ISOCHRONS FOR MARTIAN CRATER POPULATIONS OF VARIOUS AGES
William K. Hartmann Page design: Daniel C. Berman |
The Isochron System: Derivation of 2004 Iteration
Improvements can still be made by using better estimates of the ratio R_{bolide} and the gravity and impact velocity scaling relations, and by adding the effects of loss of small meteoroids in the Martian atmosphere. To understand our approach to these improvements, think (for a moment) of the size distribution as constructed of power law segments (giving straight lines in the log N vs. log D plots used here). Virtually all work before MGS dealt with only one of these segments, the shallow or so-called primary branch, involving craters in the diameter range roughly 2 km < D < 64 km, where good statistics were available at that time. How do we use the lunar data from this diameter range to estimate the number of craters formed in that size range in a given time period on Mars? Imagine this diameter segment plotted for the number of craters formed in lunar maria in the last 3.5 Ga. To get the number of craters formed on Mars in the same period, Hartmann (1999) used the Mars/Moon cratering rate correction factor R_{crater} to shift this segment vertically, because of a higher estimated cratering rate on Mars. In addition, impact velocity and scaling corrections altered the diameter of a crater produced by a given meteoroid, hence sliding the curve horizontally (to the left, to smaller sizes, because of lower Mars impact velocity and higher gravity). Taking into account the slope of the single power law segment, these two shifts were combined into an effective single vertical shift. Thus, that work assumed there was a single effective R_{crater} ratio that shifted the curve vertically by a fixed amount along the whole diameter segment, and Hartmann (1999) applied it to the whole curve (11m < D < 100,000 m), deriving a single effective R_{crater} value and shifting the whole curve vertically by that amount to convert from a lunar to a Martian isochron.
The modern extension of the Martian crater diameter distribution to both a shallow and steep branch makes this approach too simple, however. The problem, as seen in Fig. 3, is that the horizontal shift moves the intersection of the shallow branch and steep branch to the left, and has the effect of shifting the steep branch by a different net vertical amount than the shallow branch, so that the various power law branches (or parts of a polynomial fit in the style of Neukum) cannot be treated by a single vertical shift, nor does a single R_{crater} value apply to the curve. Thus, the simple R_{crater} concept loses its utility. In other words, the conversion from the lunar to Martian production functions is D dependent. These concepts were already dealt with in the treatment by Neukum and Ivanov (1994), who used scaling laws to examine the shape of the production function on other planets, and by Ivanov (2001), who discussed the Martian application. These concepts were also applied by Hartmann and Neukum (2001), and Hartmann's 2002 iteration (unpublished) to derive isochrons from Neukum's best estimate of the production function curve, and then independently from the Hartmann's best estimate of production function, comparing results from the two approaches. Here, we begin our derivation of Martian isochrons by applying thee general scaling principles to the power law form of the crater size distribution, and also improve the treatment by Hartmann (1999) by using the latest estimate of R_{bolide} as a fundamental conversion parameter instead of R_{crater}. We will present the stepwise derivation, starting with these lunar data, in Table 2, also discussed below.
[Figure 3] Schematic diagram showing the conversion of the production function measured on lunar maria to Martian conditions. The whole curve is shifted up to take into account the higher bolide impact rate on Mars. Then the curve is shifted left, to take into account the smaller crater size formed by each bolide, due to lower impact velocity and higher gravity on Mars. The net result is that the shallow or "primary" branch has a different net apparent vertical shift, or change in crater production rate, than the steep or "secondary" branch.
[Table 2]
Step (a) in the above list is to determine the lunar mare crater diameter distribution. In the past I have primarily published only graphs summarizing the data. In this paper I use in addition a tabular summary, which allows the reader to utilize the original data. A summary of the direct counts summarizing of several lunar maria is given in column 2 of Table 2. These data are collected from a range of sources over many years. The include extremely detailed, multi-year cataloging of larger lunar craters (D > 4 km) by Arthur et al. (1963, 1965a, 1965b, 1966); counts of smaller craters are added by from my sampling of different maria (esp. Tranquillitatis and Cognitum, but also including parts of Imbrium and Fecunditatis), from Ranger, Surveyor, and Apollo data. Figure 4 shows a plot of these data, showing that at D / 250 m, they fit the power laws very well. (See further discussion in subsection A below.)
