Download JUPITER AND SPEED OF LIGHT

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.
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 Figure 2: Galileo Galilei [6]
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]
Figure 3: Galileo's journal entries. [6]
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
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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.
Figure 4: Galileo's
telescope. [6]
The most important, the discovery of celestial bodies orbiting
something other than the Earth, dealt a blow to the thenaccepted 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
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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
Figure 5: Diagram of the reasoning used by Roemer to determine
the speed of light. [12]
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]
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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.
Figure 6: Latitude and longitude [13]
Devices like sextants and octants are used to
determine the altitude of Polaris. For instance, at
the North Pole, the latitude is 90N, and Polaris is at the zenith, an altitude of 90. In
Ljubljana, Polaris is 45 above the northern horizon, so Ljubljana’s latitude is 45N.
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
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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]
Figure 9 : Roemer’s drawing in
Roemer, however, agreed with Cassini’s first interpretation
his article of the Journal des
of the data and used Cassini’s work to determine the speed Sçavans. The Sun is in A, Jupiter
in B with its shadow cone, and
of light arriving at a value that was about 70% of the value
the drawing is in the reference
of c that we accept today.
system Sun-Jupiter [16]
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Roemer obtained a value of 2.14 x 10 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.
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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
Distance of Jupiter from Earth (9.11.2012) = 4.14690 AU
0.00001 AU
0.00001 AU
Change in distance ( ) = (5.26127 AU 0.00001 AU) – (4.14690 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
12
7000km/s
0.00001 AU) =
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]
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