Relative crater production rates on planets

ICARUS

260-276

31,

(1971)

Relative Crater

Production

WILLIAM

Rates on Planets

K. HARTMANN

Planetary Science Institute, 2030 East Speedway, Suite 201, Tucson, Arizona 86719 Received

July 16, 1976;

revised January

17, 1977

Dynamical histories of planetesimals in specified orbits, calculated by Wetherill (1975) and others, have given estimates of relative numbers of impacts on different planets. These impact rates, 5, are converted to crater production rates, F, by means of tables developed in this paper. Conversions are dependent on impact velocity and surface gravity. Crater retention ages can then be derived from (crater density)/(crater production rate). Such calculations of impact rates and their histories give the only basis, independent of sample dating, for establishing absolute geologic histories of the planet-,, contrary to published implications that this can be done by comparison of photos alone. A survey of the results, from orbits of interplanetary object? studied to dat,e, indicates that the terrestrial planets have crater production rates within a factor ten of each other, and that planets’ crater retention ages can probably be determined wit’h a factor of f3. Further calculations of orbital histories of additional interplanetary bodies are suggested to put photogeologic analyses from spacecraft imagery on a firmer basis. Applications to Mars, as an example, using least-squares fits to crater-count data, suggest an average age of 0.3 to 3 b.y. for two types of channels. The Tharsis volcanics are found to be slightly younger than the channels (strongly confirmed by photomorphology since they are not cut by channels) and Olympus Mons is about 0.06 to 0.6 b.y. old, contrary to recent assertions that Olympus Mons is 2.5 b.y. old and most Martian volcanic provinces older than 3 b.y. Data strongly support the hypothesis that Martian channels formed in a fluvial climate that per&ted on Mars until the Tharsis volcanism caused a change in the Martian obliquity st,ate, as outlined by Toon, Ward, and Burns (1977). I. INTRODUCTION

The purpose current

data

different

of this

paper

on crater

improving

techniques

of cratered

surfaces

is to

production

with

planets,

rates,

of

a

view

determining

crater production

reassess rates

on

rate = F,(D,

t),

that apply to other planets beyond the Earth and Moon. F,(D, t) is defined as the number of craters produced per square kilometer per unit of time in a specified diameter interval around D (km) at any specified time t in solar system history, on planet p. Only modest evidence is at hand about F,(D, t). The diameter dependence of F,(D, t) is fairly well understood, at least in recent times, with a similar powerlaw diameter dependence found in crater populations on young, undisturbed plains of the Moon, Mars, and Mercury ; this

toward ages

units.

As is well known, counts of impact crat.er frequency (number of craters of specified size per square kilometer) allow establishment of relative stratigraphic relations among provinces on any single planet where craters arc detected. Cnfort’unately, provinccs on one planet cannot be readily compared wi-ithprovinces on another planet, in terms of absolute ages, because we do not know the relative crater production 260 Copyright

0

1977 by Academic

All rights of reproduction

Press, Inc.

in any form reserved.

ISSN

0019-1035

CRATER

PRODUCTION

power law is consistent with the power-law distribution of sizes of asteroids, mctcoritrs, and other fragmented rocks (Hartmann, 1969). Whitakcr and Strom (1976), however, have found some evidence for Ddependence variations in very early crater populations. As for t dependence, various studies of terrestrial/lunar cratering, based on Apollo chronologies, establish that values of Fa(D, t) averaged at least hundreds of times higher prior to 4 aeons ago than t,hey do today, and that values of Fa(D, 1) and Fe(D, t) averaged probably well wit,hin a factor of 10 of their present values within the last’ 3 aeons (Shoemaker, 1970 ; Baldwin, 1971; Hartmann, 1972; Soderblom et cd., 1974). It, is widely assumed that this t dependence represents a smooth bransition from the actual accretion of planets t’o a nearly steady-state depletion of the asteroid and comet reservoirs; this is consistent with dynamical models showing t#hc rapid sweep-up of short-half-life objects close to planets and the slow sweep-up of remaining objects. An alternate view (e.g., Tera et al., 1974) is that the high cratering rate 4 aeons ago represents not the accretion, but a special cataclysm caused by the appearance of a group of planctcstimal fragments in the inner solar system. In cit,hcr model, all terrestrial planets would be expected to have roughly the same t dependence. The p dependence of F,(D, t) is virtually unknown and is the primary subject of this paper. Assumptions have been made about the p dependence, but, one purpose of this paper is to show that the assumptions can be tested to some extent, by calculations. In the absence of dat#able samples returned from dist,ant planets, improved knowledge of F,(D, t) is crit#ical t’o t#he understanding of absolute t,imc scalps in t,hc solar system history. If F,(D, t) is known for any single planet the absolute chronology of that, planet can be determined by the use of crat,er counts (number of craters per square kilometer) in diffcrcnt

RATES

ON PLANETS

261

geologic provinces, wit,h two exceptions. If virtually no craters are seen, then the surface is too young or too eroded and only lower-limit ages can be derived (a situation true of most, but not all, of t’he Earth). If the surface is saturated with craters, then in principle only an upper age limit can be derived; however, in practice the age is probably in the narrow interval bet,ween 4.6 and 3.9 aeons, based on accretion theory and lunar results. [Not.e that recent work cases doubt on the exact number density of craters corresponding to saturation (Woronow, in press).] If F,(D, t) were known for any pair of planets, then absolute chronologies could be compared from one planet to the other. Knowledge of F,(D, t) for all solid-surfaced plan&s and satellites (all p) would allow derivation of a relatively complete firstorder chronology of the entire solar system. The words “first order” are deliberately chosen because the precision of crater dating will always bc less than that of radioisotopic aging. However, bhis does not negate the value of ages determined from craher counts, called crater retention ages (Hartmann, 1966). Crater retention ages measure a quantity different, from radioisotopic ages; while radioisotopic ages date specific events of rock formation or m&amorphism, crater retention ages date the length of time that topographic features of a specific scale (D) can be retained on a surface in t,he face of whatever erosion occurs. As an example, the two ages could be equal in the case of an uneroded lava flow, where t,he accumulated number of crat#ers represents t.he time since rock crystallization. But if the flow is undergoing erosion, a st#eady-state number of craters occurs when the destruction rate at a certain diameter equals t,he creation rat,e at that diameter ; therefore, erosion reduces the crater retention age, especially at small diameters, while the radioisotopic age still measures the rock crystallization date. Thus, regardless of progress toward sample

