Statistics of impact crater accumulation on the lunar surface exposed to a distribution of impacting bodies

ICABUS 7~ ~ - - 2 4 2 (1967)

Statistics of Impact Crater Accumulation on the Lunar Surface Exposed to a Distribution of Impacting Bodies1 EVAN

HARRIS

WALKER 2

School of Environmental and Planetary Sciences, University of Miami, Coral Gables, Florida Communicated by S. Fred Singer Received February 14, 1967 The considerable amount of lunar crater statistics available contains much information about the origin and age of the Moon's features. In addition, these statistics can provide information concerning the relative importance of various erosion mechanisms. In this paper we have derived the statistical distribution of craters as a function of size and of the time of exposure of the surface to impacting bodies. As the time of exposure increases, the crater distribution which initially corresponds to the distribution of impacting bodies, approaches saturation, which does not reflect the distribution of impacting bodies for most cases. An equation has been derived for the distribution of secondary craters resulting from the formation of a primary crater. This allows us to take into account the effects of secondaries on the distribution of craters on the Moon. We find that secondaries do not contribute significantlyto the "Moon-wide" statistics for craters larger than 100 meters. The distribution in the lunar highlands is found to be that of a crater-saturated surface. On the other hand, Mare Cognitum is not saturated. The distribution of craters appearing in the Ranger 7 photographs suggests an age of about 1.5 X 10~years which would make the maria younger than usually supposed. For the Martian surface we feel crater saturation may have occurred, indicating that meteoric impact is the most important erosion process operating on large-scale features. It has been suggested that the Martian craters indicate that the surface of Mars is old and little water or atmospheric erosion has occurred. Others have suggested that more craters should have been observed, which would indicate a young age and extensive water and/or atmospheric erosion. We feel the age of the surface is difficult to estimate in the case of crater saturation and little can be deduced concerning the importance of water or atmospheric erosion. INTRODUCTION

The Moon's surface is an ancient record of the h i s t o r y of t h e solar system, a n d m u c h of w h a t this record has to tell us is recorded in t h e form of c r a t e r statistics. Statistical studies of t h e f r e q u e n c y d i s t r i b u t i o n of l u n a r c r a t e r s as a f u n c t i o n of size h a v e been performed b y a n u m b e r of workers. T h e

recent Ranger photographs of the Moon together with Earth-based telescopic observations p r o v i d e i n f o r m a t i o n a b o u t t h e

early h i s t o r y of the M o o n a n d t h e solar system. Considered as a n i m p a c t counter, t h e M o o n provides i n f o r m a t i o n c o n c e r n i n g the p r e s e n t mass d i s t r i b u t i o n of meteoric bodies in i n t e r p l a n e t a r y space ( 0 p i k , 1960). M a r c u s (1964; 1966a,b) has considered a Research supported in part by NASA Grant stochastic m o d e l of t h e f o r m a t i o n a n d s u r v i v a l of l u n a r craters. T h e craters are number NGR-10-007-011. a s s u m e d to be p r o d u c e d b y meteoroid 2Present address: NASA Electronics Research Center, Cambridge, Massachusetts 02139. i m p a c t . T h e y are r e m o v e d either b y i m p a c t 233

