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[O08-P27] Moon size determined from lunar eclipse images and causes of errors
Keywords:lunar eclipse, size of the moon, serving size, Image processing software
background
There was a total lunar eclipse on November 8, 2022. A lunar eclipse is a phenomenon in which the Earth's shadow is reflected on the moon, and the Earth's shadow includes a penumbra and an umbra. A lunar eclipse generally refers to an umbral eclipse, and a total lunar eclipse is a state in which the moon is completely covered by the umbra. Previous research has shown that the size of the moon can be estimated from images of lunar eclipses1). However, errors occur depending on the images used.
the purpose
The discrepancy between the calculated size of the moon and its actual size is thought to be due to the exposure at the time of photography and the influence of craters. Another new consideration was the influence of progress of lunar eclipse. Therefore, the purpose of this study was to confirm the degree of eclipse that is suitable for estimating the size of the moon.
method
The coordinates of the image were read by a computer and calculated. The coordinates were read using imaging software.
Since the Earth's shadow gradually becomes darker from the penumbra to the umbra and its outline is unclear, image processing software was used to posterize the image before reading it (Figure 1).
The center of the umbra of the moon and earth on the image is determined by the method of finding the center of a circle. This allows us to find the ratio of the radii of the moon and the earth on the image.
Figure 2 is a simplified representation of solar and lunar eclipses. From the figure, the shrinkage rate of the moon's shadow during a solar eclipse and the Earth's shadow during a lunar eclipse is the same. However, the Earth's shadow on the Moon's surface is actually smaller. In this case, the size of the moon and the smaller part of the Earth's shadow are equal. Therefore, the shadow of the Earth on the moon's surface becomes smaller by the diameter of the moon2). It was assumed that changes in the distance between the Moon and the Earth would have no effect.
The radius was calculated using spreadsheet software, with the radius of the moon on the image being R and the radius of the Earth's umbra being r, and the radius being 6,370 km. For each image, the points were read 20 times and the actual radius of the moon was calculated from the average value.
result
The calculated radius of the moon was all slightly larger than the actual radius (Figure 3). Also, there was not much difference in the size of the moon determined by the difference in eclipse.
Consideration
It is estimated that the amount of eclipses has little effect on the size of the moon, and that the exposure at the time of photography and the impact of craters are more significant. We also considered making the Earth's umbra smaller. Therefore, when we changed the proportion of the Earth's umbra, which had been calculated as 1 times the size of the Moon, to 1.1 to 1.3 times, it became closer to the actual size (Figure 4). Therefore, it is estimated that the Earth's umbra is smaller than the size of one moon. The reason for this is that sunlight refracted by the Earth's atmosphere enters the umbra, making a total lunar eclipse appear bronze-colored.
References
1) Gifu Prefectural Kamo High School Science and Mathematics Department TEAMKAGUYA (2014): “Size of the moon and distance to the moon determined from lunar eclipse”, Musashino University Mathematical Engineering Contest 1st prize winner
2) High School Astronomical Observation Network (2011): “Lunar Eclipse Observation Manual Analysis and Research Guide”
There was a total lunar eclipse on November 8, 2022. A lunar eclipse is a phenomenon in which the Earth's shadow is reflected on the moon, and the Earth's shadow includes a penumbra and an umbra. A lunar eclipse generally refers to an umbral eclipse, and a total lunar eclipse is a state in which the moon is completely covered by the umbra. Previous research has shown that the size of the moon can be estimated from images of lunar eclipses1). However, errors occur depending on the images used.
the purpose
The discrepancy between the calculated size of the moon and its actual size is thought to be due to the exposure at the time of photography and the influence of craters. Another new consideration was the influence of progress of lunar eclipse. Therefore, the purpose of this study was to confirm the degree of eclipse that is suitable for estimating the size of the moon.
method
The coordinates of the image were read by a computer and calculated. The coordinates were read using imaging software.
Since the Earth's shadow gradually becomes darker from the penumbra to the umbra and its outline is unclear, image processing software was used to posterize the image before reading it (Figure 1).
The center of the umbra of the moon and earth on the image is determined by the method of finding the center of a circle. This allows us to find the ratio of the radii of the moon and the earth on the image.
Figure 2 is a simplified representation of solar and lunar eclipses. From the figure, the shrinkage rate of the moon's shadow during a solar eclipse and the Earth's shadow during a lunar eclipse is the same. However, the Earth's shadow on the Moon's surface is actually smaller. In this case, the size of the moon and the smaller part of the Earth's shadow are equal. Therefore, the shadow of the Earth on the moon's surface becomes smaller by the diameter of the moon2). It was assumed that changes in the distance between the Moon and the Earth would have no effect.
The radius was calculated using spreadsheet software, with the radius of the moon on the image being R and the radius of the Earth's umbra being r, and the radius being 6,370 km. For each image, the points were read 20 times and the actual radius of the moon was calculated from the average value.
result
The calculated radius of the moon was all slightly larger than the actual radius (Figure 3). Also, there was not much difference in the size of the moon determined by the difference in eclipse.
Consideration
It is estimated that the amount of eclipses has little effect on the size of the moon, and that the exposure at the time of photography and the impact of craters are more significant. We also considered making the Earth's umbra smaller. Therefore, when we changed the proportion of the Earth's umbra, which had been calculated as 1 times the size of the Moon, to 1.1 to 1.3 times, it became closer to the actual size (Figure 4). Therefore, it is estimated that the Earth's umbra is smaller than the size of one moon. The reason for this is that sunlight refracted by the Earth's atmosphere enters the umbra, making a total lunar eclipse appear bronze-colored.
References
1) Gifu Prefectural Kamo High School Science and Mathematics Department TEAMKAGUYA (2014): “Size of the moon and distance to the moon determined from lunar eclipse”, Musashino University Mathematical Engineering Contest 1st prize winner
2) High School Astronomical Observation Network (2011): “Lunar Eclipse Observation Manual Analysis and Research Guide”