On the relative and absolute ages of seven lunar front face basins

ICARUS 71, !--18 (1987)

On the Relative and Absolute Ages of Seven Lunar Front Face Basins I. From Viscosity Arguments R A L P H B. B A L D W I N Oliver Machinery Company, Grand Rapids, Michigan 49503 Received July 3, 1985; revised February 27, 1987

The seven basins, Orientale, lmbrium, Crisinm, Nectaris, Humorum, and an unnamed basin between Werner and the Altai Ring show rims whose absolute and relative heights are correlated with the sharpness and crispness of the features. On the assumption that the decline in average outer rim height, not scarp height, measures the age of the basin and also that the decline represents a hot creep of rocks of very high viscosity, absolute ages were derived. Basins were found to increase in age in the sequence listed above, with a range from about 3.82 to 4.30 x 109 years. The average or effective viscosity of the surface layers down to whatever level was involved in the creep was calculated as increasing from 9.46 x 1024 poises at about 4.30 x 109 years to 1.86 x 1030 poises at present. It should be clear at the outset what the assumptions and associated observations are and why they are necessary to a solution. They will be listed in this abstract and expanded upon in the text. (1) The original rim height of each basin was a function only of basin diameter. (2) The original rim height was given by Pike's (1983) relation for fresh craters extrapolated to basin diameters. (3) The present rim height is that of the most prominent ring structure. (4) The smaller rim height of all seven basins, relative to the height predicted by (2) is due largely to creep in the lunar rocks down to some undetermined level. Other forces may contribute to the sinking of the rims, but these are considered to be of lesser importance and are discussed in the text. (5) The relative ages of the seven basins are as given in Table I. This sequence differs slightly from that of Wilhelms (1984), for example, but it is that found in Baldwin (1974, 1987) and is consistent with the results of this paper. (6) The age of formation of the lmbrium basin (3.85 x 109 years) inferred from lunar sample studies (particularly Apollo 15) is correct. (7) The age of formation of the Serenitatis basin (3.87 + / - .04) x 109 years, inferred from petrologic and geochemical studies of Apollo 17 boulders is incorrect. This is not an assumption, but is a result of the analysis of this paper. (8) The theology of the Moon may be described, for the purposes of this paper, by an effective viscosity valid throughout the layers involved in the creep. (9) This effective viscosity is used as a tool to determine basin ages and is not important in itseff. It does appear to vary in the same range as terrestrial rocks, but not the lithosphere. (10) Other factors such as isostasy, shaking due to j a r from later impacts, modification due to rim relief by ejecta, and erosion from small impacts are all close to exponential in nature, declining toward the present, and hence may be included in the determination of the effective viscosity. 1 0019-1035/87 $3.00 Copyright © 1987by Academic Press, Inc. All rights of reproductionin any form reserved.

2

RALPH B. BALDWIN (11) The rim height of the Imbrium basin subsided by 25 m in the last 2.5 x 1 0 9 years. This value was chosen arbitrarily. It could have been 250-300 m and the basin ages would not have been affected except for Orientale and that only minutely. (12) The effective viscosity of the Moon was observed to change continuously and monotonically with time. (13) Judging by Table III, the probable error of an absolute age is in the range of 10 to 50 x 106 years. It is difficult to determine exactly what this means. It will be constrained by points (14), (15), and (16). (14) An error in the age determination should not be large enough to alter the relative ages of the basins, judging by crater counts (Baldwin 1974, 1987). (15) If the viscosity of the Moon declined in the post-lmbrium period of mare formation the only basin to be affected would be Orientale and this by no great amount inasmuch as the basin is older than nearby maria. (16) If the effective viscosity were less than about 1022 poises at the time of the oldest basin then presaturation surfaces would not show the numerous craters and portions of craters that are obvious in this time span. (17) Considerably prior to the time of saturation the outer layers of the Moon had a low enough viscosity so that they could not retain the record of the then occurring cratering. (18) The approximations of this paper were adopted because it does not appear possible to make an unambiguous selection from the more elegant mathematical treatments of creep and isostasy that would lead toward reasonable ages for the giant basins. © 1987 Academic Press, Inc.

1. INTRODUCTION The primary purpose of this paper is to determine consistent absolute ages of seven lunar front face basins. On the front face of the Moon there are numerous huge basins whose fundamental structures are quite similar. However, their dimensions, horizontally and vertically, differ considerably. As examples, Imbrium is huge, sharp, and clear, with large vertical variations. Humorum is much smaller and very much subdued. Its vertical dimensions are considerably lower, absolutely and proportionately, than at Imbrium. Other basins are intermediate in sharpness of detail and in vertical dimensions. The great unnamed basin (Baldwin 1963), lying between the Altai Ring and Werner, is still more subdued. It is real, but its rim is so low that it is best seen under a low Sun. It has generally been considered that these changes in appearance represent an age sequence. Absolute ages have been approximated for these and other basins (Baldwin 1970, 1971a, 1974, Schaeffer and

Husain 1974, Nunes et al. 1974, Jessberger et al. 1974, 1976, 1977, 1978, Steiger and Jaeger 1977, Staudacher et al. 1979, Jessberger 1983), but except for Imbrium there is little agreement on how old each basin is. The Imbrium basin has been dated by returned samples of lunar rock as approximately 3.85 x 10 9 years (Steiger and Jaeger 1977, Carlson and Lugmair 1979a, Wetherill 1981). I think it must be agreed that the various basins were formed on a surface that was solid and that the original height of the rim of each basin was a function of the basin diameter. This has been accepted as valid for the ordinary lunar craters (Baldwin 1949, Pike 1980) and the basins are super craters. If the above assumption is valid, then it follows that something has happened to eliminate the sharpness of detail and to reduce the rim heights and crater depths of the older basins. Whatever the mechanism is that caused the observed changes in vertical dimensions of basins, its effects were very small in the 3.85 × 109 years since Imbrium was

AGE OF LUNAR BASINS. I. VISCOSITY born and much larger in the pre-Imbrium period dating back to the age of the oldest visible basin, a period of less than 0.75 × 109 years. II. T H E P R O B L E M

The first step was to see if there had been any subsidence of the rims of normal postImbrium craters, not basins. Pike (1980, 1983) has summarized dimensions of many craters, and 69 of them are included in my table of absolute ages (Baldwin 1985). When the calculated rim heights are determined from Eq. (1) and (2) and the ratios observed/calculated and C/O are found for individual craters and plotted in Fig. 1 vs absolute crater ages, it is found that from the present back to 2.5 x 10 9 years the points cluster around the 1.0 line, but that prior to that time most of the crater rim heights are now slightly lower than the calculated values. The points are from Class 1 craters which are all younger than Imbrium. Pike (1980) has given equations relating diameters of fresh appearing lunar craters and rim heights. They were revised in (1983). For craters up to 16.2 km diameter the revised equation is Re =

0.036D 1.014

(1)

and for 16.2+ km the equation is Re = 0.163D °472,

(2)

where Re is the rim height and D is the rim to rim diameter (each in km). Equation (2) normally is not extended into the basin range, above roughly 400 km, although the 370-km crater, Mendeleev, was used by Pike in determining the relationship. The key point in this assumption is that the equation is observed to be valid over a range of diameters from 16 to 370 km, a factor of 23, while the maximum extrapolation is a factor of 3. The exact relationship actually is not important to the determination of basin ages. If the original rim height increases in different fashion with increasing crater or basin diameter,

3

A G E (10' y) 20

,

!

