Seminar

JUPITER AND SPEED OF LIGHT

Author: Jaka Klement

Advisor: prof. dr. Tomaž Zwitter

Ljubljana, april 2012

Abstract

This seminar addresses topics of first observations of Jupiter and its moons, solving the

problem of determining longitude and determining the speed of light. The experiment at the

end of the seminar shows an extra value added on the pedagogical aspect for high school

students. To show them, that they don´t need very complex knowledge of physics in order to

be able to measure, with a simple experiment, a very interesting and nontrivial matter as the

speed of light.

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Contents 1. INTRODUCTION (ABOUT JUPITER) ...............................................................................................3

2. HISTORY OF OBSERVATIONS AND JUPITERS MOONS ...................................................................4

3. LATITUDE AND LONGITUDE .........................................................................................................7

4. DISCOVERING THE SPEED OF LIGHT .............................................................................................9

5. MEASURING THE SPEED OF LIGHT BY OUR OWN ....................................................................... 10

6. CONCLUSION ............................................................................................................................ 13

References ................................................................................................................................ 14

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1. INTRODUCTION (ABOUT JUPITER)

The Solar System consists of the Sun and planets and comets and all the other astronomical

objects gravitationally bound in orbit around it. Well over 99% of the system's mass is in the

Sun. Of the many objects that orbit the Sun, most of the mass is contained within eight

relatively solitary planets whose orbits are almost circular and lie within an ecliptic plane.

The four smaller inner planets, also called the terrestrial planets, are primarily composed of

rock and metal. The four outer planets, the gas giants, are substantially more massive than

the terrestrials. These planets are Jupiter, Saturn, Uranus and Neptune. The largest, Jupiter,

is composed mainly of hydrogen and helium. [1] Jupiter, at 318 Earth masses, is 2.5 times

the mass of all the other planets put together. Jupiter's strong internal heat creates a

number of features in its atmosphere, such as cloud bands and the Great Red Spot (first

discovered by Cassini). The average distance between Jupiter and the Sun is 778 million km

(about 5.2 times the average distance from the Earth to the Sun, or 5.2 AU) and it completes

an orbit every 11.86 years. Jupiter's rotation is the fastest of all the Solar System's planets,

completing a rotation on its axis in slightly less than ten hours. The planet is shaped as

an oblate spheroid, meaning that the diameter across its equator is longer than the diameter

measured between its poles by as much as 6.5%. [2]

Figure 1: The line up from the sun with size but not orbits to scale [3]

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2. HISTORY OF OBSERVATIONS AND JUPITERS MOONS

As we now know, Jupiter has 66 known satellites. The four largest, Ganymede, Callisto, Io,

and Europa, show similarities to the terrestrial planets, such as

volcanism and internal heating [4]. The four largest satellites, known

also as “Medician Stars” (named after Galilei's tutor Cosimo de’

Medici) or the “Galilean moons” were first discovered in January

1610 by Galileo Galilei. As he writes [5]:

“On the seventh day of January in this present year 1610, at the first hour of

night, when I was viewing the heavenly bodies with telescope, Jupiter

presented itself to me; and because I had prepared a very excellent

instrument for myself, I perceived that beside the planet there were three

starlets, small indeed, but very bright. Though I believed them to be among

the host of fixed stars, they aroused my curiosity somewhat by appearing to

lie in an exact straight line parallel to the ecliptic, and by them being more splendid than others of

their size”.

On January 7, 1610, Galileo wrote a letter containing the first mention of Jupiter’s moons. At

the time, he saw only

three of them, and he

believed them to be

fixed stars near

Jupiter. He continued

to observe them, from

January 8 to March 2,

1610. In these

observations, he

discovered a fourth

moon, and also

observed that the four

were not fixed stars,

but were rather

orbiting Jupiter. [7]

He was unable to determine their periods for lack of time and for want of a better telescope.

