

Astronomy 23/223 Homework:
Throughout our class discussions of the Moon, Mercury, Venus,
and now Mars, a recurring theme is the use of crater counts to determine
relative ages. In NASA Technical Memorandum
79730, the Crater Analysis Techniques Working Group (1978) spells out
the achievements and goals of this approach.
So, in this week’s problem set, we’ll use some real data and actually see how crater analysis is done. The only change I’ve made to actual real data is to reduce the number of craters in the small size ranges to make the problem more tractable. If you get confused, consult the inclass exercise and its solution. In a 1986 paper, K. L. Tanaka used cratering statistics to develop a detailed stratigraphy of Mars that now forms the basis for our geological stratigraphy of that planet. This stratigraphy is especially useful because it is constrained by crater abundances. So if you want to know how old any part of the martian surface is, you can simply count the numbers and sizes of craters within a given area, and estimate the age of that area as it relates to Tanaka’s time table for Mars. 
Series  N(1)  N(2)  N(5)  N(16)  N(410) 
Upper Amazonian  <160  <40  
Middle Amazonian  160600  40150  <25  <33  
Lower Amazonian  6001600  150400  2567  3388  
Upper Hesperian  16003000  400750  67125  88165  
Lower Hesparian  30004800  7501200  125200  <25  165260 
Upper Noachian  200400  25100  >260  
Middle Noachian  >400  100200  
Lower Noachian  >200 
N = cumulative number of craters greater than or equal to each
crater per one million km^{2}.
The inclass exercise we did will help you understand how this system works.
In this problem set, we use some crater count data taken from a study by George McGill, who is a faculty member at U Mass. In this study (McGill et al., 2000), George examined an area of Mars that spans the highland/lowland boundary in northcentral Arabia Terra; it is bounded by 27.5 and 47.5N, and 330 and 335W. Through meticulous study of radar images in that area, he counted 634 craters and carefully measured the size of each one. From these data, he determined the relative ages of various features in his “field area.” In this problem set, we will use a small part of his data to look at the relative ages of two types of features. We are grateful to George McGill for generously making this unpublished data available to us! From the data set, George made of list of all his craters arranged by size. I’ve truncated his list so that it contains only those craters with diameters greater than 2 km, and broken it into two groups. Group #1 contains only 5 craters, which happen to be the largest ones in the quadrangles studied. Group #2 contains the remaining craters with diameters larger than 2 km. From these 68 samples, you can create a cumulative sizefrequency distribution plot, estimate the age of the features he’s interested in, and determine which group is older. To do this, plot the following data (please use a spreadsheet; I’ll show you how if you don’t know!): 
1.  Sort the data into descending order by diameter, with the largest one first. 
2.  For each crater, calculate the cumulative number of craters per unit area with larger or equal diameters. For example, for the largest crater, this value will be 1, because the number of craters with larger of equal diameters is 1. For the next largest crater, the y value will be 2, etc. 
3. 
Normalize the cumulative number of craters to an area of 1 million km^{2}. George’s area measured 75,750 km^{2}. To determine the normalization factor, use this equation:
Now, multiply this factor times each of your values in the cumulative number of craters column. 
4.  Plot the data on an xy plot, then rightclick on each axis and covert it to log units, so you have the log of crater diameter as the x axis, and the log of the cumulative number of craters as the y axis. 
5.  Determine the y value of the line where it crosses N5 and N16, and then match this value to Tanaka’s time scale to determine when in Mars’ geologic time the units were formed. 
6.  Decide which group of samples, the small craters or the large ones, is older! 
Group #1 crater diameters  
58.00  
43.00  
45.00  
79.00  
103.00 
Group #2 crater diameters  
2.10  
4.55  
5.20  
7.10  
4.75  
2.75  
4.70  
6.15  
2.20  
2.05  
3.00  
2.02  
3.60  
3.35  
7.05  
2.95  
6.35  
4.20  
2.65  
2.60  
5.45  
2.50  
5.05  
8.25  
10.70  
2.85  
2.00  
3.50  
3.25  
7.70  
2.40  
7.35  
3.25  
7.00  
2.80  
4.85  
11.50  
3.30  
3.80  
8.95  
6.30  
2.90  
4.25  
9.60  
2.55  
5.85  
26.00  
7.75  
3.65  
18.00  
2.70  
2.30  
9.65  
4.05  
3.20  
13.65  
4.15  
9.95  
3.95  
2.45  
2.35  
3.15 
References Used:
Crater Analysis Techniques Working Group (1978) Standard Techniques for
Presentation and Analysis of Crater SizeFrequency Data. NASA Technical
Memorandum 79730, 20 pp. McGill, G.E. (2000) Crustal history of northcentral Arabia Terra, Mars. Tanaka, K.L. (1986) The stratigraphy of Mars. Proceedings of the 17^{th} Lunar and Planetary Science Conf., Part 1. Journal of Geophysical Research, 91, supplement, E139E158. 
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Last updated on 14 October, 2010. 