.

?

,

a

. •

• "

.

"

*'l°

llp.,.-~:o/c ' " " " "'~'= ''=

#"

*

i.

I* I~ °

"

2t

FIG. 1. Ratios o f observed rim heights and calculated rim heights (according to Eq. (1), (3)) vs absolute age for craters c o m m o n to the lists of Pike (1980) and Baldwin (1985). The absolute ages were measured in the latter reference, the dimensions from the former.

the calculated effective viscosities may be different than those found in this paper, but the basin ages will not differ significantly from those of Table III. This is not readily apparent, but may be verified by calculation (see Table III). The three circles near the right hand end of Fig. 1 represent average values for the older classes of craters, 2, 3 and 4, from Pike (1980). The absolute ages are not known, but Class 2 craters are close to the age of Imbrium and the Classes 3 and 4 are substantially older. They have been entered in Fig. 1 with only approximate dates. The trend line only is important here. I interpret Fig. 1 as demonstrating that there has been little settling of postImbrium craters and most of what there was occurred between 3.85 and 2.5 x 109 years, and there was substantial settling in the pre-Imbrium period. The primary assumption in this paper is that the reduction in heights of crater and basin rims is a hot creep process in highly viscous rock. The reason that I have selected rim heights rather than crater depths for this study is that most of the basin interiors are filled with ancient lavas and so we often do not know what changes in crater depth actually have occurred. It is clear that most of the reductions of the rims of craters and basins occurred shortly after each was formed and during the last 3.85 x 10 9 years little reduction occurred. As will be noted, the smaller the crater, the slower the rate of settling. Inasmuch as no creep equates to a perfectly elastic Moon or to an essentially

4

RALPH B. BALDWIN

infinite viscosity, the data presented in Fig. 1 are not inconsistent with a very small amount of settling, and a value of 25 m reduction in the rim height of Imbrium in the last 2.5 × l 0 9 years was adopted. It is well within the scatter of points in Fig. 1, derived from the smaller pits. Even if the assumption is in error by an order of magnitude it will have little effect on basin ages because only one basin, Orientale, is younger than Imbrium and it is only slightly younger. The next step was to determine the present-day external heights of the rims of the seven basins. This is complicated because in no case is the rim of a basin uniform in height. There are large variations and an "average" figure must be selected. The results are listed in Table I, column 2, along with notes describing how the results were obtained. For example, the Imbrium data from LM 41 show numerous places where the height above the interior lava surface is only 1000 to 2500 m, but there are two huge and long massifs, Mt. Hadley and Mt. Bradley, that tower about 4000 m above the present average ground level. An average figure of 2562 m was derived. As will be shown later, the exact value is not critical to the dating process. I By rim height of a basin, I refer to its height relative to the normal ground level and not to the height of the interior face of the rim. By using the height of the rim relative to ground level the rate of subsidence is proportional to the basin diameter. There has been some question as to where the rim actually is at certain basins. For the purposes of this paper, I shall assume that in each case the most prominent ring marks the crater rim. These rims are identified by the diameters in Table I. 1 In several places in this paper data values are given to several decimals. T h e y certainly are not k n o w n to such accuracy. The values given are those that resulted from the mathematical solutions and the assumptions and should be so considered.

TABLE I BASIN

DIMENSIONS

Basin

Present rim height (m)

Calculated original rim height (m/

Basin diameter (km)

Ratio present to original rim height

Notes

Orientale lmbrium Crisium Nectaris Humorum Serenitatis Unnamed

2725 2562 1758 1038 630 900 350

4105 4556 3176 3956 2837 3390 3063

930 1160 540 860 425 620 500

.664 .562 .554 .262 .222 .265 .114

1 2 3 4 5 6 7

N o t e . All original rim heights were calculated by Eq. (2). Present rim heights were determined as follows: (l) T h e r e are no adequate contour charts of the Orientale region. T e n m e a s u r e s (Orbiter IV 181 H2, 187-M oblique) o f a 45-km section o f the Cordilleras near Eichstadt gave heights ranging from 3.02 to 4.26 kin, averaging 3.5 kin. This is one of tfie highest portions of the rim and is the scarp height, not rim height, which is always lower. T h e m a p o f this half of the rim o f Orientale by H e a d et al. (1981) s h o w s that the rim slopes d o w n w a r d outside the crest, but does not c o v e r a sufficient area to permit an accurate measure of the height of the ground level. A figure of 2725 m for the rim height should be approximately correct. (2) A series of straight lines between Appenine peaks was d r a w n o n L M 41. This zig-zag line was considered to form the crest of the rim o f l m b r i u m . W h e r e v e r the line crossed a contour line the altitude was recorded. T h e average o f 276 m e a s u r e s ranging from 6000 to 10633 m was 8562 m. At the outer foot o f the A p p e n i n e s the average altitude was 6000 m. T h e difference is 2562 m. (3) In the same m a n n e r as at l m b r i u m the rim crest average altitude for Crisium was determined from 228 m e a s u r e s o n L M 62 to be 7592 m while 56 m e a s u r e s on a parallel arc that did not intrude onto Mare Foecunditatis gave 5834 m. The rim height is found to be 1758 m. (4) Similarly 139 m e a s u r e s on the AItai rim yielded 5222 m and 77 m e a s u r e s o n a parallel arc going through the Kant Plateau gave 4134 m, a difference o f 1038 m. L A C 96 was used i n a s m u c h as there are no L M charts in this area. (5) F o r H u m o r u m a series of 8 m e a s u r e s of the rim height on the south and east portions o f the basin were made on L A C 93. T h e rim height was determined to be 630 m. (6) Eighty-eight altitudes on the south and east portions o f the H a e m u s M o u n t a i n s were m e a s u r e d on L M s 41 and 42. They averaged 7048 m. A parallel arc on L M 60 which necessarily crossed shallow portions o f the flooded area to the south averaged 6243 m in altitude, a difference o f 805 m. i n contradistinction to H e a d (19791 and Pike and S p u d i s (unpublished) w h o furnished m e with their interpretation of Serenitatis as a five-ring system, 1 c o n s i d e r the ring defined largely by the H a e m u s Mountains to be the main basin rim. T h e Vitruvius ring is higher than the eastern extension o f the H a e m u s ring, but in the south it is very much lower. The 805-m rim height found from the chart is the present rim height. The H a e m u s M o u n t a i n s were seriously modified by flying debris from the younger l m b r i u m event. C o n s e q u e n t l y 1 have arbitrarily used a value of 900 m in the calculations. Later in the paper it will be s h o w n that either higher or l o w e r reasonable values o f the Serenitatis rim height will not yield a significantly different absolute age. (7) T h e rim height of the unnamed basin was estimated from c o m p a r i s o n s with rims o f nearby small craters and from the portion of the rim near W e r n e r and Aliacensis. LAC 95 was used.