[Figure 4] Comparison of crater count data in lunar maria to (a) the power law fits of Hartmann and (b) the polynomial fits of Neukum. Data come from counts in all maria in 1960's catalogs of Arthur et al., and from counts in various individual maria by the author. Mare craters go into saturation at D . 300 m and no information is available from the mare counts on shape of the production function size at the smaller sizes (see text).
Step (b) in our formulation is to improve the Mars/Moon impact ratio, R_{bolide}. A recent review of asteroid dynamics by Bottke (private communication, 2002) suggested a value of R_{bolide} ~ 3.15 (revised upward from his 2001 value of R_{bolide} ~ 2.76), and an independent review of observed Amor and Apollo asteroid statistics by Ivanov (2001) suggests R_{bolide} ~ 2.0. Bottke's value includes a variety of populations, emphasizing asteroid dynamics but also taking into account estimates of comet populations and observations of Mars crossers. Ivanov's is more empirical, based on observations of existing Mars crossers and Moon impactors of whatever origin. As a standard for our isochron diagram we adopt R_{bolide} for asteroids ~ 2.6 0.7. The uncertainty is a conservative estimate based on remaining uncertainties in the asteroidal and cometary fluxes, and is consistent with fluctuations in recent best estimates from the different authors. The uncertainty is important because it translates directly into a proportionate uncertainty in age, i.e., a factor of ~ 1.4.
Going from step (b) to step (d), the first task in correcting the lunar production function (craters/km^{2}-y) to Mars is to recognize that each crater diameter bin corresponds to a particular bolide size and thus to raise (or lower) this curve by a factor R_{bolide} to correct for the increased (or decreased) number of bolides hitting Mars, as shown in the schematic diagram, Fig. 3. In Table 2, this step is combined with the impact velocity and gravity effects from step (c), as follows. We recognize that because, on average, each asteroidal or cometary bolide hits Mars at lower velocity than the Moon, and because Mars' gravity is higher, the crater produced on Mars is smaller than the crater produced by the same size bolide hitting the Moon. These effects are treated numerically as follows.
Impact velocity effect.Crater diameter D goes approximately as impact kinetic energy E^{1/3.3} as reviewed by Baldwin (1963). Hartmann (1977) applied this, along with mean Mars and Moon impact velocities of 10 km/s and 14 km/s, respectively, to estimate that, due to this effect, a given bolide makes a crater that is
D_{Mars}/D_{Moon} = [10/14]^{2/3.3} = 0.815.
This Baldwin-based result was then used by Hartmann (1999). However, the scaling laws given by Schmidt and Housen (1987) show D scaling as E^{0.43}, so that
D_{Mars/}D_{Moon} = [10/14]^{0.43} = 0.865,
which is used here.
Gravity effect. D goes approximately as gravity g^{-0.2} as reviewed by Hartmann (1977), who applied this value and estimated that, due to this effect, a given bolide makes a crater that is
D_{Mars}/D_{Moon} = [373/162]^{-0.2} = 0.847.
This was used by Hartmann (1999). However, the updated scaling laws given by Schmidt and Housen (1987) show D scaling as g^{-0.17,}, so that
D_{Mars}/D_{Moon} = [373/162]^{-0.2} = 0.868,
which is used here.
Combining these two effects, we find that
D_{Mars}/D_{Moon} = 0.865 / 0.868 = 0.751,
so that each bolide hitting Mars makes a crater 0.751 as big as it would have if it had had an orbital history that led to a collision with the Moon (Hartmann's 1999 figure was 0.69). This effect is shown in the schematic diagram, Fig. 3.
The upshot is that if we know the crater size distribution which has accumulated on the Moon for a given period of time such as the craters formed on lunar mare surfaces in its average mare lifetime of about 3.5 Ga then we can derive the crater size distribution that would have been created on Mars during the same time by first shifting the lunar curve upward by the factor R_{bolide}, and then shifting it to smaller diameter by the factor D_{Mars}/D_{Moon} ~ 0.751.
Because power laws produce linear segments on plots of log N vs. log D, it is conceptually easy to carry out this correction for each power law segment. The leftward shift to smaller diameter of any given linear power law segment is equivalent to a constant vertical shift along the whole length of the line by
d log N = -b d log D,
where N = no. craters/km^{2} in a log /2 diameter bin, D = diameter in km, and b is the slope of the power law segment. For example, with a -2 slope, a shift of the curve to smaller diameter by a factor by one log unit causes a decrease in apparent number by two log units. A shift of the curve to half the size causes a decline in apparent number by a factor 4.