262

WILLIAM

K. HARTMANN

return missions, it is important to exploit the crater retention technique to the fullest. Even if F,(D, t) were precisely known, crater retention ages would probably always have probable errors as high as lo-20% because crater-count variations of this magnitude result from geologic complexities of individual provinces and statistical variations in crater counts. But the reader is reminded that we are currently faced with photo- or radar imagery of surfaces where formation and erosion ages are, a priori, uncertain by factors of 1000 or more. In understanding planetary geology, thermal histories, impact histories, etc., it is of critical importance to distinguish whether ages of mapped surfaces are 10fi or log years! Pioneering steps in this direction were taken by Opik (1960) in a paper entitled “The Lunar Surface as an Impact Counter,” and by Shoemaker et al. (1962) in their paper “Interplanetary Correlation of Geologic Time.” Both papers compared terrestrial and lunar meteorite-crater production rates and crater counts in order to estimate lunar mare ages ; both papers concluded that “lunar maria . . . formed at a very early period in the history of the moon” (Shoemaker et al., wording). A few years later, some researchers revised their estimates of crater production rates based on uncertain measures of current meteor activity; these led to estimates of lunar mare ages which turned out to be much too young. This capsule history, then, shows that correct geological inferences can bc made from crater-counting technique, but only if correct crater production rat,cs are known. Recently the critical importance of determining F,(D, t) for different planets has been reemphasized by interpretations of imagery of Mars and Mercury. This is especially important because some workers have recently made deliberate assumptions about the p dependence of F,(D, t) in order to discuss the chronologies of the Moon,

Mars, and Mercury. For example, Sodcrblom et cd. (1974) explicitly conclude that Fa (D, t) and Fa(D, t) have ‘(been roughly the same . . . because the oldest postaccretional mare-like surfaces on Mars and the moon display about the same crater densit,y” (emphasis by W.K.H.). In this context Soderblom has spoken of a variant of the principle of uniformitarianism, hypothesizing that the time-behavior of early intense bombardment was the same on all planets and deriving a crater production chronology F,(D, t) from that assumption. Similarly, in the case of Mercury, Murray et al. (1975) find photogeologic evidence of ‘(a sequence of events broadly similar to those recorded on the moon, im.pZying similar histories of impact bombardment” (emphasis by W.K.H.). Murray et al. (1975) explicitly use similar appearances of craters on Mercury and the Moon to argue t#hat F,(D, t) is nearly the same for all planets p. The same idea appears in other manuscripts, asserting that phot,ogeologic indications of similar crater densit,ies “suggest . . . that the impact flux on the terrestrial planets has been the same . . . .” These assertions may be true (set below) but certainly do not follow rigorously. We have every reason to suppose that thermal histories of different-mass planets are different. Lavas should be produced at different times; thus, discovery of similar crater densities on lava-like mare surfaces of different planets would not imply identical F,(D, t) for the different planets. To base our knowledge of F,(D, t), and hence entire planetary histories, on such ‘Luniformitarian” assumptions is merely to invoke an article of faith. Chapman (1976a) has crit#icized such arguments in more detail. r\‘onethelcss, t.here appears to be a current surge of papers which allege significant revisions of crater retention ages, but with little analysis of crater production rates. Malin (1976), for example, gives an excellent review of data on Martian channels, with

CRATER

PRODUCTION

newly computer-proccsscd imagery ; he reviews earlier age estimates and concludes that his “current interpretation . . . would increase these ages by an order of magnitude to estimated values of billions of years.” Most of the paper is concerned with crater counts, including (at least partly !) justifiable critiques of my own early counts of 1974 ; but Malin’s Fig. 4 shows that his final counts show virtually no significant difference from my own: Some of his data points are higher than mine and some are lower, and the error bars are comparable and overlapping for most data point’s. Since absolute ages must come from crater counts divided by crater production rates, the proposed revision in absolute age must depend on the production rates, yet these are not analyzed in the paper: Reference is made to my own 1973 estimates of the rate and the model of Soderblom et al. (1974), and the main revision is due to adoption of a rate more like the latter. Similarly, Blasius (1976) has made careful new crater counts for Martian volcanoes. He finds “unsatisfactory” disagreement between earlier ages estimated by myself (1973) and Sodcrblom et al. (1974) in spite of the fact that our two sets of estimat,ed ages for the four largest volcanoes all lie in the range of 0.07 to 0.8 aeons, and the quoted uncertainties overlap in all four instances. For the same four, Blasius suggests older ages of 0.5 to 1.2 aeons. However, a comparison of my Olympus Mons counts and Blasius’s counts (kindly provided by him) shows differences averaging less than a factor of 2 in the five G-diameter intervals from 0.25 to 1.4 km. As for the crater production rates, Blasius also rcfcrs to my 1973 estimates and the 1974 model of Soderblom et al. ; no new estimates of this rate are given. Still another example is the comprehensive review of Martian chronology by Neukum and Wise (1976), which asserts, even in its title, a “possible new time scale.” This paper asserts a major revision toward older ages,

RATES ON PLANETS

263

placing all Martian volcanism “2.5 to 3.9” b.y. ago. Yet again, after presenting careful new crater counts, the paper derives the “new time scale” with no new physical reasoning about crater production rates. There are only assumptions: “As pointed out by Soderblom et al., the highlands of Mars and the moon were cratered at the time of the early intense bombardment between 4.5 and 4 billion years ago. We assume that the martian highlands, like the lunar highlands, show the cratering record from about 4.4 billion years until the present.” In addition to the age of the Martian uplands, Neukum and Wise also assume that no upland craters have been obliterated by erosion, so that “we deal with production populations.” These assumptions may or may not bc right; the last one is highly dubious and opposed to most investigators’ results. My point here is not to criticize the careful crater counts of these authors, which indeed represent an advance Rather, I point out t.hat absolute ages involve crater counts divided by crater production rates, and it seems that the emphasis in funding, rcscarch, and publication is all on the numerator and none on the denominator. What is unfortunate is t,hat new data affecting the denominator are being virtually ignored in the current discussions of ages, seemingly from a mistaken belief that absolute ages can be derived from photocomparison and crater counting alone. A much stronger approach can be taken to determine the critical values of crater production rate F,(D, t) for different planets. This is made possible in particular by a study by Wetherill (1975), and by calculat,ions of other dynamicists, who begin with families of planet,esimals in specified t,ypes of orbits and then compute the relative number of impact,s from each family onto the surfaces of t.hc diffcrcnt plan&s. Even if a family begins near planet A, near encounters with A cause scattering onto planets B, C, etc. Since the absolute

