Craters on the Moon. Chris Kervick March, 2013

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1 Craters on the Moon Chris Kervick March, 2013 Abstract Using three supplied photographs of the moon, measurements were taken regarding the size of craters. Basic geometry was then used to calculate the corresponding depth of each crater, and a linear relationship was established between the depth of a crater and its diameter. Using formulae which relate the energy of an impacting body to the diameter of the crater it produces, an estimate was made of the masses of asteroids which collide with the moon. By counting craters of a given size, it was found that smaller impacts are far more common than larger ones. Aims To establish a relation between the diameter of craters and their depth To calculate the energy of the impacting body for a resulting diameter of given diameter. To determine the frequency and size distribution of craters on the moon. Introduction and Theory The most widely accepted theory as to the formation of the solar system is the nebular hypothesis, dating from the 18th century. The nebular hypothesis claims that the solar system formed from the gravitational collapse of a fragment of a giant rotating molecular cloud. For angular momentum to be conserved as the nebula collapsed, its rotation speed increased. Eventually it flattened into a protoplanetary disk, with the bulk of the material forming a dense protostar in the centre. 1

2 It was from this protoplanetary disk that the planets formed, via a process known as accretion. Dust grains accumulated matter over time, eventually forming planetesimals of around 10km in size. It was repeated collisions of these planetesimals which formed the current terrestrial planets. However, many planetesimals never formed planets, and it is these objects which comprise the asteroids and comets of today. Over time, many of these asteroids have collided with our moon, forming craters. There are two main reasons for the abundance of craters on the moon when compared to the Earth. Firstly, many asteroids which reach the Earth burn up in the atmosphere. The moon has no atmosphere, and thus no such protective shield. Also, owing to the low level of geological activity on the moon, craters are preserved for an extremely long time. Using basic trigonometry, it is possible to calculate the depth of a crater if the shadow length is known for a given zenith angle of the sun. Experimental Method We were provided with three separate images of the moon, and a ruler. Each image contained a scale and information on the suns zenith angle at the time the photo was taken. For each photo, three craters of varying size were chosen for examination. Their diameter (chosen to be in the direction of the shadow) was measured, and the length of their shadow was measured. Conceptual errors were noted when the precise location of the craters edge or shadows edge was difficult to determine. In each photo, the number of craters greater than the following sizes were counted: 2km, 4km, 8km and 16km. In the case of 2km, it was necessary to make an estimate based on a small region of the photo. It was also very difficult to distinguish craters of this size from natural formations. 2

3 Results and Analysis Figure 1 shows a linear relationship (on a log-log scale with an R 2 value of 0.933) between the depth of a crater and its diameter log(height) log(diameter) Fig 1: A log-log plot of crater height vs diameter Figure 2 shows the relationship between the estimated mass of a body creating a crater of a given diameter. 3

4 1.6e e e+07 1e+07 8e+06 6e+06 4e+06 2e e Fig 2: A plot of estimated mass of impact body vs diameter of resulting crater Figure 3 is a not-very-accurate plot of the number of craters with a diameter greater than a given size, vs that size. It was extremely difficult to accurately count the number of craters, particularly in the case of 2km, due to the low resolution of the photograph. 4

5 No. of Craters Greater Than Given Size Size of Crater (km) Fig 3: A linear plot of no. of craters with diameter greater than given size Figure 4 shows a log-log plot of the same data.even with our shaky measurements, there is a clear linear relationship. 5

6 7 6.5 log(no. of craters greater than given size) log(size of crater) Fig 4: A log-log plot of no. of craters with diameter greater than given size By taking the radius of the moon to be km, it was estimated that one photo represents 0.247% of the moons surface. By assuming a uniform distribution of craters over the moon, it was found that there are approximately 400,000 craters on the moon greater than 2km in diameter. Discussion and Conclusions A linear relationship was established between the diameter and depth of a crater, with a relatively high R-squared value of There was a linear relationship shown between the mass of an impacting body and the diamter of the crater it creates. Regarding the number of craters greater than a given size, our data was very crude owing to the low resolution of the photographs. Indeed, it was almost impossible to determine the difference between a small crater and a natural formation. 6

Craters of The Moon. Evan Sheridan, Tom Power, Chris Kervick March 4th 2013

Craters of The Moon. Evan Sheridan, Tom Power, Chris Kervick March 4th 2013 Craters of The Moon Evan Sheridan, Tom Power, Chris Kervick 11367741 March 4th 2013 Abstract Various properties of craters found on the moon were investigated. For a various number of craters both the