BRIEF REVIEW OF THE SHAPE OF THE CRATER SIZE DISTRIBUTION:

HISTORY OF POWER-LAW AND POLYNOMIAL CURVE-FITTING

Our basic approach is to determine the production function, or size distribution, of freshly produced craters, primarily from the well-preserved lunar mare crater populations (but augmented and confirmed by asteorid data and other crater data, as in the work of Neukum and Ivanov 1994). In the earliest research on the lunar crater diameter distributions, statistics were available only for the largest lunar craters resolvable from Earth, i.e., D 2-4 km. The best statistics were for the range 4 km < D < 100 km, which give relatively straight lines on plots of log N (no. of craters/km²) vs. log D. The earliest numerical study appears to have been that of Young (1940) who gave a value for the slope of this line as -2.5 (p. 315). Similarly, Brown (1960) analyzed the earliest crude data on log-log asteroid and meteorite size distribution, and fit straight lines to his data sets.

Hartmann (1964) used newly measured catalogs of lunar crater diameters to analyze the significance of the size distribution. Citing the work of Young (1940) and Brown (1960), he fit a slope of -2.1 to the log N vs. log D plot. (Hartmann has consistently used log increment plots where N - number of craters in diameter bins of constant D log D, usually D, and has pointed out that mathematically this plot gives the same slope as a plot of the cumulative number of craters larger than D.) Hartmann (1964) also pointed out that Brown's asteroid/meteorite size distribution curve could be used to predict a lunar crater diameter distribution with cumulative or log incremental slope of -2.4, in fair agreement with the lunar data. Hartmann used this agreement to make an early argument that the lunar craters were created by asteroid/meteorite impact. BESIDE THIS PARAGRAPH INSERT FIGURE N

Note that the straight lines adoped in this very early crater and asteroid literature (Young 1940, Hawkins 1960, Hartmann 1964) were, in effect, fitting power law relationships to the crater and asteroid diameter distributions. A power law size distribution has the form

N = k D^-b

where N is either cumulative number of craters of diameter > D, or Hartmann's log incremental number of craters in a logarithmic diameter bin (such as 1-1.4 km, 1.4-2 km, 2-2.8 km, etc.), k is a constant, and b is the power law exponent. The plot of log N (either cumulative or log-incremental) vs log D gives a straight line of slope -b.

The point of this review is to remind the reader that the original choice to put straight lines in log N-log D plots -- i.e., to fit power law curves to the data -- was essentially a way to create a convenient way of describing the available data, not a physical argument that the data fit a power law better than some other curve. Note that all theses papers were published before the discovery of the steep (so-called "secondary") branch in the size distribution (by the Ranger VII lunar impact probe in the summer of 1964), or the discovery of crater populations outside the Earth-moon system (by the Mariner IV Mars fly-by probe in the summer of 1965). Thus, at the time this convention was begun, the sharp upward turn into a steep branch at D 1.4 km was not known.

When the Ranger lunar probe revealed the universal steep branch on various terrains in the mid 1960s, the tradition of power law curve fitting was simply extended. Shoemaker (19??) counted secondary impact craters around large lunar primary craters, such as ___________, and found a slope of _______ for the cumulative or log-differential power law relation. Shoemaker (19??) as well as Kuiper et al. (19??) and Hartmann (19??) found a similar slope for the lunar steep branch (250 m D 1.4 km), and Shoemaker inferred that the steep branch was indeed caused by lunar secondary impact ejecta, falling back to the moon. This was widely accepted at the time, and it was pointed out that on the log N vs. log D plots, the sum of the two power laws (~ -2 slope for "primaries" at D > 1.4 km and ~ -3.5 slope for "secondaries" at D < 1.4 km) would give a curve with two nearly straight segments intersecting near 1.4 km, as was consistent with the data. BESIDE THIS PARAGRAPH, INSERT FIGURE A (FROM ISOCHRON SITE)

In the 1970s, Strom and coworkers (e.g., _____________________) accumulated enough additional statistics from sources such as Callisto, Mercury and Mars to argue for a third "turned-down"or steep segment at D _____ km, though the statistics of craters in this branch were poor and graded into large concentric ring basins, whose effective diameters are difficult to define.

