Enlargement of the Earth's Shadow on the Moon: An Optical Illusion

Paul Marmet and Christine Couture

Physics Department, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5

( Last modified: 2009/11/7 )

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    1- Introduction.
        Since the 1830s, crater timing has been used during lunar eclipses to determine the length of the Earth's shadow. The method is simple: one determines the time when the umbra crosses a feature of the Moon, like a crater or a limb. The Sun-Earth-Moon geometry being known quite precisely, it is then possible to calculate the length of the Earth's umbra at the Moon. However, it was noticed consistently that the Earth's umbra seems to be 2% larger than what is expected from geometrical considerations. It is generally believed that the Earth's atmosphere is responsible for this enlargement.
        In Sky and Telescope magazine (Vol. 92, No: 3 Sept 1996, page 98) it is realized that the atmospheric absorption cannot explain light absorption at a height as high as 90 km above the Earth, as required with this hypothesis, but they illustrate the phenomenon (page 98) using the phenomenon of air refraction which makes the umbra smaller (and not larger) than its geometrical size. Consequently, that explanation is not valid since such a refraction would narrow the beam and therefore produces the opposite effect. We show here that this phenomenon is an optical illusion that has been easily reproduced in a laboratory experiment.

        2- Accepted Interpretation of Umbral Enlargment.
        We know that astronomical data give us accurate values of the radii of the Sun, the Earth and the Moon. Furthermore the knowledge of their relative distances gives us accurate predictions of the exact instant when the umbra-penumbra limit sweeps some specific crater on the moon during lunar eclipses.
        However, numerous reports show that the umbra-penumbra limit appears significantly displaced on the moon during an eclipse. It is believed that the thickness of the Earth atmosphere is responsible for that displacement. The article of Roger W. Sinnott ("Readers Gauge the Umbra Again", in Sky & Telescope, April 1983, p. 387) illustrates this interpretation of the shadow's enlargement in his statement: "It [the atmosphere] always increases slightly the silhouette of our globe in forming the sharply defined central region of the shadow called the umbra." Similar conclusions are also presented by Sinnott in "A Tale of Two Eclipses" (Sky & Telescope, December 1992, p. 678). Therefore, it could be implied that crater timings during full lunar eclipses can be used as a tool to evaluate the degree of pollution of our atmosphere.
        A similar result has also been claimed by Byron W. Soulsby in "Lunar Eclipse Crater Timing Programme" (Journal of the British Astronomical Association, Volume 95, Number 1, p.18) where he writes:

"Each eclipse can exhibit oblateness variations due in the main to the conditions prevailing in the Earth's atmosphere at the time of the event, particularly when large volcanic eruptions have occurred before the observations are made."

        In order to study more deeply that phenomenon, it is important to evaluate if the reported increase of 2% of the Earth's shadow at the Moon corresponds to a reasonable value of the height at which the atmosphere is opaque. Calculations give that this amount corresponds to an altitude of 92 km on the Earth.
        This usual interpretation of the umbral enlargement forces us to believe that the atmosphere is normally opaque up to 92 km or so. But how can that be when at that altitude, the air is so extremely rarefied? It is near the altitude at which a satellite can orbit around the Earth.
        In fact, according to "Astrophysical Data: Planets and Stars" (Kenneth R. Lang, Springer-Verlag, New York, 1992, p. 36), the atmospheric pressure at 90 km above sea level is about half a million times smaller than that at sea level. Above 15 km, the atmosphere becomes relatively transparent to light, since 90% of the air and almost all the humidity and pollution are below that level. That makes an umbral enlargement due to the opacity of the atmosphere of only 0.3% which is much smaller than the 2.0 % reported.
        Furthermore, the eruption of volcanos cannot explain the umbral enlargement. According to Patrick McCormick (Sky & Telescope, October 1982, p.390), the altitude reached by some material ejected from volcano El Chichon "is in the stratosphere, some 26 kilometers (16 miles) above Earth's surface - roughly 50 percent higher than material from even the famous Mount St. Helens.". So since the atmosphere does not appear to be responsible for the umbra-penumbra limit displacement of 2% on the moon, then what causes it? We will see that the actual umbra of the Earth projected on the moon is not as big as observed and that the umbra-penumbra limit displacement is only a normal optical illusion.

