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This exercise concerns observations of eclipses of Jupiter's satellites (moons) and how these may be used to determine the speed of light.
In 1610, Galileo discovered the four biggest satellites of Jupiter: Io, Europe, Ganymede and Callisto.
The orbits of these satellites have very small eccentricities (they are nearly circular) and are close to the equatorial plane of Jupiter.
Jupiter's equator and its orbit are not much inclined over the plane of the Ecliptic.
At the end of the 16th century, the determination of longitudes could not be very accurately done for lack of stable and accurate timekeepers. Galileo then had the idea of using Jupiter's satellites as time indicators: their motions are practically circular and regular, their periods are short enough and the instants of mutual eclipses do not depend on the location of the observer.
This idea was taken up again by Cassini in 1668. It eventually met with success, due to the perfectioning of observational instruments and the invention of the clock (Huygens, 1657).
During the winter of 1671-1672, Picard and Roemer (from Uraniborg on the island of Hven, the site of Tycho Brahe's observatory, now in Sweden) and Cassini (from the Paris Observatory), observed simultaneously the moments of eclipses of Io by Jupiter. From these measurements, they measured the difference of geographical longitude between Uraniborg and Paris.
From 1672 on, Roemer worked at the Paris Observatory and continued the observation of the eclipses of Jupiter's satellites.
It was from these measurements that Roemer was able to prove for the first time that the velocity of the light is finite.
This proof was published in 1676 in the Journal des Sçavans. In what follows, we will assume that all motions occur in the same plane (that of the Ecliptic) and that Jupiter does not move (in first approximation).
The discussion refers to the figure below which shows the orbit of the Earth around the Sun (in the Ecliptic Plane).
Here, S: Sun; J: Jupiter. When the Earth is located in O', it is in conjunction (on the other side of the Sun) as seen from Jupiter; when the Earth is in O, it is in opposition.
Moreover, AJ = f; BJ = f'; CJ = d; DJ = d'
T is the period of revolution of Io around Jupiter. T = 1,769 day = 1d 18h 28min.
Let us assume that the end of an eclipse occurs at an epoch (time) t and the end of the next one at an epoch t + T. Meanwhile, the Earth has moved away from Jupiter.
If the light would reach the Earth instantaneously, we would observe from Earth a period equal to T.
However, if the velocity of light is finite, let c be its velocity. We observe the end of an eclipse at an epoch t + f / c (point A). Then we observe the end of the next eclipse at the epoch t + T + f ' / c (point B). The difference in time between the ends of two consecutive eclipses is equal to T + ( f ' - f ) / c.
Now, f' > f, so T + ( f ' - f ) / c > T.
The beginning of an eclipse occurs at epoch t' and the beginning of the next one at tha date t' + T. In between, the Earth has moved towards Jupiter.
If light would reach Earth instantaneously, we would observe on Earth a period equal to T.
However, if the velocity of light is finite, we observe the beginning of an eclipse at an epoch t' + d / c, and the beginning of the next at the epoch t' + T + d' / c. The time interval between the beginnings of two consecutive eclipses is equal to T + (d' - d) / c.
Now, d' < d, so T + (d' - d) / c < T.
Here follow some exercises which are related to Io's motion around Jupiter and which put you in position similar to that of Roemer.
Examine the figure below. The Earth is between a conjunction and an opposition. The radius of the orbit of Io is equal to six times the radius of Jupiter. The angle SJE = a is always smaller than 11°.
a) On the figure below, indicate the beginning and the end of the following four phenomena:
Eclipse: the satellite enters the shadow of Jupiter.
Occultation: the satellite, as seen from Earth, goes behind Jupiter.
Shadow: the shadow of the satellite is seen on the planet.
Passage: the satellite, as seen from Earth, passes in front of the planet.
b) Now explain why only the beginnings of the eclipses can be observed from Earth.
What happens when the Earth is close to a conjunction?
Below you will find four tables with actual timings of Io eclipses. Use these values for the calculations explained below.
Choose the date of a Jupiter opposition. (Use for instance the opposition on July 4, 1996)
Next, note the dates and the hours of:
The beginning of eclipses (before opposition) number X (the last one before opposition), as well as numbers X - 20; X - 40; X - 60.