264

WILLIAM TABLE ADOPTED

APPROACH

K. HARTMANN

I

VELOCITIES 2r_FOE

THREE SOURCE FAMILIES Local planetesimals (primitive bodies) Planetocentric circular orbits(km/set) iW?rCUTy

VIXIUS Earth MOOII Mars

1.2

Ceres

Heliocentric circular orbit@ (km/set)

Asteroids asteroids short-p comets (km/%x)

Comet@ long-p comets (km/xc)

2.1 5.1 5.G 5.6 2.5

19 15 14 14 8.6:

82 4G 38 38 31

0.29

5

23

IO Europs Ganymede Callisto

1.2 1.0 1.4 1.1

25 19 15 12

25~ 19c 15s 13

25e 196 17 17

Tethys Rhea Titan 1apetus

0.20 0.25 1.3 0.24

18 12 7.9 4.G

16~ 126 9.6 9.6

1GE 12.5 12.5 12.5

Titsnia

0.57

5.2

6.8

8.8

Triton

1.8

6.2

G.2’

7.1

0 Assume scattering by target; mostly ~0.5 z)+.~of target. *Assume scattering by planet, giving 0.5 ncJaof planet; for satellites, usually oosofrom planetocentric orbit. =Assume ~0.4 vclrbfor inner planets; r Nvrt,for outer planets. d Assume ~1.3 nur~, for planets. e Corrected for fall into planet’s potential well.

crater production rate F,(D, t) is known for the Moon, the crater production rates for the specified families can be derived for all the other planets. Each family of initial orbits, of course, provides only a subset of the total crater production rate for a given planet. If all conceivable source families were studied and properly weighted by actual relat,ivc populations and lifetimes of these families, then the relative crater production rates of different planets could be derived by summing over all families. In order to attempt this approach systematically, I have used calculations of Wetherill (1975), Weidenschilling (1975)) Gault (1975), and Shoemaker and Hrlin (1977). The first step is to compute tables

which allow the conversion of statistics of planetesimals int,o statistics of observable crater diameters. This is treated in the next section. II. CONVERSION

TABLES FOR DIFFERENT PLANETS

More explicitly we must answer the question, “If a projectile is to make a crater of diameter D on the moon, what diameter crater will it make on Mercury, Venus, etc.?” The Moon is chosen as a standard, since its crater populations are best known in terms of statistics and age. This question can be well treated from cncrgy-scaling relations if the impact velocity is known. Obviously, different orbital families corrcspond to different impact velocities on planets, and the final impact vclocit,y, VI, depends on the approach velocity, v,, which is t,he relative velocity between planetesimal and planet at large separation. For completeness, these calculations arc listed for four different velocities corresponding to four interesting families of planetcsimals. These are ancient planctesimals in circum-planetary circular orbit#s (low-velocity family applicable to satellites in circular only), ancient planetesimals orbits around the Sun but near planets’ orbits (also a low-velocit’y family), ast#eroidal and short-period cometary bodies (a medium-velocity family), and a cometary source (high-velocity family). Table I lists the adopted approach velocities, u,, for the different sources. For a planet with effect,ive radius S moving through a swarm of part,icles with velocity U, we have the hpact rate [not equal to F,(D, t)]

s,(D, 0 = =

impacts km-’ set-’ (no. of particles,/volume) x v,7rX”/4?rRZ (1) npv,Xe/4R2 ,

CRATER

PRODUCTION

RATES

where R is the geometric radius of the planet. The effective cross section, &2, is found from X2 = R2[l

+ (8?rGRzp/3v,2)]

TABLE #/4R2

II

FOR TARGET

Locrtl planetesimala Planeto-

Helio-

centric

centric

(km/set)

(km/see)

BODIES Asteroids

Comets

(km/set)

(km/%x)

Mercury

1.28

0.26

VeIlUS

1.27

0.38

0.26

Earth

1.26

0.42

0.27

MOOIl

1.22

Mars CWZS

TABLE CALCULATED

0.25

0.29

0.26

0.25

1.25

0.33

0.26 0.25

1.38

0.25

IO

1.26

0.25

0.25

0.25

Europa

1.25

0.25

0.25

0.25

Ganymede

1.24

0.26

0.26

0.26

Callisto

1.26

0.26

0.26

0.25

Tethys

1.25

0.25

0.25

0.25

Rhea

2.38

0.25

0.25

0.25

Titan

1.38

0.28

0.27

0.26

1apetus

3.20

0.26

0.25

0.25

Titania

0.75

0.26

0.25

0.25

Triton

1.04

0.32

0.32

0.30

III

IMPACT

VELOCITIES

Local planetesimals

(2)

(e.g., Hartmann, 1968). Here, p is the mean density of the planet. This gives the factor S2/4R2 in (l), as listed in Table II. The data of Tables I and II can bc used to compute relative impact rates 5,(D, t) for different target planets, as derived in (1)) if the relative numbers of planetesimals per volume, nP, and their velocities are known in the regions of each planet. Some surveys have produced these kinds of figures. However, the primary sources used in this paper tabulate results in the form of impacts per square kilometer on target planets, or as the total number of impacts on each planet, assuming a given number of projectile bodies in a selected orbit. To derive crater production rates, F,(D, t), from tabulations of the latter data, it is necessary to divide the impacts by the planet area of the various target planets.

265

ON PLANETS

Plsneto-

Helio-

centric

centric

(km/see)

(km/set)

Mercury Venus Earth MOOU

2.7

Mars CWf!!S

ol

Asteroids

Comets

(km/set)

(km/aec)

4.7

20

62

11.5

18

47

12.5

18

40

6.1

14

38

5.6

10

31

0.65

5.0

23

IO

2.7

25

25

25

Europa

2.3

19

19

19

Ganymede

3.1

15

15

17

Callisto

2.5

12

13

17

Tethys

0.5

16

16

16

Rhea

0.6

12

12

12

Titan

2.8

8

10

13

1apetus

0.5

5

IO

12

Titan&

1.3

5

7

9

Triton

4.0

7

7

8

From listings of planetesimal number densities, nP, or of estimated impact rates, 5,, Tables I and II allow one to compare impact fluxes (impacts per square likometer per second) or impact densities (impacts per square kilometer) on different target planets. However, the problem before us is to analyze and compare crater densities on the different planets. Therefore, we must provide tables to convert from relative impact rates for planetesimals of specified sizes to relative numbers of craters of specified sizes. The point is that a specific planetesimal from any specified starting family will make different-sized craters on different planets. The energy released by an impacting planetesimal depends on its impact velocity, VI. From conservation of energy, vi2 = VW 2 + v,,,2. Table impact planets We tesimal

