- Email: [email protected]
Severe insulin resistance associated with subcutaneous amyloid deposition
:caaus 5, 590-605 (1966)
A Stochastic Model of the Formation and Survival of Lunar Craters V. Approximate Diameter Distribution of Primary and Secondary Craters A L L A N H. MARCUS Case Institute of Technology, Cleveland, Ohio and University of Cambridge, Cambridge, England Communicated by A. G. Wilson Received March 29, 1966 Approximate upper and lower bounds are obtained for the expected number density of lunar craters by means of a model which takes into account the formation of primary and secondary craters and the destruction of craters by obliteration and filling. Some numerical examples considered are relevant to primary and secondary craters formed by meteoroidal impacts. Predicted number densities are compared with crater-diameter distributions from photographs taken by the Ranger 7 spacecraft. The shape of the observed distribution is the same as that of the predicted distribution, but the observed density of small craters is about 15 times larger than that predicted by analogy with terrestrial explosion craters. If the observed excess is real, then either some primary craters produce an unusually large number of secondaries, or else many of the smaller lunar craters are of internal origin. The excess may be explained in part by incompleteness of secondary crater counts for terrestrial explosion craters upon which the model functions are based. further approximation for the expected number density of all observable craters with the In previous papers (Marcus, 1966a and assumption that both types occur and cannot 1966b) the author developed formulas for be distinguished from each other. (approximately) the expected number denIn an earlier paper (Marcus, 1964) the sity of all observable craters as a function of the crater diameter x. It was assumed in author obtained an exact formula for the these papers that only primary craters can expected number density of "clean" craters, occur. This m a y be nearly correct for craters i.e., those whose perimeters are not damaged larger than 5 km in diameter. But the prob- by other craters. These results were late," able "secondary impact" origin of m a n y revised to include the effects of spontaneous smaller craters, for example, those surround- disappearance of craters due to filling (Marcus, 1966b), still retaining the exac! ing Copernicus • (Shoemaker, 1962; Dodd, nature of the solution. When secondary craSalisbury, and Smalley, 1963), was already ters can occur, various auxiliary functions known b y the time photographs transmitted appearing in that formula become complib y the Ranger 7 spacecraft resolved lunar cated nmltiple integrals which cannot be re" r a y s " into regions with extremely high denduced to elementary functions. This result sity of small craters. Much preliminary stais therefore of little practical use in studying tistical data is already available for these small craters (Miller, 1964; Shoemaker, the effects of secondary crater formation. t965; H a r t m a n n , 1965; Osborn, 1965; Brink- Nevertheless, these earlier papers contain mann, 1966). These data will be discussed most of the notions and notation to be used later. I t now seems worthwhile to develop in the remainder of the present paper. 590 1. INTRODUCTION
STOCHASTIC MODEL OF LUNAR CRATERS V. 2. BASIC APPROXIMATIONS
The basic model used here is described, in several earlier papers (Marcus, 1964 Secs. 2, 3, and 7; 1966b, Secs. 2 and 3), and hardly bears repeating here in great detail. The expected number ~*(x;t) of observable craters, both primary and secondary, per unit area per unit diameter interval at time t, whose diameters are x, is given by ~*(x;t)
= for X*(x;u)f(u) t
X exp {-- f~ ~
,
(x;v)f(v)dv}du;
(1)
h*(x ;t) is the expected number per unit diameter interval of observable x-sized craters, primary or secondary, from a new primary formed at time t; f(t) is the expected number of new craters of a n y size born per unit area per unit time at time t; u*(x;t) is the rate at which x-sized craters are obliterated by new craters arriving at time t and m a y be interpreted as the average area within which an x-sized crater can be destroyed b y a new arrival; to, or with crater filling to(x,t), is the birthdate of the oldest possible x-sized crater not completely filled at time t. With these functions defined, Eq. (1) is derived in the same way as Eq. (15) of Part I, (Marcus, 1964). The function ~,*(x;t) can be easily approximated by assuming that there is no chance that a primary crater can be obliterated by its secondaries, and that no secondary crater can be completely destroyed b y other secondaries from the same primary. In this case we have an upper bound k**(x;t) for ~*(x;t), namely X*(x;t) ~ X**(x;t) = p(x;t) ÷ f~M q(x;y,t)E(S;y,t)p(y;t)dy;
(2)
p(x;t) is the probability density of diameter x of primary craters born at time t, truncated at the minimum diameter x0 and maximum diameter xM of observable craters; q(z;y,t) is the probability density of the diameter z of each secondary crater from a primary of diameter y born at time t. E(S;y,t) is the average number (S) of secondaries larger than x0 from a primary of size y born at time t. The first term of the right side of Eq. (2)
591
is the expected number of x-sized primaries per unit diameter interval from each primary born at time t, or simply p(x;t). The secondterm is the contribution of the secondary craters, namely, the average number of secondary craters of diameter x, per unit interval, from a primary of diameter y formed at time t, averaged over all values of y. Alternatively, Eq. (2) can be derived from Eq. (12) of Part I, (Marcus, 1964). Let Q(c,b,y,x,t) be the probability that a second ary from a primary of diameter y born at time t whose center is at c destroys an earlier x-sized crater whose center is at b. If Q is set identically equal to zero in Eq. (12) of the first paper, we obtain Eq. (2). This corresponds exactly to our assumption that there is no chance of a secondary destroying either its primary or other secondaries from the same primary. The small craters observed in the Ranger 7 photographs appear to be widely dispersed relative to their sizes, with little overlapping. If this is generally true then the upper bound h**(x;t) is in fact a close approximation to h*(x;t). On the other hand, some of the very elongated Copernican secondaries possibly consist of several smaller secondaries which have formed close to each other. If the latter situation is conunon, then h**(x;t) m a y substantially overestimate ~,*(x;t) for small values of x; that is to say, Q m a y not be negligibly small. The author has not found a satisfactory lower bound for h*(x;t). If we ignore secondaries altogether we get just p(x;t), but this is merely a crude estimate since a sufficiently peculiar distribution of secondaries could reduce ~*(x;t) even below this value. In the remainder of the present paper we assume that ~,**(x;t) is in fact a close approximation to ~*(x;t). A good approximation for p*(x;/) is not so easily obtained. We content ourselves with upper and lower bounds. The basic assumption here is that a crater of diameter x can be destroyed only b y a larger crater, of diameter y say, whose center is within a distance (y -- x)/2 of the center of the x-sized crater. This assumption is discussed in (Marcus, 1966a). Given this assumption, a lower bound to ~*(x;t) is obtained by assuming that only a primary crater can destroy a
592
ALLAN H. MARCUS
given x-sized crater, and that it is impossible for a secondary--any secondary from any primary--to destroy an x-sized crater. Thus ~*(x;/) _> A~(x;t) =
~XM o -~ (y -- x)'2p(y;t)dy,
(3) where A~(x;t) is the damage rate due to primaries. An upper bound is obtained in the following way: Suppose that a primary forms S secondaries. Let E~ be the event that the ith secondary destroys a given x-sized crater, where i -- 1, 2, . . . , S, and E0 denotes the event that the given primary destroys a fixed x-sized crater. Then Prob
E~ i =0,
/
_<
Prob (Ei) i =0
= Prob (E0) + S Prob (El)
(4)
assuming Prob. (E~), the probability of the event. E~, the same for each i >__ 1. In the same manner as Eq. (3) we obtain
.*(x;t) < ,**(x;t) = h~(x;t)
ft~ X**(x;u)f(u)exp l - - f t g**(x;v)f(v)dvf du
_< ~*(x;t) < ]i ~h**(x;u)f(u)
If the functions p(x;t), q(z;y,t), and E(S;y,t) are not explicitly functions of time t, then neither are )¢*(x;t), Ax(x;t), or ~**(x;t). Therefore in this case (suppressing t) we have ~,**(x) (1 - exp {--u**(x)[F(t) -- F(t0)]})
x**(x) __< ~*(X;t) <: AI(X----~
X (1 -- exp
+ f : ~ E(S;y,t)p(y;t)dy
f ~ ~ (z -- x)'2q(z;!I,t)dy.
A region small with respect to such fluctuations m a y be large in absolute area, say Copernicus surrounded by a region several Copernican diameters in size. In applying our analysis to actual crater data, it will be necessary to multiply E(S;y,t) by an unspecified scaling factor to be determined empirically for each region of interest. Upon combining Eqs. (1), (2), (3), and (5), we obtain
(5)
The inequality in Eq. (4) becomes equality only if the events E~ are nmtually independent, which is certainly not exactly true since secondary craters tend to cluster among themselves in the neighborhood of the prim a r y crater which forms them. We will later consider some numerical examples which suggest that the ratio (xit)/Ai(x,t) is never much different from unity for most values of x. In that case we conclude that A~(x;t) and g**(x;t) are very " t i g h t " lower and upper bounds for g*(x;t) and that either is satisfactory approximation. The average number of secondaries from a primary of diameter y born at time t, E(S;y;t), appears in Eq. (2) and Eq. (5). It has been implicitly assumed that the region of interest has very large area. Otherwise, crater counts are likely to fluctuate very considerably according as the region of interest is or is not close to a large primary crater which has produced m a n y secondaries.