The end of eclipses (after opposition): numbers 0; 20; 40; 60.
Then find the total duration (in days, hours and minutes) of 40 observed periods:
Before opposition:
from X - 40 to X; let D be this duration
from X - 60 to X - 20; let D' be this duration
After opposition:
from 0 to 40; let F be this duration
from 20 to 60; let F' be this duration
Explain why F > D and F' > D'.
Explain why F - D > F' - D'.
What is the approximate velocity of light, expressed in km/sec? (This one is not so easy!)
Hints: The mean distance between the Earth and Jupiter and the time of an opposition is 4.2 Astronomical Units (AU). 1 AU = 149,600,000 km (radius of the Earth's orbit). Where is the Earth in its orbit at the times corresponding to the measurements of D' and F'? How much did the distance from the Earth to Jupiter decrease/increase during the measurement? You may try to repeat the calculation with D and F.
00 14/12/94 08h37 34 12/02/95 12h41 68 13/04/95 16h41 01 16/12/94 03h06 35 14/02/95 07h09 69 15/04/95 11h09 02 17/12/96 21h34 36 16/02/95 01h37 70 17/04/95 05h37 03 19/12/94 16h03 37 17/02/95 20h05 71 19/04/95 00h05 04 21/12/94 10h31 38 19/02/95 14h34 72 20/04/95 18h34 05 23/12/94 04h59 39 21/02/95 09h02 73 22/04/95 13h02 06 24/12/94 23h28 40 23/02/95 03h30 74 24/04/96 07h30 07 26/12/94 17h56 41 24/02/95 21h58 75 26/04/95 01h59 08 28/12/94 12h25 42 26/02/95 16h27 76 27/04/95 20h27 09 30/12/94 06h53 43 28/02/95 10h55 77 29/04/95 14h55 10 01/01/95 01h21 44 02/03/95 05h23 78 01/05/95 09h23 11 02/01/95 19h50 45 03/03/95 23h51 79 03/05/95 03h52 12 04/01/95 14h18 46 05/03/95 18h19 80 04/05/95 22h20 13 06/01/95 08h46 47 07/03/95 12h48 81 06/05/95 16h48 14 08/01/95 03h15 48 09/03/95 07h16 82 08/05/95 11h17 15 09/01/95 21h43 49 11/03/95 01h44 83 10/05/95 05h45 16 11/01/95 16h11 50 12/03/95 20h12 84 12/05/95 00h14 17 13/01/95 10h40 51 14/03/95 14h41 85 13/05/95 18h42 18 15/01/95 05h08 52 16/03/95 09h09 86 15/05/95 13h10 19 16/01/95 23h36 53 18/03/95 03h37 87 17/05/95 07h39 20 18/01/96 18h05 54 19/03/95 22h05 88 19/05/95 02h07 21 20/01/95 12h33 55 21/03/95 16h33 89 19/05/95 20h36 22 22/01/95 07h01 56 23/03/95 11h02 90 22/05/95 15h04 23 24/01/95 01h30 57 25/03/95 05h30 91 24/05/95 09h32 24 25/01/95 19h56 58 26/03/95 23h58 92 26/05/95 04h01 25 27/01/95 14h26 59 28/03/95 18h26 93 27/05/95 22h29 26 29/01/95 08h55 60 30/03/95 12h55 94 29/05/95 16h58 27 31/01/95 03h23 61 01/04/95 07h23 95 31/05/95 11h26 28 01/02/95 21h51 62 03/04/95 01h51 29 03/02/95 16h19 63 04/04/95 20h19 30 05/02/95 10h48 64 06/04/95 14h48 31 07/02/95 05h16 65 08/04/95 09h16 32 08/02/95 23h44 66 10/04/95 03h44 33 10/02/95 18h12 67 11/04/95 22h12
