## Determination of the Speed of Light

### Contents

Goals of this project:

• To learn how to use a telescope
• To understand the geometry of the Solar System
• To understand and apply Kepler's Laws and Newton's Laws
• To write a simple program in IDL for the data analysis.
• To measure the speed of light

• Performing the visual observations
• Analyzing the data, including the IDL program
• Understanding all sources of error
• Arriving at a reasonable estimate of c, or explaining why you did not
• Answering the questions at the end of the lab

### I. Introduction

In late 17th century there was a pressing need for an accurate clock, because a good clock is necessary to determine one's longitude while at sea. This was before the invention of the chronometer. Galileo suggested that the moons of Jupiter could serve as such a clock.

Galileo, using his new telescope, had discovered the 4 large "Galilean" moons of Jupiter in 1610. The Galilean moons are Io, Europa, Ganymede, and Callisto. Observations showed that their orbits are nearly circular, are nearly in the ecliptic plane, and that the satellites eclipsed regularly. The moons are bright, and easily viewed in a small telescope. The idea was that one could make tables of the predicted times of eclipses, and then use observations to calibrate the shipboard clocks. The short orbital periods (from 1.7 days for Io to 16.7 days for Callisto) meant that one had a good chance of being able to observe an eclipse every few days.

The system actually worked in practice, and was used prior to the invention of the chronometer.

However, there was a problem. The eclipses are most easily observed near quadrature, and most timings were consequently obtained near that phase. However, by the early 1670's, Ole Rømer had amassed a sufficient number of eclipse timings to note that the eclipses were not exactly regular, contrary to the predictions of Newtonian mechanics (which had just come along). In particular, eclipse timings obtained close to conjunction showed eclipses about 10 minutes later than expected, and eclipse timings obtained close to opposition were up to 10 minutes early. Rømer showed that the irregularity correlated with the distance between the Earth and Jupiter, and concluded that the speed of light was finite. Rømer published his results in 1676. His value was uncertain, because at the time the length of the Astronomical Unit was poorly known.

In this exercise, you will attempt to confirm Rømer's observation, and measure the speed of light.

### Orbits of the Galilean Satellites

Sidereal
Name Period (d) Eccentricity Inclination
Io 1.769138 0.0041 0.040
Europa 3.551181 0.009 0.047
Ganymede 7.154553 0.002 0.183
Callisto 16.68902 0.007 0.281

These inclinations are in degrees with respect to the Jovian equator; the inclination of the Jovian system with respect to its orbit about the Sun is 3.13o.

### II. What you will do

This lab is straightforward. You will observe the times of eclipses of the moons of Jupiter, and compare them to the predictions from Newtonian mechanics. Because you only have 3-4 weeks in which to complete the lab, and you need data over half a synodic period of Jupiter (the synodic period of Jupiter is 399 days), you will supplement your visual observations with tabulated eclipse times.

You will correlate the deviations of the times of eclipse with the distance of Jupiter from the Earth in order to estimate the speed of light.

### III. How to do this lab

1. Determine whether or not Jupiter is visible during the month you will be doing the lab. You can look in the American Ephemeris (available in the library), or you can run SKYCAL. If Jupiter cannot be observed because it is too close to the Sun, you may do this lab as an archival exercise. To improve the likelihood of being able to observe an eclipse, Jupiter should be more than 5 hours of Right Ascension from the Sun, and visible for at least 3 hours each night.

The eclipes are best observed near quadrature.

When Jupiter is in conjunction with the Sun, and observing is not possible, this lab may not be offered. Please check with the instructor.

2. Determine when an eclipse is likely to occur. Times of umbral disappearance and reappearance (see below) are listed for Io, Europa, Ganymede, and Callisto. These have been taken from the monthly listings in Sky and Telescope. You can use these times and the periods in the table to predict when future eclipses will occur. Make sure you use the synodic period (the period as seen by a observer on the Sun), and not the sidereal period, which is listed in the table. If no eclipses will occur during the month you do the lab, and you can prove this to the TA or instructor, then you can do this lab purely as a data analysis exercise.

3. Read the tutorials on times and coordinate systems.

4. You will need to get the exact time. Get this from WWV. Either find a radio that receives WWV, or go to the the US Naval Observatory time site or the The NIST WWV site. The web page is less precise than WWV radio, since there are transmission delays over the web, and these may be significant. Nonetheless, it should be good to a second or so, which is certainly adequate for your purposes.

US Naval Observatory time: (Reload to update: the clock stops after 60 seconds.)

5. You will need to understand the geometry of the solar system. If you have never taken an astronomy course, read the appropriate section in a basic astronomy text. You may assume that the Earth has an orbital eccentricity of 0, and you may ignore the 1.3o inclination of Jupiter's orbit to the ecliptic. For the first approximation, you may also ignore Jupiter's eccentricity and the 3o inclination of the Jovian rotation axis. You will need to determine the angles between the Earth, Jupiter, and the Sun.

To determine the location of the Sun and Jupiter in the sky, you will need to find an ephemeris. There are many ways to do this, depending on the level of precision you want.