{--~l(x)[F(t) - F(t0)]l).
(7)
It is now necessary to say a few words about the time-dependent terms in the square brackets. We consider two limiting cases, F(t) - F(to) very large corresponding to a great deal of crater-forming activity, and F ( t ) - E[to(x,t)] very small corresponding to a great deal of crater-filling activity with relatively little crater-forming activity. In the case
F(t) -- F(to) >> 1 the terms in brackets in Eq. (7) approximate to unity and the resulting lower and upper bounds to ~*(x;t) are simply h**(x)/g**(x) and X**(x)/A,(x), respectively. In the second case,
F(t) -- F(to) --~ 0 and the lower and upper brackets in Eq. (7) approximate to ~**(x) {F(t) -- F[to(x,t)]}, Al(x) {F(t) -- F[to(x,t)]}, respectively, yielding at approximation
both ends the
593
STOCHASTIC MODEL OF LUNAR CRATERS V.
~*(x;t) = )¢*(x){F(t) -- F[to(x,t)]}
(8)
Under some assumptions a b o u t the rate of crater-forming and crater-filling and the initial d e p t h - d i a m e t e r relation, Eq. (8) assumes a v e r y simple form which we deal with later. I n the remainder of the paper we will use only Eq. (7). I n order to do so we m u s t specify p(x), q(z;y), and E ( S ; y ) . Towards this end we review the results of some terrestrial cratering experiments. 3.
E X P E R I M E N T A L SECONDARY C R A T E R S
We now consider observations on terrestrial experiments relevant to secondary craters formed b y the ejection of material from a p r i m a r y explosion crater. The d a t a are t a k e n from (Roberts, 1964) and (Moore, Kachadoorian, and Wilshire, 1964) who consider chemical and nuclear explosion craters. In Table I we list the diameter x, diameter z* of the largest secondary, and ratio c* -z*/x for the Sedan, Scooter, and Tulalip D6.5a craters. F o r comparison purposes we
In Table I I we list the n u m b e r S of seconda r y craters whose diameters are larger t h a n x0 ( = 10 feet) from the Sedan and Scooter craters. The d a t a of Tables I and I I are t a k e n f r o m (Roberts, 1964). Roberts suggests t h a t (in our notation) S = Constant W r, where r is a positive constant equal to or slightly less t h a n unity, and W is the energy of the explosion or as applies to his data, weight of explosive. Using the scaling law between the diameter x of a p r i m a r y crater and the energy of the explosion which caused it (Vaile, 1961) W = constant X 3"4, where the constant 3.4 is not known with certainty, but m a y possibly be within the range 3.0 to 4.1. Upon combining these two results, we obtain [replacing S b y E(S;x)] E(S;x) = Mo(cx/xo)"
= 0 TABLE I RATIO OF DIAMETER OF LARGEST SECONDARY CRATER TO DIAMETER OF ITS PRIMARYa
Crater
Sedan Scooter Tulalip D6.5a Copernicus
x (Diameter of primary ) (Feet)
1200 340 5 12
z* (Diameter of largest
secondary) (Feet)
120 39 0.2
2.52 X 10 s
0.103
After Roberts (1964). obtain the same for Copernicus, b u t do not imply a similar origin for the lunar crater. I t is noted t h a t for the larger craters, c* -0.10 to 0.11. TABLE II NUMBER, ~, OF SECONDARIES LARGER THAN X0 ~ 10 FEET a Crater
Sedan
Scooter
Diameter
(Feet) 1200
340
a After Roberts (1964).
cx > xo
cx ~_xo
(9)
with 0 < c < 1, approximately c = 0.11
(10)
m = 3.4r, slightly less t h a n 3.4,
(11)
c* = z * / x
0.100 0.115 0.039
2.6 X 104
for for
S 465
16 to 18
where these results are suggested b y Roberts' data. T h a t is to say, no secondary from a p r i m a r y of diameter x is bigger t h a n cx, and the n u m b e r of secondaries larger t h a n x0 is given approximately b y the power law, Eq. (9). I n fact, from Table I I with x0 -- 10 ft a n d c = 0.11, we obtain m = 3.0, thus r = 0.88 M0 = 0.27,
(12)
but these estimates, based on just two points, are highly uncertain. F r o m (Dodd, Salisbury, and Smalley, 1963; Moore, Kachadoorian, and Wilshire, 1964) we obtain the fact t h a t the size distribution of secondaries is an inverse power law, q(z;x) -- ~x0~z-~-l(1 -- (cx/xo)-~) -1 forx0 < z <=cx = 0 otherwise,
(13)
594
A~LI~N H. MARCUS
where the constant w is rather large, perhaps ~o = 3.0 to 4.0. This distribution is obtained only for the Sedan experiment, but similar results a p p l y to the lunar craters Langrenus and Copernicus. The author's own analysis suggests co = 2 for the Sedan secondaries. Some additional data has recently become available. Shoemaker, quoted b y B r i n k m a n n (1966) uses functions similar to our Eqs. (9) and (13) with numerical values (in our notation) of c = 0.06, w = 3.56, and m = 3.56. In a letter to the author, Roberts (1966) points out t h a t the exponent in the e n e r g y diameter relationship is possibly less t h a n 3.4, probably between 2.5 and 3.4, and t h a t the value 3 m a y be the best choice. H e believes t h a t c = 0.11 is satisfactory. In Eqs. (9) and (11) the value m = 3 would be a minimum, he feels. Some additional important remarks on incompleteness of the secondary crater counts for terrestrial explosion craters are contained in a letter b y Roberts which is quoted in the Appendix. I t is clear t h a t the available data do not adequately specify the needed model functions. The author has not seen a n y further studies, however, and is thus forced to rely on material available as described in this Section. 4. ]~XAMPLES: METEOROIDAL IMPACT I-IYPOTHESIS
A form of Eq. (9) which permits a smooth transition of E ( S ; y ) to zero as cy decreases to one is given b y "f.
£OS,y) = Mo[(cy) m - 11 for cy > l = 0 otherwise.
Eq. (9) and Eq. (16) differ only slightly, with Eq. (16) preferred for physical reasons. I t m a y or m a y not be permissible to let XM become infinitely large. I n this connection the sign of ~, - m is crucial. If ~, -- m > 0, the n u m b e r of primaries decreases faster with increasing diameter t h a n the n u m b e r of secondaries per primary; thus the total n u m b e r of secondaries from large primaries will be negligible and we m a y take XM as large as we like. On the other hand, if 3' - m ~ 0, the n u m b e r of secondaries per p r i m a r y increases faster t h a n the n u m b e r of primaries decreases with increasing diameter; thus most of the secondaries come from the largest primaries and we m u s t retain a finite value of XM to guarantee convergence of integrals in Eq. (5). On the Moon, it is possible t h a t most of the observable secondaries come from the largest primaries, therefore ~/ - m < 0 is intuitively more plausible. With the above interpolation formulas, Eqs. (14)-(16), we consider these two cases. CASE 1 : 2 < - y < m < w + ~ - -
We will now work out some numerical examples relevant to the meteoroidal impact hypothesis. We assume x0 = 1 and
06)
2
I t is easily found by direct integration t h a t for 1 < x < cy, fcu 4~ (z -- x)2~z ~-1(1 -- (cg)-~)-~dz
p(:c) = ' y x - ' r - l ( l --3'.11 -v)
1
l-
for I ~ x < XM (14)
__ 7r¢o4 [1 -- (cy)-~] -I
= 0 otherwise, where ~/ > 2. As the preceding Section suggests, the diameter of the largest seconda r y from a p r i m a r y of diameter y is only a fraction of y, say cy, where 0 < c < 1 and c is the same for all y. The distribution of the diameter z of a secondary from a p r i m a r y of diameter y is an inverse power law
q(z;y) = ozz-'~ it1
--
= 0 otherwise.
(C~J)--¢°)-1 for 1 < z < cy
+
2(cy)l-'°X w-I
2X2_~
w(o~ -- 1)(~o -- 2)
(Cy)2-~ ~o--2
(cY~ -'°x2]
(17)
We assume t h a t XM is very large, but finite. Since all relevant powers of y are negative except for m - % the contribution of XM call be neglected everywhere except in connection with t h a t term. We therefore obtain approximately t
(15)
i The approxinmtion is obtained by also assuming 1
--
(cy)
-'~
=
1.
STOCHASTIC MODEL OF LUNAR CRATERS V. & ( x ) = ~3,/2~,(~)x ~--~ { coioc" [ (CXM)~f y(a, u**(x) = & ( x ) I + - - - 7 - - L m -- */ co(a) x~ */(2)
X m
(5" + ~ -- m) (3) m -- 3, 3,(3) 1]'}
(18)
(5" + co)(~) 7y . ,
F o r m o d e r a t e l y large values of x, the t e r m s in the d e n o m i n a t o r of X**(x)/~**(x) which involve x are of m u c h smaller order t h a n unity. T h u s for m o d e r a t e l y large values of x, we o b t a i n the a p p r o x i m a t i o n h**(x)
23,(2) 1 + coM0c______~ x~
~**(x)
~x~
where for a n y n u m b e r a, a (a) = a(a -
595
X
l ) ( a - 2).
