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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:
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.
"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."
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 DxB or DxC . 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.
The above model gives only a qualitative prediction of errors DxB or DxC 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 % .
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.
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.
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.
Geometry between the different elements used in the construction of a shadow.
data give a theoretical umbral radius (sthe) of
In our experiment we have measured the required displacement of the cardboard (the Earth) (uexp) 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 (sexp) as seen on Fig. 2 and 3. We also calculated the theoretical length of the umbra (sthe) which then gave us the umbral enlargement in percentage.
Relation between the Earth movement and the other variables.
The Earth's displacements measured experimentally (uexp±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 uexp, its corresponding sexp 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.
Enlargment ± 0.2%
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:
limit at 9.69 cm can be joined smoothly with the two
for Tycho as expected from the model that the displacement is
the threshold of sensitivity of the eye. We can then put the
point for Plinius on the curve. This gives us the intensity of
relative to the intensity of Tycho (we have to remember that
in an area of the Moon that reflects less light than Tycho). We
experimentally that Tycho is approximately 2.7 times brighter
(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).
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.
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.
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 Lacoursičre, 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.
Micrometric system (with vernier) to move the Earth's shadow
Just before the simulated moon eclipse