III gives calculated values of the velocities, VI, for different target using data of Table I. now ask the question: If a planefrom one of our previously identified

WILLIAM

266 TABLE

K. HARTMANN

IV

FACTOR Y TO COIU~ECT~01% RELATIVE IMPACT VELOCITIEB

CORRECTION

Local planet&m& Planeto-

Helio-

centric

centric

(km/set)

(km/see)

Asteroids

comets

(km/set)

(km/sea)

Mercury

0.73

1.54

1.81

VWIUS

2.15

1.36

1.29

Earth

2.38

1.36

1.06

1.00

1.00

1.00

Mars

0.90

0.66

0.78

Cere.3

0.067

0.29

0.44 0.62

Moon

1.M)

IO

1.00

5.51

2.02

Europa

0.82

3.95

1.45

0.43

Ganymede

1.18

2.97

1.09

0.38

cauisto

0.91

2.27

0.91

0.38

Tothys

0.13

3.21

1.18

0.35

Rhea

0.16

2.27

0.83

0.25

Titan

1.04

1.39

0.66

0.27

1apetus

0.13

0.79

0.66

0.25

Titania

0.41

0.79

0.43

0.18

Triton

1.61

1.18

0.43

0.15

a Y =

(u&l,

c

)I.=.

families (defined by the approach vclocitics in Table I) strikes a given target planet at a speed higher (or lower) than the speed at which it would have hit the Moon, how much bigger (or smaller) will the crater it makes be than the crat.er that would be made by the same-sized planetesimal, from the same family, st,riking the Moon? According to energy (E) scaling laws (D/DC)

= (E/E<)“=

= (V/VC)2’3.3.

[Use of a different energy exponent such as $ or $, discussed by some authors, has been examined and does not change the basic conclusion of this paper, other uncertainties being comparable; see Baldwin (1963).] Because the cumulative frequency’ of primary impact craters (D > 2 km) has 1If numbers of craters in log increments of D are plotted in a log N-log D incremental frequency diagram, the same power law happens to apply to straight segments, i.e., N = kD-*. The author prefers this form of plotting, since deficiencies of craters in certain D increments are more pronounced in incremental plots than in cumulative plots. Some

been found to vary approximately as D-2, the crater production rate appears to increase (or decrease), consequently, by a factor Y, defined as Y = (D/D<)2 = (V/Va)‘.21. Table IV gives a correct,ion factor Y for various relative impact velocities. Further discussion has been given by Hartmann (1973). The factor Y is the number by which relative impact rates must be multiplied to produce relative crater densitics corrected for impact velocity differences from one plan& to another. It is normalized to the Moon since t#hc Moon will be the basis of further comparisons. Table V lists an additional correct.ion factor, 2, to correct for the fact that larger-diameter craters are produced on bodies with smaller surface gravities, since more cjccta is removed. The relation of crater diameter D and gravity is DIDu = (al&", where g is the surface gravity on the indicated planet. Obrrbcck and Aoyagi (1972) give a value of Ic = 0.12, and Gault and his co-workers (private communication, Feb. 1975) have found cxperimcntal results of lc = 0.2 to 0.25. Here we adopt k = 0.2. Therefore, the correction factor, 2, is 2 = (D/‘Dc)~ = g/qc-o.4. In summary, if WCarc given a dynamical calculat,ion indicating that the s,(D, t) influx rate (objects km-2 yr-I) from a certain orbit onto a designated planet p is X times that on the Moon, Tables IV and V give the appropriate corrections for computing tho crater production rate F,(D, t) for a given-sized crater, rclativc to the lunar rate, under the assumption that other authors (such as Chapman) have used incremental log-log plots with linear increments in D, in which straight segments have power law relations with exponents more negative by unity, i .e ., N’ = kD-3.

CRATER

PRODUCTION

the cumulative primary crat,er production follows a - 2 power law in diamctrr D : F,(D,

t) = XYZ,

where X = Sp(D, t)/Fa(D,

RATES

TABLE

If all reasonable orbits of all conceivable small bodies in the solar system had been investigated, we could simply calculate the carter production rate Fi = XiYiZ< for each orbit i, and then compute the total carter production rate, F, by an appropriate weighting of orbits according to their degree of population. To take a simpler case (to some degree true), if a reasonable subset of important orbits had been studied and if it were found that the resulting crater production rates hardly varied from one planet to another, then we could comfortably conclude that the actual crater production rates are similar on all planets; then we could proceed to interpret planetary chronologies. The truth is somewhere between these extremes. Not very many orbits have been studied, and data on the outer planets are especially sparse. However, Wetherill (1975) has tabulated histories of bodies from nine initial orbits that allow us to treat terrestrial planet cratering. These data are augmented by a summary of comet statistics by Gault (1975), and studies of Jupiter-scattered planetesimals by Kaula and Bigcleiscn (1975) and Weidcnschilling (1975), and studies of Apollo asteroids by Shoemaker and Helin (1977). These studies allow a preliminary assessment of the rclative cratering histories of planets on a firmer basis than has been available in recent years. Table VI divides available source orbits int#o two groups, early planetesimals and modern planetesimals. The former have circular orbits near planets ; they might represent the solar system during planet

V

CORRECTION FMTOR Z TO CORRWT FORSURFACE GR~WTY~ g = 4G(iW/D2)

t).