234

EVAN HARRIS WALKER

obliteration or by filling. The filling is as- time t = 0 we obtain for a solution of (1) sumed to proceed at a constant rate, either N = (1/TrR~)([1 - exp (--~rR2gt)], (2) as a layer of accreted meteoric material or as a layer of lava: Margushasnot'considered which is simply an asymptotic exponential the effects, of~ secondary cratering. T h e growth curve. As the time t approaches treatment we give here considers obliteration infinity the number of craters per unit area (large craters covering small craters) and N approaches a value of erosion (small craters eroding and filling N® = 1/~rR ~. (3) large craters). The effects of secondary cratering on the resulting distribution of This is the total number of craters and parts craters has also been included in this treat- of craters that could possibly occupy an area. ment. Our mathematical approach is also The number is obviously too large, since simpler than that used by Marcus. when only a small portion of a crater remains A basic question to be answered is whether it is very difficult to identify the area as or not the Moon exhibits an impact-satu- a separate crater. It will therefore be necesrated surface. It is generally assumed that sary for us to assume that the impact of a the distribution of craters on the Moon crater destroys older craters over a larger reflects the distribution of meteors in space area than t h a t area occupied b y the new a n d / o r of secondary ejecta material. For a crater. If we designate this area as ~rR 2, surface exhibiting only a f e w craters this is Eq. (2) will become true, but for an impact-saturated surface N = (1/~R2)[1 - exp (-~TrR2gt)]. (4) where large craters have covered smaller ones and small craters have eroded away Although this result is quite simple, we will larger ones, it is not possible to relate the find that the more detailed calculation given distribution of craters to the distribution of below is quite similar to (4). bodies forming the craters without Considering the statistics of crater accumulation; THE CRATER~SIzE DISTRIBUTION INCLUDING EROSION AND OBLITERATION EFFECTS without such a theory, it is not clear how one would identify a saturated surface. Let us now consider the detailed calculaThe study of cratering statistics also holds tion of the statistics of impact crater acthe possibility of determining the ages of cumulation including a distribution in the different areas of the :Moon, F u r t h e r m o r e , size of the impacting bodies. Let us also the relative importance of other erosion include the erosive effects produced b y the mechanisms will probably be judged i n i m p a c t of small bodies on large craters and relation to this mechanism which is, perhaps, the obliterative effect of large bodies imthe domil~anli~mechanism in forming lunar pacting in the vicinity of small craters; the surface features. latter effect has already been included in the treatment of the simple problem given 2~k SIMPLE TREATMENT above. If we assume that the rate at which Let us first consider a simple problem in craters in the range from R to R A - d R are which all the impacting bodies produce formed per unit area is given" b y G, then Craters of the same size. We will assume that neglecting the obliterative and erosive effects a crater formed on top of an old crater re- mentioned above, we would have moves that crater from the surface. If the d 2 y / d t = G(R)dR. (5) rate at which the bodies impact on the surface is a constant g per unit area, the rate of T o take into account the obliterative effect change d N / d t of the number of craters per of large craters formed in the vicinity of unit area is given b y small craters, we must consider the probd N / d t = g - 7rR2gN, (1) ability t h a t the impact of a body form!ng a cra~er with radius R ' > R remo;ccs a crater w~ere R is t h e radius of the craters. A s already present on the surface having a suming the surface to be free of craters at radius R. This will be the integral over the

STATISTICS

OF IMPACT

intersection of the area of the craters formed per unit time with the area of the craters having a radius R, divided b y ~rR~ to give us the number of these craters. If we designate the distribution of craters already on the surface b y F, then we will have for the obliterative effect

-

ffi°" [G(R')eTrR'2F(R, t)dR]dR',

(6)

where Rma~ is the radius of the largest crater formed b y the distribution of impacting bodies. Let us now consider the rate at which the erosive effects of small bodies impacting on large craters will remove these large craters. This expression will be the same as (6) except that the integration will be from Rm~. to R and we must multiply the integrand b y the ratio of the volume of the small crater to that of the crater being eroded. If the volume of the crater is uR3, where ~ is a proportionality constant, then the erosive term will be R

The coefficient e has been replaced b y t in (7). The quantity d is the ratio of the volume of a crater to the volume of material that must be eroded in order to fill in and remove that crater from the statistical count. The use of the constant e' is quite useful. If one must include all craters that can be counted then systematic errors in the identification of marginal features severely affect the statistics. The total number of craters having a radius in the range R to R + dR that will accumulate on a surface in a time t is given by dN =

dt = F ( R , 0 d R .

(a)

Substituting (5), (6), and (7) into (8) and assuming G is independent of the time, we obtain

F(R, t) = G(R)t - e~r fot F(R, t)dt X f : ~ " G(R')R'2dR ' P

ACCUMULATION

235

This expression for F neglects secondary craters; we will return to this subject later. If we define S to be

(10)

x

(9) can be w ~ t t e n in the form

F(R, t)--- G(R)t - S(R) fot F(R, t)dt.