There is a series of observations, all interrelated, that when properly analyzed, should yield more information concerning the pre-Imbrium history of the M o o n than has heretofore been forthcoming.

A G E O F L U N A R B A S I N S . I. V I S C O S I T Y

1. Imbrium has been dated as approximately 3.85 × 109 years old. 2. The Moon is approximately 4.6 × 109 years old. 3. The period under discussion is thus about ~ of geologic time. 4. Imbrium is the largest of the seven front face basins. 5. There is an observed progression in relative basin ages. I have previously placed them in order of increasing age (Baldwin 1974) as Orientale, Imbrium, Crisium, Nectaris, Humorum, Serenitatis and unnamed basin. 2 Not all authors agree with the Serenitatis position (Wilhelms 1984). 6. The present absolute height of the highest quadrant of the most prominent ring bordering each basin declines in height from basin to basin with increasing age. These rings are very low for the oldest three basins and relatively high for the others. 7. It is clear that the terra surfaces of the Moon are saturated with impact craters. (Hartmann 1980, 1984) 8. Older craters in these areas exhibit lower and less prominent rims than do newer craters and this effect extends to craters at least as small as 10 km in diameter. 9. Many craters or portions of craters are older than the date of saturation. 10. The slope, A, in the equation for the larger craters: log Arc = A log D + B

(3)

relating cumulative numbers of craters larger than diameter, D (in km), is about - 1 . 8 instead of exactly - 2 (Baldwin 1971a, 1985). 11. All of the seven basins are younger than the date of saturation with the possible exception of the oldest, the unnamed basin. From the above observations it has 2 This basin is shown very nicely on a Lick photograph of Sept. 6, 1936, with the Moon's phase at 20.40 days. It is reproduced as Plate XXIV in "The Measure of the M o o n " (Baldwin 1963).

5

proven possible to determine a dating of events on an absolute time scale. The accuracy of the dating is a function of the assumptions, but as will appear, rather wide variances in the assumptions will yield quite comparable and internally consistent dates. The observed changes in appearance tell us that some process operated, particularly in the early days, on the Moon to reduce the heights of the rims of craters and the great basins. I have been able to imagine only five ways of accomplishing this. The first is by erosion by smaller impacts, but it appears (Baldwin 1963, p. 193) that erosion is not able to account for the simultaneous reductions in rim heights and crater depths for craters up to at least the diameter of Clavius or Baily. Erosion is a real process, but in reducing rim heights in absolute dimensions it should not affect a large crater rim as much proportionately as a small crater rim. The second is that a slow process of creep of lunar materials was responsible. In this paper, I shall treat matters as though all of the effects were produced by creep in a Moon with viscosity increasing with time. A certain amount of erosion must have occurred, but its effects would run approximately parallel to and very much less than the effects of creep in the sense that erosion must have been greatest in the very early times when the flux of planetesimals was highest. A third possibility is that the Moon is elastic, but that the original rims of ancient basins were lower than they would have been if formed at a later time. This solution is considered to be highly improbable. There is no known mechanism to cause this. A crater formed in water is instantaneously not far different (Schmidt 1987) from one formed in rock and recent craters do not differ greatly from those 3.5 × 109 years old. The fourth possibility is that the Moon was layered in the sense that the viscosity was highest at the surface and the underlying layers had lower viscosity. This is the model used by Passey and Shoemaker (1982) in their study of

6

RALPH B. BALDWIN

morphological changes on Ganymede and Callisto. It undoubtedly is a model affecting the early changes on the Moon, but I consider that with the caliber of the data with which we must work, this is a refinement beyond the scope of the present paper. Its inclusion would change the calculated viscosities without significantly altering the determined basin ages. Last, the shock waves from subsequent impacts may have tended to reduce crater and basin rims, but it is difficult to understand how they could raise crater bottoms in nearly elastic rock. The viscosity as measured in this paper is an effective viscosity, necessarily including all the other roughly parallel processes that worked in conjunction with the major process, that of creep. As such, it is used as a tool and not as a single physical process. It may not accurately represent any single process at any time, but can be used to determine absolute basin ages. It is because several phenomena affected the Moon, rather than just one, that for dating purposes we cannot select any one of the several elegant mathematical models of Dane~ (1962, 1965, 1968, 1971), Solomon et al. (1982), or Passey and Shoemaker (1982). Inasmuch as many of the processes operative on the Moon declined in nearly an exponential fashion a simple exponential equation (4) was selected as appropriate. Scott (1967) worked with equations by Haskell (1937) and put them into a form suitable for use with lunar craters. He then tested the equations by producing small craters of three different sizes in asphalt heated to 60°C and maintained at that temperature so that the viscosity would be 10 6 dyne-sec/cm2. Under the influence of gravity the craters decayed as expected. Scott stated that " a time of about 4 x 10 6 years is required for the complete disappearance of a 10 km diameter crater in rock of viscosity I0 zz poises on the Moon. {The analytical model assumes, of course, purely viscous flow, and the analysis develops from this; other results would be obtained if other time-dependent material behaviors were to

be postulated. In particular, a residual crater would be left in a viscoplastic material when shearing stresses had decayed to the threshold value.}" I shall assume that changes in the lunar layers involved in the distortion of craters and basins can be represented as consistent with an effective viscosity at any single time and that this viscosity changed inversely with time. If the extrapolated time of 4 x l 0 6 years found by Scott is in error it will not change the determined absolute ages found in this paper. It will change the viscosities found, because the viscosity and time are inversely related and time is closely constrained by the present, the absolute age of Imbrium, and the fact that the Moon is about 4.6 x 109 years old, and the oldest visible craters are somewhat younger. Scott's equation may be approximated very closely by the equation Y = 1.007e -4~x - .007,

(4)

where the vertical dimension, Y, varies from 1 to almost 0 and represents a height dimension while the horizontal dimension, X, varies from 0 to 10 and represents time. I f X = 0, Y = 1, while i f X = 10, Y = .001287. X and Y represent dimensionless numbers, but they are easily converted into meters for Y and time for X. For every corresponding increment in X, Y declines by a constant percentage. Y represents the remaining rim height. The equation may be used with any applicable viscosity on any lunar crater of whatever size and age, provided certain conditions are met. It has been known for some time (Baldwin 1968) that the Moon possesses an elastic limit or threshold stress and that the threshold is lower for earlier epochs than later. It is not well defined, but was found to approximate 3 to 5 × 107 dyne/cm 2 with the smaller value at an earlier time. Its effects should be included in any analysis of lunar viscosity changes and determinations of absolute crater ages, although wide