III. APPLICATIONS TO PLANETARY CHRONOLOGIES

267

ON PLANETS

Z

372 887 981 162 373

0.72 0.51 0.49 1.00 0.72

29

1.99

10 Europa Ganymede Callisto

111 133 145 97

1.00 1.08 1.05 1.23

Tethys Rhea Titan Iapetus

16 21 149 12

2.52 2.26 1.03 2.83

Titania

72

1.38

252

0.84

Mercury Venus Earth fi oon Mars Ceres

Triton a Z = (sPlg
= (gQ/gP)+o.~.

growth, since planets probably grew from small bodies condcnscd from near-circular orbiting gas. The latter have orbits based on observed asteroids and comets ; they represent the solar system today and probably at any time after gravitational scattering of planetesimals by grown plan&s commenced. Table VI lists the craters produced per square kilometer, F,(D, t), from an arbitrary population of planetesimals placed in each source orbit, normalized to the Moon. It is of interest that about 74% of the values in Table VI lie, within range of a factor of 10, between 0.6 and 6 times the lunar rate. Therefore, any reasonable calculation of total crat,ering rate, by summing over different orbital families, tends to produce similar crater production rates on different planets, since effects of unusual orbits are averaged out. This leads to some confidence that we can

0.33 0.67 0.98 1.27

Few 1.36 0.34 1.01 1.27 0.70 0.19 Few

(comets and asteroids)

G w2 w3 Wl W6c W5 w4 SH

VI

Many 4.67 4.18 4.00 1.89 3.61 1.97 Few

0.41 0.82 0.99 1.89

(AU)

Aphelion

2.6 0.89 5 . t5 0.37 1.2 1.7 8.1 0.26

53 000 1.0 1.2 1.3

B

0.079 0.84 1.6 0.62 1.8 1.7 1.4 0.74

2.7 2.2 1.9

0

0.160 0.96 0.66 1.3 1.9 0.92 0.74 1.45

0.93 4.9 1.9

@a#

1 1 1 1 1 1 1 1

1 1 1 1

Target planet

a W, Wetherill (1975) ; G, Gault (1975) ; SH, Shoemaker and Helin (1977). * Estimated by W.K.H. from additional figures and cross-section considerations. c Same m W6 above, with different readings of Table VI.

Tabulated comets Comet Temple 2 Comet Encke Mars crosser Mars crosser 1959 LM Icarus 19 Known Apollos

Modern planetesimals

W9a W8 W7 W6

(AU)

Perihelion

(circular orbits)

Mercury planetesimal Venus planetesimal Earth planetesimal Mars planetesimal

Early planetesimals

Remarks

Initial orbit

TABLE

0.94 0.38 0.21 0.57 7.7 0.13 0.24 0.66

0.10 0.50 8.0

<0.002 ?

<0.14 <0.14
-

JIII*

99.9 99.8 80 62 14 85 29 -50

0 16 19 14

Percentage escaping solar system

EXAMPLES OF RELATIVE CRATER PRODUCTIONRATES FROM SPECIFICORBITS (Craters km+, Normalized to Moon)

0.03 0.41 0.51 0.05

A

0.25 0.30 0.10 0.05 0.30

B

0.20 0.10 0.10 0.15 0.15 0.20 0.10

C

0.50 0.40 0.10

11

Weighting in Table VII

0.30 0.50 0.10 0.10

E

CRATER

PRODUCTION

269

RATES ON PLANETS

TABLE VII CR.ITK:H PK~DLJCTI~NRITES FOR MOUKL SOL~K SYSTEMS(Craters km-?) Applicable time

Modela

Description

Total crater production B

Early

A

Modern

B C D E

Early planetesimals in circular orbits near terrestrial planets Asteroidal 407, Cometary 60% Asteroidal Cometary Mars crossers favored

0

@

rates

a

fl

JIII -

1

3

3

1

0.5

0.8 2

1 1

0.9 1

1 1

2 2

0.00007 0.0007

5 2

2 1

0.7 2

1 1

0.3 4

0.02 0.00009

(1Different models are approximations derived from weights in Table VI. The weights are treated as relative populations of craters in the different orbits. (Weights in model A are proportional to planetary masses, and selected as illustrations in other models.) Models are calculated as follows. First the crater production rates in Table VI are normalized to equal numbers of impacters in each orbit (after multiplying craters/km2 by planet area). Then these populations of craters are multiplied by the weights and the fraction from that orbit that does not escape the solar system. Finally, results are renormalized to the lunar crater production rate and list,ed to one significant figure.

specify average crater production rates on all terrest#rial planets to better t,han an order of magnitude, if we consider the net effect of all sources of cratering. For a certain planet to receive an extreme rate of cratering outside of t.his range, two conditions would have to be met: (i) One source orbit would have to overwhelm the cffect#s of other source orbits for that planet, e.g., have a much larger population of planetcsimals ; and (ii) that source orbit would have to dump many more planetcsimals on that planet than on any other planet. Thcrc is a good counterargument that (i) is not true: the Earth receives a variety of meteorite types and it is very unlikely that they all come from one source orbit. Instead, they appear to represent a variety of parent bodies originally in the asteroid belt (Chapman, 1976b), broken at different times, and fed toward Earth on different orbits by different perturbative mechanisms. On the other hand, about 32% of meteorite falls are H-group chondritcs (t,he second-mostcommon type) and about 45% of these appear to be involved in a nonrandom clustering of cosmic ray ages (Wasson,

1974). Thus, about 147, of our present meteorites may come from a single disruptive event, raising the strong possibilit,y that at certain intervals in the past certain planets may have had episodes of cratering dominated by fragments from particular disruptive events. Since half-lives of terrestrial planet-crossing orbits are typically of the order 107-lo8 yr, it is unlikely that such episodes were long-lived or dominant in the planets’ histories. In summary, it appears unlikely that conditions (i) and (ii) would have been true for any one planet for most of solar system history; t,hereforc, it is likely that the averaging of effect’s of different source orbits t,hrough time caused similar average crater production rates on different terrestrial planets, based on the orbits studied so far. Several appropriat#e averagings are att.empted in Table VII, where five model solar systems are constructed from the available data. These models are specified to consist of the target planets (plus Ganymede, for which only crude limits are available) plus populations of bodies drawn from Table VI, according to different weightings. Model A is a primeval solar

270

WILLIAM

K. HARTMANN

fluct,uations in t,he cratering rat,cs during the last 3 acons (once the early intense bombardment was complete), and thus one is tempted t#o argue that planetary fcat’ures dating from the last two-thirds of geologic time can be dated within an uncertainty factor of 3 in either direction, from crater counts. However, it is useful to consider the individual models B through E more closely. If one type of model could be selected, more orbit calculations could be made, and t’he uncertainty could be reduced still further. Conversely, if there is uncertainty about the proper type of model, then w-e need to be concerned about special types of orbit,s such as Mars-crossing asteroids, that might be underrcpresented. That most meteorites appear t,o come from asteroids (Chapman, 1976b) favors the asteroidal model B, which has a variation

system, with only bodies in near-circular orbits. The other models arc modern systems with variable emphasis given to asteroid-like and comet-like source orbits. The array of modern planetesimal models (B through E) illustrates several interesting points. First, in spite of the uncertainties in choosing among the models, the resultant crater production rates for all the terrestrial planets in Table VII vary by less than a factor of 17, ranging from 0.3 for one Mars entry to 5 for one Mercury entry. Second, the range of variation (uncertainty) for each individual planet ranges from 2 for Venus to 13 for Mars. Of the 20 entries, 80% are within a factor of 2.5 of each other. In other words, the table confirms Wetherill’s original conclusion that the rates on terrestrial crater production planets are probably similar. Most researchers agree that there have not been major

