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

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1 Craters of The Moon Evan Sheridan, Tom Power, Chris Kervick March 4th 2013 Abstract Various properties of craters found on the moon were investigated. For a various number of craters both the crater height,which utilized knowledge of the Sun s zenith angle, and crater diameter were measured which suggested a linear relationship of the form : h = αd. For each crater measured we found the range of kinetic energies of the impacting asteroids to be E kinetic 1.87x Finally, the number of craters on the moon was found to be of order 100,000 with less craters appearing in the mare regions. 1

2 1 Aims To measure the length of shadows cast by craters and thus deduce the height of the crater. To measure the diameter of a number of craters a deduce a relationship between crater height and crater width. Deduce certain properties of the bodies that impacted the Earth and the Moon during the early Solar System. Estimate the number of craters on the moon. 2 Backround and Theory In the early formation of the Solar System both the Earth and the Moon were bombarded with many asteroids. Due to the geological activity of the Earth at the time these asteroids didn t leave long lasting craters that are visible to the extent that they are on the Moon. The experiment focuses on these relics of the early formation of the Solar System and attempts to garner some idea about the nature of these impacts. The main theory of how the solar system formed is labelled the nebular hypothesis. It states that some 4.5 billion years ago a giant cloud of molecular dust collapsed upon itself due to it s own gravitational pull. Most of the mass became concentrated in the centre of this cloud while due to the conservation of angular momentum the kinetic energy of the cloud was converted into heat at the centre giving rising to the formation of the Sun. However, not all the mass was concentrated at the centre. Some of it formed what is known as a protoplanetary disk that surrounded the Sun. Essentially what happened then is that a bunch of orbiting mass around the Sun came in direct contact with more and more mass and formed planetesimals. Under the force of gravity these orbiting planetesimals formed into planets. This process is known as accretion. Stray orbiting matter around the Sun that never accumulated into one of these planets or moons of a planet are deemed as asteroids and there were quite a few of these asteroids at that time. Thus resulting in a lot of collisions with the Earth and Moon, inevitably leading to craters on the moon. Unlike 4.5 billion years ago we don t experience a bombardment of asteroids anymore. It is thought this is because that after the formation of the large gas giants many of these gravitationally flung these asteroids out of the solar system resulting in the Oort Cloud and Kuiper-Edgeworth Belt. In order to calculate the height of a crater on the moon we consider the following situation: h θ l Crater Width D The Height of the crater is given by : 1

3 where l = Length of the Shadow. h = l tan(θ) For the Diameter of the crater we have the following: ( ) 1 E 4 D = 2.5 ρg M This formula is derived from the results we obtain in the experiment. (2.1) 3 The Experiment The first part of the experiment requires one to use a ruler to measure 9 craters of varying sizes using 3 images of the lunar surface. The shadow length must also be measured and from both this and the Sun s angle with respect to the crater the height of the crater can be found. After collecting all this data one a log-log plot will be plotted of the crater wall height against the crater diameter to see if any relationship can be established. The second part of the experiment requires one to investigate the equation (2.1) and to figure out the range of kinetic energies and masses of the impacting bodies from the data obtained in the first part of the experiment. As well as this as physical explanation must be given for (2.1). The final part of the experiment requires one categorize the craters over a certain area in terms of craters > 16km,8km,4km and 2km. Then the number of craters in each respective category shall be counted up. The data will be plotted on both linear and log plots in order to extrapolate a relationship. The number of craters on the moon will be estimated. Finally, the mare regions of the moon will be then investigated by deriving the crater ration. 4 Results and Analysis We picked 9 craters of varying sizes from the 3 pictures given. It was noted that if the images were taken at a time of full moon then there would been no shadows cast by the craters and no heights would be able to be calculated. We then got the shadow length and diameter successfully. The height of the crater was easily found. We then plotted the following log plot: 2

4 As we can see this suggests a linear relationship between the Crater Height and Crater Diameter, i.e : h = αd with α some constant of proportionality. Intuitively, this is what we expect because if we have a huge impact it will tend to both dig deep and dig wide resulting in the height and width being somehow proportional. For each of the craters we analysed we found the following kinetic energies for the asteroid by modifying (2.1) such that : E = D4 ρg m 2.5 With ρ 2x10 3 kgm 3 and g m = 1.62ms 2 From this we got the following results: Diameter (Km) E kinetic (Joules) x x x Thus the range of kinetic energies for the craters that we sampled is: E kinetic 1.87x10 10 Using dimensional analysis it was found that the units of the RHS go like : ( ) kgm 2 s kg ms m (m) 1 4

5 Getting rid of the 4 th root = m. Thus the units are correct. Given that above we have found that the crater height and crater diameter have a linear relationship this give rise to the following situation: Consider: i.e the potential energy at a height h. Now from out results : and E = mg m h (4.1) D = αh ρ = m D 3 where m is the mass of the crater and we are now approximating the volume of the crater by a cube. Thus subbing into (4.1): E = ρd3 Dg m α = E = D4 ρg m 2.5 where α is the constant of proportionality. E can be thought of as the amount of energy needed to remove the crater mass m from by the height of the crater h. If we assume an elastic collision then we can equate this energy to the kinetic energy of the asteroid. This, however, is a simplification. Nevertheless, it is a physical explanation. We assumed that asteroids typically moved at 55kms 1 when they hit the surface of the moon. We then used : E = 1 2 mv 2 to solve for the mass of each body hitting the crater. We got the following: Diameter (Km) E kinetic (Joules) Mass asteroid (Kg) x x x Thus the range of masses of the impacting bodies was found to be : 124 m We then investigated the number of impacts on over a specific area, categorizing the impacts by size. The data we found is illustrated in the following plot : 4

6 We found that there were very few craters of large size (i.e greater than 16 Km ) whereas as the craters became smaller the number of craters increased significantly. We can see that the diameter of craters over a specific area is inversely proportional to the number of craters over the same area. Even from the images this made sense because it was clear that there were far more smaller craters than bigger ones. This suggests that not many large asteroids actually impacted the moon during the early solar system. Perhaps this also meant that bigger bodies were actually going far too fast to be captured by the moon s gravitational attraction. Finally the number of craters on the moon were calculated to be on the order of 100,000. Given that the images were blurry in some instances and almost impossible to read anything off the mare regions the crater ratio was found to be 4 : 1 in general. 5 Conclusions Although a linear relationship was extrapolated between the crater height and the crater depth the images were quite difficult to analyse. For instance, it was assumed that the impact was a uniform collision whereas in reality it is not and this definitely affected the results when measuring shadow distances and crater diameters. Also, various simplifications may distort the outcome. Nevertheless, the results give a general gist of what is going on. It was found that smaller craters were far more common than larger craters on the surface of the moon, suggesting that not many large asteroids collided with the moon. The result concerning the crater ratio and the mare regions is subject to conjecture for it was quite difficult to discern what was a crater and what wasn t, especially in the mare regions. 5