But the great discovery has been made: four tiny moons move around the larger planet in

Figure 3: Galileo's journal entries. [6]

Figure 2: Galileo Galilei [6]

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circular orbit of different amplitude, and with a velocity the

greater, the smaller the distance of the satellite from Jupiter [8].

The fact, that it was hard to distinguish one satellite from another

in Galileo´s days, made the determination of their periods very

difficult.

Galileo’s discovery proved the importance of the telescope as a

tool for astronomers by showing that there were objects in space

to be discovered that until then had remained unseen by the

naked eye.

The most important, the discovery of celestial bodies orbiting something other than the Earth, dealt a blow to the then-accepted Ptolemaic world system, which held that the Earth was at the center of the universe and all other celestial bodies revolved around it [9].

Galileo had special interest in observing Medicean stars, for the main purpose of

determining their periods of revolution. By April, 1611, a little more than a year after his

discovery, he was able to distinguish one satellite from another, and he had approximately

determined their periods. Galileo soon noticed that if he succeeded in calculating these

tables with sufficient accuracy, he would have at hand a new method for determining the

longitude of various places on the earth, a very important problem for navigators but very

difficult to solve.

Clocks had not yet been invented by which the time could be carried from one place to

another. Therefore, for the determination of longitude, it was necessary to resort to some

celestial phenomenon, like eclipses, which once predicted, could be observed from

everywhere. The travelers, by knowing on the one hand the moment at which phenomenon

was to take place in the time of standard meridian, and by the determining on the other

hand the local time of phenomenon from the position of celestial bodies visible on their

location, could arrive at a determination of their longitude. Till then, this determination was

limited to solar and lunar eclipses and to a few occultations of the brightest stars. They were

therefore too rare to be of any practical use. But the rapid motion of the Medician stars

around Jupiter multiplied the observable celestial phenomena. If these events could have

been predicted with sufficient accuracy, they would have enabled the navigators to obtain

their position in longitude at all times when Jupiter was visible. Since Galileo was unable,

because of his poor health, to continue his observations and to find a solution of the

problem which bothered him, he asked for the help of Father Renieri. He was able to

complete the ephemerides. [10]

Half a century later, ephemerides, written by Renieri, enabled Gian Domenico Cassini to

compute the first table that was sufficiently accurate to predict the configuration of

Medician stars. Io orbited in about 1.8 days. Europa took 3.5 days. Ganymede’s orbital

Figure 4: Galileo's

telescope. [6]

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period was just over 7 days. And Callisto, the most distant of the Galilean moons, went

around Jupiter in 16.7 days. If the periods were indeed regular, and could be determined

with high precision, the positions of the moons around Jupiter could be predicted for any

given time in the future. The disappearance or reappearance of the moons at certain points

in their orbit was the easiest position of the moons to time exactly. Astronomers knew that

moons only shine because they reflect light. So, when the moons pass into Jupiter’s shadow,

they go dark. Jupiter’s shadow is a cone of darkness that extends out from Jupiter into space

in the direction opposite to the sun. So the position of Jupiter’s shadow depends on where

Jupiter is in its orbit. Astronomers had to take this into account, as well as the orbital period

of the moons, when they calculated an ephemeris for eclipses of the moons. But once they

had done these calculations, the ephemeris could be used to forecast the times of future

eclipses. This table led the

Danish astronomer Olaus

Roemer three years later

(1676) to discover the speed of

light. [11]

Roemer was not initially trying

to find the speed of light. Like

many scientific investigations

today, speed of light was just a

by-product. His original goal

was to see if the orbiting

moons of Jupiter could be

used to find the longitude of

places on earth (something

that Galileo proposed half a

century before), something

that was of paramount

importance to ships’ navigators. In a sense, the Jupiter moons were some sort of a very

precise celestial clock. [11]

Figure 5: Diagram of the reasoning used by Roemer to determine the speed of light. [12]

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3. LATITUDE AND LONGITUDE

The main purpose of navigation is to

determine a ship’s position on the surface of

the earth. The system of latitude and longitude

is used for this.