TABLE LEAST-SQUARES CRATER-COUNT FITS Terrain

Diameter interval

AND

VIII

DERIVED AGES OF MARTUN FEATURES Slope

Y Int,ercept

Correlation coefficient

(km)

Derived crater retention age for 2-km craters” (lo9 yr)

Average of most lunar front-side maria Selected channels (not associated with chaotic terrain) Selected channels (associated with chaotic terrain) Tharsis plains (broad region surrounding four major volcanoes ; about 10’ km*) Surface of Olympus Mons (estimated impact craters only)

2

-128

(3.4)

-1.96

-2.658

0.991

11

-1.26

-2.965

0.955

1

0.3

- 16

-1.81

-2.832

0.868

1

0.5

-128

-1.96

-2.991

0.994

0.8

-2.40

-3.536

0.898

0.2

0.3 -

0.25-

2

a Ages estimated as in text, based on assumpt,ions : (i) average lunar mare age is 3.4 b.y. (based on dated samples), and (ii) average Martian cratering rate in last aeon was twice lunar rate during postmare period (based on results of this paper; average of estimates in Table VII). Crater-count data include material published by Hartmann (1973, 1974), augmented by some new data of my own. Counts by other workers appear to be similar, and a synthesis with data of Blasius, Chapman, and Jones is in progress.

CRATER

PRODUCTION

of only 2.5 among terrestrial planrts. However, the most cncrgctic tcrrcstrial impact of the last wntrry, the Tunguska event, which is the only event with energy sufficient to produce crater sizes usually discussed in planetary data, was probably a cometary phenomenon. This favors a model such as C or D. Still greater uncertainty is caused by Mars-crossing asteroids. Noneccentric Mars crossers can strongly favor Mars impacts (Table VI, orbit WS), but the number of Mars crossers is highly uncertain. Several facts suggest, but do not prove, that Mars crossers may be abundant enough to give Mars a higher crater production rate than permitted by some Martian crater chronologies (Soderblom et al. 1974; Neukum and Wise, 1976), which asswne a certain close relation between Martian and lunar crater production histories. For instance, initial finds of the Apollo search program of Shoemaker and Helin (Helin et al., 1976) include three new Mars crossers, two new Apollos, and three possible Mars crossers. The Mars-crosser inventory is thus expanding. Furthermore, a naive listing of known Mars crossers and Apollos shows 50 times as many Mars crossers, with similar size distributions (Fig. l), suggesting the possibility of a very high Martian cratering rate, which would imply geologic youth for many important MarGan features. However, Wethrrill (privato communication) points out that many of these cat#aloged Mars crossers, as defined by James Williams, spend only a small fraction of t’heir lives actually in orbits that cross Mars. Because of secular variations of their orbital elements, many of the orbits often do not, cross Mars orbit ; furthermore, Wetherill (1975) calculates lifetimes of two Mars crossers averaging 28 times longer t,han lifetimes of two representative Earth crossers. These results suggested that n Eart,h-crossing asteroids could give higher impact rates on t’he Earth and Moon that the same n Mars crossers

271

RATES ON PLANETS

0.5

I

32

FIG. 1. Comparison of size distributions (number per log increment of mass) of known (1975) Apollo and Mars-crossing asteroids, based on typical albedo and size-magnitude relations for asteroids studied through mid 1975. Fits to parallel power laws at large diameters, consistent with other asteroid populations, suggest large-size populations may be relatively complete. At comparable sizes roughly 50X more Mars-crossers are known than Apollos, but see text, for qualifications regarding definition of Mars crossers.

would give on Mars. Thus, the problems of rclativc lifetimes, collision probabilities, and application of Mars-crosser statistics to Mars’ cratering rate need further study. This question of Mars crossers’ contribution to the cratering of planet,s is illustrated in Table VII by model E, in which Marscrossing orbits are given heavier weight than in models B through D. The result is a Martian cratering production rate four times higher than the lunar rate, which could bring ages for Martian volcanoes and channels from fashionable values of about l-2 b.y. down to less than 0.5 b.y.-a change with important consequences for concepts of Mars’ early massive atmosphere or for episodes of recent climate change, as

272

WILLIAM

K. HARTMANN

discussed in the next section. Pending further examination of t,ho Mars-crosser problem, the possibility is not excluded that the Mars crater production rate might be even higher than in model E, though most dynamicists now favor a lower value. Obviously more observations and more Monte Carlo orbit calculations would be useful. In spite of these uncertainties in choosing among models A through E, it is impressive that all of the average crater production rat’es for all terrestrial planets in Table VII are well within a factor of 10 of the lunar rate. IV.

EXAMPLE:

APPLICATIONS

TO

MARS

As an example, some of the stratigraphically young Martian features mentioned in Section I can be reexamined with the present results. Absolute ages of such features can be determined in principle by dividing crater number densities, N (craters km-2) by crater production rates F (craters km-2 yr-l). In practice, this calculation is best carried out by scaling to the average of typical lunar maria, which have the bestknown crater density and age in the solar system. Thus, age of Martian feature = 3.4 b.y. (N$/Na)

(Fa/Fg),

where the N ratio comes from crater counts and the F ratio comes from results such as those shown in Table VII. This method is especially suitable for stratigraphically young features, since lunar and terrestrial evidence indicates that the crater production rate (at least in the Earth-Moon system and probably elsewhere) is better determined and more constant in the last few aeons than in the first half-aeon. Table VIII gives least-squares solutions for crater counts from my own data on the average of most front-side lunar maria, selected channels not associat,ed with chaotic terrain, channels associated with chaotic terrain, the broad Tharsis plains

