Speed of Light – Roemer’s method

Jasjeet Singh Bagla

Centre for Science Education and Communication,
10, Cavalary Lane, University of Delhi, Delhi 110007

Abstract: Roemer’s experiment was the first experimental proof of the now well excepted fact that the velocity of light is finite. The experiment can be performed using very simple equipment.  I performed this experiment and obtained reasonably good results. Here we concentrate on the description of the sources of error and ways of avoiding them, apart from a general discussion the experiment.

Introduction: Speed of light is a fundamental constant of nature. A number of methods have been employed to measure it.   Oleus Roemer was the first person to measure the speed of light, and the method he used is now known as the Roemer’s method.

Before Roemer measured the speed of light, it was assumed to be infinite.  It was quite natural of ancients to think so because the speed of light is several magnitudes larger than anything they were used to.  Soon after Renaissance, ships started traveling to faraway lands and need for maintaining a time standard arose.  As clocks were not available on ships and there was no radio, the only possibility left was in the sky, you could predict some event and by observing that, a reasonably accurate time could be maintained.  Cassini observed the eclipses of the Galilean satellites of Jupiter and found that they occurred at regular time intervals and could easily be predicted without resorting to any esoteric geometrical arguments. Later Roemer at Paris Observatory discovered that his predictions based on near opposition data were off by a large amount, in fact as much as ten minutes near conjunction.  To explain this observed discrepancy, he hypothesized that the speed of light is finite and the problem arises because of variation in Earth- Jupiter distance as Earth alternatively approaches and recedes from Jupiter as it goes about in its orbit. On these basis when he performed a simple calculation, he found that the velocity of light is around 2.2 x 10⁸ meters/second.  This value has a large error because of approximations made like Jupiter is stationary. In this way the speed of light was brought down to a finite value from infinity. 

The present accepted value is 2.998 x10⁸ m/s.

Roemer’s method is a simplest method to estimate the velocity of light.  All you need a telescope and an accurate clock.  To find the velocity of light, eclipses of the Galilean satellites are observed.  Then by comparing the observed time period with known time periods of the satellites and using the change in distance between earth and Jupiter, the velocity of light is found.

Consider the position of the Earth and Jupiter when the first eclipse is being observed.  The eclipse is seen from the Earth at the time t₀ which is t₁ later than the actual eclipse as seen from the satellite concerned. If the Earth is approaching Jupiter then for a later eclipse the light time correction t₂ will be less than t₁ because of reduced distance.  The difference t₁ – t₂ can be easily found by comparing the observed period.  The difference of two eclipse timings subtracted from the standard time period which has been multiplied by the number of cycles the satellite has completed gives the difference in light travel time.  The difference in distance can be calculated by assuming circular orbit for the two planets and taking the time of opposition as the reference time.  The ratio of the two gives the velocity of light.

  dr = r₁ –r₂

  dt = n x T – (t₂ – t₁ ) 

  c = dr / dt

The calculations done here are only approximate, the approximations used are

1. The orbits of the two planets are assumed to be circular and coplanar.

2. The radius of Jupiter’s orbit is assumed to be same as its distance from the sun at the time of opposition, which is our reference time.

3. No account of error induced because of change in the eclipse duration has been made as it is not significant.

These approximations have been made to make calculations simpler.

Calculation of Earth Jupiter distance

To calculate the Earth –Jupiter distance, we use opposition as our reference time.  We assume the Earth’s orbit to have the radius 1 A.U. and the radius of Jupiter’s orbit is assumed to be its distance from the sun at that time.  Then finding out their relative positions and distance is very simple as motion of each body is given by a uniformly rotating vector

  r_j = j₀ cos (ω_jt) î + j₀ sin (ω_jt) ĵ

  r_e = e₀ cos (ω_e t) î + e₀ sin (ω_et) ĵ

  r_ej =   √ [ ( (j₀ cos(ω_j t) – e₀ cos (ω_e t))² + (j₀ sin(ω_j t) – e₀ sin (ω_e t))² ]

r_j is the position vector of Jupiter where origin is at the Sun and the x axis is along the opposition position. r_e is the position vector of Earth.  ω_e and ω_j are the angular frequencies of the Earth and Jupiter respectively.  j₀ and e₀ are the assumed radii of orbits of Jupiter and Earth, respectively.  î and ĵ are unit vectors along the two axis.

