ISOCHRONS FOR MARTIAN CRATER POPULATIONS OF VARIOUS AGES

UPDATING OF MARS ISOCHRON DERIVATION: 2002 ITERATION

Below you will find a text from July 2000 that describes the basic principles of our derivation of the crater chronology diagram in the PSI format, used to date crater retention ages on Mars in our papers on Mars Global Surveyor results. That text gives values used at that time. The following material updates that earlier discussion.

The diagrams at the end of this page are the latest iteration of the diagram (valid as of Oct 2002) and we hope they may be useful to other researchers. Remember the diagram works like a histogram: determine the number of craters/km2 in each square root 2 diameter bin (D = 500m to 1 km, 1 - 1.4 km, 1.4 - 2 km, etc); then plot the number in that bin. The data points show the incremental number of counts in each square root 2 log D bin.

The updates of the discussion are as follows. First, the R bolide value estimated by Bottke from dynamical calculations of resonance effects has been raised to 3.2, and this is coupled with an empirical value of 2.0 from Boris Ivanov, based on observed numbers of Mars-crossing asteroids. As a result, in the new iteration we use Rbolide = 2.6. This is a higher Martian cratering rate than the old value of 1.44, which tends to make Martian ages younger (but not by a uniform amount over the whole diagram, because of effects involving the slope of the curve and the gravity/velocity scaling; see text below).

The second major change is in the handling of the steep branch. In the 2000 iteration we used the Hartmann power law, based on lunar data, as described below. This was basically an extrapolation along the slope -3.82 that had been determined from lunar data. Gerhard Neukum's data, however, shows a "downward" curvature at smaller diameters, in a size range where our old lunar data did not give good values (because of saturation effects). For these reason, in the new iteration, we grafted the Neukum curve onto our power law curve for the shallow branch. The effect of this change is that the isochrons bend down very slightly at MGS-MOC sized craters (<100m scale). This in turn means that the smallest craters give somewhat older ages than in the previous iteration.

This second change helps resolves a problem that we and others have raised about the earlier isochrons -- that the small craters gave younger ages than the large craters. As we have commented in our papers, we believe some of that effect is real. As on earth, 30-m scale relief is apt to be younger than 3-km scale relief. For example, a 10-m thick lava flow on a 100 MY-old surface would wipe out D=30m craters and make ages at that scale young; but 3-km craters seen at Viking resolutions would still appear relatively fresh and would be counted. Hence large craters give older ages than young craters on such surfaces.

The net effect of the 2002 update is that the typical ages do not change by large amounts, although the general shape of the crater counts fits the isochrons better.

We still view the uncertainty of ages as about a factor 2 to 3 in either direction (mostly due to uncertainty in Rbolide). Thus, a model age of 30 MY may be 10 to 90 MY, but is still very young in terms of Martian history. In geophysical terms, such a result is very important. Of course, no ages are expected to be > 4.0 to 4.5 Gy, and so all ages are truncated at that end of the scale. Indeed, old model ages of around 3-4 Gy are fairly well constrained to be early in Martian history, because of the high crater densities involved. The weakest and most frustrating model age measurements fall around 1 to 2 Gy, because the uncertainties mean that it is hard to specify whether the surface dates from the first or last third of Martian history.

MARS ISOCHRON DERIVIATION: JULY 2000

Suppose we have the production function crater diameter distribution (N = craters/km²-y as a function of diameter D) for the moon, as determined from lunar crater counts and dates of lunar provinces, e. g. as measured by Hartmann et al. (1981, 1999) and Neukum and Ivanov (1994). The first step in converting this to Mars is to correct for the difference in the rate of bolides hitting Mars and the moon. Suppose we know from dynamical studies of asteroid an comet populations that the ratio of impactors/km²-y hitting Mars is R_bolide times the number of impactors/km²-y hitting the moon, for bolides of fixed size. Generally, with Mars closer to the asteroid belt than the moon, R_bolide is believed to be > 1. A recent review of asteroid dynamics by Bottke (private communication, 1999) suggests

R_bolide for asteroids ~ 1.44 +/- 0.4.

The value for R_bolide is Bottke's; the value for the uncertainty is our conservative estimate of the remaining uncertainties int he asteroidal flux. The additional flux of short period and long period comets is believed not to make a dramatic change in this number. Thus, if we plot the lunar production function diameter distribution (craters/km²-y) the first step in correcting 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 hits on Mars.

The second step in converting to Mars is to 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 can be treated as follows.

Impact velocity effect. Baldwin (1963) and other authors have found that crater diameter D scales approximately as impact kinetic energy E^1/3.3. 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

as big on Mars as on the moon.

Gravity effect. Hartmann (1977) reviewed several authors' results and found that D goes approximately as gravity g^-0.2. He applied this to estimate that, due to this effect, a given bolide makes a crater that is

D_Mars/D_moon = [373/162]^-0.2 = 0.847

as big on Mars as on the moon.

Combining these two effects, we find that

D_Mars/D_moon = 0.815 x 0.847 = 0.690,

so that each bolide hitting Mars makes a crater 0.69 as big as it would have if it had had an orbital history that led to a collision with the moon.

Assumptions used to create isochrons. To create isochrons for Mars, we make several other assumptions that should be specified:

1). The size distribution hitting Mars is the same as that hitting the moon. We believe this should be true, since most dynamical effects are essentially size-independent. However, bolides populations in the belt in the size range < ~10 m and especially < ~ 1 m may be significantly affect by the Yarkovsky and other effects, producing possible anomalies in Martian crater diameter distributions below ~ 50 m and especially < ~ 5 m.