        3- On the Threshold of Sensitivity of the Eye.
        There is an important fact that has been overlooked to explain the umbral enlargement on the moon. It is linked to the sensitivity of the eyes. It is commonly known that under a certain threshold of light intensity, light cannot be detected by the eye. This limiting threshold is quite general and must be applied especially when observing a dark limit during a lunar eclipse.
        During a lunar eclipse, we see that the light intensity goes from zero intensity (at A on Fig. 1) at the umbra-penumbra limit to an increasing intensity when moving across the penumbra. In order to illustrate that increase of illumination we have plotted on Fig. 1 two independent lines corresponding to two different intensities of light. The two intensities plotted correspond to two different coefficients of reflection (albedo) of solar light on the Moon at different locations. Due to the sensitivity of the observer's eyes, we see on Fig. 1 that the location of the detection threshold is either observed in B or in C depending on the amount of light reflected on the particular location observed on the Moon. On Fig. 1, we see that the location where the detection threshold B or C is observed is at a closer distance to the umbra-penumbra limit (in A) for a brighter feature than for a darker one.
        Those considerations show that the locations B or C determined by the observer as the detection threshold are not the umbra-penumbra limit at A. Due to the inherent threshold of sensitivity of the eye the umbra-penumbra limit appears at B or C. Therefore, the observations give an error Dx_B or Dx_C . In the case of a lunar eclipse, the distance (or time) between the umbra-penumbra limit (in A) and the point of detection B or C of the umbra corresponds to a displacement called Dx. On the opposite side of the Earth umbra, this displacement Dx is symmetrical so that the total width of the umbra on the moon appears to be widened by 2Dx.

4- Apparatus.
The above model gives only a qualitative prediction of errors Dx_B or Dx_C from the well known phenomenon of the threshold of sensitivity of the eye. In order to prove that the displacements actually observed on the Moon during eclipses are really due to the threshold of sensitivity of the eye, one must actually perform the experiment. We have built an apparatus in order to test the model above in conditions that can be measured with the total absence of atmosphere. This was realized in a laboratory, using a well calibrated circular source of light for the Sun and an accurately known occulting disk to simulate the Earth. The shadows were projected on an exact photograph of the Moon. In the controlled conditions of a lab, after repeated observations, it is not too difficult to measure the displacement of the umbra-penumbra limit with an accuracy of the order of 0.1 % .

Figure 1

Difference of illumination between points B and C in the penumbra and the umbra-penumbra limit in A versus the distance between them for two sources of different intensity.

        In our experimental set up, light is projected onto a rectangular black piece of cardboard which creates an umbra that is intercepted on the high quality picture of the Moon. Observers locate themselves as close as they wish to the Moon (picture). Placing the observer closer or further to the Moon would just correspond to having a more or less powerful telescope. The piece of cardboard (the Earth) is moved slowly with a micrometer mechanism. The moving umbra is observed crossing the Moon (picture) and the observer is asked to report at what point a certain feature (a crater) of the Moon is being crossed by the umbra. This measured displacement of the Earth between the positions where the shadow is entering and leaving the relevant crater on the Moon enables us to measure the length of the umbra at the Moon.
        In Fig. 1, we see that for a bright source of light, the distance between the theoretical (A) and the observed beginning (B and C) of the umbra is smaller than for a source of lower intensity. Therefore, we expect a smaller value (x on a bright feature.
        We used two different intensities for our "dummy" Sun (by changing the voltage on the lamp) in order to test the variation of location of the detection threshold as illustrated on Fig 1. We also used two different craters:
        - Tycho, in the southern part of the Moon, is in a very bright region that reflects more light;
        - Plinius, in Mare Tranquillitatis, is in a darker region of the Moon that reflects less light.
        That last region was much darker on our photograph of the Moon.
        Since there is less light reflected in the area where Plinius is located than in the area where Tycho is located, for equal intensities of the Sun (same voltage on the lamp), we could make another independent test of our hypothesis using that variation of albedo between Tycho and Plinius.
        The Sun was simulated using an overhead projector. At the place where the transparency is usually placed, we installed a frosted glass to diffuse the light. At the prime focus of the projector, we used a diaphragm in the shape of a disk that acted as the Sun, so the radius of that diaphragm corresponds to the size of our Sun. We had that diaphragm cut in an aluminum sheet so being so near to the projector's lamp, it would not overheat. Light intensity was measured with a calibrated photometer.
        To simulate the Earth, we used a rectangular piece of black cardboard. It was neither necessary nor preferable to have a disk because we only wanted to project an umbra. If we had used a disk, only its diameter would have been of any importance so we used a rectangle which was easier to cut. We found that this rectangular shape had the important advantage of eliminating the problem of centering the disks.
        For the Moon, a black and white picture was used and placed on a blackboard. The contrast on the photograph was larger than with the real Moon so that the change of light intensity in different regions was easier to record.
        As can be seen from the photograph of our apparatus, we had to use a lot of diaphragms to cut scattered light. We used many cylindrical pieces of black cardboard between the projector and the screen in order to cut any further stray light. We also had to put the projector in a partially closed box so there would be no residual light coming from the projector into the room. The whole room was then transformed into a dark room.
        We appreciate the collaboration of 15 volunteers which accepted to spend much time to get their eyes adapted to darkness and then report their observations on the "timing" of Tycho and Plinius at different light intensities. Crater Plinius was observed only at the highest light intensity due to the difficulty of observation of that dark region (on the photograph) at low light intensity.
        All our observers provided us with 116 observations. Observations could be repeated to the satisfaction of the observer. Furthermore, the observer was not made aware of the reading on the micrometer recorded by the operator.