1 -- ( 5 " q _ ~ _
m)(3~ ~ - - m
./(2)
Sinfilarly X**(x) = X
.
1+
Again in the ease t h a t F(t) -- F(to) >> 1, we o b t a i n lower a n d u p p e r b o u n d s for N ( x ) :
x--- z -
(cx~)~f-~ x"
m -- 5"
(19)
m -- 5"
2coMoc~
*/(a) [
~x ~1+
1 --
xco
- (3,+~-m)(~;
Thus X**(x)
~**(x)
2~ (2) X I + - - - T J - - L
m--5" m--3, ,),(3) Xm coMoc~ [ (CXM)~-~ 5"(3) X T -1--k ~ k m - - 5" co(a) ( * / + co - m)(~)m - 5"
~x 3
3,(3)
-]
(3, -V~)(~)5"J
a n d as x~ - ~ ~ we obtain the finite limit X m
x**(~)/~**(x) - , 2co(~)/~x~.
(20)
X
I n the case t h a t F(t) -- F(t0) ~ 1 a n d the t i m e - d e p e n d e n t b r a c k e t in Eq. (7) is a p p r o x i m a t e l y unity, we h a v e the following lower a n d u p p e r b o u n d s for N ( x ) , the c u m u l a t i v e n u m b e r of craters per unit area whose d i a m eter is larger t h a n x:
w(3)
f c =)
-7rX2 - -< N ( x ) = h
2coMoc~ 1+
x~
-- ( 1 - _<
5"(a) { 2coM0c~ N ( x ) _< - -7rx 2 l q x~
[ ×
5"(3)
m)
*/Ca) ~ 1 (3"-I-co)(3),5"(coq-2)1}
x~ (*/ -- m)(co + 2 -- m)
1]} 5"(~0 -t- 2) " (24)
~*(y;t)dy _< -,fix - 2
(CXM)'~--~'X~ ( m - - 5")(co + 2 - - 3')
x 'n
1
(m -- ~/)(co + 2 -- m)
]
3"(co + 2)J" (21)
I t is a s s u m e d here that. XM is v e r y large. CASE 2 : m < 3 , < ~ q - 3 , - -
2
W e again consider XM--~ ~ , as above, X**(x) p**(x)
(*/--m)(¢o+2--
25"(2) ~x 3
CASE 3: S a m e as either Case 1 or Case 2, except t h a t crater-filling is n o w the domin a n t factor. W e t h e n use Eq. (8) a n d need consider only h** (x). I t is necessary to specify the crater-filling a n d c r a t e r 4 o r m i n g rates, a n d the initial d e p t h - d i a m e t e r relationship. W e use the model of (Marcus, 1966b, Sec. 8). T h e crater-filling rate a(t) is given b y
proceeding
~ 5" -- m coMoc~ [ */(3) 1 q- ~ [ ( 5 ' q - ~
t > 0,
a(t) = aoe - ° '
x m
m ) ( 2 */) _- m
.t(a)
(*/ + co)(a)
(25)
~]
•
(22)
596
ALLAN H. MARCUS
the crater-forming rate f(t) b y f(t) = boe-~t
t > 0
(26)
and the initial d e p t h H ( x ) of a crater of diameter x b y H ( x ) = Cox~
x > 0,
(27)
where a, f~, ~, ao b0, Co are positive constants. We assume all craters are initially paraboloidal in shape. I t is then easily found t h a t F(t) -- F[to(x,t)] = C~[(1 + C.~x~)~/" -
1] (2s)
or with sufficient precision for our purposes, approximately F(t) -- F[to(x,t)] = C3x ~/",
(29)
where C1 = bo~3-1e-~t C2 = acoao-le"t/2 C3 = C1C2~l".
Upon combining Eqs. (19) and (28) we obtain the following approximation for the cumulative number N (x) per unit area larger t h a n diameter x, assuming s = ~ , ~/a >0: N (x)
~/C _____3 Xs
+
x~
(CXM) m--'~X'Y
X
(m--
~)(s+o~--
~,)
X 7tt
(m -- ~,)(~o + s - - m)
~(s + ~)
" (30)
If -y > m we need merely delete the t e r m involving XM. 5. NUMERICAL STUDIES OF THE IMPACT HYPOTHESIS
We will now illustrate some of the conclusions of the previous sections. In Case 1 of Section 4 the q u a n t i t y of greatest interest, from which other results m a y be derived is the ratio between the upper bound t~**(x) and lower bound Al(x), say R ( x ) = t,**(x)/hl(x).
We see t h a t Eq. (18) specifies R ( x ) when % m, ~, Mock, and CXM are specified. For application to the impact hypothesis we shall set ~f -- 2.25, m = 3, and c = 0.11. In connection with the Ranger data we will consider a m i n i m u m crater diameter x0 of 2 meters; with the plausible value of the maxi m u m crater diameter XM t a k e n as 180 km, this gives CXM as 104 in units of x0 and justifies the approximations used in Section 4. We consider two of the most likely values of ~, w = 3 and ~ = 4 . Because the nonuniformity of the distribution of centers of secondaries from a given p r i m a r y induces considerable variation in the density of small craters from point to point, we consider several possible values of Moc2"25: O, 10-4, 10-~, 10-2, 10-1. The results are shown in Fig. 1 and 2. I n both cases, the crater-destroying capability of the secondaries is very important when the density of secondaries is high, even though the n u m b e r of large secondaries is small. The ratio R ( x ) is not significantly greater t h a n 1 for x larger t h a n 20 or 30 (40 to 60 meters diameter) when ~0 = 4. But when ~ = 3, R ( x ) is conspicuously larger t h a n I even when x = 200 (400 meters diameter). This nleans t h a t when crater counts from unflooded continental regions become available, it will be necessary to take into account the effects of obliteration caused b y both p r i m a r y and secondary craters. Case 3 of Section 4 is relevant to the s t u d y of the Ranger 7 data, since most of the observed craters are in the flooded Mare Cognitum. As in (Marcus, 1966b) we will use an index s = 1.75 for the observed distribution of primaries after extensive flooding. Figures 3 and 4 show the expected cumulative number densities N ( x ) for the same set of parameters used in Figs. 1 and 2, with allowance for disappearance of primaries and secondaries due to filling. The distributions N ( x ) are all normalized b y setting the constant C3 in Eq. (30) equal to one. To correspond to terrestrial telescopic observations of the maria, C3 must be adjusted to give a density of a b o u t 1.8 craters larger t h a n 1 k m diameter per 103 k m 2 on the p r i m a r y crater curve.
I00"
W • 3
>¢ tt IO.
!
i
IO
!
Ioo
Iooo
g
Fzo. 1. Ratio R(z) between upper and lower bounds of the damage rate ~*(~) as a function of diameter x. Parameters are those of impact hypothesis with to = 3. Secondary density parameter is (top to bottom) Moc~-~ = 10-~, 10 -~, 10 -~, 10 -4. W.=4
R(x} f
,o
io °
,
!
i0 ~
I0'
I0 z
It}
X
FIG. 2. Ratio R(z) between upper and lower bounds of the damage rate ~*(z) as a function of diameter z. Parameters are those of impact hypothesis with ~ = 4. Secondary density parameter is (top to bottom) M'od.~ = lO-l, lO'-a, 10-3, 10-4. 597
5.08
ALLAN H. MARCUS
,o ~
,o ~
,d
,d
N(x) io"
i 0 -~"
, o -3
nd ~
2.
.
.
5.
IO
20
5' 0
I 0, 0
2 0
X
Fro. 3. C u m u l a t i v e n u m b e r d e n s i t y N(x) for m a r i a (theoretical) as a f u n c t i o n of d i a m e t e r x. P a r a m e t e r s a r e t h o s e of i m p a c t h y p o t h e s i s w i t h ~ = 3 a n d e x t e n s i v e crater filling. S e c o n d a r y d e n s i t y p a r a m e t e r is (top to b o t t o m ) M0c2.25 = 10 -1, 10 -~, 10 -3, 10 -4, 0. 6. COMPARISON WITH R A N G E R 7 CRATER DISTRIBUTIONS
We now compare the predicted distributions of Section 5 with actual crater-diameter distributions from Ranger 7 photographs. A v e r y complete crater count has been made b y H a r t m a n n (1965) and is shown in Fig. 5 b y the horizontal solid bars. The points at the end of the bars represent N ( x ) in its usual form, the n u m b e r of craters per square kilometer whose diameter exceeds x. The lower solid curve is' the predicted distribution of primaries. The upper solid curve is
the best fit obtainable with ~ = 4, namely 1.0. The dashed curve is the best fit obtainable with ~ = 3, namely M o c ~.25 = 10-1. T h e curve with ~ -- 3 seems definitely better than t h a t with w = 4. However, the distribution seems distinctly " r o u g h " and one might imagine t h a t an appropriate "smoothing" procedure would ease analysis of the data. This has been done b y K o p a l (1965b). The resulting distribution is shown in Fig. 6 (points) and the best fit is the upper solid curve ~ = 4, M o c 2"~5 = 1. Shoemaker's (1965) counts have already M o c 2"25 ~-
STOCHASTIC MODEL
OF LUNAR
CRATERS
599
V.