2). The average cratering rate ratio R_bolide between Mars and the moon remains the same, if averaged over geological periods of time > ~ 10 My. Individual asteroid breakup events could produce variations over short time periods but these should average out over periods longer than the dynamical half life of fragments against ejection from the belt and collisions with planets, of the order of a few My.

The upshot of this discussion 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 that have accumulated on lunar mare surfaces in the average mare lifetime of about 3.6 Gy, 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 ~ 0.69.

A MARS ISOCHRON FOR 3.6 Ga SURFACE AGE: THREE BRANCHES OF THE CURVE

The production function crater diameter distribution can be fitted very well to a curve of log N vs log D plotted as three power law segments, shown in Figure A.

Because power laws produce linear segments on our plots of log N vs log D, it is conceptually easy to carry out this correction for each power law segment. This is because the leftward (smaller diameter) shift of any given power law segment, which is linear on the log-log plot, is equivalent to a constant vertical shift along the whole length of the line by

d log N = -b d log D

where -b is the slope of the power law segment. For example, with a -2 slope, a shift to smaller diameter by a factor by one log unit causes an decrease in apparent number by two log units. A shift to 1/2 the size causes a decline in apparent number by a factor 4.

Similarly,

log [N_mars/N_moon] = -b log [D_Mars/D_moon]

Thus, if we have a value for the shift in diameter from D_moon to D_Mars, we can derive the corresponding vertical displacement of that whole power law (straight line) segment of the production function on the log N - log D plot. In the lunar curve, three branches have been identified (Hartmann, 1999): the steep branch (or "secondaries") with slope -3.82 on the cumulative or log-differential plot, at D < 1.4 km; the shallow branch with slope -1.80, at 1.4 km < D < 64 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.

Steep branch. For the lunar mare, Hartmann (1999) derived

log N_{steep, moon mare} = -3.82 log D - 2.616.

Note again that N here is defined as the incremental number of craters in each log D increment to base square root 2, e.g. 707 m to 1 km, 1 to 1.414 km, 1.414 to 2 km, etc. According to the above we must raise this curve by R_bolide = 1.44 and shift it to smaller D by 0.69. Note that the latter shift means that instead of turning up at D ~ 1.4 km, as on the moon, the steep branch turns up at D ~ 975 m on Mars. From the above, the shift to smaller D is equivalent to a vertical shift downward by 4.12. The combination of these two shifts gives

delta log N_steep = -0.457.

This shift will give us the number of craters formed on Mars in the steep branch in a time equivalent to the average lunar mare age, about 3.6 Ga. Thus we have

log N_{steep, Mars 3.6 Ga} = -3.82 - 3.073.

Shallow branch. For the lunar maria, the Basaltic Volcanism Project (Hartmann et al., 1981) derived

log N_{moon, shallow, mare} = -1.80 log D - 2.920.

Using the slope -1.80, the combination of shift upward and leftward corresponds to net downward shift by

delta log N_shallow = -0.132.

Thus we have

log N_{Mars, shallow, 3.6 Ga} = -1.80 log D - 3.052.

Turned-down branch. On the moon this branch begins at about 64 km, but the shift to smaller D on Mars by 0.69 gives a beginning diameter of D ~ 44 km. Having assumed a slope of -2.2 for this branch, we have

log N_{Mars, turned-down, 3.6 Ga} = -2.2 log d - 2.393.

Using these equations we can easily create isochrons for Mars, after the style described by Hartmann 1999.

ISOCHRONS FOR DIFFERENT AGES

In the system we have adopted, we assume that the cratering rate was still slightly declining at 3.6 Ga ago, but that it was constant since then. Thus we used the accumulated N value at 3 Ga not as 3.0/3.6 times the 3.6 Ga value, but as

N_{Mars, 3.0 Ga} = 0.95 (3.0/3.6) x N_{Mars, 3.6 Ga}.

It follows from the above that the isochron for 1 Ga is at 1/3 times the crater density as the 3.0 isochron, the isochron for 100 My is at 1/30 of the 3.0 isochron, and so on.

The actual numbers of craters we have predicted in each diameter bin for a surface of age 3.0 Ga is given in the table below.

The following figure is a blank template that shows the isochrons we have currently derived for Martian cratered surfaces (updated 2002). Also shown are some versions of this figure with sample data sets that allow reading ages from the diagrams.

REFERENCES

Hartmann, W. K. 1984. Does Crater "Saturation Equilibrium" Exist in the Solar System. Icarus, 60: 56-74.

Hartmann, W. K., R. Strom, S. Weidenschilling, K. Blasius, A. Woronow, M. Dence$ (Elmsford, NY: Pergamon Press).

Hartmann, W. K. and Gaskell 1997. Planetary Cratering 2: Studies of Saturation Equilibrium. Meteoritics and Plan. Sci. 32: 109-121.

Malin, M. C. and 15 others (MOC-MGS imaging team). 1998. Early Views of the Martian Surface from the Mars Orbiter Camera of Mars Global Surveyor. Science, 279:1681-1692.

Tanaka, K. L. 1986. The Stratigraphy of Mars. Proc. Lunar Planet. Sci. Conf. 17, Part 1.; J. Geophys. Res. 91, suppl. E139- 396.