        5- Data and Analysis.
        Fig. 2 and 3 show us the ray tracings leading to the limits of the umbra and the penumbra. The relevant corresponding parameters used in the lab are given. We have:
        R = radius of the Sun = (2.8915 ± 0.0005) cm
        R = radius of the Earth = (7.05 ± 0.01) cm
        s = radius of the Earth's umbra on the Moon
        v = distance between the end of the umbra cone and the Sun
        r = distance between the Sun and the Earth = (92.7 ± 0.1) cm
        d = distance between the Earth and the Moon = (296.6 ± 0.1) cm
        u = displacement of the Earth between the points where the crater enters and exits the umbra.

Figure 2
Geometry between the different elements used in the construction of a shadow.

These data give a theoretical umbral radius (s_the) of (20.36 ± 0.04) cm.
In our experiment we have measured the required displacement of the cardboard (the Earth) (u_exp) between the location where the umbra just enters the observed crater and the location where the umbra leaves the crater. Using simple geometry, it was then easy to relate that distance to the experimental length of the Earth's umbra on the Moon (s_exp) as seen on Fig. 2 and 3. We also calculated the theoretical length of the umbra (s_the) which then gave us the umbral enlargement in percentage.

Figure 3
Relation between the Earth movement and the other variables.

        6- Results.
        The Earth's displacements measured experimentally (u_exp±0.01) cm in the laboratory by fifteen independent observers in different experimental conditions gave the following diameters in the case of the crater Plinius illuminated with high light intensity (in cm):
10.05, 10.00, 10.12, 10.09, 10.06, 10.01, 9.90, 9.90, 9.95, 10.08, 10.07, 10.04, 9.95, 10.01, 9.99, 9.87, 9.87, 9.85, 9.91, 9.88, 9.87, 9.84, 9.84, 9.85, 9.94, 9.92, 9.89, 9.87, 9.98, 9.91, 9.94, 9.90, 9.86, 9.93, 9.92, 9.90, 9.88, 9.91, 9.89
        In the case of crater Tycho illuminated at high intensity, the following data gave the Earth displacement between the two positions, on each side of the Earth, corresponding to the detection threshold (in cm):
10.01 9.97, 10.01 9.97, 9.90, 9.96, 9.89, 9.89, 9.93, 9.95, 9.97, 9.97, 9.94, 9.86, 9.84, 9.82, 9.82, 9.82, 9.85, 9.82, 9.80, 9.92, 9.88, 9.88, 9.89, 9.87, 9.88, 9.91, 9.89, 9.93, 9.79, 9.82, 9.82, 9.87, 9.84, 9.85, 9.85, 9.88, 9.80
        In the case of crater Tycho illuminated at low intensity, the observations led to the following results (in cm):
10.10, 10.12, 10.02, 10.07, 10.08, 10.07, 9.95, 9.99, 9.94, 10.12, 10.13, 10.09, 9.90, 9.87, 9.91, 9.89, 9.85, 9.92, 9.90, 9.94, 9.92, 9.93, 9.92, 10.01 9.92, 9.91, 10.03, 10.05, 9.99, 9.88, 9.79, 9.85, 9.89, 9.90, 9.86, 9.91, 9.92, 9.92
        Table 1 gives the mean value of u_exp, its corresponding s_exp and the umbral enlargement percentage for each crater as a function of light intensity. The light intensity (L) in lux was measured on a photometer.

Table 1: Umbral Enlargement.

L (lux)	Crater Name	u_exp _(cm)	S_exp _(cm)	Observed Enlargment ± 0.2%
5.5	Plinius	9.94 ± 0.01	20.87	2.5
5.5	Tycho	9.866 ± 0.001	20.76	1.9
1.4	Tycho	9.96 ± 0.01	20.91	2.7

7- Discussion.
On Fig. 1 we see that the distance between the theoretical point A (giving the exact umbra-penumbra limit) and the observed thresholds B or C is inversely proportional to the brightness of the source. One then expects that the umbra-penumbra limit must show a displacement with respect to the detection threshold that is proportional to the inverse of the illumination. Since our eyes have a logarithmic response to light intensity, we use the log scale. This is illustrated on Fig. 4 which defines F as the inverse of the log intensity seen by human eyes. The numerator of the function and A are the calibrating constants satisfying the characteristics of photometers. The value of F is then given by the relation:

where L is the illumination in lux, A is the constant 11.43 and P is another undefined constant of proportionality. We can see that if the Sun's brightness were infinite, the line giving the brightness would be vertical on figure 1. In that case, the observed distance (Dx would be zero. This inverse brightness relationship gives us the possibility to get a quantitative relationship between light intensity and the apparent displacement of the beginning of the umbra.