• Sky and Telescope provides a monthly ephemeris, with the position of Jupiter and the Sun at mid-month. You can interpolate from this. Sky and Telescope is published monthly, and back issues are available in the Math-Physics-Astronomy Library.
• The Astronomical Almanac provides the apparent positions of the Sun and Jupiter every day. The Astronomical Almanac is published annually, and is available in the Math-Physics-Astronomy Library.
• You can use the SKYCAL ephemeris generator to generate positions for the Sun and major planets with an accuracy of 0.1o.
• You can go directly to JPL on the web and generate a customized ephemeris using the HORIZONS system. This is generally overkill.

6. You will use Kepler's third law to determine the relative distances between the Sun, the Earth, and Jupiter.

### IV. The Mount Stony Brook Observatory

The Mount Stony Brook Observatory (MtSB) is the name our undergraduate observers have given to the University's 14" telescope atop the ESS building. Details of the telescope and its operation are given in a postscript document. You must read this manual.

This lab does not use the CCD camera, so you may ignore that part of the writeup.

For this lab, you can also use one of the 8" reflectors, if the 14" is in use or out of commission. Ask the TA to get one out for you.

Prior to using the telescope, you will be given a lesson by a TA or other experienced observer. Note that no one is allowed to observe alone: for safety reasons you must have an observing partner. For practical reasons, it is handy to have one observer looking through the telescope and the other keeping time.

The 14" telescope can be heavily booked. You must sign up on the board outside room ESS 443C.

### V. What to Measure

Note that Jupiter has 4 bright moons. The monthly chart of Jupiter's Satellites in Sky and Telescope is handy for identifying them.

There are 6 specific points in the satellite orbit that you can measure. These are:

• The beginning and end of transit, where the satellite crosses in front of the planet. This is hard to measure accurately because of the poor contrast between the satellite and the much brighter planet.
• The beginning and end of the eclipse of the satellite behind the planet. These are difficult to measure because of contrast with the bright planet. One may occur in shadow, in which case it would not be observable.
• The beginning and end of the passage of the satellite through the umbra of the planetary shadow. These are the best, because the contast will be good. One is likely to occur while the satellite is eclipsed by the planet, at least for the inner satellites.
It is recommended that you plan to observe the disappearance or reappearance of one of the satellites in the planetary shadow.

The best satellite for your purposes is Io. It is the closest to Jupiter, so it has the most eclipses. Because it is closest to Jupiter, and has the lowest eccentricity, the path length through the umbra is the most constant, which minimizes uncertainties (this is a serious problem with Callisto). On the other hand, Io is never far from the very bright planet, and it can be hard to see on nights with poor seeing.

Europa is the second best satellite to work with. It is further from Jupiter, but eclipses only half as frequently. It too has a low inclination.

How many eclipses should you measure? As many as possible, because this will help you beat down your errors. In principle, one observation near opposition or conjunction will suffice (given timings made near quadrature). However, you are doing the visual observations mainly to gain the experience of taking observations, and your observing time is constrained by the length of the lab. You should observe at least 2 eclipses so that both observers can witness an eclipse. This will let you measure the synodic orbital period.

### VI. Data Analysis

You are required to write an IDL program to analyze the data for this lab. Use of EXCEL spreadsheets is strictly prohibited.

Once you measure the time of eclipse, compare it to the prediction. Use the data in the eclipse timing files to determine the O-C (observed - computed) difference as a function of the distance between The Earth and Jupiter. How much data should you use? Use at least enough to go from opposition to conjunction.

You may use the IDL routine rd_jupecl to read in the eclipse times. Right-click on one of the eclipse timing files (Io, Europa, Ganymede, or Callisto) to download it to a file. Use rd_jupecl to read the array into a file (run with no arguments for on-line help). The array contains the year, month,day, hour, and minute of the eclipse, along with a flag indicating whether this is an appearance or a disappearance. Write an IDL program to convert these into running times, and to select the data you want to use.

The O-C deviation will depend on the distance between the Earth and Jupiter, and you will be able to explain it in terms of a finite value of the speed of light c. Estimate c in terms of the astronomical unit

Note, however, that there are complications. If you plot up the O-C deviations, you will see a tremendous amount of scatter. The individual data segments show smooth trends, and there are data segments where the O-C deviation gives you a reasonable answer for the speed of light. There are also segments that appear to give a negative answer for the speed of light. You should demonstrate this.

The main reason for these apparently non-sensical results is due to the assumption of a circular orbit for Jupiter. It has an eccentricity of nearly 5%. The main consequences of this are that

• the distance from Earth to Jupiter varies between 3.96 and 6.44 AU, and
• the synodic period of Io is not constant.
Correct for these factors. Which is more important? You can look up the details of Jupiter's orbit in the Astronomical Almanac, but all you really need to know is that Jupiter was a aphelion near Julian day 2449153.

Consider also the effect of the non-zero inclination of Io's orbit. Determine the magnitude of this effect. Is it important?

Do a complete error analysis. Discuss the uncertainties in your measurements, and in the tabulated eclipse times. Discuss other uncertainties arising from the non-zero orbital eccentricity of the Earth, as well as the non-zero inclinations of Jupiter's orbit and rotation axis. Are any of these important? be sure to distinguish systematic uncertainties from measurement uncertainties.