Description of an eclipse: Consider an eclipse of a satellite of Jupiter. The geometry is described in the diagram give here.  As the satellite enters the shadow, its brightness, as seen from the Earth, starts decreasing.  As the satellite has a finite size.  It takes some time for it to disappear for the observer. This is the description of the disappearance of a satellite.  In case of reappearance the sequence of the event is reversed.  In a disappearance, the last speck is timed, for reappearance, first speck is important.  A good observer can time an event with accuracy of about 0.1 seconds.  In doing this experiment, we decided to focus our attention on Io the innermost Galilean satellite of Jupiter as its eclipse duration is small.  Eclipses of all other satellites are of longer duration and hence the error of measurement is correspondingly large.  Another reason for choosing Io was its short time period, which means that eclipse of Io occurs more frequently, and a large number of observations imply smaller statistical error.

Sources of Error and Precautions: The number of sources of errors for this experiment is very large, especially as we are doing only an approximate calculation.

We are assuming the orbits of the Earth and Jupiter to be circular.  This gives an error of the order of twenty seconds over six months.

Variation in transparency of the sky and altitude of Jupiter at time of eclipse also causes some error.  The magnitude of this depends on the variation of the observing conditions.  To reduce error of this kind it is advisable to note down the condition as remarks with each observation and to do final calculation with observations taken under similar conditions. 

Because of change in the Earth-Jupiter-Sun Geometry, the distance of the point where the eclipse takes place, from Jupiter, as seen from Earth keeps on changing. As Jupiter is very bright compared to any of its satellites, so its effect on the timing keeps on changing.  To remove this to some extent, pairs of observations equally spaced around quadrature should be used.  For such pairs, the distance from the limb is same.  Quadrature is the relative position when the Sun-Earth-Jupiter angle is ninety degrees.

As observing skill is a crucial factor, it is advisable not to use first few observations for calculations.

Although it has been suggested that in doing calculation a straight line should be fitted between Earth Jupiter distance and the time delay, and the slope should be used to calculate the velocity of light¹, we consider it better to take a weighted average of the result from individual calculation.  The value should be weighted with the number of orbits the satellite has undergone between the two observations.

Results obtained in the experiment

We performed the experiment in the pre-opposition and post-opposition apparitions of Jupiter in 1989 -1990.  In the two sets we obtained the following values of the velocity of light

Pre opposition data  c= 6.15 x 10⁸ m/s

Post opposition data c=2.45 x 10⁸ m/s

The value obtained in the first set is very different from the actual value.  Apparently this is because of short term perturbations as calculations using the predicted timing³ also yield the same result.

Acknowledgements:

I had done the experiment as my B.Sc. (H) Physics final year project during 1989-90.  The project was done mainly at the Centre for Science Education and Communication (CSEC) at Delhi University.  A 100mm Newtonian reflector owned by CSEC was used for the project.

The theme was suggested by Professor P. K. Srivastava (Department of Physics and Astrophysics, University of Delhi, also the Director CSEC) and Dr. Ravi Bhattacharjee (Department of Physics, S.G.T.B. Khalsa College, Delhi).  Discussions with Professor T. P. Prabhu (IIA, Bengaluru) were very helpful in understanding errors.

Tarun Deep Saini (now at IISc Bengaluru) and Divya Oberoi (now at NCRA-TIFR) sometimes stayed for company and sky watching.

Special thanks are also due to Mr. Nanda and his family, who often provided a nourishing breakfast to help me through the day after observations at night.  Mr. Nanda was the caretaker of CSEC at the time.

References

1.  A modern version of Roemer’s experiment, R S  McMillan and J D Kirszenberg : Sky and Telescope Nov. 1972, vol. 44,  p. 300.

2. Timing the Eclipses of Jupiter’s Galilean Satellites, John E. Westfall : ALPO information brochure [This is a useful document for an observer.  Also available at: http://www.alpo-astronomy.]

3. ALPO (Association of Lunar and Planetary Observers) Hand book, 1989

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