and the surface of Olympus Mons. The least+squares fit is to the straight line in log-log coordinates 1ogN = (slope) log D + (y intcrcopt), where N is the incremental number of craters per square kilometer in the logarithmic diameter interval (D, D112). The slopes of - 2.0 found for lunar maria and the Tharsis plains indicat’e little disturbance of the primary crater distribution, believed to have about this slope. Slopes of - 1.3 and - 1.8 for craters in channels suggest some erosive losses ; the slope of -2.4 for Olympus Mons is steeper than expected, possibly indicating some admixture of small volcanic pits (see also Hartmann, 1973). To determine the rat.io N$/Na, a diameter of 2 km was selected because all curves overlap their D range at this point, which should bc relatively free of contamination by volcanic pi@ secondaries, etc. The age solution of the first group of channels, for example, is as follows : N#/Na is found to bc O.SOl for a diameter of 2 km. If the Martian cratering rate has been twice the lunar, based on Table VII, then the crater rctent#ion age of these features is 0.401 times the lunar age. If t,he lunar mare age is 3.4 b.y., then the channel ages are found to be about 1.4 b.y. Figures are given to one significant, figure in Table VIII because of the large uncertaint,y, primarily in crat,ering rate (Table VII), estimated to be a factor of 3. The ages arc understood to be ages during which 2-km crat,crs have been retained on these surfaces. Thcrc! are two points t,o t,his exercise. First, it illustrates that the estimated age of Martian features depends strongly on what we believe is t,he Mart,ian crater production rate: It cannot be derived simply by comparing Martian and lunar photos, as some aut,hors seem to imply. Second, it illustrates t,hat the “old-age scale” of some authors is difficult to reconcile with est,imated crater production rates. For

CRATER

PRODUCTION

example, Neukum and Wise (1976) place most Martian volcanic landforms at ages of 2.5 to 3.9 b.y. and Olympus Mons at 2.5 b.y., based on assumptions about similarity of lunar and Martian cratering. These results make Martian volcanics considerably older than most investigators have supposed. As shown by Table VIII, they would require a Martian crater production rate less than the lunar rate by a factor about 1.5 to 5. In view of Table VII this seems unlikely, although it might be possible if unusual orbits dominate or if the number of Mars crossers is unexpectedly low and/or they have much longer half-lives against collision than Apollos have. Neukum and Wise’s placement of the most recent major Martian volcanism, Olympus Mons, at 2.5 b.y. is also difficult to reconcile with cvidcncc that the Tharsis uplift is not isostatically supported and would require at least lunar-like viscosities (higher than Earth interior viscosities) just to support Tharsis for the last 1 b.y. (Phillips and Saunders, 1975). Finally, it is difficult to reconcile with reported Viking detection of possible seismic activity. On the other hand, the shorter time scale reported here agrees in an interesting way with several results : (i) Morphological evidence suggests that Martian fluvial conditions persisted until the Tharsis volcanics formed, but not long afterward. Related evidence suggests that the highly erosive conditions that degraded early craters declined rather suddenly into the present conditions, on a time scale short compared to Martian history (Hartmann, 1973 ; Malin, 1976 ; Chapman, 1976a). These results suggest that the climate change was not due to a slow exponential decay of a massive atmosphere over most of Martian history ; and the proposed young ages suggest that it was not a rapid decay of a massive atmosphcrc early in Martian history. Instead, a rela-

273

RATES ON PLANETS

tively sudden favored.

and

recent

transition

is

(ii) Burns et al. (1977) have found that the creation of the Tharsis bulge of accumulated volcanics changed the obliquity state of Mars to the present state from an earlier state when the obliquity reached 45”. Toon et al. (1977) have shown that in this earlier state the entire summer pole could bc kept at daily average temperatures higher than the melting point of ice, producing higher temperature by greenhouse effects and allowing fluvial erosion. Tharsis volcanism, then, may be the short-lived event that changed the Martian climate. Evidence from item (i) is consistent with this. (iii) Fairly rapid development of the Tharsis complex in an interval about 1 b.y. ago agrees with the time scale given here and with the age adopted in Phillips and Saunders’ (1975) gravitational analysis, and provides a mechanism for the rapid climate change evidenced by crater degradation states and the channels. The Tharsis bulge is still coming into adjustment (and may even be still volcanically active). Olympus Mons probably was active in the interval 70 to 600 m.y. ago, consistent with Viking seismic results. (iv) The time scale derived here is more consistent with current data on asteroids and comets, as developed in this paper, than a very long time scale is. According to this hypothetical scenario, then, liquid water may have been much more abundant on Mars before the Tharsis volcanics began to accumulate. This accumulation caused a change in obliquity and Obliquity varied for a period climate between states allowing and prohibiting greenhouse effects (Toon, et al., 1977). Permafrost layers formed and remelted, explaining why many channels emanate from chaotic collapsed terrain. (Note that their existence requires a cold period of frozen permafrost prior to the period

274

WILLIAM

K. HARTMANN

when the water flowed, which in turn is prior to the present cold period.) By the time t,hc current Tharsis surface was forming (0.3 to 2.5 b.y. ago according to the highly uncertain dates in Table VIII), the obliquity had changed to the current state, with nearly permanent cold dryness. This marked a relatively quick transition to less erosive conditions. (Local water flow due to geothermal melting or other processes might still occur.) Major volcanic cones were added on top of the Tharsis bulge; the most recent was probably Olympus Mons (surface as young as 0.07 to 0.6 b.y. according to Table VIII). The reader should note that these ages are considered to have a formal uncertainty of a factor of ~3, due mainly to the present uncertainty in Martian crater production rate. V. CONCLUSIONS

AND CONSEQUENCES

Estimates of relative rates of impacts on planets have been converted to estimates of relative production of craters by means of the tables given here. The results suggest that we can specify the crater production rate F,(D, t)to within a range of a factor of 10 for most terrestrial planets p, crater diameters D, and times 2. It appears unlikely that any terrestrial planet would have a crater production rate as much as lo2 times different from the rate on the other terrestrial planets. To refine knowledge of the chronologies of other planets, further information is needed in two areas. First, it is extremely important for more Monte Carlo calculations to be done on the final impact sites of planetesimals in other orbits. A better statistical sample of representative Marscrossing orbits, Earth-crossing orbits, Venus-crossing orbits, comet orbits, and orbits near resonances in the asteroid belt is needed to understand the lifetimes and distributions of planetesimals that may be gravitationally scattered and make craters on various terrestrial planets. Second, further searches for small bodies in such

