Lunar Production Function
The Moon is the main test site to study the impact crater chronology. Active volcanic activity ended in most areas >3 Ga ago (with the exception of small possibly younger basaltic lava flows; Hiesinger et al. 2000). During the last 3 Ga the lunar surface has been modified by newly formed impact craters. The moon was relatively well studied during the Apollo and Luna space missions. Lunar rock samples returned to Earth provide a unique opportunity to correlate the number of craters, accumulated in the vicinity of sampling sites, and isotopic ages of rocks (Neukum et al. 2001a; Stoffler and Ryder 2001). Hence, one can estimate the rate of cratering on the moon; i.e., the number of impact craters accumulated on a unit area per unit of time.
Figure 1 demonstrates the correlation between number of counted craters with diameters >1 km per unit area N(1) and the measured age of lunar rocks. The main features here are: (1) approximately linear growth of N(1) with surface age up to ~3 Ga, and (2) the exponentially increased cratering rate for older surfaces. This period of recorded high impact rate on the young moon is often called the late heavy bombardment (LHB) period. (See Hartmann et al., 2002, for the history of this term and hypothesis of the LHB nature.) Neukum (1983) proposed to present the cratering rate decay in time in the form of the equation for the number of impact craters accumulated at the surface of the age T. Expressing the number of craters through the point N(D = 1 km) one can write this Equation as:
where N(1) is the crossing point of the N(D) curve with the D = 1 km axis, normalized per area of crater counting (in km2), and T is the age of surface in billions of years (Ga).
The modern (T< 3 Ga) constant flux is believed to be mainly the flux of Mooncrossers, derived from the Main Belt of asteroids. The nature of the enhanced ancient (LHB) flux (T> 3.3 Ga) is still under discussion. Most probable hypotheses include (Hartmann 2002; Gomes et al. 2005):
• The remnants ("leftovers") of planetesimals not incorporated into planets before their differentiation and crust formation (now of concern; Bottke et al. 2006)
• The flux of comets from the formation zone of Neptune and Uranus
• The flux of asteroids from the ancient asteroid belt due to the migration of the orbits of Jupiter and Saturn ~0.5 Ga after the Solar System formed.
As illustrated in the following, observed impact craters allow investigators to conclude that LHB projectiles have an SFD similar to the modern asteroid belt. Hence, one can assume that LHB bodies have experienced a similar collision evolution as asteroids.
To study SFD of planetary impact craters quantitatively, one can introduce the socalled production function (PF). The PF is the SFD of craters that would be observed at the once completely renovated (erased) planetary surface before the area was saturated with impact craters. (Saturation or equilibrium area density impact craters are observed in old planetary surfaces, when each newly formed crater destroys one or several previously formed craters. On lunar mares the equilibrium state is observed for craters <200300 m in diameter.) The PF in the whole range of possible impact crater diameters rarely may be observed because of the different geologic histories of different planetary areas. However, the PF may be restored piece by piece by combining data for large craters in the oldest areas and smaller craters in the younger areas. Here the PF is illustrated using two mostly detailed approaches, by W. Hartmann and G. Neukum (see Neukum et al. 2001a for a review).
In the early days of lunar crater counting (Opik 1960) researchers used telescopic observations of the near side of the moon. Observations allowed them to count craters >510 km in diameter. It was found that craters have an SFD close to the power law N(> D) ~ D2. Later, spacecraft images of the lunar surface enabled researchers to count impact craters as small as 10 m in diameter. After a few years of discussions about the best applicable power law, after more data had been accumulated, it was recognized that the lunar crater PF is not a simple power law, where in all craters the diameter range would have SFD presented in the form N(> D) ~ Dm. Thorough analysis of telescopic data (Chapman and Haefner 1967) has shown that the local slope of the SFD curve (the derivative dlogN/dlogD) varies with the crater size. Later Shoemaker (1965) found from highresolution spacecraft images that small craters have steeper SFD than large craters, N(> D) ~ D29. Different researchers have used different approximations to take into account the "wavy" nature of the impact crater SFD, undulating around the general trend; in any surface small craters are less numerous than larger craters.
Observed impact crater SFDs on various planets are similar in their main details to a crater population produced by the modern asteroids leaving the Main Belt. Just as the early Solar System could have different small body populations, it can be proved that these ancient populations also experienced some collision evolution, but it cannot be proved that the Main Belt asteroids were the main populations during the LHB period.
Separated impact crater counts in various size ranges may be generalized as a universal relation, the PF. Production function is defined as the impact crater SFD accumulated on an ideal planetary surface once obliterated and preserved all impact craters formed after the starting obliteration event (e.g., lava flow deposition, the floor of a large crater, and its continuous ejecta zone). The real cratering records may be quite different from the ideal PF due to a set of surface processes, individual for each terrestrial planetary body.