Thus, if we have a value for the shift in diameter from D_{Moon} to D_{Mars}, we can derive the corresponding vertical displacement (on the log N axis) of that whole power law segment of the production function on the log N - log D plot, as indicated in Fig. 3. Adopting R_{bolide} = 2.6 and the leftward shift in D by .751, we have d log N = log (2.6) - b log (0.751), or
d log N = 0.4150 + 0.1244 b
In the power law fits to the lunar curve, three branches have been identified (Hartmann, 1999): the traditional "shallow branch" (or "primaries") measured from Earth-based photos, with slope -1.80, at 1.4 km < D < 64 km; the "steep branch" (or "secondaries," as identified by Shoemaker, 1965) with slope -3.82 on the cumulative or log-differential plot, at D < 1.4 km; and the "turned-down branch" with slope -2.2 at D > 64 km.
We now give the equation defining each branch for the Moon and Mars. We will start with the shallow branch and the turned-down branch, which are easiest to derive, and then discuss how we fit the steep branch onto the shallow branch.
A. Shallow branch. For the lunar maria, the Basaltic Volcanism Study Project (Hartmann et al., 1981) derived a power law fit to available data:
log N_{Moon, shallow, mare} = -1.80 log D - 2.920.
Neukum (1983) used a different approach and fit a polynomial function over a wide range of D, which produced more curvature in the shallow branch itself. Neukum continued to use a polynomial fit, deriving a relation that fit craters in lunar mare surfaces, lunar young crater interiors, and asteroids. Neukum and Ivanov (1994) used principles similar to those in this paper to convert this Neukum "universal" polynomial function to Mars, and Neukum et al. (2001) and Hartmann and Neukum (2001) discussed in detail the further application of polynomial and power law functions to Mars.
In the present work, we critically examined the fit of the data for the lunar mare shallow branch to the polynomial and power law functions proposed earlier. Figure 4a shows a comparison of observed lunar mare crater diameters data from the Arthur catalog of the 1960s (Arthur et al., 1963, 1965a, 1965b, 1969) and from my own later counts on Maria Cognitum, Tranquillitatis, and to a lesser extent Imbrium, Oceanus Procellarum, and other maria, to the power laws used here. Figure 4b shows a comparison of the same data to the Neukum universal polynomial curve. The Arthur et al. and Hartmann data show less curvature than the Neukum function in the multi-km range, and fit a power law (straight line in this format) in this range. To develop the Martian curve, we thus adopt the power law with a slope of -1.80 for this diameter region to give a mean lunar mare function for our step a.
Using the slope -1.80, the combination of shift upward by 2.6 and leftward by 0.751 in D corresponds (by the above equation) to a net upward shift in N by Δ log N_{shallow} = +0.1911. Thus we have
log N_{Mars, shallow, 3.5 Ga} = -1.80 log D - 2.729
B. Turned-down branch. On the Moon this branch begins at about 64 km, but the shift to smaller D on Mars by 0.751 gives a beginning diameter of D ~ 48.1 km. With an assumed slope of -2.2 for this branch, the intersection with the above law at D = 48.1 km gives:
log N_{Mars turned-down,} 3.5 Ga = -2.2 log D - 2.056
C. Steep branch. In the 1999 iteration of the isochrons, Hartmann (1999) used simply a power law with a slope of -3.82, a slope which had been measured for the Moon (Hartmann and Gaskell, 1997, p. 113) an approach similar to that used above for the shallow branch. This was applied at D #1.414 km, and the 1999 equation was
log N_{Moon, steep,, mare} = -3.82 log D - 2.616 (D < 1.414 km).
However, the measurable data for the lunar mare steep branch apply primarily only down to D ~ 250 m, because the lunar maria are saturated with craters below that size (Hartmann, 1984; Hartmann and Gaskell, 1997). Thus the shape of the production function curve below 250 m is not well determined from the lunar mare data and deserves more attention. Also, the 1999 lunar equation gave values systematically lower than lunar mare counts in the specific region 250 m-1 km (by a factor around 1.9), though the fit is better when including the D bins just outside that region). Close examination of the fit in that region leads us to match the two curves at D = 2.0 km rather than 1.414 km, and adopt a new equation for the lunar steep branch in the range 250 m < D < 2.0 km,
log N_{Moon, steep, mare} = -3.82 log D -2.312 (Valid at 250 m < D < 2.0 km)
This curve is shown in Table 2, column 3.