(11)

If we solve (11) for F(R, t) we obtain

F(R, t) - G(R) { 1 - exp [-S(R)t]}

~(12)

The similarity between (12) and readily apparent. Since dN is given b y

(4) is

~(R)

dN = FdR,

(13)

p ~Rl3

(7)

--~ ~R

CRATER

--3

t

R

!~ F(R, t)dt f;~,o G(R')R'SdR':

(9)

the cumulative number of craters greater than R0 per unit area at any given time t will be N =

~ (1 - e-St)dR.

(14)

In the limit as t approaches infinity, F approaches the saturation value F., where

F . = G/S.

(15)

Since S is an integral of G, the saturation distribution F . of craters on the surface will probably not show a very strong dependence oil G. Thus the distribution of craters on the lunar surface is very likely to b e a poor guide in trying to determine the distribution of meteoric bodies in space. Contrary to the statistical treatment given this problem b y 0 p i k (1960) one must consider the question of whether the surface used as an impact counter is a saturated surface and the extent to which erosion has played a part in shaping the surface. If we assume t h a t G can be expressed b y a simple power law

G = aR p

(16)

236

EVAN HARRIS WALKER

then (10) will yield S ( R ) = ~ra [ p - - ~

is evaluated in terms of the total mass ejected from the crater

Rp+3)

(R v+3 max

eVR-3

(R:,,++

+ ~

~:,,+6",] +mln)J (17)

-

so that F= becomes

.-: "-++,

+ - 1 '

+

+>

It should be noted that values for the exponent p in (16) between - 3 and - 6 allow us to neglect the terms involving Rmaxand Rmi.. In this case (16) becomes (

g

)-1 R-~

p+6

p~-3

~;

--6 < p < --3.

(19)

Integrating F~ from Ro to R ~ yields the cumulative number of craters per unit area for a saturated surface

[ (.~ 6

N~= 2~

p-~ X

THE

(R~-2

-2 - R~=).

(20)

DISTRIBUTION INCLUDING

M=

M~v~ B 2k =

v-Odv,

(22)

where M+v~/2 is the kinetic energy of the impacting body and k is the cratering efficiency which depends on the composition of the surface. For basalt and granite k has a value of about 5.5 X 10s ergs/gm. The quantities Vband va are the upper and lower limits of the ejecta velocity. The lower limit of the ejecta velocity for basalt targets is about 6 X 103 cm/sec; the upper limit will be comparable or possibly somewhat greater than the incident velocity v~. The relative dimensions of the fragments ejected from the crater will be inversely proportional to the shears acting on the material during the formation of the crater compared to the shear strength of the material. The magnitude of the shears will be proportional to the energy density across the shock front that moves through the material which, in turn, will be proportional to the square of the velocity of ejection. Thus, we can relate the linear dimension of the blocks r ejected from the crater to the velocity of ejection (Walker, 1965) by (23)

SECONDARY CRATERS

r / R = C S / p v ~ = Cv -2,

The solution of the problem of the statistical distribution of craters subjected to a distribution of impacting bodies which we have obtained in (12) neglects the effects of secondary cratering. The effects of secondary cratering, however, are not difficult to incorporate into the solution. Since the quantity G represents the distribution of primary craters obtained if removal effects are omitted, it will only be necessary to replace this function with a function H that takes account of both primary craters and secondary craters. Let us now consider what the function H will look like. The relation between the mass of material ejected to the velocity of the ejecta is given by d M = Bv-adv, (21)

where R is the radius of the crater that is formed, S and p are the crushing strength and density of the target material, and C is a constant of proportionality. This ad hoc equation is in good agreement with the experimental cratering work of Gault et al. (1963). For impacts in basalt targets a value of C = 3.6 X 107 cm3/sec ~ gives a good fit to the data. The value of C therefore is about 0.49. Substituting from (23) into (21) yields d M = -- i B ( R C ) o-~)/2r(~-a)/2dr" (24)

where B has a value of 2.55 (see Gault et al., 1963; and Walker, 1965). The coefficient B

