a ``full'' Venus ever be seen from the
Earth in this model? Why?
}

\item{{\bf Q12:} What observational phenomena did the Ptolemaic model account for?
}

\item{{\bf Q13:} What predictions did the Ptolemaic model make in regard to the
phases of Venus?
}

\item{ {\bf Q14: Draw} a geocentric Ptolemaic cosmology. Label the objects in the
figure. (Yes, this is just a re-sketch of Fig. 5.)
}

\end{itemize}

\section{The Copernican Model}

Sometimes a seemingly inconsequential observatoin has a profound
implication. Individuals who first recognized the consequence of these
observations stand out as the geniuses of their time. One such person
was Galileo Galilei (1564-1642). Galileo was the first person to use a
telescope to observe Venus. Over time, he noted that Venus, like the
Moon, exhibited phases, changing from a small, disk-like object to a
large crescent-shaped object. See panels a-e in Figure 7.

Galileo was aware of the recently published heliocentric model
developed by Nicolaus Copernicus (1473-1543). This model was
extremely controversial because it removed the Earth from its
privileged position at the center of the Universe. Nevertheless,
Galileo found that the heliocentric model was able to completely
explain the observed phases of Venus.

\begin{itemize}

\item{Drawing on Figure 8, place Venus in the positions needed to
produce the phases seen in panels a and e of Fig. 7. Label these
positions with the letters {\bf a} and {\bf e}.
}


\item{{\bf Q15:} Explain how the heliocentric model accounts for the observed
phases of Venus.
}

\item{{\bf Q16:} Explain how the heliocentric model accounts for the observed
sizes of the various images of Venus.
}

\item{Apparent size is related inversely to its distance. The farther
away an object is, the smaller it will appear. At two distances, {\bf
a} and {\bf e}, the apparent size changes as
$$ {{\rm distance\ from\ us\ in\ panel\ a}\over{\rm distance\ from\
us\ in\ 
panel\ e}} = {{\rm apparent\ angular\ size\ at\ e}\over{\rm apparent
\ angular\ size
\ at\ a}}$$
or, with astronomical brevity,
$$ {{D_a}\over{D_e}} = {{\theta_e}\over{\theta_a}} $$
Using a ruler, measure in millimeters the apparent size of the disk of
Venus in panels {\bf a} and {\bf e}.

{\bf Q17:} Record the two measurements ($\theta_e$, $\theta_a$, in mm) in your report.

Assume that Venus was located 1 unit from the Earth in panel {\bf
e}. Use the equation above to compute how far from Earth Venus is in
panel {\bf a}. {\bf Q18:} Show your work.
}

\item{{\bf Q19:} Why does this result seem reasonable?}

\vskip 0.2in
\hrule
\vskip 0.2in

\item{{\bf Q20:} If the distance from Earth to the nearby Venus is 1 unit, and
from the Earth to the far away Venus is 7 units, then how many units is
Venus from the Sun? How many units is the Earth-Sun distance? See
Fig. 9 if you need help visualizing this.
}


\item{{\bf Q21:} 
If the Earth-Sun distance is defined as {\it one astronomical unit},
1 A.U., what is the Venus-Sun distance in astronomical units?
}

\item{Now let's think again about Mars. In the heliocentric picture,
Mars orbits around the sun in an orbit wider than that of Earth.  See
Fig. 10. 

Since Mars is located farther from the Sun than the Earth, it moves
more slowly in its orbit. For instance (in Fig. 10) in the time it
takes the Earth to move from its points 1 to 2, Mars will have moved
only from its points 1 to 2.

Using your ruler, connect the same-numbered points along the two
orbits, and extend the line to the stars. Trick: extend each subsequent line
a bit farther to the right --- to avoid crossing your lines. Connect the
end of each line to show the path that Mars would make among the stars.
}

\item{{\bf Q22:} Explain how retrograde motion is produced in this model.
}

\item{{\bf Q23:} Why couldn't the Ptolemaic model be modified to account for
Galileo's observation that Venus exhibited a full phase?
}

\item{{\bf Q24:} How is Venus's phasing behavior explained in the heliocentric model?
}

\end{itemize}
\clearpage

%%%% figures go here . . . . 

\begin{figure}
\epsscale{1.0}
\plotone{heliolab-fig1.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.45}
\plotone{heliolab-fig2.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.45}
\plotone{heliolab-fig3.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.6}
\plotone{heliolab-fig4.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.9}
\plotone{heliolab-fig5.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.9}
\plotone{heliolab-fig6.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.9}
\plotone{heliolab-fig7.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.45}
\plotone{heliolab-fig8.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.9}
\plotone{heliolab-fig9.ps}
\caption{}
\end{figure}

\begin{figure}
\epsscale{0.75}
\plotone{heliolab-fig10.ps}
\caption{}
\end{figure}



\end{document}