,o'] I O'
t0°
N(x) i 0 -~
!
g
2
5
!
fO
•
e
20
50
e
fO0
200
X "r"
FIG. 4. Cumulative number density N ( x ) for maria (theoretical) as a function of diameter x. Parameters are those of impact hypothesis with w = 4 and extensive crater filling. Secondary density parameter is (top to bottom) M ~ .u = 10-I, 10-3, 10-3, 10-4, 0. been smoothed. T h e y are shown in Fig. 7, the upper points representing the observed "within r a y " distribution and the lower points the observed "between r a y " distribution. The lower solid curve is the predicted distribution of primaries, the dashed curve is ~0 = 3, M o c ~'25 = 0.1, a n d t h e dotted curve is ~ -- 3, M o c ~'25 = 0.4. These give an adequate fit of the data, but only for x > 50 meters. K o p a l (1965b) has also smoothed these data and obtained the distribution shown in Fig. 8 (points) which is well fit b y the curve w = 3, M o c 2"25 = 0.1.
The extensive crater counts b y Osborn (1965) are shown in Fig. 9. The plotted points are crater densities derived from the Ranger 7 photographs indicated in the Figure. The roughness of the curve is merely due to statistical fluctuation, as was the roughness of H a r t m a n n ' s data. Osborn's data is also well characterized b y the curve = 3, M o c 2"25 = 0.1 but appears to be incomplete for x < 100 meters. The same incompleteness is suspected in the data of Miller (1965) and B r i n k m a n n (1966) whose counts are not reproduced here.
600
ALLAN H. MARCUS
I°5"
x%
.~ io'i=
0,,
Z
I0"
%6~,
\
id 5.
;o"
I
i 0 "~
crater
diameter,
i
td
I0 ~
x, km
Fro. 5. Solid points on bars are observed cumulative number density N(x) on Mare Cognitum, after Hartmann (1965). Lower solid curve is predicted from primary craters. Upper solid curve is predicted from primary and secondary craters with ~ = 4, Moc~.~ -- 1.0. Dashed curve is predicted from primary and secondary craters with w = 3, Moc~.~ = 0.I. I n s u m m a r y , the R a n g e r 7 d a t a are well predicted b y the theoretical Eq. (30), in connection with numerical parameters used in Section 5, b y setting ~ = 3 a n d M o c ~'~5 = 0.1. These results will be more fully i n t e r p r e t e d in the next Section. 7. CONCLUSIONS: ORIGIN OF SECONDARY CRATERS
While the shape of the d i a m e t e r distribution of small craters agrees with t h a t predicted b y t h e p r i m a r y a n d s e c o n d a r y i m p a c t hypothesis, the density of small craters is
a b o u t an order of m a g n i t u d e larger t h a n t h a t predicted b y a n a l o g y with terrestrial explosion craters. Using the values -y - 2.25 a n d s = 1.75 deduced previously, the observed distribution requires m = 3, ¢0 = 3, a n d for x0 = 2 meters, M0c~.~5 = 0.1. W i t h c -- 0.11 we h a v e M o = 14, or for x0 = 3 meters, M0 = 14 (2/3) 3 = 4.1, which is 15 times Iarger t h a n the value 0.27 suggested in Section 3 for terrestrial explosion craters. We have seen t h a t crater density estimates are likely to be inaccurate b y a large factor, b u t the a b o v e factor of 15 seems t o o large to be
STOCHASTIC
MODEL
OF
LUNAR
CRATEI:~
V.
601
I0 7 •
i06
.
I0~=
104=
|0 s.
~
10 z =
I~
I0'"
~I0
"-
= Io"-
,o"i 0 "s,
Id*. jo -r. _'%.
lO'* ..
;O.L
:0.|
!
Io"
,
,o'
,,
,o'
id' crater
diameter,
x , kin.
Fro. 6. Points are smoothed cumulative number density N(x) after Kopal (1965b) based on Hartmann (1965). Lower curve is predicted from primary craters, upper curve is predicted from primary and secondary craters with ~ -- 4, M0c2.~ = 1.0. explained by observational loss or inaccuracy. In the first place, it was noted in Section 2 t h a t we completely ignore geographical variation in the density of secondary craters. The regions photographed b y Ranger 7 are not large compared to the scale of geographical variation of secondary crater density. In particular, most of the last photographs seem to be of a region of especially high
crater density associated with a " r a y . " While Shoemaker's data suggest a difference in density of a factor of 4 between "on-ray" and "between-ray" craters, this is still not large enough to account for the observed excess of a factor of 15. A second possibility is that at least some of the lunar primary impacts have produced exceptionally large numbers of secondary craters. Perhaps some of the recent primary
~02
ALLAN H. MARCUS
,o,J
'l
•
"% %; % °° %. *% ",. .. • "o
i0 II,
• • I 0 i,
~ ~ ' ° . o ~I=: I0', v
0. I0°. X z IO".
I
i~
,'0"~
I'0•
I0'
I O "3
crater
diameter,
x, km
Fro. 7. Upper points are "within ray" cumulative number density N (.x) oa Mare (3og[fitt~n, lower points are "between ray" cumulative number density on Mare Cognitum, after Shoemaker (1965). Solid curve is predicted from primary craters, dashed curve is predicted from primary and secondary craters with ~ = 3, M0c2.~ = 0.1, dotted curve is predicted from primary and secondary craters with ~ = 3, M0d .~5 = 0.4. projectiles impacted on the Moon with exceptionally high velocity, tile violent explosion producing unusually large quantities of ejeeta. Although m u c h of this ejecta would escape from the Moon, it is possible t h a t an unusually large q u a n t i t y would be retained, creating a locally high density of secondary craters. This possibility is further developed ill a letter b y Roberts which is quoted in the Appendix. Finally, the surplus of small craters m a y be due to the fact t h a t m a n y of t h e m are, in
one sense or another, of internal origin. T h e y m a y be caused by degassing (Kuiper and H a r t m a n n , 1964; Miyamoto, 1964; Fielder, 1965) or b y collapse (subsidence) as has been argued b y Kopal (1965a). The two sources of uncertainty in our analysis n m s t always be kept in mind. First of all, the formation of secondary impact craters is not really well understood in a quantitative sense and the model functions we have been using nlay be quite inaccurate. Secondly, the crater-density estimates are based on small numbers of craters and it is
STOCHASTIC MODEL OF LUNAR CRATERS V.
603
,o.t ,o'-[ I 0 ~' .~
Id E 2¢
L
10'
~.
~--Io" X
z
io" i 0 -L
i 0 "s '
i0"i. .6
I0
i .I
!
I0
/O °
I 0 "~
crater
I
IO'
d~am¢ter,
x,km.
hG. 8. Points are smoothed cumulative number density N(x) after Kopal (1965b) based on Shoemaker (1965). Lower curve is predicted from primary craters, upper curve is predicted from primary and seconda~craters with o = 3, Moc2.2~= 0.1. thus possible that these estimates are inaccurate b y a factor of 2 or so. Detailed empirical s t u d y of crater counts from the Ranger 7 photographs is still needed to clear tip this last difficulty.
to be at least as complete as those reported b y others and yield about the same diameter distribution. Crater distributions from the Ranger 8 and 9 have recently become available (Jet
Note added in proof. In a letter to the
Propulsion Lab., Calif. Inst. Technol. Tech. Rept. 32-800, Pt. II). The diameter distribu-
author (May 22, 1966), W. H. Osborn has reported a revision of the crater counts in his Scope paper, based on improved scale and altitude determinations from the Ranger 7 photographs. The revised counts appear
tion is similar in shape to that found in Mare Cognitum, with similar density in Alphonsus but rather higher density in Mare Tranquillitatis. This suggests that somewhat less crater-filling activity has occurred in
604
ALLAN H. MARCUS [] - 3 8 0 + -38! A - 372 Z - 379 O - 373 • - 359 4~- 3f7
+ Q
0 - 24~
IO ~ .I
+
I O*
i 0 "~.
vt
E
Q. x
z \ \ 0
i0 "b.
16"
to-'
:o. crater
diameter,
x.