The number of separate bodies ejected with a size in the range r to r + dr will be given by dn = d M / p r 3,

(25)

where p is the density. We have assumed the individual bodies to be cubic in shape. Substituting from (24) into (25) will give us

237

STATISTICS OF IMPACT CRATER ACCUMULATION

the differential expression for the number of ejected bodies as a function of their size r

d2N ' = H(R)dRdt = G(R)dRdt + ] R ='' G(R')dtdndR', (34)

B (RC)U_~)l~r(~_o)f2dr (26) d n = -- 2p The total mass of material ejected from a crater is proportional to the total amount of energy carried by the impacting body. Thus, the mass of material M , ejected by the impact of a secondary body of linear dimension r m a y be expressed by

M, = pr3v2/2lc,

(27)

where k is the constant of proportionality given in (22). The mass of the secondary crater will be proportional to the cube of the radius of the secondary crater M , = vR~,

(28)

where v is a proportionality constant. If we use (23) to substitute for v in (27) and use (27) and (28) to eliminate M,, we obtain

vR~-~ (pRC/2k)r ~

(29)

which relates the radius of the secondary crater to the dimension r of the impacting body. Using (29) to substitute for r in (26) we obtain

dn -

3B (RC)(9_~)14 4p X ~,--~-)

R,(~-~)~tdR,.

(30)

The integral of (21) yields, on solving for B, B = (~-- 1)M(v~- ~ -

v~-~)-'.

(31)

The total mass M of the primary crater can be related to the radius of the primary crater by [see Eq. (28)] M = vR~.

(32)

Substituting (31) and (32) into (30) yields

dn = 3~ (1 - t~)[vF' - v~-~]-~C (~-3')" X(~)(~-7)I4R(21-st~)ItR,(~'-'5)I4dR,.

(33)

In order to take into account secondary cratering in (5) we replace R' for R in (33) and write

where R replaces R8 in the expression for dn. Defining }, to be 3v (1 -- 2)[v~-~ --v~-~]-'C (9-8~)/4

× (2:---k) (~-7~" (35) (34) becomes

H ( R ) = G(R) + hR (3~-25)14 × f : ~ " G(R')R'(2~-3~)I4dR '.

(36)

Using (36) to represent the crater distribution, including both primaries and secondaries, (10) and (12), become

p'(R, t) = ~H(R) - ~ {1 - exp [-S'(R)t]} =

+ The exponent of R, in (33), on substituting for the value of/~, is -4.34. The cumulative distribution would have an exponent of -3.34. It is interesting to compare here the size-frequency distribution of secondary craters produced in the vicinity of the Sedan nuclear explosion crater (Nevada). Shoemaker (1965) gives a curve showing the cumulative number of secondary craters associated with the Sedan crater as a function of the diameter of the secondary craters. "The exponent of the function that best fits the estimated size distribution is slightly greater that --4. (The absolute value of the exponent is slightly less than 4.)" This result is in very good agreement with the results obtained here. Brinkmann (1966) gives an exponent of --3.56 (obtained from E. M. Shoemaker) which compares very well with our theoretically derived value of - 3 . 3 4 . More interesting is a comparison of Brinkmann's expression (after Shoemaker's data) for the cumulative number of secondaries which can be expressed as N, = 4.43 X lO-5(R'/Ro) 3"~6,

(39)

238

E V ~ N HARRIS WALKER

where R' is the radius of the primary crater and R0 is the radius of the largest secondary. Integrating our Eq. (33) to obtain the cumulative number of secondaries in the range R0 to R', R'>> R0, and evaluating k for alluvium targets (Shoemaker's data is derived in part from the Sedan nuclear explosion crater in alluvium) we obtain N8 = 4.90 X IO-4(R'/Ro) ~~4.