Latitude, the angular distance north or south

of the equator, is easy to determine. The

angular altitude of the north celestial pole is

equal to a person’s latitude. Polaris is close to

the north celestial pole so for accuracy to a

degree, Polaris’ altitude gives the latitude.

Devices like sextants and octants are used to

determine the altitude of Polaris. For instance, at

the North Pole, the latitude is 90N, and Polaris is at the zenith, an altitude of 90. In

Ljubljana, Polaris is 45 above the northern horizon, so Ljubljana’s latitude is 45N.

Figure 7: determining latitude

Longitude is the angular distance east or west of some arbitrary zero longitude meridian,

known as the prime meridian. Longitude begins at 0 at the Greenwich prime meridian and

goes 180 East and 180 West.

But unlike latitude, Longitude is difficult to determine. One would need a very accurate clock

in order to be able to determine longitude. An error of a few minutes can put you kilometers

Figure 6: Latitude and longitude [13]

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away from where you really are. The sailors didn’t have a reliable clock back in 1700. The

difficulty of determining longitude in the absence of a reliable sea-going clock was one of the

most important challenges facing maritime societies. [11]

How does timekeeping help determine longitude? Simply put, you can determine your

longitude if you know the difference between the time on a local sundial, and the time at the

prime meridian. Since it takes the earth 24 hours to turn 360, it follows that to turn

through 15 takes one hour. Knowing this is the basis for finding longitude.

Figure 8: determining longitude [14]

For example, imagine that you are sailing from New York, going east to London. After several

weeks nothing is visible on any horizon. You would like to know, what your longitude is. As

the mean sun is over head your clock shows 12:00. But you know that is 2 pm at Greenwich.

A one hour difference would indicate that you were 15 West of the prime meridian. Two

hours means you are 30 West, and that is your longitude.

It is interesting to know how precisely one can measure longitude and latitude. If your watch

kept time to within one second and you marked the sun’s rising to within one second, you

would be within one kilometer of your true location in longitude. With a good sextant that is

skillfully used and knowing how much Polaris is shifted from the north celestial pole, your

latitude position would similarly be about 1 kilometer in error. [11]

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4. DISCOVERING THE SPEED OF LIGHT

Cassini had been timing the period of Io around Jupiter

marking each orbit by when the moon left Jupiter’s shadow.

His data showed that when Jupiter and earth were close, the

observed times of Io’s eclipse agreed with those predicted by

the ephemeris. But when Jupiter and earth were far apart,

the observed eclipse times occurred 10 to 12 minutes later

than predicted.

He originally thought that this was a result of light taking a

finite time to travel from Jupiter to earth. However he

changed his mind, believing that light could not have a finite

speed, and thought that something else was responsible for

the inconsistent timings. [15]

Roemer, however, agreed with Cassini’s first interpretation

of the data and used Cassini’s work to determine the speed

of light arriving at a value that was about 70% of the value

of c that we accept today.

Roemer obtained a value of 2.14 x 108 m/s for velocity of

light. For this reason, he is credited as the person to make the first modern measure of

speed of light.

The significance of Roemer’s determination was finding that light had a finite speed and was

not instantaneous, as many people had thought before him. Roemer made this discovery

only with good clock and a telescope only 60 years more advanced than what Galileo had

used to discover Jupiter’s moons. [11]

Distance between Earth and Jupiter can be calculated using trigonometry, knowing two sides

(distance between the Sun and Earth; distance between the Sun and Jupiter) and one angle

(the angle between Jupiter and Earth as formed at the Sun) of a triangle. The distance from

the Sun to Earth was not well known at the time. It was, however, very well estimated in

1639 by observing transit of Venus to 0.7 AU. But Roemer was taking it as a fixed value a, the

distance from the Sun to Jupiter can be calculated as some multiple of a from Kepler´s third

law.

Model that Roemer used left just one adjustable parameter – the time taken for light to

travel a distance equal to a, the radius of Earth's orbit. Roemer had about thirty observations

of eclipses of Io from 1671–73 that he used to find the value which fitted best: eleven

minutes.