Figure 4
Displacement of the limit of the umbra as a function of the inverse of light intensity (log scale), as observed in the laboratory.

        The theoretical limit at 9.69 cm can be joined smoothly with the two experimental points for Tycho as expected from the model that the displacement is caused by the threshold of sensitivity of the eye. We can then put the experimental point for Plinius on the curve. This gives us the intensity of Plinius relative to the intensity of Tycho (we have to remember that Plinius is in an area of the Moon that reflects less light than Tycho). We then deduce experimentally that Tycho is approximately 2.7 times brighter than Plinius (on the photograph).
        In agreement with the fact that the contrast of the Moon used in our experiment is greater than in real life and our room darker than what people often observe in eclipses (i.e. our conditions of contrast are a lot better than conditions in a real lunar eclipse), we observe an umbral enlargement inversely proportional to the intensity of the Sun (as illustrated on figure 4).
        There is another factor about the atmosphere that has not been discussed. Light rays passing through the atmosphere are naturally bent because the atmosphere acts like a prism. This is why, during an eclipse, the Moon surface is never completely black but reddish: the red part of the solar spectrum passing through the low atmosphere is the only part scattered on the Moon in the region of totality before being reflected back to us on Earth.
        An hypothetical observer located on the Moon would see those rays being refracted by the Earth atmosphere and the Sun would appear bigger. Consequently, this second effect makes the Sun rays converge due to a lensing effect of our atmosphere. Therefore, due to that lensing effect, the umbra projected on the Moon would be smaller. This refraction by the Earth atmosphere gives an effect that is contrary to the observations claiming that the Earth's shadow must be larger due to the thickness of the atmosphere (Antonín Rükl, Atlas of the Moon, Aventinum, Prague, 1990, p.214).

        8- Second Experiment.
        There is another more elementary way to see the effect of the threshold of sensitivity of the eyes. It can be done by the simple projection of light through an aperture. For example, let us consider that the light emitted by a projector reaching a white screen is limited by a circular aperture between the source and the screen. We will observe some fuzziness on the screen around the image of the luminous disk. This fuzziness is the penumbra. Due to the optical illusion explained above, our eyes will not perceive completely the full size of the illuminated disk. Its size will appear to vary as a function of the light intensity of the source. We will see the bright disk apparently shrinking on the screen as light intensity decreases. This corresponds to an increase in the umbra's length.
        To realize that experiment with an overhead projector, one has to cover it so no stray light can escape. This demonstration was readily shown using our experimental setup. The disk projected on the screen seems to grow bigger in the region of fuzziness as the intensity of the projector is increased. However, solely the light intensity and no geometrical data is changed. This effect is quite similar to the one taking place during lunar eclipses.

        9- Conclusion.
        It is perfectly clear that a shadow appears smaller when the amount of light is increased as illustrated on Fig. 4. This is exactly the case for the crater timing on the Moon during a lunar eclipse. The occulting diaphragm in our lab was certainly not surrounded by any atmosphere. However, it seemed that the umbra was about 2 % larger than the real value due to the optical illusion just as in the case of astronomical observations.
        We have seen that the atmosphere may be opaque for 15 km, but certainly not 90 km. We have also proved that the sensitivity of the eyes is a factor leading necessarily to an umbral enlargement. Therefore, the accepted interpretation of umbra-penumbra limit displacement according to which the atmosphere is the major factor is not compatible with the well known phenomenon that characterizes the human eye. We therefore believe that almost the totality of the reported umbra-penumbra limit displacement is an optical effect that has nothing to do with the thickness of the Earth atmosphere.

10- Acknowledgment.
We would like to thank Élise Milot for the picture of the Moon as well as all our observers: Isabelle Broussell, Roger Chagnon, Serge Desgreniers, Pierre Gauthier, Saeed Hadjifaradji, Karin Hinzer, Martin Krzywinski, Yves Lacoursière, Ken Lagarec, Pascal Lauzon, Dave Leblanc, Rick Legault, Moussa Mousselmal, Grant Nixon, Serge Oliveira. We also wish to acknowledge the financial assistance of the Natural Science and Engineering Research Council of Canada.

Apparatus.

Preparing the projection of the Earth's shadow (when the lab was still illuminated)

Micrometric system (with vernier) to move the Earth's shadow

Just before the simulated moon eclipse

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