,'o'
km
Fro. 9. Points are cumulutive number density N(x) in Mare Cognitum from indicated Ranger 7 photographs, after Osborn (1065). Lower curve is predicted from primary craters, upper curve is predicted from primary and secondary craters with oJ - 3, M0c2-~ -- 0.1. Mare Tranquillitatis than in the other two regions. A detailed analysis of these data will be published separately. APPENDIX
Excerpts from a letter by W. A. Roberts (September 12, 1966): "Any counts of secondary craters for modelling studies, such as Tulalip, will be incomplete due to the discrete boundaries of the test pad. Incomplete counts for Sedan are related to available time, both from a
contractual point of view and because the e]ecta blanket was radioactive, requiring that field work be limited and performed iu haste. "In your conclusions, you suggest that exceptionally high velocity impacts may have produced unusually large quantities of ejecta. Such large quantities of ejecta would have been expelled from an appropriately large crater volume. On the moon, the lack of atmosphere and the low gravity will probably result in higher impact (terminal)
STOCHASTIC MODEL OF LUNAR CRATERS V.
velocities and longer trajectories for the missies. The higher impact velocities will result in correspondingly wider secondary craters for given conditions of missle and target since the diameter would be a function of the kinetic energy or about V 2/a. This could m e a n that, for a given lower limit of resolution, a missle of a given size which would not produce a visible secondary crater from a p r i m a r y on earth could produce a resolvable structure on the moon tending to increase the n u m b e r of secondaries above some diameter limit. Most of the large secondaries seen adjacent to Sedan a p p e a r to have been formed b y composite missies. One such missle, completely winnowed on a longer trajectory, m a y have resulted in eleven small craters within a radius of about 75 feet west of Sedan. On the moon, the longer trajectories of the composite missies will allow more time for the composite missles to disperse. Although there is no atmospheric drag to break up these missies, small differences in ejection angle and velocity in these masses (composite missies, or unconsolidated masses) would result in more dispersion for the longer flight times and larger numbers of secondaries. " B o t h Sedan and Scooter were deeply buried charges, and the ejection velocity is obviously a function of scaled burst depth. Missle velocities near Sedan were of the order of 0.1 km. Missle velocities of these masses resulting in secondary craters on the moon could easily be an order of magnitude higher from the v e r y shallow scaled burst depths of the p r i m a r y craters. This would yield craters (given missle mass) a b o u t 5 X larger. I do not have enough d a t a to resolve the problem of the ejecta blanket when the parameters scaled depth of burst, yield, explosion medium, and explosive type are considered . . . . "
605
V. G. (1963). Crater frequency and the interpretation of lunar history. Icarus 2, 466-480. FrsLVER, G. (1965). "Lunar Geology." Lutterworth Press, London. HAIrr~ANN, W. K. (1965). Secular changes in meteoritic flux through the history of the solar system. Icarus 4, 207-213. KoPAL, Z. (1965a). The nature of secondary craters photographed by Ranger VII. Math. Note 430. Boeing Scientific Research Lab., Seattle, Washington. KOPAL, Z. (1965b). Topography of the Moon. Space Sci. Rev. 4, 737-855. KvlPER, G, P., A~D HARTMA~r, W. K. (1965). Interpretation of Ranger VII Records. Jet Propulsion Lab., Cali]. Inst. Technol. Tech. Rept.
32-700, Pt. II, Chapter 3, 32-74. Maacvs, A. H. (1964). A stochastic model of the formation and survival of lunar craters. I. Distribution of diameter of clean craters. Icarus 3, 460-472. MARCUS, A. H. (1966a). A stochastic model of the formation and survival of lunar craters. II. Approximate diameter distribution of all observable craters. Icarus 5, 165-177. MARCUS, A. H. (1966b). A stochastic model of the formation and survival of lunar craters. III. Filling and disappearance of craters. Icarus 5, 178189. MILLER, B. P. (1965). Distribution of small lunar craters based on Ranger VII photographs. J. Geophys. Res. 70, 2265-2266. MIYAMOTO,S. (1064). Morphological aspects of the lunar crust. Icarus 3, 486-490. MOORE, H. J., KAC~-ADOORL~,R., A~D WmSHI~E, H. G. (1964). A preliminary study of craters produced by missle impacts. U. S. Geol. Survey, Astrogeol. Studies Ann. Progr. Rept., July 1, 1963 to July 1, 196/~, Pt. B, 58-92. OSBORN, W. H. (1965). Lunar crater counts. Scope (J. Fed. Univ. Astron. Soc.) 7, 63-76. ROBERTS, W. A. (1964). Secondary craters. Icarus
3, 348--364. ROBERTS, W. A. (1966). Personal communication.
SHOEMAKEr, E. M. (1962). Interpretation of lunar craters. In "Physics and Astronomy of the Moon" (Z. Kopal, ed.), pp. 283-359. Academic Press, New York. SHORMArd':R,E. M. (1965). Preliminary analysis of REFERENCES the fine structure of the lunar surface in Mare BRINKMANN,R. T. (1966). Lunar crater distribuCognitum. Jet Propulsion Lab., Call]. Inst. Techtions from the Ranger 7 photographs. J. Geonol. Tech. Rept. 32-700, Pt. II, Chap. 4, 75-134. phys. Res. 71, 340--342. VAILR, R. B. (1961). Pacific craters and scaling DODD, R. W., JR., SALISBURY, J. W., AND SMALLEY, laws. J. Geophys. Res. 66, 3413--3438.
A Stochastic Model of the Formation and Survival of Lunar Craters V. Approximate Diameter Distribution of Primary and Secondary Craters A L L A N H. MARCUS Case Institute of Technology, Cleveland, Ohio and University of Cambridge, Cambridge, England Communicated by A. G. Wilson Received March 29, 1966 Approximate upper and lower bounds are obtained for the expected number density of lunar craters by means of a model which takes into account the formation of primary and secondary craters and the destruction of craters by obliteration and filling. Some numerical examples considered are relevant to primary and secondary craters formed by meteoroidal impacts. Predicted number densities are compared with crater-diameter distributions from photographs taken by the Ranger 7 spacecraft. The shape of the observed distribution is the same as that of the predicted distribution, but the observed density of small craters is about 15 times larger than that predicted by analogy with terrestrial explosion craters. If the observed excess is real, then either some primary craters produce an unusually large number of secondaries, or else many of the smaller lunar craters are of internal origin. The excess may be explained in part by incompleteness of secondary crater counts for terrestrial explosion craters upon which the model functions are based. further approximation for the expected number density of all observable craters with the In previous papers (Marcus, 1966a and assumption that both types occur and cannot 1966b) the author developed formulas for be distinguished from each other. (approximately) the expected number denIn an earlier paper (Marcus, 1964) the sity of all observable craters as a function of the crater diameter x. It was assumed in author obtained an exact formula for the these papers that only primary craters can expected number density of "clean" craters, occur. This m a y be nearly correct for craters i.e., those whose perimeters are not damaged larger than 5 km in diameter. But the prob- by other craters. These results were late," able "secondary impact" origin of m a n y revised to include the effects of spontaneous smaller craters, for example, those surround- disappearance of craters due to filling (Marcus, 1966b), still retaining the exac! ing Copernicus • (Shoemaker, 1962; Dodd, nature of the solution. When secondary craSalisbury, and Smalley, 1963), was already ters can occur, various auxiliary functions known b y the time photographs transmitted appearing in that formula become complib y the Ranger 7 spacecraft resolved lunar cated nmltiple integrals which cannot be re" r a y s " into regions with extremely high denduced to elementary functions. This result sity of small craters. Much preliminary stais therefore of little practical use in studying tistical data is already available for these small craters (Miller, 1964; Shoemaker, the effects of secondary crater formation. t965; H a r t m a n n , 1965; Osborn, 1965; Brink- Nevertheless, these earlier papers contain mann, 1966). These data will be discussed most of the notions and notation to be used later. I t now seems worthwhile to develop in the remainder of the present paper. 590 1. INTRODUCTION
STOCHASTIC MODEL OF LUNAR CRATERS V. 2. BASIC APPROXIMATIONS
The basic model used here is described, in several earlier papers (Marcus, 1964 Secs. 2, 3, and 7; 1966b, Secs. 2 and 3), and hardly bears repeating here in great detail. The expected number ~*(x;t) of observable craters, both primary and secondary, per unit area per unit diameter interval at time t, whose diameters are x, is given by ~*(x;t)
= for X*(x;u)f(u) t
X exp {-- f~ ~
,
(x;v)f(v)dv}du;
(1)
h*(x ;t) is the expected number per unit diameter interval of observable x-sized craters, primary or secondary, from a new primary formed at time t; f(t) is the expected number of new craters of a n y size born per unit area per unit time at time t; u*(x;t) is the rate at which x-sized craters are obliterated by new craters arriving at time t and m a y be interpreted as the average area within which an x-sized crater can be destroyed b y a new arrival; to, or with crater filling to(x,t), is the birthdate of the oldest possible x-sized crater not completely filled at time t. With these functions defined, Eq. (1) is derived in the same way as Eq. (15) of Part I, (Marcus, 1964). The function ~,*(x;t) can be easily approximated by assuming that there is no chance that a primary crater can be obliterated by its secondaries, and that no secondary crater can be completely destroyed b y other secondaries from the same primary. In this case we have an upper bound k**(x;t) for ~*(x;t), namely X*(x;t) ~ X**(x;t) = p(x;t) ÷ f~M q(x;y,t)E(S;y,t)p(y;t)dy;
(2)
p(x;t) is the probability density of diameter x of primary craters born at time t, truncated at the minimum diameter x0 and maximum diameter xM of observable craters; q(z;y,t) is the probability density of the diameter z of each secondary crater from a primary of diameter y born at time t. E(S;y,t) is the average number (S) of secondaries larger than x0 from a primary of size y born at time t. The first term of the right side of Eq. (2)