Figure 9 : Roemer’s drawing in his article of the Journal des Sçavans. The Sun is in A, Jupiter in B with its shadow cone, and the drawing is in the reference system Sun-Jupiter [16]

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5. MEASURING THE SPEED OF LIGHT BY OUR OWN

We can determine the speed of light by ourselves if we fallow next few steps.

Step 1: Find two dates when Jupiter and Earth are far and near, preferably about two to

three months after Jupiter is in conjunction and a about month before Jupiter is at

opposition.

Conjunction means that, as seen from Earth, two celestial bodies appear near one another in

the sky. For our purpose conjunction is phenomena, when Sun and Jupiter appear near one

another, so Earth and Jupiter are as far away from one another as possible.

Opposition means, that two celestial bodies are on opposite sides of the sky, viewed from a given place. When two planets, for our purpose Earth and Jupiter, are in opposition, Jupiter is visible almost all night, its orbit is roughly closest to the Earth and half of the planet (Jupiter) visible from Earth is completely illuminated.

Figure 10: figure shows position of Jupiter and Earth during opposition and conjunction. [17]

We can find dates for conjunction and opposition in various different tables obtained on the

internet. For this year, Jupiter and Earth are in conjunction on May 13, and in opposition on

December 3. So the best dates to do our observations are 13.8.2012 (three months after

conjunction) and 3.11.2012, one month before opposition.

On these dates the orbital geometry allows Jupiter to be easily observed from earth. In

August, the local time for observing Jupiter would be a few hours before dawn. In November

observing would be around midnight. On these dates we will observe eclipses of Io.

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The relative positions of Jupiter, earth and the Sun for these two dates are shown in the

diagram in Figure 9 along with the distances between Jupiter and earth.

Figure 11: Relative positions of Earth and Jupiter at conjunction (August) and opposition (November) dates.

Step 2: Observe and time an eclipse of Io on the conjunction date

The eclipse happens at the moment Io moves into Jupiter’s shadow. If there is no eclipse on our

conjunction date, there will be one the next day or the day after next. We shall observe that one. In

our case eclipse happens on 16.8.2012 at 2:59:30. Distance between Jupiter and Earth is 5.26127 AU.

Measurement error of time of eclipse is 10 seconds and distance between Jupiter and Earth is

uncertain to 0.00001 AU.

Step 3: Observe and time an eclipse of Io on the opposition date

If there is no eclipse on our opposition date, there will be one the next day or the day after next. We

shall observe that one. In our case eclipse happens on 9.11.2012 at 00:58:30. Distance between

Jupiter and Earth is 4.14690 AU.

Again, measurement error of time of eclipse is 10 seconds and distance between Jupiter and Earth is

uncertain to 0.00001 AU.

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Step 4: Estimate how many orbits Io will make from the conjunction date to the opposition date

Number of orbits Io will make is quotient of difference between opposition date and conjunction

date and synodic period of Io. Synodic period of Io is 1.76925 days.

Number of orbits = (9.11 – 16.8) / 1.76925 days per orbit Number of orbits = 85 days / 1.76925 days per orbit Number of orbits = 48.044 orbits We have to round this value down, since the number of orbits must clearly by a whole number. In

the time between our two observations of Io´s eclipses, Io rounds Jupiter 48 times.

Step 5: Calculate the time interval from the first eclipse to eclipse predicted by step 4. Time to predicted eclipse = number of orbits * synodic period of Io Time to predicted eclipse = 48 orbits * 1.76925 days per orbit = 84.924 days Step 6: Calculate the time interval from the first eclipse to second eclipse.