(40)

These results are surprisingly close considering the complexity of the problem. The discrepancy between the coefficients is actually less than appears to be the case. T h e difference arises from our different values for the exponent of R'/Ro. Allowing for this we can state t h a t both the exponent and the coefficient for our theoretically derived expression agree very well with the experimentally obtained expression. THE DISTRIBUTION OF CRATERS ON THE MOON

The highland areas of the Moon present an almost obviously crater-saturated appearance. The sharp outline of the large craters makes it possible to recognize craters t h a t still have only a portion of their walls intact. For this reason, small values for and e' are indicated. Assuming older features are obscured within about 1.5 times the radius of a new crater, ~ is approximately 2. For the value of p, (~pik's (1960) statistics for large craters on Mare Imbrium yield a value of p = - 2 . 6 . Since Mare Imbrium has obviously not become saturated with craters in the size range considered b y 0pik, his value of p is probably accurate. Since this value of p lies outside the range - 3 to - 6 Eq. (20) does not hold. If we assume R .... >> R0 >> Rmin, Eq. (18) will give for N~,

N~ = (Rv~°~'2/4~e)R~~6.

(41)

McGillem and Miller (1962) have obtained the statistical distribution of highland craters in the vicinity of Maurolycus. They obtain N = 2.877 X 10-~D -1"7°, (42) where D is the crater diameter in km and N is the cumulative number of craters per km -~. Setting R,~,~ equal to the radius of

Maurolycus, the largest crater in the statis tics, we obtain N~ = 2.44 X 10-2D -1"6.

(43)

Clearly, (42) and (43) are in good agreement. Figure 1 shows a plot of N as a function of D for (42) and (43). The maria may also be considered to have originated as impact craters. If so, one would expect them to fit the general statistics for impact craters. But we find that for p = 2.6, as used for the lunar highlands, the saturation density is too great. About 60 maria greater than 100 km in radius are predicted for half the surface of the Moon, whereas only 12 to 14 are observed on the side facing the Earth. To form a crater with the dimensions of Mare Imbrium would require the impact of a body of asteroidal dimensions. The value of p for asteroids having a radius from 1 to 45 km is - 2 . 5 9 0 p i k (1960). This also gives a result that is too large. For comets Opik (1960) gives a value of p = - 3 . 1 . With e = e' = 2, N~ becomes N= = 7.70 X 10-SR~ 2.

(44)

This gives 14.6 maria with a radius greater than 100 km on half the Moon's surface. This result can only be considered as suggestive, not conclusive. The result is sensitive to the value of p assumed for comets. Also, the fact that a value of p = 2.6 gives too m a n y large craters m a y simply mean that there have not been enough large impacts for saturation to have occurred for the craters larger than 100 km in radius. The photographs made b y the Ranger vehicles have provided information that makes it possible to obtain the distribution of craters down to craters 1 meter in diameter. Shoemaker (1965) obtains a distribution from the Ranger 7 pictures which, when expressed analytically, gives for the cumulative number of craters N per km 2 with diameter greater than D, N = 3.56 X 10-2D-I"~,

(45)

where D is expressed in kin. Miller (1965) obtains from the Ranger 7 pictures N = 9.69 X 10-3D-1'856

(46)

239

STATISTICS OF IMPACT CRATER ACCUMULATION

O

id 3 t~J

{D nr W [Z (3 t~-

o

\

Z

'2

hJ

-3

(D

\

Id5

,b CRATER

' DIAMETER

:,loo

KM.

FIa. 1. The cumulative number of craters per k m ~in the region of Maurolycus is piotted against the crater diameter, in kin. The data of McGillem and Miller together with their curve fitting this data are compared with the theoretical curve, Eq. (44) ( -., N® theoretical) (- - -, N = 2.877 X 10-2/)-1-7°; McGillem a n d Miller data).

and Brinkmann (1966) obtains N = 8.56, X 10"3D-2"12.

(47)

Figure 2 shows a plot of the data used for each of these analyses of the Ranger 7 pictures. Brown (1960) has studied the frequency of meteorite falls in highly populated areas. He obtains'a value of p = -3.409. He gives a minimum value for a [see Eq. (16)] On the Moo n of 9.57 X 10-19 (cgs) a n d a maximum value of 2.86 X 10-18 (cgs). The ~i~es of ithe meteorites considered fall in the range needed to produce the craters photographed by Ranger 7. Since p is smaller than -- 3 and

larger than - 6 , the saturation density will be given by (20). Taking e -- e' = 2, (20) becones N= = 1.128 X 10-1D -2.