AGE OF LUNAR BASINS. I. VISCOSITY

7

noted that for very ancient large craters to remain visible, the viscosity must have been very much greater than 1022 poises throughout the entire period. Weertman (1970) derived an "effective viscosity" for the outer layers of the Moon ranging from 1025 poises at 100 km depth to 1027poises at about 60 km and higher values nearer the surface. Baldwin (1970, 1971a, 1971b) found an average viscosity of at least 1026.5 poises increasing with time from 1024 to more than 1027poises, while Arkani-Hamed (1973a, 1973b) determined a minimum H = -253T + 1405, (5) value of 1027 poises. Scott (1967), Haskell (1937), and Dane~ where T is the absolute age in units of 109 years and H is equal to the height of a (1962, 1965, 1968, 1971)each recognized column of lunar rock that could be sup- that the structure of a lunar crater would ported by the threshold strength at each become modified due to slumping at a rate corresponding time. The present-day limit proportional to crater diameter and also according to Eq. (5) is equivalent to about proportional to the absolute height and in1405 m of lunar rock of density 2.9 and to versely with the viscosity. Figure 1 is evidence that little subsidence 300 m at about 4.36 x 109 years (arbitrary, but early). The present-day value cor- of crater rims occurred after 2.5 x 109 years responds to 5.75 × 107 dyne/cm2 or 57.5 ago, while the low and subdued contours of bars, while at 4.36 × 109 years the figure is very old craters indicate that the lunar viscosity was low enough in the earliest 12.7 bars. Arkani-Hamed (1973a, 1973b) derived a times to permit even rather small craters to strength of about 50 bars for the upper 600 become diminished, but not so low as to km of the lunar interior for the last 3.3 × 109 cause large craters and basins to vanish. The work by Scott confirms that even at the years. These values seem reasonable because time of saturation the lunar viscosity was the rim of the oldest basin has decayed to very much greater than 1022 poises. an approximate 350 m while the youngest III. THE PROCEDURE basins have not yet reached the limiting The first step was to confirm that the stress. The rims of very ancient craters at, or even before, the age of saturation, are viscosity of the Moon was variable with low, but have not disappeared. As Table I time. The history was broken up into shows, rims of Nectaris, Serenitatis, Hu- several long periods: morum, and the unnamed basin have rea1. the last 2.5 x 109 years; ched intermediate heights and the older 2. f r o m 3.85 x 109 years (the formation of basins have declined more than Nectaris. Imbrium) to 2.5 x 109 years; This is interpreted to mean that the four 3. from 4.17 × 109 to 3.85 X 109 years; oldest basins subsided until their rims 4. from the unknown date of formation of reached the heights corresponding to the the unnamed basin to 4.17 x 109 years. then threshold strength of lunar materials and thereafter did not sink farther. A value of the present rim height of Is there any other evidence concerning Imbrium of 2562 m is listed in Table I. the viscosity of the Moon's outer layers? Judging from the smaller normal craters Yes. It is very high. Baldwin (1963, p. 423) little, if any, subsidence occurred in the last

variations in the threshold strength would produce only relatively minor changes in derived absolute ages (see Appendix). Such a residual strength is actually a valid approximation to non-Newtonian flow, in which relaxation proceeds at a progressively slower rate as stress differences decline. A recent unpublished revision of the 1968 work has reduced the limiting stresses slightly. For calculational purposes I have specified a linear equation

8

RALPH B. BALDWIN

2.5 x 109 years. Crisium and Nectaris have considerably lower present rim heights than Imbrium. The threshold strength of lunar rocks as defined cannot have been greater than the present rim height of Crisium, around 1758 m. If it were as great as that figure the age o f Nectaris would be found to be improbably high. Therefore, it was concluded that the threshold strength was not the limiting factor for Orientale, Imbrium, and Crisium. Based on analysis of Fig. 1 it was arbitrarily assumed that the Imbrium rim sank only 25 m in the last 2.5 x l09 years. Selection of any reasonably larger amount of sinking at Imbrium would only affect the viscosity determinations in the post-Imbrium period. Ages of the older basins would not be affected, while the age of Orientale would be slightly increased. We thus have values for the rim of Imbrium as follows: 1. 4556 m at 3.85 × 109 years; 2. 2587 m at 2.5 x 109 years; 3. 2562 m at 0. These data will be used to determine lunar viscosities since 3.85 x 109 years and then an absolute age for Orientale. Similarly several arbitrary dates will be selected for the unnamed basin and corresponding viscosities calculated for the period prior to 3.85 x 109 years. To do this we must evaluate in Eq. (4) the X and Y variables derived from Scott. This requires a calculated time Xi = ((4

× 10 6) years × N poises × I0 km)/(10 22 poises x Dkm),

(6)

where N is the viscosity at each point in time and Dkm is the rim to rim diameter of the crater or basin being evaluated. XI then is the numerical equivalent of X, the actual time in years for a structure of a given size, with a viscosity of N, to b e c o m e almost fully eliminated, i.e., for X to go from 0 to I0. Next, an incremental time, X2, was calculated as a decimal of Xj,

Xz = Q x X1,

(7)

where Q is .00001 from 0 to 2.5 x 109 years, .0001 from 2.5 to 3.85 x 10 9 years, .001 from 3.85 to 4.4 x 109 years. A continuous curve would have served equally well. It proved necessary to allow Q to vary in some such m a n n e r as a function of time to permit the iteration to be completed on the computer in a reasonable time. Last, the absolute age of each increment in units of 10 9 years was calculated for each iteration as T = T1 - (X2/I09),

(8)

where Tr is the value of T in the previous iteration. Since for each instant of time, T, we know the value of X, we know the corresponding value of Y. Due to the nature of Eq. (4), Y subsides by a constant proportion for each equal increment in X. In the present calculation no rounding was used because of the thousands of iterations, so the calculational errors were minimized in the three cases and the three factors corresponding to each incremental time, X2, were .999951665158 for the period 0 to 2.5 x 109 years, .999516755986 for 2.5 to 3.85 x 109 years, .99517798210 for 3.85 to 4.4 x 109 years.