The problem now is how to fit the steep branch curve below 250 m to Mars (down to the MGS crater resolution limit of 11 m). In the 1999 work, we simply extrapolated the -3.82 power law to these small diameters, as is done in Table 2, column 4. In many Martian areas, however, we have found the observed diameter distribution curving slightly toward shallower slopes at small diameters, i.e., a deficiency at small crater sizes, which we attributed to obliteration losses among small craters by erosion, deposition, lava flows, etc. (Hartmann and Berman, 2000; Berman and Hartmann, 2002). Neukum, however, has argued that he can trace the steep branch production curve downward into the decameter range by counting on interiors of young craters and fitting those data to segments defined by older surfaces. By fitting the entire diameter range of data to a single polynomial, Neukum thus found a downward curvature toward small diameters below 250 m. The Martian data suggest that including Neukum's curvature in the production function gives a better description of the isochrons than a simple power law extrapolation at D < 250 m.
[To put this in historical context, the power laws were utilized by Hartmann in the 1960s as a convenient approximation, partly because power laws had been introduced to lunar crater discussions by Young (1940) when only the shallow branch was known, and partly because power laws had been used to fit asteroid and meteorite size distributions (Hawkins, 1960; Brown, 1960), and also because experiments gave approximate power law size distributions, as discussed by Hartmann (1969). No claim was intended that a power law is necessarily or a priori better than another fit. In any case, it is always risky to assume that downward curvature is due to the production function, because preferential losses of small craters are expected due to erosion. Nonetheless, considering all these influences, Hartmann et al. (2001) and the "2002" iteration (Berman and Hartmann, 2002) attempted to utilize aspects of the Neukum curvature in the steep branch. Here again, I have chosen the Neukum data, as converted to Mars by Neukum and Ivanov, for an improved shape of steep branch below D = 250 m.]
In the lunar maria, the intersection of the steep and shallow branch was at 1.4 to 2.0 km, but this should move to the left by a factor 0.751 on Mars, giving an intersection at D = 1.05 to 1.50 km. To fit the Neukum-Ivanov curve onto the Hartmann curve we could have forced an intercept exactly at D = 1.05 or 1.50 km, but instead we adopt the measured slope of -3.82 at 250 m < D < 1.4 km (Hartmann and Gaskell, 1997, p. 113), and give weight to all the data points from my own lunar crater counts (D > 250 m). Thus we derive
N_{Mars, steep, 3.5 Ga} = -3.82 log D - 2.372 (for the range 250 m < D < 1.4 km),
We fit the Neukum-Ivanov Mars curve onto this Hartmann predicted Mars "stem" curve, giving a steep branch that fits our observed lunar diameter distribution down to 250 m, but preserves the curvature of the Neukum "universal" production function diameter distribution at D < 250 m. The resulting isochron shape is listed in Table 2, column 5.
Adjustment of the steep branch for losses of small meteoroids in the Martian atmosphere. Step (e) is a final improvement, made for the first time in this iteration, is made by allowing for breakup and loss of small meteoroids in the Martian atmosphere. They are thus considered lost to the Martian cratering record. As Popova et al. (2003) show, the breakup is dependent on meteoroid taxonomic type, and is more important for icy comets and weak stones than for strong stones and irons. Small fragments are reduced to subsonic speeds and do not create "normal" hypervelocity impact explosion craters. Using estimated data for numbers of different meteoroid types (3% irons with very high strength, 29% ordinary chondrites with strength 25 bars, 33% weak stones like carbonaceous chondrites with strength 5 bars, and 35% weak icy bodies with strength ~ 1 bar), Popova et al. (2003) graphed a corrected size distribution of craters expected under the current Martian atmosphere. Popova (private communication, 2003) used this formalism to gave a detailed calculation for these effects as a function of D, used here. Table 2, Column 6, gives Popova's calculated data for losses of small craters as a result of meteoroid loss in the Martian atmosphere, and column 7 [= (column 5) / (column 6)] gives the final estimated production function of craters on a Martian surface of age 3.5 Ga.
