Let us measure the excentricity of the terrestrial orbit

Roland Szostak
"EAAE Summerschools" Working Group
University of Münster, Germany

Most textbooks of physics present the terrestrial orbit by drawings which show an ellipse of high excentricity. This suggests a remarkable variation of the distance between Sun and Earth during the year. But in fact in nature the numerical excentricity of the terrestrial orbit is only e = e/a = 0,01675. Remember ! a = long elliptical axis, b = short elliptical axis, e = focal length. The value of e is so tiny, that such an ellipse cannot be distinguished from a circular orbit in a drawing using a normal pen.(b/a = 1,00014)*.

By which procedure may it be possible under these circumstances to measure the excentricity in the classroom?

A first approach could be the idea to take photographs of the Sun throughout the hole year. The angular width of the solar disc varies by about 3 % within this period. The focal length f = 50 mm of a normal camera produces an image of the Sun, which is 0,4 mm in diameter on the film. Trying to determine the excentricity from these pictures better than by 10 % would mean to measure differences in size on the film better than 1µm. So this procedure will not work. It will be necessary to use a camera with a considerably larger focal length or a telescope with a magnifying factor which has to be kept highly constant throughout the year. But we can be independent of this delicate condition by measuring the angular width of the solar disc via the daily rotation of the Earth. In this case it is sufficient to measure the motion of the solar image on a screen, produced by a telescope at rest. The Sun exhibits its angular width by the time elapsed for passing a line on the screen. This happens within about 2 minutes. For determining the variations of the angular width by better than 10 %, it is necessary to measure the time of this passage by 0,2 s exactly. This is not just easy. In addition there is the problem that the brightness of the solar disc drops near its contouring edge. So we have problems to determine the excentricity of the terrestrial orbit.

But there is another way to determine it; a way which produces results with a few percent error only by almost no technical equipment at ail, as we will see: There is a phenomenon which was well known already by the Greeks in ancient times. The seasons, whose beginning and ending are given by the dates of the equinoxes and solstices, are not equally long. They differ by up to 5 days, as you may find in any odd calender. In ancient times people thought that the Sun was moving around the Earth, which was at rest. So these deviations were understood as an excentric motion of the Sun. Changing to the heliocentric view due to Copernicus this is caused by the excentricity of elliptical orbits due to Kepler. The excentric motion of the Earth has an interesting consequence.


Fig. 1

Look at figure 1: It produces a periodic deviation of the Sun's position for a terrestrial observer compared to the case of a circular motion. The drawing shows the Earth on its orbit at four positions belonging to intervals of a quarter of a year each. At the positions of perihel P and aphel A there are no deviations. But at B and C a terrestrial observer perceives the Sun at a position which deviates by the angle DFE compared to the case if the terrestrial orbit were circular. By virtue of the daily rotation this results in a deviation of day time. At the positions P and A the Sun is passing the meridian at 12 h noon. But at position B the Sun has not yet reached the meridian at 12 h; that means that the Sun is somewhat "late". At position C the Sun will pass the meridian earlier than 12h, correspondingly. Figure 2 shows this deviation in time throughout the year. We have good clocks now, which measure exactly the continuous flow of the time. So we can measure these differences of "Solar time" against the independently running time easily.


Fig. 2

This variation of solar time can be checked easily in the classroom by observing a Sun beam which passes a little hole in a card board which has been fixed at a window towards south. We simply register the time, when the centre of the Sun's image passes a line on the floor (figure 3).


Fig. 3

Recently there are very cheap radio controlled clocks in the shops, which display the time by 1 s exact reliably. Keeping attention to an accurate observation technique we obtain this meridional time by 3 s exact. We need no lenses, only this piece of cardboard. It is sufficient to repeat this simple measurement in the classroom a few times once a week, to obtain remarkable deviations of about a few minutes for demonstrating this effect. A longer sequence of measurements will not be very good for school practice. But having been convinced of the Sun's delay by this short own experience, the students may collect the data from other sources, then. There is one data bloc, which is very cheap and available to everybody: In a normal calender you may find the data for Sun rise and Sun set, at least once a week(²). By simply taking the mean value between Sun rise and Sun set one obtains the meriodional time. This way one finds a deviation throughout the year as shown in figure 4 which is called the "time equation".