00 02/06/95 08h06 34 01/08/95 12h20 68 30/09/95 16h40 01 04/06/95 02h34 35 03/08/95 06h48 69 02/10/95 11h09 02 05/06/95 21h03 36 05/08/95 01h17 70 04/10/95 05h37 03 07/06/95 15h31 37 06/08/95 09h56 71 06/10/95 00h06 04 09/06/95 10h00 38 08/08/95 14h15 72 07/10/95 18h35 05 11/06/95 04h28 39 10/08/95 08h44 73 09/10/95 13h04 06 12/06/95 22h57 40 12/08/95 03h12 74 11/10/95 07h33 07 14/06/95 17h25 41 13/08/95 21h41 75 13/10/95 02h01 08 16/06/95 11h54 42 15/08/95 16h10 76 14/10/95 20h30 09 18/06/95 06h22 43 17/08/95 10h39 77 16/1095 14h59 10 20/06/95 00h51 44 19/08/95 05h08 78 18/10/95 09h28 11 21/06/95 19h20 45 20/08/95 23h26 79 20/10/95 03h57 12 23/06/95 13h48 46 22/08/95 18h05 80 21/10/95 22h26 13 25/06/95 08h17 47 24/08/95 12h34 81 23/10/95 16h54 14 27/06/95 02h45 48 26/08/95 07h03 82 25/10/95 11h23 15 28/06/95 21h14 49 28/08/95 01h32 83 27/10/95 05h52 16 30/06/95 15h43 50 29/08/95 20h00 84 29/10/95 00h21 17 02/07/95 10h11 51 31/08/95 14h29 85 30/10/95 18h50 18 04/07/95 04h40 52 02/09/95 08h58 86 01/11/95 13h18 19 05/07/95 23h09 53 04/09/95 03h27 87 03/11/95 07h47 20 07/07/95 17h37 54 05/09/95 21h56 88 05/11/95 02h16 21 09/07/95 12h06 55 07/09/95 16h25 89 06/11/95 20h45 22 11/07/95 06h35 56 09/09/95 10h54 90 08/11/95 15h13 23 13/07/95 01h03 57 11/09/95 05h22 91 10/11/95 09h42 24 14/07/95 19h32 58 12/09/95 23h51 92 12/11/95 04h11 25 16/07/95 14h01 59 14/09/95 18h20 93 13/11/95 22h40 26 18/07/95 08h30 60 16/09/95 12h49 94 15/11/95 17h09 27 20/07/95 02h58 61 18/09/95 07h18 95 17/11/95 11h37 28 21/07/95 21h27 62 20/09/95 01h47 96 19/11/95 06h06 29 23/07/95 15h56 63 21/09/95 20h15 97 21/11/95 00h35 30 25/07/95 10h24 64 23/09/95 14h44 98 22/11/95 19h04 31 27/07/95 04h53 65 25/09/95 09h13 32 28/07/95 23h22 66 27/09/95 03h42 33 30/07/95 17h51 67 28/09/95 22h11
00 14/01/96 19h09 34 14/03/96 23h15 68 14/05/96 03h18 01 16/01/96 13h38 35 16/03/96 17h44 69 15/05/96 21h46 02 18/01/96 08h06 36 18/03/96 12h12 70 17/05/96 16h15 03 20/01/96 02h35 37 20/03/96 06h40 71 19/05/96 10h43 04 21/01/96 21h03 38 22/03/96 01h29 72 21/05/96 05h11 05 23/01/96 21h03 39 23/03/96 19h37 73 22/05/96 23h40 06 25/01/96 10h00 40 25/03/96 14h05 74 24/05/96 18h08 07 27/01/96 04h29 41 27/03/96 08h34 75 26/05/96 12h36 08 28/01/96 22h57 42 29/03/96 03h02 76 28/05/96 07h05 09 30/01/96 17h25 43 30/03/96 21h30 77 30/05/96 01h33 10 01/02/96 11h54 44 01/04/96 15h59 78 31/05/96 20h01 11 03/02/96 06h22 45 03/04/96 10h27 79 02/06/96 14h30 12 05/02/96 00h51 46 05/04/96 04h55 80 04/06/96 08h58 13 06/02/96 19h19 47 06/04/96 