591
is the expected number of x-sized primaries per unit diameter interval from each primary born at time t, or simply p(x;t). The secondterm is the contribution of the secondary craters, namely, the average number of secondary craters of diameter x, per unit interval, from a primary of diameter y formed at time t, averaged over all values of y. Alternatively, Eq. (2) can be derived from Eq. (12) of Part I, (Marcus, 1964). Let Q(c,b,y,x,t) be the probability that a second ary from a primary of diameter y born at time t whose center is at c destroys an earlier x-sized crater whose center is at b. If Q is set identically equal to zero in Eq. (12) of the first paper, we obtain Eq. (2). This corresponds exactly to our assumption that there is no chance of a secondary destroying either its primary or other secondaries from the same primary. The small craters observed in the Ranger 7 photographs appear to be widely dispersed relative to their sizes, with little overlapping. If this is generally true then the upper bound h**(x;t) is in fact a close approximation to h*(x;t). On the other hand, some of the very elongated Copernican secondaries possibly consist of several smaller secondaries which have formed close to each other. If the latter situation is conunon, then h**(x;t) m a y substantially overestimate ~,*(x;t) for small values of x; that is to say, Q m a y not be negligibly small. The author has not found a satisfactory lower bound for h*(x;t). If we ignore secondaries altogether we get just p(x;t), but this is merely a crude estimate since a sufficiently peculiar distribution of secondaries could reduce ~*(x;t) even below this value. In the remainder of the present paper we assume that ~,**(x;t) is in fact a close approximation to ~*(x;t). A good approximation for p*(x;/) is not so easily obtained. We content ourselves with upper and lower bounds. The basic assumption here is that a crater of diameter x can be destroyed only b y a larger crater, of diameter y say, whose center is within a distance (y -- x)/2 of the center of the x-sized crater. This assumption is discussed in (Marcus, 1966a). Given this assumption, a lower bound to ~*(x;t) is obtained by assuming that only a primary crater can destroy a
592
ALLAN H. MARCUS
given x-sized crater, and that it is impossible for a secondary--any secondary from any primary--to destroy an x-sized crater. Thus ~*(x;/) _> A~(x;t) =
~XM o -~ (y -- x)'2p(y;t)dy,
(3) where A~(x;t) is the damage rate due to primaries. An upper bound is obtained in the following way: Suppose that a primary forms S secondaries. Let E~ be the event that the ith secondary destroys a given x-sized crater, where i -- 1, 2, . . . , S, and E0 denotes the event that the given primary destroys a fixed x-sized crater. Then Prob
E~ i =0,
/
_<
Prob (Ei) i =0
= Prob (E0) + S Prob (El)
(4)
assuming Prob. (E~), the probability of the event. E~, the same for each i >__ 1. In the same manner as Eq. (3) we obtain
.*(x;t) < ,**(x;t) = h~(x;t)
ft~ X**(x;u)f(u)exp l - - f t g**(x;v)f(v)dvf du
_< ~*(x;t) < ]i ~h**(x;u)f(u)
If the functions p(x;t), q(z;y,t), and E(S;y,t) are not explicitly functions of time t, then neither are )¢*(x;t), Ax(x;t), or ~**(x;t). Therefore in this case (suppressing t) we have ~,**(x) (1 - exp {--u**(x)[F(t) -- F(t0)]})
x**(x) __< ~*(X;t) <: AI(X----~
X (1 -- exp
+ f : ~ E(S;y,t)p(y;t)dy
f ~ ~ (z -- x)'2q(z;!I,t)dy.
A region small with respect to such fluctuations m a y be large in absolute area, say Copernicus surrounded by a region several Copernican diameters in size. In applying our analysis to actual crater data, it will be necessary to multiply E(S;y,t) by an unspecified scaling factor to be determined empirically for each region of interest. Upon combining Eqs. (1), (2), (3), and (5), we obtain
(5)
The inequality in Eq. (4) becomes equality only if the events E~ are nmtually independent, which is certainly not exactly true since secondary craters tend to cluster among themselves in the neighborhood of the prim a r y crater which forms them. We will later consider some numerical examples which suggest that the ratio (xit)/Ai(x,t) is never much different from unity for most values of x. In that case we conclude that A~(x;t) and g**(x;t) are very " t i g h t " lower and upper bounds for g*(x;t) and that either is satisfactory approximation. The average number of secondaries from a primary of diameter y born at time t, E(S;y;t), appears in Eq. (2) and Eq. (5). It has been implicitly assumed that the region of interest has very large area. Otherwise, crater counts are likely to fluctuate very considerably according as the region of interest is or is not close to a large primary crater which has produced m a n y secondaries.
{--~l(x)[F(t) - F(t0)]l).
(7)
It is now necessary to say a few words about the time-dependent terms in the square brackets. We consider two limiting cases, F(t) - F(to) very large corresponding to a great deal of crater-forming activity, and F ( t ) - E[to(x,t)] very small corresponding to a great deal of crater-filling activity with relatively little crater-forming activity. In the case
F(t) -- F(to) >> 1 the terms in brackets in Eq. (7) approximate to unity and the resulting lower and upper bounds to ~*(x;t) are simply h**(x)/g**(x) and X**(x)/A,(x), respectively. In the second case,
F(t) -- F(to) --~ 0 and the lower and upper brackets in Eq. (7) approximate to ~**(x) {F(t) -- F[to(x,t)]}, Al(x) {F(t) -- F[to(x,t)]}, respectively, yielding at approximation
both ends the
593
STOCHASTIC MODEL OF LUNAR CRATERS V.
~*(x;t) = )¢*(x){F(t) -- F[to(x,t)]}
(8)
Under some assumptions a b o u t the rate of crater-forming and crater-filling and the initial d e p t h - d i a m e t e r relation, Eq. (8) assumes a v e r y simple form which we deal with later. I n the remainder of the paper we will use only Eq. (7). I n order to do so we m u s t specify p(x), q(z;y), and E ( S ; y ) . Towards this end we review the results of some terrestrial cratering experiments. 3.
E X P E R I M E N T A L SECONDARY C R A T E R S
We now consider observations on terrestrial experiments relevant to secondary craters formed b y the ejection of material from a p r i m a r y explosion crater. The d a t a are t a k e n from (Roberts, 1964) and (Moore, Kachadoorian, and Wilshire, 1964) who consider chemical and nuclear explosion craters. In Table I we list the diameter x, diameter z* of the largest secondary, and ratio c* -z*/x for the Sedan, Scooter, and Tulalip D6.5a craters. F o r comparison purposes we
In Table I I we list the n u m b e r S of seconda r y craters whose diameters are larger t h a n x0 ( = 10 feet) from the Sedan and Scooter craters. The d a t a of Tables I and I I are t a k e n f r o m (Roberts, 1964). Roberts suggests t h a t (in our notation) S = Constant W r, where r is a positive constant equal to or slightly less t h a n unity, and W is the energy of the explosion or as applies to his data, weight of explosive. Using the scaling law between the diameter x of a p r i m a r y crater and the energy of the explosion which caused it (Vaile, 1961) W = constant X 3"4, where the constant 3.4 is not known with certainty, but m a y possibly be within the range 3.0 to 4.1. Upon combining these two results, we obtain [replacing S b y E(S;x)] E(S;x) = Mo(cx/xo)"
= 0 TABLE I RATIO OF DIAMETER OF LARGEST SECONDARY CRATER TO DIAMETER OF ITS PRIMARYa
Crater
Sedan Scooter Tulalip D6.5a Copernicus
x (Diameter of primary ) (Feet)
1200 340 5 12
z* (Diameter of largest
secondary) (Feet)
120 39 0.2
2.52 X 10 s
0.103
After Roberts (1964). obtain the same for Copernicus, b u t do not imply a similar origin for the lunar crater. I t is noted t h a t for the larger craters, c* -0.10 to 0.11. TABLE II NUMBER, ~, OF SECONDARIES LARGER THAN X0 ~ 10 FEET a Crater
Sedan
Scooter
Diameter
(Feet) 1200
340
a After Roberts (1964).
cx > xo
cx ~_xo
(9)
with 0 < c < 1, approximately c = 0.11
(10)
m = 3.4r, slightly less t h a n 3.4,
(11)
c* = z * / x
0.100 0.115 0.039
2.6 X 104
for for
S 465
16 to 18
where these results are suggested b y Roberts' data. T h a t is to say, no secondary from a p r i m a r y of diameter x is bigger t h a n cx, and the n u m b e r of secondaries larger t h a n x0 is given approximately b y the power law, Eq. (9). I n fact, from Table I I with x0 -- 10 ft a n d c = 0.11, we obtain m = 3.0, thus r = 0.88 M0 = 0.27,
(12)
but these estimates, based on just two points, are highly uncertain. F r o m (Dodd, Salisbury, and Smalley, 1963; Moore, Kachadoorian, and Wilshire, 1964) we obtain the fact t h a t the size distribution of secondaries is an inverse power law, q(z;x) -- ~x0~z-~-l(1 -- (cx/xo)-~) -1 forx0 < z <=cx = 0 otherwise,
(13)
594
A~LI~N H. MARCUS
where the constant w is rather large, perhaps ~o = 3.0 to 4.0. This distribution is obtained only for the Sedan experiment, but similar results a p p l y to the lunar craters Langrenus and Copernicus. The author's own analysis suggests co = 2 for the Sedan secondaries. Some additional data has recently become available. Shoemaker, quoted b y B r i n k m a n n (1966) uses functions similar to our Eqs. (9) and (13) with numerical values (in our notation) of c = 0.06, w = 3.56, and m = 3.56. In a letter to the author, Roberts (1966) points out t h a t the exponent in the e n e r g y diameter relationship is possibly less t h a n 3.4, probably between 2.5 and 3.4, and t h a t the value 3 m a y be the best choice. H e believes t h a t c = 0.11 is satisfactory. In Eqs. (9) and (11) the value m = 3 would be a minimum, he feels. Some additional important remarks on incompleteness of the secondary crater counts for terrestrial explosion craters are contained in a letter b y Roberts which is quoted in the Appendix. I t is clear t h a t the available data do not adequately specify the needed model functions. The author has not seen a n y further studies, however, and is thus forced to rely on material available as described in this Section. 4. ]~XAMPLES: METEOROIDAL IMPACT I-IYPOTHESIS
A form of Eq. (9) which permits a smooth transition of E ( S ; y ) to zero as cy decreases to one is given b y "f.