(48)

This equation is plotted in Fig. 2. Although the exponent on D cbmpares well with the experimental data, (48) predicts too large a value for N=. It appears, therefore, that the surface of Mare Cognitum is hot saturated. .... .If the time of exposure of the surface is assumed short, the cumulative density of craters N will be !'. :

240

EVAN HARRIS WALKER at

N =

( R ~~-] -

- -

p+l

R~+I).

Nn,ln = 2.98 × 10-1D-~'4°9

(49)

(5{))

and the maximum value will be

If the time t is taken to be 4.5 X 109 years and R ~ >> R0, then the minimum value of N expressed in terms of the diameter will be, using Brown's data,

N~

= 8.92 X 10-1D-2"4°9.

(51)

Equation (50) is plotted in Fig. 2. Although the results agree well with the data, there

\

\'..

,o4

%

\' ,o3

\

v

w

PRIMARY

CRATERS

DOMI NATE

\

io2

rr (9 Lu 0

10

e"...\

W Z

SE

W J

OARY

CRATERS

\

"....'\

DOMINATE

161

0

-2

I0

~ "" \

(', \

-4

I0

I

-t

IOM

IM.

i

I

I0 M.

lOOM.

CRATER

I KM.

|

IOKM.

I00 KM.

DIAMETER

FIG. 2. The cumulative number of craters per km 2 on Mare Cognitum is plotted against the crater diameter, in lea. The data from the crater counts b y Miller, Brinkmann, and Shoemaker are shown. The theoretical saturation curve from Eq. (49) is shown, which lies well above the data. The nonsaturation curve from Eq. (51) which gives the minimum cumulative density of craters in 4.5 X 109 years is also shown; it also lies above most of the data, indicating t h a t Mare Cognitum is much younger than this age. A vertical line is shown separating the region where secondary craters will be important from t h a t where they are not important . . . . . . . , Min. theoretical values of No for T = 4.5 X 109 yrs, (D Miller's data. - - - - - , Brinkmann's curve. , Shoemaker's curve (ray areas). - . . . . . . , Shoemaker's curve (areas between rays). ~ , Theoretical saturation curve N® = 0.1128 D -2.

STATISTICS OF IMPACT CRATER ACCUMULATION

is too much spread in the data to establish the age of Mare Cognitum. The data suggest t h a t 4.5 X 109 years is too long. A value of 1.5 X 109 years is probably more accurate. It should be noted that Brown's minimum value for a leads to an upper limit on t. Since the lower limit on a is based on an actual count of meteorites, this upper limit on t is quite definite. The ages of the maria will be most important in establishing their origin, and crater statistics offer the opportunity of establishing these ages. The disparity in the crater counts obtained b y different authors represents the biggest uncertainty that exists in the determination of the age of Mare Cognitum. For this reason, more accurate crater counts are needed. T H E EFFECT OF SECONDARIES

We have so far neglected the effect of secondary craters on the distribution of lunar craters. Making use of (36) and substituting from (16) we obtain H ( R ) = aR"

1 +25_

3fl+4p

X [(Rmax/R) (~a~+'~)/'- 1] I.

(52)

If we substitute 1.64 X 10-~ for ), [the value in (40)], --3.409 for p, and 2 X 107 cm for R ~ , we obtain H ( R ) = aR-3.~(1 + 1.087 X 10'R -°93)

(53) in cgs units. From this result we see t h a t if R >> 104 cm we can neglect the effect of secondaries when considering the "Moonwide" statistics. Of course, in certain areas, such as along rays, this cannot be done. In Fig. 2 we have indicated this cutoff point. T H E AGE OF THE MARTIAN SURFACE AS INDICATED BY THE PRESENCE OF IMPACT CRATERS

The22 photographs of the Martian surface taken b y Mariner IV clearly show the presence of a large number of craters on the Martian surface strongly resembling the craters on the Lunar surface. Leighton et al. (1965) have argued that the presence of these craters indicates t h a t the terrain of Mars is

