Craters of the Moon Nikki Truss 09369481 Abstract: In this experiment, the craters of the moon were examined. By measuring the diameter of the craters and the lengths of their shadows, their height was calculated. It was found that there is a linear relationship between the height of a crater and its diameter. Using a formula relating the diameter of a crater to the energy of the impacting body, it was possible to estimate a range of masses for bodies colliding with the moon, which was found to be 5×10!" kg to 1×10!" kg. By assuming uniform distribution of craters, the number of large craters on the moon (d >16km) was found to be approximately 3100. Similarly the total number of visible craters on the moon was found to be approximately equal to 15000. Aims: Our aims in this experiment were; •
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To calculate the height of craters on the moon by measuring the length of the shadows cast in them To relate this data to the size of the craters To calculate the energies of the impacts that caused these craters To reach some conclusions about the distribution of certain size craters on the Moon To derive some properties of bodies that impacted on the Moon and Earth during the early phases of the Solar System Introduction and Theory: For years, astrophysicists have been struggling to discover the origins of the Solar System, the most commonly accepted theory is the nebular hypothesis, which was formulated in the 18th century. According to this model, stars form in giant clouds of molecular hydrogen, which are unstable, then within these clouds matter begins to gather together into lumps. These lumps then coalesce into denser clumps and eventually collapse and form stars. Following the formation of a star, the young star develops a gaseous accretion disk around it, known as a protoplanetary disk. Once again, matter gathers into small clumps, which eventually coagulate to form bodies known as planetesimals. If the disk is large enough then more accretion occurs and planetesimals may impact and combine to form terrestrial planets. Any planetesimals which did not accrete enough matter formed comets, meteors and asteroids. These comets, meteors and asteroids travel throughout the solar system, occasionally impacting with other bodies, as evidenced by the craters left by such impacts on our Moon. In this experiment we approximated these craters as troughs and thus calculated the height of the crater wall using the length of the shadow and the Sun’s zenith angle, θ, (note that all images are taken at a time other than the full moon, as then there would be no shadow) from the diagram below; Fig. 1 We can see from this diagram that ℎ =
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We now want to relate the diameter of a crater to the energy of its impact. We know that the crater height, h, is proportional to its diameter, D, so ℎ ∝ 𝐷 We also know then, that its volume, V, is proportional to its diameter cubed, 𝐷 ! , so From this, knowing that 𝑉 =
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𝑉 ∝ 𝐷 ! , where ρ is the density of the Moon, and m is its mass, we get 𝑚 ∝ 𝜌𝐷 ! The kinetic energy, E, of the impact can be approximated as the potential energy needed to lift the mass up to the height, h, of the crater, so then 𝐸 = 𝑚𝑔! ℎ Where 𝑔! is the acceleration due to gravity on the Moon. Combining our two equations, we get the following 𝐸 ∝ 𝜌𝐷 ! 𝑔! D → 𝐸 = 𝑘𝜌𝑔! 𝐷 ! where k is a constant. Rearranging this relationship, and knowing it is found from experiment that k=2.5, we get our final expression; 𝐷 = 2.5
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Looking at this formula from a physical point of view it makes perfect sense that the diameter of the crater is proportional to the collision energy, and inversely proportional to the density of the rock and the acceleration due to gravity. Experimental Method: §
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A number of craters were selected from three photographs of the Moon. For each crater the diameter and the length of the shadow were measured using a ruler. Using the scale given on the photograph to convert all distances to kilometres, and the trigonometric diagram above, the height of each crater was calculated from the length of their shadows. A log-­‐log plot of the height of the craters vs the diameters was then plotted (as seen below). Then using the above formula relating the crater height to impact energy, the energies of the collisions causing each of these craters was calculated. A range of values of impact velocity were given (10 – 100 𝑚 𝑠 !! ) and from this a range of values for the masses of the impacting bodies could be found. The number of craters of diameters larger than 2km, 4km, 8km, and 16km, were counted and the total surface area represented by the photograph calculated. From this, the number of each of these sizes of craters on the total surface area of the Moon was estimated. A graph of the number of craters vs the diameter was then plotted. Results and Analysis: From the following graph, Fig 2, it was evident that the relationship between crater height and crater diameter is linear, meaning deep craters are almost always wide, and shallow craters almost always narrow. Fig. 2 As discussed above, using the formula relating D to E, it was possible to calculate the energies of each collision causing a crater. From this range of energies, it was possible to find a range of masses of impacting bodies for the given range of impact velocities (10 – 100 𝑚 𝑠 !! ). This range of masses was found to be 5×10!" 𝑘𝑔 to 1×10!" 𝑘𝑔. The number of craters of diameter greater than 2km, 4km, 8km, 16km was then counted, and comparing the surface of the photograph to the entire surface area of the Moon (about 0.225%) it was possible to find the number of such craters on the entire Moon’s surface. It was found that there are approximately 3100 craters with diameter greater than 16km, and approximately 15000 craters with diameter greater than 2km on the Moon. A graph of the number of craters vs diameter was then plotted for the Moon in total, as shown below, Fig 3. Number of craters vs diameter (d) 16000 R² = 0.99732 No. of Craters 14000 12000 10000 8000 6000 4000 2000 0 0 2 4 6 8 10 12 14 16 18 Diameter d (km) Fig. 3 From this, it is obvious that there are many more impacts from small bodies than there are from large bodies. Discussions and Conclusions: o
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From our plot of crater height vs diameter, it is obvious that there is a linear relationship between the two, as crater height increases, so too does the diameter. From the range of impact velocities, it was found that the range of masses of impacting bodies is 5×10!" 𝑘𝑔 to 1×10!" 𝑘𝑔. It was found that there are many more smaller craters on the Moon than larger ones, and there are approximately 15000 craters on the lunar surface. This implies that small bodies collide with the Moon relatively frequently. The errors in this experiment were quite large as the resolution of the images was poor, thereby making it difficult to accurately measure the diameter and length of the shadow of any given crater as the edges were unclear. More accurate results could have been obtained with a higher resolution and a smaller scale on the photographs.