First eclipse (conjunction) = 16.8.2012 at 2:59:30 10s Second eclipse (opposition) = 9.11.2012 at 00:58:30 10s Time difference = second eclipse – first eclipse = 84.9160 days 0.0002 day

Step 7: Calculate time difference between predicted eclipse and observed one. Time difference ( ) = Predicted time – observed time Time difference ( ) = 84.924 days – 84.916 days = 0.008days 0.0002 day Time difference ( ) = 691 s 20s Step 8: Find the change in distance between Jupiter and Earth from the date of first eclipse

to the date of second one.

Distance of Jupiter from Earth (16.8.2012) = 5.26127 AU 0.00001 AU Distance of Jupiter from Earth (9.11.2012) = 4.14690 AU 0.00001 AU

Change in distance ( ) = (5.26127 AU 0.00001 AU) – (4.14690 AU 0.00001 AU) = 1.11437 AU 0.00002 AU Change in distance ( ) = 167155500km 3000km

Step 9: Calculate the speed of light. Speed of light (c) = ( ) / ( )

Speed of light (c) =

= 241904km/s 7000km/s

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Obtained result is quite accurate, taking in to account, all the simplifications we make during our 9

steps towards assessment of speed of light. We didn’t take into account eccentricity, a little different

position and geometry of Jupiter’s shadow in our first and second observation of eclipses and maybe

something else. But nevertheless results serve their purpose; we showed that speed of light is finite

and we can very easily measure it by our self in our back yard, using telescope, just powerful enough

to observe eclipses of Io.

Measurement error is very small, less than 3%, which is not surprising, since the

measurements, obtained by observing eclipses of Jupiter's closest moon Io, served for a very precise

positioning on the earth's sphere. Other effect, listed above, have more decisive importance, for our

value for speed of light to differ from the correct value.

The correct value for the speed of light is 300000km/s. Our value differs from the right one for less

than 20%.

6. CONCLUSION

In short, we view the history of observations of Jupiter's moons. We started with Galileo

Galilei, and then continue to Cassini and finely Roemer, who is also the first officially

stated value for the speed of light. We also look at how important Jupiter's

moons were in navigation and determination of longitude in the past. In the end, we made a

rough estimation our self, of the value for the speed of light, with a simple observation

of the eclipse of Jupiter's closest moon, Io.

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References

[1] http://en.wikipedia.org/wiki/Solar_System [29.3.2012]

[2] http://ase.tufts.edu/cosmos/view_chapter.asp?id=9&page=3 [15.4.2012]

[3] http://www.cosmicelk.net/asteroids.htm [29.3.2012]

[4] Pappalardo, R T (1999). "Geology of the Icy Galilean Satellites: A Framework for Compositional

Studies".Brown University.

[5] Drake, Stillman (1957). “Discoveries and opinions of Galileo”. Page 50-60

[6] http://solarsystem.nasa.gov/scitech/display.cfm?ST_ID=2283 [29.3.2012]

[7] Galilei, Galileo, Sidereus Nuncius. Translated and prefaced by Albert Van Helden. Chicago &

London: University of Chicago Press 1989, 14–16

[8] Galilei, Galileo (1967):”Dialogue Concerning the Two Chief World Systems”. 117-119

[9] "Satellites of Jupiter". The Galileo Project. Rice University. 1995.

[10] Schuman, Henry (1952): “The history of astronomy”. 90-130

[11] http://www3.gettysburg.edu/~marschal/clea/roemerlab.html [29.3.2012]

[12] http://physics.ucr.edu/~wudka/Physics7/Notes_www/node65.html [29.3.2012]

[13] http://en.wikipedia.org/wiki/Latitude [29.3.2012]

[14] http://www.physics.miami.edu/huerta/class/mls603/Table_1.html [15.4.2012]

[15] Bobis, Laurence andLequeux, James (2008) :«CASSINI, RØMER AND THE VELOCITY OF LIGHT«.

Journal of Astronomical History and Heritage, 11(2), 97-105 (2008).

[16] http://www.rundetaarn.dk/engelsk/observatorium/light1.html [29.3.2012]

[17] http://en.wikipedia.org/wiki/Conjunction_(astronomy_and_astrology) [29.3.2012]