IV. THE VISCOSITIES

The first requirement was to determine the values of c o n s t a n t viscosity in each time period, using the data for Imbrium, that would yield the observed rim settling. To implement, I determined the single viscosity that would allow the original rim height of Imbrium to reach 2587 m at 2.5 x l09 years. It was found that for the constant viscosity case the required value was 6.309 x 1027 poises. A similar procedure gave a constant viscosity of 3.610 x 1029 poises for

AGE OF LUNAR BASINS. I. VISCOSITY 1

31

2

3

4

AGE (109'y)

30 29 28

ct~ v

27 26

>

25 24

FIG. 2. T h e c o n s t a n t viscosities within each time period that would be required for I m b r i u m and Hum o r u m to reach their present-day rim heights. Five a s s u m e d ages o f H u m o r u m are s h o w n . T h e curved line is the accepted solution.

the period from 2.5 x 109 years to the present. The great difference between the two makes it clear that the viscosity was variable in the post-Imbrium period. This is illustrated in Fig. 2. This result does not follow from the choice of 25 m-settling at Imbrium after 2.5 x 109 years. Any possible amount of settling would result in widely different viscosities. Only in the case of an elastic Moon would the two lines be identical (at infinity), and this implies no settling in 3.85 x 10 9 years. The next attempt was to see if the unnamed basin could be brought into a consistent picture. Its calculated original rim height was 3063 m and at present it averages about 350 m. Its age is great, but unknown. The basin is covered with craters at a density roughly equal to that of a saturated surface. Consequently a series of five arbitrary dates for the age of this basin were assumed. They were 4.40, 4.35, 4.30, 4.25, and 4.20 × 10 9 years. Values of constant viscosity were calculated for the unnamed basin from each of the five assumed dates for the period to 4.17

9

× 109 years, the date according to Eq. (5) when the threshold strength of the Moon corresponded to 350 m of lunar rock. These viscosity values are plotted as horizontal lines in Fig. 2 and in every case they are very much lower than subsequent viscosities. When Fig. 2 is examined carefully it is amply clear that the viscosity of the outer layers of the Moon varied tremendously throughout geologic history. However, it is inconceivable to expect a series of discontinuities in viscosity, so the actual changes must have been progressive throughout time. Consequently the main effort was to determine a series of interconnected linear equations relating changes in viscosity from 3.85 to 2.5 × 10 9 years and from 2.5 x 10 9 years to the present and then to find if it were possible to discriminate among the theoretical ages assigned to the unnamed basin for test purposes. Linear equations were selected for two reasons. First, the equations depart from a cubic spline curve by relatively small amounts because of the large viscosity differences in each time interval. Second, the use of a cubic spline would have involved a very much larger number of approximations and iterations and it was not felt that a significant increase in accuracy would result. From Fig. 2 it was determined that the probable value for the viscosity at 2.5 x 10 9 years lay between 6.5 x 10 27 and 3 x 10 29 poises. Numerous test values within these limits were selected. For each of them there is a corresponding viscosity at 3.85 x 10 9 years that would permit the Imbrium rim to sink from 4556 to 2587 m. Six selected sets of results are listed in Table II, columns 2 and 3. The value of the viscosity at 2.5 x 109 years defines that at 0 time under the assumptions of this paper. Table II illustrates the results of a great many trials. I next solved for the various possible combinations of viscosity at 4.17 x 109 years for each of the five selected ages of the unnamed basin, using the

10

RALPH B. BALDWIN TABLE II DERIVED VISCOSITIES FOR NINE POSSIBLE SOLUTIONS

0

2.5

A g e s (in units of 109 years) 3.85

4.17

1.902 x 1030 p

1.066 x 1028

5.000 x 1026

9.210 X 1025

1.775 X 1030 1.830 X 1030 1.855 X 1030

1.441 × 1028 1.276 × 1028 1.213 × 1028

2.000 X 1026 3.000 × 1026 3.500 X 1026

9.210 × 1025 9.210 × 1025 9.210 X 1025

(4.35) 2.36 x 1025 (4.30) 9.46 x 1024 9.46 × 1024 9.46 × 1024

1.880 x 10 30 1.902 x 103° 2.020 × 1030

1.150 x 1028 1.066 x 1028 7.830 X 1027

4.000 X 10 26 5.000 X 1 0 2 6 1.000 X 1027

9.210 x 1025 9.210 x 1025 9.210 X 1025

9.46 x 1024 9.46 x 10 24 9.46 x 1024

1.902 x 1030 2.020 × 1030

1.066 x 1028 7.830 × 1027

5.000 x 1026 1.000 × 1027

6.718 × 1025 6.718 × 1025

(4.25) 4.00 x 10 24 4.00 × 10 24

dimensions o f this basin. Only the m o s t p r o b a b l e solutions are s h o w n in Table II. M a n y others were discarded. If each of the several viscosities determined for 4.17 x 109 years are joined with those found for 3.85 x 109 years we would obtain a series of linear equations covering the entire postsaturation period on the Moon. It remains to be seen if m o s t of these equation sequences can be eliminated and a m o r e definitive solution reached. As a first step nine possible combinations were decided u p o n and then for each case the evolution of the I m b r i u m rim resulted in the first three viscosities of each line. The values in line 4 were selected as best fitting the data, for reasons that will soon be shown. Although any of the derived solutions for an age o f the u n n a m e d basin of 4.4 x 109 years would lead to the o b s e r v e d ending rim heights of this basin and I m b r i u m and ages for the other basins could be derived, they would tend to be substantially older than those listed in Table III. The initial viscosity would h a v e been quite a bit larger than that for 4.17 x 109 years. Such a conclusion, while possible, does not seem probable. It is doubtful that the outer layers

were m o r e viscous at a time so close to the origin of the M o o n than a quarter of a billion years later. H e n c e it was concluded that the u n n a m e d basin could not be as old as 4.4 x 10 9 years. The best solution for an age of the unn a m e d basin of 4.35 x 109 years is given in the first line. The second through the seventh lines give possible solutions for the viscosity at each date if the u n n a m e d basin were f o r m e d 4.30 x 109 years ago. T h e r e are two possible solutions shown for a starting date o f 4.25 x 109 years and no reasonable solution was found if the basin were as recent as 4.20 x 109 years because the starting viscosity would h a v e had to be lower than 1021 poises. It is clear f r o m this analysis that the effective viscosity increased substantially f r o m the time the u n n a m e d basin was formed. A viscosity of 1021 poises or less at the time the u n n a m e d basin was formed would have caused m o s t prior craters to have essentially disappeared. This did not happen and so the presaturation viscosity was at least l022 poises. A m u c h lower viscosity in m u c h earlier times would be consistent with the conclusion of Davis and Spudis (1985) that there had been t r e m e n d o u s

AGE OF LUNAR BASINS. I. VISCOSITY widespread early liquid rock covering much of the lunar surface. With the viscosity solutions of Table II we may determine the corresponding absolute ages for each solution. They are listed in Table III. For each of the nine viscosity solutions the procedure was to find for each basin the only absolute date that would allow the original rim height to subside to the observed present-day height, bearing in mind that for the four oldest basins they would only subside until the threshold limit was reached. TABLE

III

A B S O L U T E A G E S OF S E V E N BASINS (10 9 YEARS) Unnamed

Humorum

Serenitatis

Nectaris

Crisium

lmbrium

Orientale

4.35 4.30 4.30 4.30

4.26 4.19 4.22 4.23

4.17 4.06 4.12 4.14

4.11 4.00 4.05 4.07

4.03 3.94 3.97 4.00

3.85 3.85 3.85 3.85

3.82 3.83 3.83 3.82

4.30 4.30 4.30

4.24 4.25 4.26

4.15 4.17 4.21

4.08 4.11 4.17

4.00 4.03 4.11

3.85 3.85 3.85

3.82 3.82 3.79

4.25 4.25

4.21 4.23

4.16 4.19

4.10 4.16

4.03 4.11

3.85 3.85

3.82 3.79

The following solution was derived using Pike's (1980) earlier crater dimensions with considerably lower rim heights. 4.30

4.22

4.15

4.08

4.00

3.85

3.82

This comparison was placed in Table III to illustrate the fact that as long as the larger basins are assumed to have had higher initial rim heights than smaller basins, similar to values derived from Pike's (1980) or (1983) equations, little difference in the absolute age of each basin may be derived. Pike's (1983) equation calls for absolute initial rim heights greater than those from the earlier equation by amounts ranging from 203 to 615 m. Although Table III gives the derived absolute ages for each basin for each set of viscosities, line 4 represents the accepted solution.