orbits are necessary in order to understand the populat,ions and relative importance of the various types of source 0rbit.s. Present statistics are poor; for example, Whipple (1973) estimates that there are about 100 Apollo ast.eroids larger than 1 km, about twice the number shown by the size distribution in Fig. 1. To summarize the planets, Table VII suggests that the Mercurian crater production rate is about 0.8 to 5 times the lunar rate, indicating that Mercury’s lava plains and the Caloris basin, which have crater densities about equal to their lunar counterparts, might be younger than their lunar counterparts. This would be consistent with a somewhat more extended thermal period on Mercury than on the Moon, in keeping with Mercury’s larger mass. Table VII suggests that Venusian and t’errestrial cratering are both about one to two times the lunar rate. This indicates, on the basis of radar-imaged crater statistics on Venus, that the surface of Venus has accumumlated and retained lOO-km-scale craters longer than the Earth, and has a crater retention age on the order of aeons for craters this size Table VII also suggests that Martain crater production rates are most likely greater than the lunar rates, probably between the value of 6.S estimated by Hartmann (1973) and the value of about 1 estimated by Sodcrblom et al. (1974). (Case D, without any Mars crossers, seems unlikely for Mars.) This would place crater retention ages in the first half of geologic time for large craters (D = 100 km) in the Martian cratered uplands, but in the second half for small craters (D = 2 km) in the cratered uplands and all craters in the sparsely cratered plains. Major volcanoes and the most recent major channels would have estimated ages of within the last aeon, while some units of polar strata, some chaotic units, and perhaps some channels would be younger, having ages of only a few lo8 yr.

CRATER

PRODUCTION

Table VII includes a very rough estimate of the cratering environment on outer planet satellites, based on some considerations of impacts on Ganymede. The general conclusion is that the types of objects considered here, including cometary objects, would crater the outer planet satellites at a rate less than a tenth the rate of lunar cratering. Unless another source of crat,ering objects is very important in that environment (e.g., Trojan-like objects?), the accumulation of craters on those satellites may have been much slower than on the inner solar system. If this is correct, and if thermal events or isostatic relaxation of icy soils have erased the original accretional surfaces, outer satellite surfaces would not be saturated with craters. Further Monte Carlo-type calculations of cometary and Trojan-like orbits, and further inventorying of small bodies in the outer solar system may be required to make any firm prediction, however. Although this paper is influenced by, and heavily dependent on, the work of Wetherill (1975), it extends Wetherill’s results in several ways. Wetherill calculated only impact rates, 5, defined above. This paper : (i) gives correction factors Y and 2 to correct for velocity and gravity differences, allowing comparison of crater production rates F; (ii) shows that different models of the solar system createring environment arc possible by different interpretations of t’he small number of studies published to date; (iii) emphasizes that Mars-crossing asteroids are a major area of uncertainly, affecting crater production rate estimates for terrestrial planets; and (iv) includes new Martian crater-count data. More dctailcd discussions of cratering data from tcrrcstrial planets are envisioned on the basis of the logic developed here. The tables are presented to aid further modcling in the event of (i) future Monte Carlo-type calculations of statistics of orbits, and (ii) future discovcrics of inter-

RATES

ON PLANETS

275

planetary bodies, especially Mars crossers. Efforts in both directions are urged. ACKNOWLEDGMENTS I thank Karl Blasius, Clark R. Chapman, Donald R. Davis, Richard Greenberg, L. A. Soderblom, and George Wetherill for helpful discussions during various phases of this work. This publication, but not the original research, w&s supported by the NASA Planetology Program. REFERENCES BALDWIN, R. (1963). The Measure of the Moon. Univ. of Chicago Press, Chicago. BALDWIN, R. (1971). On the history of lunar impact cratering: The absolute time-scale and the origin of planetesimals. Icarus 14, 36-52. BL~SIUS, K. R. (1976). The record of impact cratering on the great volcanic shields of the Tharsis region of Mars. Icarus 29, 343-361. BURNS, J., W.~RD, W., BND TOON, 0. (1977). Mars before Tharsis: Much larger obliquity in the Past? Abstract, 8th DPS Meeting, Honolulu. CHAPMSN, C. R. (1976a). Chronology of terrestrial planet evolution: The evidence from Mercury. Icarus 28, 523-536. CHAPMAN, C. R. (1976b). Asteroids as meteorite parent-bodies : An astronomical perspective. Geochim. Cosmochim. Acta 40, 710-719. GAULT, D. E. (1975). Private communication. HARTMANN, W. K. (1966). Martian cratering. Icarus 5, 565-576. H_&RTMANN,W. K. (1968). Growth of asteroids and planetesimals by accretion. Astrophys. J. 152, 337. HARTMANN, W. K. (1969). Terrestrial, lunar, and interplanetary rock fragmentation. Icarus 10, 201-213. HARTMANN, W. K. (1972). Paleocratering of the Moon: Review of post-Apollo data. Astrophys. Space Sci 17, 48-64. H,~~TM~NN, W. K. (1973). Martian cratering, 4: Mariner 9 initial analysis of cratering chronology. J. Geophys. Res. 78, 4096-4116. HELIN, E. F., Bus, S. J., .~NDPRYOR, C. P. (1976). Discovery of 1976AA. Bull Amer. Astron. Sot., 8, 458. KAULA, W. M., AND BIGELEISEN, P. E. (1975). Early scattering by Jupiter and its collision effects in the terrestrial zone. Icarus 25, 18-33. MALIN, M. C. (1976). Age of Martian channels. J. Geophys. Res. 81, 4825~4845. MURRAY, B., STROM, R., TR~SK, N., AND GAULT, D. (1975). Surface history of Mercury: Implications for terrestrial planets. J. Geophys. Res. 80, 2508-2514.

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crater distributions : Implications for geologic history and surface processes. Zcarus 22, 239-263. TERA, F., PAPANASTASSIOU,D., AND WASSERBURG, G. (1974). Isotopic evidence for a terminal lunar cataclysm. Earth Planet. Sci. Lett. 22, 1-21. TOON, O., WARD, W., AND BURNS, J. (1977). Climatic change on Mars: Hot poles at high obliquity. Abstract 8th DPS Meeting, Honolulu. W.~SSON, JOHN T. (1974). Verlag, New York.

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WEIDENSCHILLING,S. J. (1975). Mass loss from the region of Mars and the asteroid belt. Icarus 26, 361-366. WETHERILL, G. (1975). Late heavy bombardment of the Moon and terrestrial planets. Proc. Lun. Sci. Conf. 6, 1539-1561. WHIPPLE, F. (1973). Note on the numbers and origin of Apollo asteroids. The Moon 8, 340-345. WHITAKER, E. A., AND STROM, R. G. (1976). Populations of impacting bodies in the inner solar system. Lunar Sci. VII, 933. WORONOW, A. (1976). Crater equilibrium: A Monte Carlo Geophys. Res., in press.

saturation and simulation. J.