As can be seen in Fig. 5, the main variation between the 1999 and later models occurs at D < 250 m, and becomes important for surfaces younger than a few Ga. Older surfaces reach saturation at D < 250 m, masking these differences. Figure 2 emphasizes that the main uncertainties are for small craters on young surfaces. For surfaces younger than a few Ga, change from -3.82 power law in the 1999 steep branch to the Neukum curvature in the 2002 and 2004 iterations was the major change, increasing inferred ages by an order of magnitude for crater counts at the smallest scale, below about 60 m. Hearteningly, the change from the 2002 to 2004 iteration is minimal, being within the typical uncertainties of the crater counts at small sizes. The correction to the 2002 Hartmann curve shape is only ~ 10% at D = 31-63 m, but grows to a sizeable factor 2 difference in the smallest bin at which we count craters (11 m < D < 16 m). This illustrates the need for counts over a wide D range whenever possible. As shown in Fig. 5, the Popova effect begins to dominate below that, since we predict that hypervelocity cratering will cease below D ~ 0.1 to several meters (being from irons at the smallest sizes). In order to dramatize the loss of small meteoroids, Fig. 5 is the only figure in this paper to extend the diameter distribution down to D = 4 m. The most dramatic effects on crater loss occur below D = 11 m, i.e., below the normal cutoff for our crater counts, based on MGS resolution.
[Figure 5] Comparison of isochron shapes from current and previous iterations, showing isochrons for 3 Ga and 1 Ma. Differences between the iterations are less than or comparable to the scatter in crater counts for typical stratigraphic units, except for young surfaces at small diameters, below D ~ 125 m. The difference from 1999 to 2002 iteration is due mainly to introduction of the "Neukum curvature" in the steep branch at small sizes, and the smaller difference from 2002 to 2004 is due mainly to the "Popova effect" loss of small meteoroids in the Martian atmosphere, affecting craters primarily at D . 31m.
The above approach (Table 2, column 7) gives an isochron for Mars for any surface with age equal to the effective average age of lunar maria. In Hartmann and Neukum (2001) we took this average age to be 3.4 Ga. I adopt here a mean age of 3.5 Ga for the surfaces I have counted, as discussed in step (a) above. [Note: The Neukum time dependence (see equation below), shows ~ 14% difference in total crater density accumulated since 3.5 Ga as opposed to 3.4 Ga, a small change compared to the uncertainty in R.)
Step (f) is to convert the 3.5 Ga isochron to isochrons for other ages, using the dependence of cratering as a function of time, as measured in the Earth/Moon system. This time dependence has been interpreted as a product of scattering of the earliest planetesimals among the planets throughout the inner solar system, and since planetesimals are scattered by planetary encounters on timescales ~ 20 Ma, Mars is unlikely to have had a radically different time dependence than that measured in the Earth/Moon system. Hartmann (1965a, 1965b) showed that the Canadian shield cratering implied an age of the order 3-4 Ga for lunar maria, and that this in turn required a much higher cratering rate prior to 3.5 Ga, with a decay since then, and this result was confirmed by lunar samples. Neukum (1983) and Neukum et al. (2001) give a particularly useful quantitative version of this decay of flux as a function of time, based on analysis of crater populations at different lunar landing sites. Neukum's resultant accumulated number of craters/km^{2} formed since time T (in Ga) is assumed to have the same time dependence at all sizes, and as expressed for lunar craters larger than 1 km the time dependence is
N_{D>1 km} = 5.44 (10^{-14)} [(e^{6.93T}) -1] + 8.38 (10^{-4})T
This formulation shows that the total accumulated crater density for 3.5 Ga lunar maria should be 1.86 the density for a 3.0 Ga surface, and 5.76 the density for a 1.0 Ga surface. The 3.5 Ga isochron derived above is thus converted to isochrons for 3.0 and 1.0 Ga using those ratios, as shown in Table 2, Columns 8 and 9. Because the measured cratering rate (averaged over geologic time) has been essentially constant since 1.0 Ga ago, the ages for younger surfaces are proportional to these data. In columns 7, 8, and 9 the isochrons at the smallest diameters, D . 31 m, are regarded as an approximation based on extrapolation of the Neukum production function.
Figure 6 shows the final diagram, the 2004 iteration of the Martian crater diameter distribution isochrons for ages from 10,000 y up to 4 Ga.
[Figure 6] Final 2004 iteration of Martian crater-count isochron diagram. Upper solid line marks saturation equilibrium. Heavier short solid lines (1 km < D < 16 km) mark divisions of Amazonian, Hesperian, and Noachian eras; lighter nearby solid lines mark subdivisions of eras all based on definitions by Tanaka (1986). Uncertainties on isochron positions are estimated at a factor ~ 2, larger at the smallest D . 100 m (total uncertainties in final model ages, derived from fits at a wide range in D, including uncertainties in counts, are estimated a factor ~ 3).
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