11

There is one other factor that aided in the determination of which solution was deemed the most probable. Present-day rim heights of ancient craters are poorly known. I have selected from the LM charts as first choice, and LAC charts where nothing better was available, a list of 13 craters that appear to be about as ancient as the unnamed basin, i.e., almost back to the date of saturation. In each case the crater is so old that its origin is way back in the low viscosity period. Most of the sinking of the rims occurred early. Because the rim of a crater sinks at a rate that is proportional to the crater diameter, craters of differing sizes, but similar ages, will show present rim heights that differ considerably. Figure 3 shows as dots the observed present rim heights of the 13 craters plotted against crater diameter. Because of the faster rate of sinking, the larger craters have subsided by a bigger actual and percentage amount than the smaller. The curved line represents the calculated ending rim heights for craters of varying sizes if they were all the same age as the unnamed basin, 4.30 × 10 9 years, and the viscosities were those of the accepted solution. Although the points on Fig. 3 scatter rather widely, it appears that the same trend is followed. A similar relationship was determined for each of the rejected viscosity solutions and although the closer the viscosity equations were to the accepted values the closer the fits were to the points of Fig. 3. None of the other equations gave quite as close a fit as did the accepted set of equations (also see Table IV). V. COMPARISON AGES

OF DERIVED

BY VARIOUS

ABSOLUTE

AUTHORS

Other scientists who have endeavored to assign absolute dates to the great basins have done so based on the ages derived for lunar materials brought back to earth. Some of their results are listed in Table V with only one comment. They may be compared to a 1974 set of approximate absolute ages of mine and to the values of Table III.

12

RALPH 100 ,w " ~ " , \\

?_~

2OO

400

300

B. B A L D W I N 500

TABLE V

Grater Diameter (km)

ABSOLUTE BASIN AGES(109yEARS)

7®i Basin

\\ \

Baldwin (1974)

~...°

Jessberger

Maurer

e t al.

et al.

(19741"

(19781

E if: 400-

SchaeferHusain (19741

Nunes e t al.

(19741

Old constants Orientale lmbrium Crisium Nectaris Serenitatis Humorum

200~

FIG. 3. Present-day rim heights of 13 craters from Table IV vs diameter. Each of these craters is approximately the same age as the unnamed basin, 4.3 x l 0 9 years.

3.80 3.95 4.05 4.21 4.25

-3.82 3.84 3.85 3.88

-3.88 b 3.98

3.85 4.00 4.13 4.20

3.85 3.99 4.13 4.2 4.45?

4.25

3.89

--

4.13~1.20

4.13 4.2

a Jessberger wrote to me on N o v e m b e r 8, 1983, as follows: "Concerning m y list o f absolute basin ages 1 first will state the following caveat that from geochronological information alone it cannot be excluded that at all landing sites the 3.82-3.89 ages are related to the lmbrium event. 1 do not, however, favor this interpretation as we argued in our 1974 paper." b Between lmbrium and Nectaris. ' Older than Nectaris.

VI. O T H E R I N T E R P R E T A T I O N S

As has been made clear throughout this paper, the selection from viscosities that were specifically identified in Table II are considered to be the m o s t probable o f a series o f solutions. Other interpretations T A B L E IV ANCIENT

Name

Unnamed basin Deslandres Unnamed Hipparchus Unnamed Hevelius Romer P Descartes Romer S Macrobius M Lindsay B Andel A Wilhelm J

CRATER

Diameter (km)

RIM HEIGHTS

Present rim heights (m)

Notes

500

350

1

234 183 150 143 106 60.7 48 43.7 41.7 37 27 19.2

500 450 600 600 796 600 600 600 750 600 450 357

2 3

can be made, but in m o s t cases the ages found do not differ greatly from those accepted as most probable. I regard any late age for H u m o r u m and the' unnamed basin as improbable. Table VI s h o w s in the last c o l u m n the accepted values o f absolute ages for the s e v e n basins. It also s h o w s what would have been the effects on ages if the present rim heights were s o m e w h a t different than the accepted values. VII. C O N C L U S I O N S

4 5

6

Note. (1) Unnamed basin between Werner and the Altai ring. (2) Average of two measures. (3) Includes the Straight Wall. (4) Apollo landing site is centered on this crater. (5) Average of 15 measures. (6) Average of three measures.

It has been found that the M o o n ' s surface layers p o s s e s s e d a threshold strength and an effective viscosity, each variable with time. The viscosity w a s calculated in various w a y s and o n e solution in Table II was selected as fitting the observations well. It is not the only possible solution, but I consider it to be the most probable. Other possibly viscosity solutions will yield similar absolute basin ages. From the data of Table II, absolute ages were derived by a long iterative program on a c o m p u t e r for s e v e n basins. The accepted results are in Table III and the last column o f Table VI. T h e s e absolute ages are quite comparable to those found from interpretations o f radiometric ages of returned rock

AGE OF LUNAR BASINS. I. VISCOSITY TABLE ABSOLUTE

AGES

VI (10 9 YEARS)

Basin

Calculated original rim height (m)

Possible present rim height (m)

Mea s ured age

Calculated age

Ac c e pt e d age

Orientale

4105 4556 3176 3956

Serenitatis

3390

--3.85 -----

3.82 3.88 3.85 4.00 4.07 4.08 4.08

3.82

lmbfium Crisium Nectaris

2725 2500 2562 1758 1038 950 1000

-------

4.14 4.16 4.23 4.30 4.34 4.37

Humorum Unnamed

2837 3063

900 805 630 350 325 300

3.85 4.00 4.07

4.14 4.23 4.30

Note. C a r l s o n and L u g m a i r (1979b) have detected a " l a r g e cataclysmic e p i d s o d e " from various Apollo 17 highland samples. The age was determined as 4.18 + / - 0.04 x 109 years. T h e Apollo landing was immediately adjacent to Sereoitatis. T h e similarity in their age and that derived in this p a p e r is interesting.