Fig.4

But surprisingly this curve is not the same as expected due to figure 2. With some imagination the expected curve can recognized in the two big extrema at left and at right. Obviously there is another effect of twice periodicity superimposed, as shown in figure 5.


Fig. 5

In order to find out the origin of this additional effect let us regard a normal Sun dial. If we install a gnomon vertically on a horizontal ground the full hours will not occur equidistantly on the clock face. An equidistant division will be obtained only, if the gnomon is inclined to be in coincidence with the terrestrial axis and the clock face in coincidence with the equatorial plane. So we see, the nonequidistant characteristic is an artefact of oblique projection, if the Sun dial is inclined against the rotating axis.

A similar artefact, but with regard to the period of a year, results from the fact that there is an inclination between the orbital plane and the equatorial plane of the Earth. For a terrestrial observer the Sun does not move in the equatorial plane throughout the year but in the ecliptical plane which is inclined by 23,50° (figure 6).


Fig. 6

If the Sun were continuously running in the equatorial plane, an observer on the daily rotating Earth would see the Sun passing his meridian all the year in equidistant intervals of average solar days. But the Sun, which runs continuously along the equatorial plane, will pass the meridian of this observer somewhat earlier, as shown in figure 6, on its way between F and S. Correspondingly the Sun passes the meridian somewhat later than the average time, if it is on its way between S and H.

This projection error DFp has four zeros along the year: the two equinoxes F and H for springtime and for autumn, and the solstices S and W for summer and winter. These are the beginnings of each of the seasons. So this function DFp possesses the double periodicity, where we looked for. From figure 6 one can find by geometrical rules.

This function can be completely calculated with e = 23,50° and converted into time difference DtP, by the angular velocity of the terrestrial rotation, where 1° corresponds to about 4 min. As the dates of the zeros are well known by the equinoxes and solstices, the values of this function can be subtracted from the empirical data in figure 4. By this we obtain the empirical curve in figure 7, which corresponds to the expected behavior. This contains only the influence of the excentricity. The zeros of figure 7 reveal the dates of perihel and of aphel immediately. It is surprising that we find these dates within an error of less than 2 days by these simple means, keeping in mind, that a drawing of this elliptical orbit cannot be distinguished from a circular orbit.


Fig. 7

Let us now calculate the value of the numerical excentricity. Let us take a point C, which is reached exactly at the half time between A and P in figure 8. Due to the second law of Kepler the areas of the elliptical parts SAC and SCP must be exactly equal. To calculate the areas is tedious.


Fig. 8

So we exchange the shadowed parts in figure 8 and get two equal areas which are simply a quarter of an ellipse each. These two shadowed areas are two congruent triangles except for the little curved part which causes an error of only 10-4 in our case because of the small value of the excentricity. Then the triangle containing the angle DFE offers the relation

where the term e2 = 10-4 can be neglected. DFE can be converted to DtE again by 1° = 4 min. Examination of figure 7 and 8 reveals that the date of point C will be close to the maximum of the curve. So it justified practically to take the amplitude ( DFE )0 in figure 7 for calculating
Comparing the value e = 0,01768 obtained from figure 7 with the exact value e = 0,01675 we have a result, which is by about 5 % exact. Reminding that we used simply dates of Sun rise and Sun set in an odd ordinary calender, this result is very satisfying. If we took simple annual data books of amateurs (3), we obtain results with an error of about 1 %.

There is still another correction to be made, which is rather trivial but to be considered on school level. The dates of Sun rise and Sun set given in the calender (2) are valid only for a certain city, in our case for Kassel, which is situated at 9,5° eastern length. At this geographical place the Sun passes the meridian generally somewhat later than the reference meridian for the Middle European Time which is situated at 15° eastern length. The difference 15° - 9,5° results in a delay of 22 min. This had been taken into account already in our data given in figure 4 without mentioning. If this correction is not made accurately, the zero line in figure 7 will be shifted up or down. But for calculating e from this figure, this shift is not so Important. We can omit this shifting in the end because we need only the amplitude, which can be read between the minimum and the maximum of figure 7 simply.

1) W. Schlosser, Th. Schmidt-Kaler: Astronomische Musterversuche für die Sekundarstufe Il, Hirschgraben-Verlag, Fft./M.

2) Wochenkalender 1994, Deutscher Sparkassenverlag GmbH, Stuttgart.

3) Ahnerts Kalender für Sternfreunde 1994, Johann Amborsius Barthe-Verlag.