23h24 81 06/06/96 03h26 14 08/02/96 13h48 48 08/04/96 17h52 82 07/06/96 21h55 15 10/02/96 08h16 49 10/04/96 12h20 83 09/06/96 16h23 16 12/02/96 02h45 50 12/04/96 06h48 84 11/06/96 10h52 17 13/02/96 21h13 51 14/04/96 01h17 85 13/06/96 05h20 18 15/02/96 15h41 52 15/04/96 19h45 86 17/06/96 23h48 19 17/02/96 10h10 53 17/04/96 14h13 87 16/06/96 18h17 20 19/02/96 04h38 54 19/04/96 08h42 88 18/06/96 12h45 21 20/02/96 23h07 55 21/04/96 03h10 89 20/06/96 07h14 22 22/02/96 17h35 56 22/04/96 21h38 90 22/06/96 01h42 23 24/02/96 12h03 57 24/04/96 16h07 91 23/06/96 20h11 24 26/02/96 06h32 58 26/04/96 10h35 92 25/06/96 14h39 25 28/02/96 01h00 59 28/04/96 05h03 93 27/06/96 09h08 26 29/02/96 19h29 60 29/04/96 23h31 94 29/06/96 03h36 27 02/03/96 13h57 61 01/05/96 18h00 95 30/06/96 22h04 28 04/03/96 08h25 62 03/05/96 12h28 96 02/07/96 16h33 29 06/03/96 02h54 63 05/05/96 06h56 30 07/02/96 21h22 64 07/05/96 01h25 31 09/03/96 15h50 65 08/05/96 19h53 32 11/03/96 10h19 66 10/05/96 14h21 33 13/03/96 04h47 67 12/05/96 08h50
00 04/07/96 13h16 34 02/09/96 17h31 68 01/11/96 21h51 01 06/07/96 07h45 35 04/09/96 12h00 69 03/11/96 16h20 02 08/07/96 02h13 36 06/09/96 06h28 70 05/11/96 10h49 03 09/07/96 20h42 37 08/09/96 00h57 71 07/11/96 05h18 04 11/07/96 15h10 38 09/09/96 19h26 72 08/11/96 23h47 05 13/07/96 09h39 39 11/09/96 13h55 73 10/11/96 18h16 06 15/07/96 04h08 40 13/09/96 08h24 74 12/11/96 12h44 07 16/0796 22h36 41 15/09/96 02h52 75 14/11/96 07h13 08 18/07/96 17h05 42 16/09/96 21h21 76 16/11/96 01h42 09 20/07/96 11h33 43 18/09/96 15h50 77 17/11/96 20h11 10 22/07/90 06h02 44 20/09/96 10h19 78 19/11/96 14h40 11 24/07/96 00h31 45 22/09/96 04h48 79 21/11/96 09h09 12 25/07/96 18h59 46 23/09/96 23h17 80 23/11/96 03h37 13 27/07/96 13h28 47 25/09/96 17h45 81 24/11/96 22h06 14 29/07/96 07h56 48 27/09/96 12h14 82 26/11/96 16h35 15 31/07/96 02h25 49 29/09/96 06h43 83 28/11/96 11h04 16 01/08/96 20h54 50 01/10/96 01h12 84 30/11/96 05h33 17 03/08/96 15h22 51 02/10/96 19h41 85 02/12/96 00h02 18 05/08/96 09h51 52 04/10/96 14h10 86 03/12/96 18h31 19 07/08/96 04h20 53 06/10/96 08h38 20 08/08/96 22h48 54 08/10/96 03h07 21 10/08/96 17h17 55 09/10/96 21h36 22 12/08/96 11h46 56 11/10/96 16h05 23 14/08/96 06h15 57 13/10/96 10h34 24 16/08/96 00h43 58 15/10/96 05h03 25 17/08/96 19h12 59 16/10/96 23h32 26 19/08/96 13h41 60 18/10/96 18h00 27 21/08/96 08h09 61 20/10/96 12h29 28 23/08/96 02h38 62 22/10/96 06h58 29 24/08/96 21h07 63 24/10/96 01h27 30 26/08/96 15h36 64 25/10/96 16h56 31 28/08/96 10h04 65 27/10/96 14h25 32 30/08/96 04h33 66 29/10/96 08h54 33 31/08/96 23h02 67 31/10/96 03h22