13

of observations that must be accounted for before any lunar "cataclysm" can be seriously considered. No answers have been forthcoming. The results of the present study are consistent with my 1974 paper. On the present paper's results, the seven major basins are distributed over about 500 myr and they clearly are of widely differing ages. There are still older basins, and some younger, visible from the Earth that are generally regarded as real. They include Humboldtianum, Nubium, Smythii, Foecunditatis 1 and 2, Grimaldi, the unnamed basin including Schiller, and probably others. They were not included here because of the lack of data on present-day rim heights. Crater counts by Hartmann (1966, 1967, 1968, 1969), Baldwin (1969a, 1969b, 1970, 1971a, 1985), and Neukum et al. (1975, 1976) indicate that the smaller normal craters declined in rate of formation as time progressed toward the Imbrium date of 3.85 x 109 years. There is no indication from the counts of normal craters or from counts of the large basins that, except for Orientale and Imbrium, there was any clumping of impacts as would be required under the "Terminal Lunar Cataclysm" hypothesis.

samples by Schaeffer and Husain (1974), Nunes et al. (1974), as well as my earlier work, Baldwin (1970, 1971a, 1981). They differ drastically from the conclusions of Jessberger et al. (1974) and Maurer et al. (1978). It should be noted that if the age of the VIII. COMPARISON WITH THE EARTH unnamed basin is not 4.30 × 1 0 9 years, but is older or younger, the relative ages of the The Earth, of course, is much larger than basins will not be affected. The absolute the Moon. It developed more internal heat ages will be changed proportionately less than the Moon and to this day the Earth has than the change in the date of the unnamed a hot interior extending close to the surface basin. The dates for Imbrium and Orientale while the Moon has a deep, cool set of will not be changed. outer layers extending down to perhaps 800 As has been argued previously (Baldwin km, the region of the microseismic shocks. 1974, 1981, Neukum et al. 1975, Hartmann The inner parts of the Moon are hot and 1975), the suggestion by Tera et al. (1973, there possibly may be a small liquid core 1974a, 1974b) that there was a "Terminal (Runcorn 1983). Lunar Cataclysm" simply cannot be corThe reason for this digressign is that rect. The concentration of observed ages of unless the Moon and Earth were formed in lunar rocks at nearly 4 x 1 0 9 years ago is widely differing environments and time, the not a determinant of a "Terminal Lunar outer layers of the Moon must have hardCataclysm." It must be a result of the ened sooner than those of the Earth and if formation of the huge Imbrium basin possi- flooded by very early liquid rock as seems bly aided and abetted by the formation of probable (Baldwin 1963, p. 427, Hartmann Orientale. In Baldwin (1974) there is a list 1980, Spudis and Schultz 1983, Davis and

14

RALPH B. BALDWIN

Spudis 1985), these layers also would have cooled and solidified rather more quickly than they did on Earth. In this paper the ages of the basins have been found back to 4.30 × 109 years with other older structures listed, but undated. The question may properly be raised: Are these early dates on the Moon consistent with what we know about the early Earth? The geological activity of the Earth has destroyed or buried deeply most of the evidences of the first few hundred million years. When the exposed early Archaean rocks are found and dated we continually are impressed by the fact that the early Earth was subjected to much erosion, presumably by water and thus the outer layers were not only solid, they were cool, but with much igneous activity. As late as 1966 the oldest known rocks on the Earth were about 2.8 x 10 9 years old (Moorbath 1977, summary paper). In that year a young New Zealand geologist, V. R. McGregor, working with the Geological Survey of Greenland, found a distinctive sequence of geological events including some he felt were considerably older than 2.8 X 10 9 years. These Am~tsoq gneisses were dated at Oxford as 3.75 x l 0 9 years. Subsequently it was found that there had been erosion of volcanic lavas and deposits of sedimentary rock, some under water, about 3.8 z 10 9 years ago and portions of them had formed certain inclusions in the Amitsoq gneisses. There the matter rested for a few years until Froude et al. (1983) measured the ages of four detrital zircons from quartzites at Mt. Narrayer in Western Australia. The measured ages range from 4.090 to 4.190 x 10 9 years and they stated that "all four claims may be interpreted as having the same age, 4150 M a . " This age is greater than those found for five of the lunar basins. No evidence of a "Terminal Terrestrial Cataclysm" has been found and I do not expect that one will be found. After all, it would take an impact much greater than that of Imbrium to leave

widespread massive traces in early Archaean rocks. Cameron and Ward (1976) have suggested an even more massive collision with the Earth, but, if real, it was much earlier than the date of the unnamed basin. Van Niekerke and Berger (1983) have recently reported some African rock of about the same apparent age, 4.1 x 10 9 years. Because of the great age of the zircons and the fact that they were eroded from their environment and reincorporated into slightly younger rock, it is clear that the Earth's surface more than 4.1 × 10 9 years ago was hard, cool, and subject to erosion. Thus there is no conflict between the terrestrial data and the ages found here for the lunar basins. IX. THE NATURE OF"THE VISCOSITY Slow settling of crater and basin rims or rising of their interiors under the relatively low loads impressed by lunar gravity implies a form of viscosity of the Moon's rocks that cannot (as yet) be adequately tested under laboratory conditions. Analyses of changes in altitudes on the Earth such as the Fenno-Scandian postglacial uplift (Haskell 1937) called for a viscosity of the aesthenosphere that is from 3 to 8 orders of magnitude less than those derived for the Moon in this paper and elsewhere (Baldwin 1970, 1971b, Weertman 1970, Arkani-Hamed 1973a, 1973b). There is no near unanimous opinion that the viscous flow is Newtonian or nonNewtonian (Melosh 1980). Numerous papers have researched these matters (Robertson 1964, Weertman and Weertman 1975). All that can be said is that the Moon's outer layers over a large span of billions of years have moved to reduce altitude differences in a manner that suggests that the viscosity is very high and has been increasing throughout geologic time. This interpretation has allowed the determinations of the ages of the seven basins. One question, in particular, remains. Why is the Moon's viscosity so much