There is no general agreement yet about the detailed structure of the production function on terrestrial planets. However, all proposed PFs are similar in their main features. The following illustrates the issue of PF construction with the most widely cited PFs proposed by W. Hartmann and G. Neukum (see the review by Neukum et al. 2001a).
2.1 Hartmann Production Function (HPF)
HPF has been designated as a table of N(D) data, selected by W. Hartmann from many sources to present N(D) dependence for an average lunar mare surface. Here the resurfacing event, erasing the surface, is assumed to be the mare basalt emplacement. It was believed that the time period of basalt emplacement was short in comparison with the whole geologic history of the moon (as measured from returned lunar basalt samples; Stoffler and Ryder 2001).
Recent papers about photogeologic analysis of nonsampled lunar basalt flows (Hiesinger et al. 2000, 2003) estimate the relative area of younger and older flows of mare basalts. Figure 2 illustrates the age distribution model of mare basalt (Hiesinger et al. 2003). One can see that visible volcanic activity in some areas could have ended as recently as 1.5 Ga ago. However, the median age of basalt flows is about 3.4 Ga. About 60% of visible mare surfaces were covered by basalts; the model age was ~2.8 to ~3.5 Ga. According to the modeled accumulation rate of impact craters (Fig. 1) one can assume that looking back in time during this basalt emplacement period, the number of craters increased from a factor 0.65 to 1.35 relative to the number of accumulated craters with a median surface age of 3.4 Ga. Hence, the accuracy (synchrony) of HPF values, combined areas with a slightly different age, may be estimated as a factor of 1.35.
Original HPF is presented in a socalled incremental form. It represents the number of counted craters normalized to 1 km2, NH, with diameters in the diameter bin interval DL < D < DR, where DL and DR are left and right bin boundaries, and Dr/Dl = 21/2. The right boundary of one interval is equal to the left boundary of the next diameter bin (going from smaller to larger diameters). The data can be plotted
versus the geometric average of the bin boundary values Dav = (DL x DR)1/2. Graphic representation of the latest version of the HPF (Hartmann 2005) is presented in Fig. 3a.
In Fig. 3a one can see that the number of craters per bin varies 8 to 10 orders of magnitude for the whole range of crater diameters. The cratercounting community has proposed a standard way to present the data in a relative form (Rplot), where the original data are normalized to a power law function ~ D3. It helps to decrease the vertical range of plotted values. The standard definition of R is as follows
Here the differential number of craters (AN/AD) is multiplied by a "streamline" function, Dav. Hence, the Rrepresentation of the SFD presents the deviation of the differential SFD of the simple unique power law. If the differential number of craters, AN/AD, decreases with the crater diameter as D3, it is represented in the Rplot by a horizontal line of a constant Rvalue. SFD, steeper than D3, is a decreasing function in the Rplot, and SFD, less steep than D3, is an increasing function in the Rplot. This is demonstrated in Fig. 3b for the HPF. In the following, D « 300 m, the HPF, describes an equilibrium state of small craters on mare surfaces, the interval 300 m < D < 1.4 km HPF in the Rplot is a decreasing function, and in the range 1.4 km < D < 64 km the Rplot for HPF is an increasing function of D.
Figure 3. Hartmann's Production Function (HPF) in the incremental form (a) and Rplot (b) is shown as tabulated data points (squares). Thick lines corresponds to the piecewise power law representation (Equation 3), dashed curve in (b) presents here for comparison Neukum's Production Function, discussed later in this chapter. Dashed line 1 in (b) corresponds to the "empirical saturation" SFD, proposed by Hartmann (1984). Reprinted from Neukum et al. (2001a) with permission from Springer
Figure 3. Hartmann's Production Function (HPF) in the incremental form (a) and Rplot (b) is shown as tabulated data points (squares). Thick lines corresponds to the piecewise power law representation (Equation 3), dashed curve in (b) presents here for comparison Neukum's Production Function, discussed later in this chapter. Dashed line 1 in (b) corresponds to the "empirical saturation" SFD, proposed by Hartmann (1984). Reprinted from Neukum et al. (2001a) with permission from Springer
To work logically with HPF, Hartmann (2005; see also Ivanov et al. 2001) proposes piecewise exponential relations, which use 10base logarithms:
log = 2.616  3.82 log DL, 0.3km < D < 1.41km (3)
log = 2.920  1.80 log Dl, 1.41km < D < 64km (4)
For D < 0.3 km lunar mare craters are in equilibrium, and the HPF should be assumed from data for younger surfaces, where small craters have not reached yet saturated area density (Hartmann 2005). Graphic representation of Equations (3)(5) is shown in Fig. 3.
2.2 Neukums' Production Function (NPF)
Following the idea of Chapman and Haefner (1967) about the variation of the local SFD slope depending of the crater diameter range, G. Neukum (1983; see also
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"Old" N(D) (Neukum 1983) 
"New" N(D) (Neukum et al 2001a) 
Coefficient "sensibility"' 
R(Dp) (this work)" 
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