Astronomy Online

III.2 Roemer and the Velocity of Light

This exercise concerns observations of eclipses of Jupiter's satellites (moons) and how these may be used to determine the speed of light.

Jupiter's satellites and the measurement of longitudes in the 17th century

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.

Roemer proves that the velocity of the light is finite.

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.

The Earth is between an opposition and a conjunction

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 Earth is between a conjunction and an opposition

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.


Exercises

Here follow some exercises which are related to Io's motion around Jupiter and which put you in position similar to that of Roemer.

The Io Phenomena

Some calculations with data from the Ephemerides

Below you will find four tables with actual timings of Io eclipses. Use these values for the calculations explained below.


IO: Begin of Eclipse between the conjunction on November 17, 1994 (20h00) and the opposition on June 1, 1995 (11h00).

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	


IO: End of Eclipse between the opposition on June 1, 1995 (11:00) and the conjunction on December 18, 1995 (22:00)

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


IO: Begin of Eclipse between the conjunction on December 18, 1995 (22h00) and the oppostion on July 4, 1996 (12h00).

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	


IO: End of Eclipse after the opposition on July 4, 1996 (12h00)

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	

This exercise was elaborated by CLEA (Comite de Liaison Enseignants Astronomes). Please direct any related questions or remarks to: Josee Sert (France) who will pass them on to CLEA.


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