AGE OF LUNAR BASINS. I. VISCOSITY

15

higher than that of the Earth? There are laboratory and at high temperatures the two indications. It is generally agreed in strength of dunite. They state, "The analyses of the viscosity of different amounts of water that have been found to materials (Weertman and Weertman 1975) be of importance for the strength of dunites that the hotter the material the lower the are of the order of 0.01 wt %, amounts viscosity. On Earth the temperature rises much less than most estimates of the water rapidly with increasing depth to a zone of content of the earth's undepleted upper maximum fluidity at a depth of about 100 mantle. Such water may have an important km and a temperature of 1200°C. Melosh role in determining upper mantle flow be(1980, p. 327) states, "The concept of the havior." It is not known that the Moon ever lithosphere must be treated with care. We had any water in its rocks. The increase in observed effective visuse the term here to denote the elastic portion of the upper mantle or crust. Its cosity of the Moon's outer layers throughthickness is thus a function of load du- out geologic time could be a result of the ration: for very-short-term loads of less passage of time, the slow cooling of the than a few years the entire mantle is elastic rocks coupled with the gradual elimination and the ((lithosphere)) extends to the of traces of water by exposure to a vacuum. outer core. On a time scale of 104 y the It appears that as Melosh suggested, a elastic portion is about 100 km thick (it new phenomenon is indicated. This is the extends to the depth where the Maxwell ultraslow settling of altitude differences in time equals the load duration), while for the "elastic" portion of a viscoelastic loads lasting 106 y or more, the elastic material over geologic time. lithosphere may be only 15 to 30 km thick APPENDIX (Walcott, 1970). These ideas, which involve temperature as a regulator of rheology, are It has been suggested that the straight incomplete: If the Maxwell time is a func- line of Eq. (5) might not accurately reption of stress as well as temperature, then resent the variations in threshold strength the effective viscosity of the Earth and the over time and that this might affect the lithosphere's thickness are functions of determinations of age. stress as well as duration. The idea that I believe that the threshold strengths at stress plays a role in determining the me- each end of Eq. (5) are fairly good and have chanical properties of the lithosphere has used 1405 and 300 m respectively in each of not been much explored: This is a result of the following analyses. the Newtonian emphasis of most of the Four equations have been derived; two for curves lying below the straight line of current models." If the Moon were structured as the Earth, Eq. (5) and two for curves lying above it. the ancient basins and craters would be gone long since. The so-called "elastic portion" must be very much thicker than on T A B L E VII the Earth and the temperature gradient far Basin Upper curves L o w e r curves lower. Any changes in the rim height must U n n a m e d 4.30 x 109 years 4.25 4.30 4.25 necessarily occur in the "elastic portion" H u m o r u m 4.25 4.22 4.26 4.22 which thus cannot be strictly elastic, but Sereni4.15 4.13 4.15 4.13 must react over geologic time as if it were tatis Nectaris 4.08 4.07 4.08 4.06 viscous. Crisium 4.00 3.99 3.99 3.99 A second cause of the higher viscosity of Imbrium 3.85 3.85 3.85 3.85 lunar rocks may be suggested by Chopra Orientale 3.82 3.82 3.82 3.82 and Paterson (1984) who investigated in the

16

RALPH B. BALDWIN

The end points in each case were 4.25 or 4.30 × l 0 9 years. The results o f the age determinations are listed in Table VII. As m a y be seen, the results are quite comparable to the accepted solution in line 4 o f Table III and on this basis alone no c h o i c e could be made. H o w e v e r , if we m a k e the c o m p a r i s o n with the rim heights o f smaller craters as w a s done in the article proper, it is found that the lower curves do not agree at all with the observations o f rim heights s h o w n in Fig. 3. The line is completely a b o v e the o b s e r v e d points for each starting age. The case is different for the two upper curves. Each is similar to the accepted result s h o w n in Fig. 3, but the curve for a starting age o f 4.25 × 109 years is not as good a fit as the one starting at 4.30 × 109 years. I cannot c h o o s e b e t w e e n Eq. (5) and the following equation effective from 4.30 × 109 y e a r s .

H=

-33.721T 2-

111.98T+

1405.

(9)

In no case do the ages found with Eq. (9) differ from the accepted values by more than 20,000,000 years or 0.02 × l09 years. Therefore the ages found in line 4 o f Table III will remain as the accepted values. ACKNOWLEDGMENTS I am grateful to Richard Pike, who furnished data on lunar craters and basins, and to H. J. Melosh for the numerous reprints and references to pertinent articles and for welcome suggestions for improving the text. Paul Spudis recommended improved methods of data collection and furnished numerous maps and charts. 1 particularly want to thank Z. F. Dane~ who sent reprints and much correspondence. He graciously allowed me to visit him for 2 days in Tacoma during the middle of one of his busiest periods to discuss the problems and approximations made in this paper. REFERENCES ARKANI-HAMED, J. 1973a. Viscosity of the Moon. I. After mare formation. Moon 6, 100. ARKANI-HAMED, J. 1973b. Viscosity of the Moon. I1. During mare formation. Moon 6, 112. BALDWIN, R. B. 1949. The Face o f the Moon, p. 136. Univ. of Chicago Press, Chicago.

BALDWIN, R. B. 1963. The Measure o f the Moon. Univ. of Chicago Press, Chicago. BALDWIN, R. B. 1968. A determination of the elastic limit of the outer layers of the Moon. Icarus 9, 401-404. BALDWIN, R. B. 1969a. Absolute ages of the lunar maria and large craters, I. Icarus 11, 320-331. BALDWIN, R. B. 1969b. Ancient giant craters and the ages of the lunar surface. Astron. J. 74, 570. BALDWIN, R. B. 1970. Absolute ages of the lunar maria and large craters. II. The viscosity of the Moon's outer layers. Icarus 13, 215-225. BALDWIN, R. B. 1971a. On the history of lunar impact cratering: The absolute time scale and the origin of planetesimals. Icarus 14, 36-52. BALDWIN, R. B. 197lb. The question of isostasy on the Moon. Phys. Earth Planet. Inter. 4, 167-179. BALDWIN, R. B. 1974. On the accretion of the Earth and Moon. Icarus 23, 97-107. BALDWIN, R. B. 1981. On the origin of the planetesimals that produced the multi-ring basins. Proc. Lunar Planet. Sci. Multi-Ring Basin Cot~[~ 12A, 19-28. BALDWIN, R. B. 1985. Relative and absolute ages of individual craters and the rate of infalls on the Moon in the post-lmbrium period. Icarus 61, 63 -91. BALDWIN, R. B. 1987. On the relative and absolute ages of seven lunar front face basins. II. From Crater Counts. Icarus 71, 19-29. CAMERON, A. G. W., AND N. R. WARD 1976. The origin of the Moon. Lunar Sci. VII, 120-122. CARLSON, R. W., AND G. W. LUGMAIR 1979a. Sm-Nd constraints on early differentiation and the evolution of KREEP. Earth Planet. Sci. Lett. 45, 123-132. CARLSON, R. W., AND G. W. LUGMAIR 1979b. Early lunar history recorded by norite 78236 (abstract). In Papers Presented to the Conference on the Lunar Highlands Crust, pp. 9-11. Lunar and Planetary Inst., Houston. CHOPRA, P. N., AND M. S. PATERSON 1984. The role of water in the deformation of dunite. J. Geophys. Res. 89, 7861-7876. DANEg, Z. F. 1962. Isostatic Compensation o f Lunar Craters. Res. Inst. Rept. RIR-GP-62-1. University of Puget Sound, Tacoma, Washington. DANEg, Z. F. 1965. Rebound Processes in Large Craters. U.S. Geol. Surv. Astrogeol. Stud. Ann. Prog. Rept. July l, 1964, to July l, 1965. Part A. Lunar and Planetary Investigations, pp. 81-100. DANEg, Z. F. 1968. On slow thermal convection in a layer of variable viscosity. Icarus 9, 8- I 1. DANE~, Z. F., AND D. R. MCNEELY 1971. Possibility of a layered Moon. Icarus 15, 314-318. DAVIS, P. A., AND P. D. SPUDIS 1985. Petrologic province maps of the lunar highlands derived from orbital geochemical data. Proc. Lunar Planet.

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