The use of power laws in cratering and asteroid fragmentation work was given an additional boost when Hartmann (1969) showed experimentally and emprically that power law segments gave good fits to size distributions of fragments in systems such as fractured basalt blocks, gravel in a dry streambed, rocks blown out of volcanic craters, and lunar rocks gave good fits to power laws, often over two to three orders of magnitude in diameter. This work and Hawkins (1960), as well as an earlier literature on mineral grinding cited by Hawkins (Gaudin, 1944) also suggested that the b value was lower (shallower slope on the plots) when fragments were produced from bodies broken in low energy-density events, and higher (steeper slope) for systems produced by repeated grinding or high energy density. The quantitative match between the steep slope of debris from high energy cratering events and secondary debris blown out of lunar craters in supported Shoemaker's (____) conclusion that his "secondary" steep branch of lunar craters (D < ~1.4 km) was produced by rocks blown out of craters, which Shoemaker assumed were the lunar craters themselves. However, more recent work of Neukum and Ivanov (1994) and Hartmann (1999) suggests that many or most of the projectiles causing lunar secondary craters may be debris from craters on asteroids in the asteroid belt, rather than entirely debris ejected from lunar craters. BESIDE THIS PARAGRAPH INSERT FIGURE O.

Also, more recent laboratory experimental work suggests that the fragments from collisional breakup and cratering events can have more complex size distributions than single power law segments, being fitted by several power law segments or, alternatively, by a more complex curve, especially if the diameter distribution is traced over several orders of magnitude (Fujiwara???, Ryan et al.). NEED TO FIND REFERENCES AND FIGURE HERE.

Therefore, in the 1980s, Neukum and coworkers took two new steps. Given the three proposed "segments" of the production function, Neukum abandoned the fitting of straight-line power law segment on the log-log plots, and simply fitted a polynomial curve to the data, giving a somewhat S-shaped curve that fit the data. In some ways this procedure is more scientifically neutral or defensible than using power laws, since a polynomial can fit any curve, while there is no a priori guarantee that a power law gives the best fit. In this way, Neukum compared branches of the size distribution from different sources, and constructed a universal polynomial "production function" size distribution that appeared to represent the initial or "input" size distribution of craters on a wide variety of surfaces, from the moon to asteroids and other bodies.

In early Mars Global Surveyor image analysis, Hartmann (1999, 2000) compared his power-law fit to the Neukum/Ivanov polynomial fit and concluded that, at a general first order level, both fits were reasonably consistent. The slopes in the steep segment are nearly identical, though the curvature in the shallow branch and transition region is somewhat different. In the 2000 paper, Hartmann attempted a synthesis by using a logarithmic average of his power law curve (made of power law segments) and the Neukum/Ivanov polynomial (a revised version kindly provided by Neukum) to study Martian impact gardening. However, he noted with some surprise that his own lunar mare data seemed to fit the power law straight line segments better than they fit the polynomial curve (1999, Figure ___). BESIDE THIS PARAGRAPH, INSERT FIGURE FROM NEUKUM IVANOV PAPER, OR MAYBE ONE OF OUR PLOTS SHOWING THE NEUKUM IVANOV CURVE. MAYBE A FIGURE FROM HARTMANN 1999 SHOWS THIS BEST. (MAPS ISOCHRON PAPER)

Neukum has refined the polynomial curve, based on continued accumulation of crater count data from asteroids and other sources. The latest version, May, 2000, is somewhat flatter and closer to the power law segments than earlier versions. Neukum, Ivanov, and Hartmann are also working together to review the crater size distributions, and several papers will be published as review chapters in a planned work on Mars Chronology, to be published in Space Science Reviews and in book form by the International Space Science Institute in Bern, Switzerland, in 2001. In summary, the various styles of treating the crater populations are essentially consistent and are being synthesized.

The overall goal of that work, and the work reported on this web site, is to test and demonstrate the degree of consistency between the hitherto independent approaches, and use this body of knowledge to derive isochrons and measure the approximate ages of Martian surface features, and make a comparison with Martian meteorite data.