£OS,y) = Mo[(cy) m - 11 for cy > l = 0 otherwise.
Eq. (9) and Eq. (16) differ only slightly, with Eq. (16) preferred for physical reasons. I t m a y or m a y not be permissible to let XM become infinitely large. I n this connection the sign of ~, - m is crucial. If ~, -- m > 0, the n u m b e r of primaries decreases faster with increasing diameter t h a n the n u m b e r of secondaries per primary; thus the total n u m b e r of secondaries from large primaries will be negligible and we m a y take XM as large as we like. On the other hand, if 3' - m ~ 0, the n u m b e r of secondaries per p r i m a r y increases faster t h a n the n u m b e r of primaries decreases with increasing diameter; thus most of the secondaries come from the largest primaries and we m u s t retain a finite value of XM to guarantee convergence of integrals in Eq. (5). On the Moon, it is possible t h a t most of the observable secondaries come from the largest primaries, therefore ~/ - m < 0 is intuitively more plausible. With the above interpolation formulas, Eqs. (14)-(16), we consider these two cases. CASE 1 : 2 < - y < m < w + ~ - -
We will now work out some numerical examples relevant to the meteoroidal impact hypothesis. We assume x0 = 1 and
06)
2
I t is easily found by direct integration t h a t for 1 < x < cy, fcu 4~ (z -- x)2~z ~-1(1 -- (cg)-~)-~dz
p(:c) = ' y x - ' r - l ( l --3'.11 -v)
1
l-
for I ~ x < XM (14)
__ 7r¢o4 [1 -- (cy)-~] -I
= 0 otherwise, where ~/ > 2. As the preceding Section suggests, the diameter of the largest seconda r y from a p r i m a r y of diameter y is only a fraction of y, say cy, where 0 < c < 1 and c is the same for all y. The distribution of the diameter z of a secondary from a p r i m a r y of diameter y is an inverse power law
q(z;y) = ozz-'~ it1
--
= 0 otherwise.
(C~J)--¢°)-1 for 1 < z < cy
+
2(cy)l-'°X w-I
2X2_~
w(o~ -- 1)(~o -- 2)
(Cy)2-~ ~o--2
(cY~ -'°x2]
(17)
We assume t h a t XM is very large, but finite. Since all relevant powers of y are negative except for m - % the contribution of XM call be neglected everywhere except in connection with t h a t term. We therefore obtain approximately t
(15)
i The approxinmtion is obtained by also assuming 1
--
(cy)
-'~
=
1.
STOCHASTIC MODEL OF LUNAR CRATERS V. & ( x ) = ~3,/2~,(~)x ~--~ { coioc" [ (CXM)~f y(a, u**(x) = & ( x ) I + - - - 7 - - L m -- */ co(a) x~ */(2)
X m
(5" + ~ -- m) (3) m -- 3, 3,(3) 1]'}
(18)
(5" + co)(~) 7y . ,
F o r m o d e r a t e l y large values of x, the t e r m s in the d e n o m i n a t o r of X**(x)/~**(x) which involve x are of m u c h smaller order t h a n unity. T h u s for m o d e r a t e l y large values of x, we o b t a i n the a p p r o x i m a t i o n h**(x)
23,(2) 1 + coM0c______~ x~
~**(x)
~x~
where for a n y n u m b e r a, a (a) = a(a -
595
X
l ) ( a - 2).
1 -- ( 5 " q _ ~ _
m)(3~ ~ - - m
./(2)
Sinfilarly X**(x) = X
.
1+
Again in the ease t h a t F(t) -- F(to) >> 1, we o b t a i n lower a n d u p p e r b o u n d s for N ( x ) :
x--- z -
(cx~)~f-~ x"
m -- 5"
(19)
m -- 5"
2coMoc~
*/(a) [
~x ~1+
1 --
xco
- (3,+~-m)(~;
Thus X**(x)
~**(x)
2~ (2) X I + - - - T J - - L
m--5" m--3, ,),(3) Xm coMoc~ [ (CXM)~-~ 5"(3) X T -1--k ~ k m - - 5" co(a) ( * / + co - m)(~)m - 5"
~x 3
3,(3)
-]
(3, -V~)(~)5"J
a n d as x~ - ~ ~ we obtain the finite limit X m
x**(~)/~**(x) - , 2co(~)/~x~.
(20)
X
I n the case t h a t F(t) -- F(t0) ~ 1 a n d the t i m e - d e p e n d e n t b r a c k e t in Eq. (7) is a p p r o x i m a t e l y unity, we h a v e the following lower a n d u p p e r b o u n d s for N ( x ) , the c u m u l a t i v e n u m b e r of craters per unit area whose d i a m eter is larger t h a n x:
w(3)
f c =)
-7rX2 - -< N ( x ) = h
2coMoc~ 1+
x~
-- ( 1 - _<
5"(a) { 2coM0c~ N ( x ) _< - -7rx 2 l q x~
[ ×
5"(3)
m)
*/Ca) ~ 1 (3"-I-co)(3),5"(coq-2)1}
x~ (*/ -- m)(co + 2 -- m)
1]} 5"(~0 -t- 2) " (24)
~*(y;t)dy _< -,fix - 2
(CXM)'~--~'X~ ( m - - 5")(co + 2 - - 3')
x 'n
1
(m -- ~/)(co + 2 -- m)
]
3"(co + 2)J" (21)
I t is a s s u m e d here that. XM is v e r y large. CASE 2 : m < 3 , < ~ q - 3 , - -
2
W e again consider XM--~ ~ , as above, X**(x) p**(x)
(*/--m)(¢o+2--
25"(2) ~x 3
CASE 3: S a m e as either Case 1 or Case 2, except t h a t crater-filling is n o w the domin a n t factor. W e t h e n use Eq. (8) a n d need consider only h** (x). I t is necessary to specify the crater-filling a n d c r a t e r 4 o r m i n g rates, a n d the initial d e p t h - d i a m e t e r relationship. W e use the model of (Marcus, 1966b, Sec. 8). T h e crater-filling rate a(t) is given b y
proceeding
~ 5" -- m coMoc~ [ */(3) 1 q- ~ [ ( 5 ' q - ~
t > 0,
a(t) = aoe - ° '
x m
m ) ( 2 */) _- m
.t(a)
(*/ + co)(a)
(25)
~]
•
(22)
596
ALLAN H. MARCUS
the crater-forming rate f(t) b y f(t) = boe-~t
t > 0
(26)
and the initial d e p t h H ( x ) of a crater of diameter x b y H ( x ) = Cox~
x > 0,
(27)
where a, f~, ~, ao b0, Co are positive constants. We assume all craters are initially paraboloidal in shape. I t is then easily found t h a t F(t) -- F[to(x,t)] = C~[(1 + C.~x~)~/" -
1] (2s)
or with sufficient precision for our purposes, approximately F(t) -- F[to(x,t)] = C3x ~/",
(29)
where C1 = bo~3-1e-~t C2 = acoao-le"t/2 C3 = C1C2~l".
Upon combining Eqs. (19) and (28) we obtain the following approximation for the cumulative number N (x) per unit area larger t h a n diameter x, assuming s = ~ , ~/a >0: N (x)
~/C _____3 Xs
+
x~
(CXM) m--'~X'Y
X
(m--
~)(s+o~--
~,)
X 7tt
(m -- ~,)(~o + s - - m)
~(s + ~)
" (30)
If -y > m we need merely delete the t e r m involving XM. 5. NUMERICAL STUDIES OF THE IMPACT HYPOTHESIS
We will now illustrate some of the conclusions of the previous sections. In Case 1 of Section 4 the q u a n t i t y of greatest interest, from which other results m a y be derived is the ratio between the upper bound t~**(x) and lower bound Al(x), say R ( x ) = t,**(x)/hl(x).