241

very old and has not been subject to erosion. Their work, if correct, would strongly argue against atmospheric or water erosion having ever been important on Mars. I t would further argue against a n y dynamic processes taking place in the interior of the planet. Anders and Arnold (1965) have made an effort to compare the surface density of craters on Mars with the density of craters on the lunar surface. These authors find the number of craters greater than 20 km in diameter per 106 km ~ of Martian surface, as obtained in the Mariner IV pictures, is less than he would expect based on the density of craters on the lunar surface and on statistical studies of the relative rate of impacts on the lunar surface and Mars. The fact that the density of craters they caluclate for the Martian surface is more than that recorded by the Mariner IV is taken b y Anders and Arnold to mean that the surface of Mars is only about 800 million years old. This fact, according to Anders and Arnold, would indicate that erosion processes have played a significant role in shaping the surface features of Mars and would suggest the presence of considerable amounts of water in a liquid form a n d / o r a substantial atmosphere as recently as 1 billion years ago. The calculation of the surface density of craters on Mars is based on the density of craters on the Moon and the expected higher rate of asteroidal and cometary impacts on Mars. The number of craters expected by Anders and Arnold is the density of craters on the Moon times the ratio of the rate of impact on Mars to the rate of impact on the Moon. However, as we have shown, the lunar continental areas present us with a surface that has been saturated with craters of all sizes. For such a surface, further bombardment b y meteoric bodies will destroy older craters as rapidly as new craters are formed. Thus the density of craters on the lunar continents is the maximum possible density and further bombardment of the Moon would not increase the density of craters there. Furthermore, if the maria were formed b y the impact of large bodies, as sometimes assumed, then the entire lunar surface can be taken as an example of a saturated or near-saturated surface. I t

242

EVAN HARRIS WALKER

should be noted that the observed density of c r a t e r s on M a r s , 37 p e r 106 k m 2 ( A n d e r s a n d A r n o l d , 1965), is c o m p a r a b l e to w h a t would be obtained from a random sampling of t h e l u n a r surface, t h e m a r i a a n d c o n t i n e n t s t a k e n t o g e t h e r . On t h e m a r i a t h e d e n s i t y of c r a t e r s g r e a t e r t h a n 20 k m in d i a m e t e r is 11 p e r 106 k m -2. T h e a v e r a g e for t h e m a r i a a n d c o n t i n e n t s t o g e t h e r is a b o u t 56 p e r l 0 s k m 2. I t is q u i t e l i k e l y t h a t t h e n u m b e r of c r a t e r s shown in t h e M a r i n e r I V p i c t u r e s is s o m e w h a t lower t h a n t h e n u m b e r of c r a t e r s a c t u a l l y existing in t h e p i c t u r e areas. T h e a n g l e of i l l u m i n a t i o n b y t h e S u n was g r e a t e r t h a n 40 ° . U n d e r such l i g h t i n g conditions, m o s t l u n a r c r a t e r s w o u l d b e difficult t o o b s e r v e p h o t o g r a p h i c a l l y . T h u s t h e v a l u e of 37 c r a t e r s g r e a t e r t h a n 20 k m in d i a m e t e r p e r 106 k m 2 s h o u l d b e considered as a lower l i m i t on t h e d e n s i t y of M a r t i a n craters. Since M a r i n e r I V r e c o r d e d a d e n s i t y of c r a t e r s on M a r s t h a t is c o m p a r a b l e w i t h t h e d e n s i t y of c r a t e r s on t h e l u n a r surface, w e can o n l y a s s u m e t h e M a r t i a n surface is also s a t u r a t e d w i t h c r a t e r s a n d t h a t t h e surface of M a r s is old enough t o h a v e b e c o m e s a t u r a t e d w i t h craters. I t is p r o b a b l e t h a t t h e m o s t i m p o r t a n t erosion m e c h a n i s m o p e r a t i n g on M a r s is t h e i m p a c t of m e t e o r i c bodies. ACKNOWLEDGMENT

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