Sindbad~EG File Manager

\documentclass[11pt]{amsart}
\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   % ... or a4paper or a5paper or ... 
%\geometry{landscape}                % Activate for for rotated page geometry
%\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}
\def\insertimage#1#2{\includegraphics[width=#2\textwidth]{#1}}

\title{R\o mer and the Speed of Light}
 

\begin{document}
\maketitle

\section{R\o omer measures the orbital period of Io}

The date was April 29, 1676.  The place was the Royal Observatory in Paris.
The Danish astronomer Ole R\o mer was observing Io,  one of the moons of Jupiter.
He had been measuring the orbital period of Io since at least March 7.  This 
can be done quite accurate to the nearest second, by noticing the exact moment 
at which Io enters or leaves the shadow of Jupiter.   In other words, 
for half of each orbit, Io is eclipsed by Jupiter;  one second you see it, 
the next second it enters Jupiter's shadow and you don't see it.  Or,  one 
second it is invisible, and the next second you see it, as it emerges from 
Jupiter's shadow.  These two events are called {\em eclipses} and {\em emergences}.

Io takes about 42 hours to orbit Jupiter.
From Earth,  in any given week,  you can see several emergences, or several eclipses,
but usually not both,  as if an emergence is visible, the eclipse will take place 
when Io is on the other side of Jupiter from Earth.   To measure the orbital period,
you measure the timing of successive eclipses or emergences.   To improve the accuracy,
you measure, for example,  ten successive eclipses and take the average time.

The reason R\o mer was measuring the orbital period of Io was this:  he wanted to 
show that the speed of light was finite.  The idea is this:  sometimes Earth is 
moving {\em towards} Jupiter,  and sometimes Earth is moving {\em away} from Jupiter,
depending on the time of year.   Let $d$ be the distance that Earth moves (toward or away from)
Jupiter during one orbit of Io.   If the speed of light is finite, it will take light
a certain time $T$ to travel the distance $d$, and our measurement
of the orbital period of Io will be off by the amount $t$.  It will be shorter by $t$ 
when Earth is moving towards Jupiter, and longer by $t$ when Earth is moving away from Jupiter.
So if we compare the measurements at different times and get different answers, we can 
conclude that the speed of light is finite.

R\o mer did this,  at two different times of the year, sometime before 1676, and found 
no difference in the orbital periods.  Therefore, the speed of light is very fast!  
If it took one second to travel an Earth diameter, we would expect the two measurements
to differ by seven minutes,  so light must travel hundreds of Earth diameters per second
(or perhaps the speed is infinite).

But R\o mer realized that, if the speed of light is finite, the effects should pile up 
over many orbits.  There is a certain time when the Earth is exactly between the Sun 
and Jupiter;  then Jupiter is said to be ``in opposition'' (it is on the opposite side of 
Earth from the Sun).   Right at the opposition, Earth is not moving either towards or 
away from Jupiter, but in a perpendicular direction.  Therefore, right after the opposition,
you should get an accurate orbital period measurement, regardless of the speed of light.
But if you wait a while and measure again, you should get a different orbital period,
due to the accumulation of errors due to the speed of light. 

R\o mer made these measurements after the opposition of March 2, 1672.  The first 
four orbits gave an average period of 42 hours 28 minutes and $31 \frac 1 4$ seconds.
After thirty orbits, the average period was 42 hours 28 minutes and 3 seconds.  That 
was 32 seconds shorter, or about one second per orbit.  Thus, luckily for R\o mer, the 
error that was too small to be seen in one orbit could be seen after 30 orbits.

R\o mer then calculated (using trigonometry, especially the law of cosines, on the triangle
formed by the Sun, Jupiter, and Earth)   that light would travel a distance equal to 
the radius of Earth's orbit in eleven minutes, and would reach Earth from Jupiter in about 
ten minutes (at the distance Jupiter was in April, 1672).   The actual speed of light 
could not be determined at that time, since the radius of the Earth's orbit was not yet known.
But at least, it was finite!  

\section{Longitude and the Moons of Jupiter}
How did a Danish astronomer come to be working at the Royal Observatory in Paris,
and why was he being paid to measure the moons of Io?   It was {\em not} because of 
curiosity about the speed of light--that was what we now call a ``spin-off''.  The real 
issue was the search for a way to determine longitude at sea, which was a vital issue 
for navigation in the Age of Discovery.

To determine longitude, it would be enough to observe the time of sunset or sunrise, 
which can be predicted a year in advance;  if the sun where you are is going down one 
hour earlier than in Greenwich Observatory, then you are 15 degrees of longitude west
of Greenwich (because 15 is one twenty-fourth of 360).   So all you need is a clock..
But in 1672, there were no clocks that could keep accurate time on board a tossing, turning 
ship.  Pendulum clocks worked only on land.  So accurate navigation was extremely difficult.
You could try to estimate the ship's speed by letting out ropes and counting the knots
going by, but if there were ocean currents, which there were, you would not account for their effects.

Galileo had proposed using the moons of Jupiter for a clock; but it seemed to be too difficult
to observe the eclipses of the moons from on board a ship.  However, the method might work 
on land, and this method had been pioneered by the Italian astronomer Giovanni Cassini.
Cassini was invited to France by King Louis XIV to establish the Royal Observatory.  It
opened in 1671, with Cassini as director.  He immediately sent  Jean Picard to Uraniborg,
Denmark, the island where Tycho Brahe's observatory was.  The idea was that Cassini in Paris
and Picard in Uraniborg would measure the times of the eclipses of Io, and the difference of 
times would enable them to compute the difference in longitude of Paris and Uraniborg.
That was the reason for the interest in the accurate measurement of the orbital periods.
R\o mer was a young assistant to Picard, and he must have been a good one, since Picard 
arranged for him to come to Paris and work at the Royal Observatory.  R\o omer had 
already been observing the eclipses of Io since at least 1668, so he probably showed 
Picard how to do it!    

\section{Reception of R\o mer's work} Cassini was not immediately convinced, and apparently 
neither was Picard.  Therefore R\o mer, who was in a junior position,  did not publish
his work.  The reason for Cassini's caution was that there are 
several other possible causes of discrepancies in the orbital periods.  The orbits of Earth
and Jupiter are not circular but elliptical, and each of these causes a variation in the 
orbital period.  Those could be calculated and corrected for;  and even after correcting 
for the speed of light there still remain discrepancies;  those turned out later to be because
of ``orbital resonance'' with the other moons of Jupiter (Europa and Ganymede in particular),
but that explanation didn't come for another century.

 In any event, after four years further measurements must have established the discrepancies
 more convincingly.  Cassini announced 
on August 22, 1676, that henceforth the tables of predicted eclipse times would incorporate 
a detection for this ``new, not previously detected, inequality.''   However, the corrections
that Cassini made were not the same for each of Jupiter's moons (as they should have been 
if they were due to the speed of light), so apparently Cassini still regarded this as an 
empirical correction.  But in England,  R\o mer's  idea was accepted by Astronomer Royal
John Flamsteed, as well as Edmond Halley and Isaac Newton, although Robert Hooke did not 
accept it, believing that the movement of light was ``virtually instantaneous''.  Final 
confirmation of R\o mer's work did not come until 1727, when James Bradley succeeded to 
measure ``stellar aberration''.   That, however, is another story.

 

\end{document}