We see t h a t Eq. (18) specifies R ( x ) when % m, ~, Mock, and CXM are specified. For application to the impact hypothesis we shall set ~f -- 2.25, m = 3, and c = 0.11. In connection with the Ranger data we will consider a m i n i m u m crater diameter x0 of 2 meters; with the plausible value of the maxi m u m crater diameter XM t a k e n as 180 km, this gives CXM as 104 in units of x0 and justifies the approximations used in Section 4. We consider two of the most likely values of ~, w = 3 and ~ = 4 . Because the nonuniformity of the distribution of centers of secondaries from a given p r i m a r y induces considerable variation in the density of small craters from point to point, we consider several possible values of Moc2"25: O, 10-4, 10-~, 10-2, 10-1. The results are shown in Fig. 1 and 2. I n both cases, the crater-destroying capability of the secondaries is very important when the density of secondaries is high, even though the n u m b e r of large secondaries is small. The ratio R ( x ) is not significantly greater t h a n 1 for x larger t h a n 20 or 30 (40 to 60 meters diameter) when ~0 = 4. But when ~ = 3, R ( x ) is conspicuously larger t h a n I even when x = 200 (400 meters diameter). This nleans t h a t when crater counts from unflooded continental regions become available, it will be necessary to take into account the effects of obliteration caused b y both p r i m a r y and secondary craters. Case 3 of Section 4 is relevant to the s t u d y of the Ranger 7 data, since most of the observed craters are in the flooded Mare Cognitum. As in (Marcus, 1966b) we will use an index s = 1.75 for the observed distribution of primaries after extensive flooding. Figures 3 and 4 show the expected cumulative number densities N ( x ) for the same set of parameters used in Figs. 1 and 2, with allowance for disappearance of primaries and secondaries due to filling. The distributions N ( x ) are all normalized b y setting the constant C3 in Eq. (30) equal to one. To correspond to terrestrial telescopic observations of the maria, C3 must be adjusted to give a density of a b o u t 1.8 craters larger t h a n 1 k m diameter per 103 k m 2 on the p r i m a r y crater curve.
I00"
W • 3
>¢ tt IO.
!
i
IO
!
Ioo
Iooo
g
Fzo. 1. Ratio R(z) between upper and lower bounds of the damage rate ~*(~) as a function of diameter x. Parameters are those of impact hypothesis with to = 3. Secondary density parameter is (top to bottom) Moc~-~ = 10-~, 10 -~, 10 -~, 10 -4. W.=4
R(x} f
,o
io °
,
!
i0 ~
I0'
I0 z
It}
X
FIG. 2. Ratio R(z) between upper and lower bounds of the damage rate ~*(z) as a function of diameter z. Parameters are those of impact hypothesis with ~ = 4. Secondary density parameter is (top to bottom) M'od.~ = lO-l, lO'-a, 10-3, 10-4. 597
5.08
ALLAN H. MARCUS
,o ~
,o ~
,d
,d
N(x) io"
i 0 -~"
, o -3
nd ~
2.
.
.
5.
IO
20
5' 0
I 0, 0
2 0
X
Fro. 3. C u m u l a t i v e n u m b e r d e n s i t y N(x) for m a r i a (theoretical) as a f u n c t i o n of d i a m e t e r x. P a r a m e t e r s a r e t h o s e of i m p a c t h y p o t h e s i s w i t h ~ = 3 a n d e x t e n s i v e crater filling. S e c o n d a r y d e n s i t y p a r a m e t e r is (top to b o t t o m ) M0c2.25 = 10 -1, 10 -~, 10 -3, 10 -4, 0. 6. COMPARISON WITH R A N G E R 7 CRATER DISTRIBUTIONS
We now compare the predicted distributions of Section 5 with actual crater-diameter distributions from Ranger 7 photographs. A v e r y complete crater count has been made b y H a r t m a n n (1965) and is shown in Fig. 5 b y the horizontal solid bars. The points at the end of the bars represent N ( x ) in its usual form, the n u m b e r of craters per square kilometer whose diameter exceeds x. The lower solid curve is' the predicted distribution of primaries. The upper solid curve is
the best fit obtainable with ~ = 4, namely 1.0. The dashed curve is the best fit obtainable with ~ = 3, namely M o c ~.25 = 10-1. T h e curve with ~ -- 3 seems definitely better than t h a t with w = 4. However, the distribution seems distinctly " r o u g h " and one might imagine t h a t an appropriate "smoothing" procedure would ease analysis of the data. This has been done b y K o p a l (1965b). The resulting distribution is shown in Fig. 6 (points) and the best fit is the upper solid curve ~ = 4, M o c 2"~5 = 1. Shoemaker's (1965) counts have already M o c 2"25 ~-
STOCHASTIC MODEL
OF LUNAR
CRATERS
599
V.
,o'] I O'
t0°
N(x) i 0 -~
!
g
2
5
!
fO
•
e
20
50
e
fO0
200
X "r"
FIG. 4. Cumulative number density N ( x ) for maria (theoretical) as a function of diameter x. Parameters are those of impact hypothesis with w = 4 and extensive crater filling. Secondary density parameter is (top to bottom) M ~ .u = 10-I, 10-3, 10-3, 10-4, 0. been smoothed. T h e y are shown in Fig. 7, the upper points representing the observed "within r a y " distribution and the lower points the observed "between r a y " distribution. The lower solid curve is the predicted distribution of primaries, the dashed curve is ~0 = 3, M o c ~'25 = 0.1, a n d t h e dotted curve is ~ -- 3, M o c ~'25 = 0.4. These give an adequate fit of the data, but only for x > 50 meters. K o p a l (1965b) has also smoothed these data and obtained the distribution shown in Fig. 8 (points) which is well fit b y the curve w = 3, M o c 2"25 = 0.1.
The extensive crater counts b y Osborn (1965) are shown in Fig. 9. The plotted points are crater densities derived from the Ranger 7 photographs indicated in the Figure. The roughness of the curve is merely due to statistical fluctuation, as was the roughness of H a r t m a n n ' s data. Osborn's data is also well characterized b y the curve = 3, M o c 2"25 = 0.1 but appears to be incomplete for x < 100 meters. The same incompleteness is suspected in the data of Miller (1965) and B r i n k m a n n (1966) whose counts are not reproduced here.
600
ALLAN H. MARCUS
I°5"
x%
.~ io'i=
0,,
Z
I0"
%6~,
\
id 5.
;o"
I
i 0 "~
crater
diameter,
i
td
I0 ~
x, km
Fro. 5. Solid points on bars are observed cumulative number density N(x) on Mare Cognitum, after Hartmann (1965). Lower solid curve is predicted from primary craters. Upper solid curve is predicted from primary and secondary craters with ~ = 4, Moc~.~ -- 1.0. Dashed curve is predicted from primary and secondary craters with w = 3, Moc~.~ = 0.I. I n s u m m a r y , the R a n g e r 7 d a t a are well predicted b y the theoretical Eq. (30), in connection with numerical parameters used in Section 5, b y setting ~ = 3 a n d M o c ~'~5 = 0.1. These results will be more fully i n t e r p r e t e d in the next Section. 7. CONCLUSIONS: ORIGIN OF SECONDARY CRATERS
While the shape of the d i a m e t e r distribution of small craters agrees with t h a t predicted b y t h e p r i m a r y a n d s e c o n d a r y i m p a c t hypothesis, the density of small craters is
a b o u t an order of m a g n i t u d e larger t h a n t h a t predicted b y a n a l o g y with terrestrial explosion craters. Using the values -y - 2.25 a n d s = 1.75 deduced previously, the observed distribution requires m = 3, ¢0 = 3, a n d for x0 = 2 meters, M0c~.~5 = 0.1. W i t h c -- 0.11 we h a v e M o = 14, or for x0 = 3 meters, M0 = 14 (2/3) 3 = 4.1, which is 15 times Iarger t h a n the value 0.27 suggested in Section 3 for terrestrial explosion craters. We have seen t h a t crater density estimates are likely to be inaccurate b y a large factor, b u t the a b o v e factor of 15 seems t o o large to be
STOCHASTIC
MODEL
OF
LUNAR
CRATEI:~
V.
601
I0 7 •
i06
.
I0~=
104=
|0 s.
~
10 z =
I~
I0'"
~I0
"-
= Io"-
,o"i 0 "s,
Id*. jo -r. _'%.
lO'* ..
;O.L
:0.|
!
Io"
,
,o'
,,
,o'
id' crater
diameter,
x , kin.
Fro. 6. Points are smoothed cumulative number density N(x) after Kopal (1965b) based on Hartmann (1965). Lower curve is predicted from primary craters, upper curve is predicted from primary and secondary craters with ~ -- 4, M0c2.~ = 1.0. explained by observational loss or inaccuracy. In the first place, it was noted in Section 2 t h a t we completely ignore geographical variation in the density of secondary craters. The regions photographed b y Ranger 7 are not large compared to the scale of geographical variation of secondary crater density. In particular, most of the last photographs seem to be of a region of especially high
crater density associated with a " r a y . " While Shoemaker's data suggest a difference in density of a factor of 4 between "on-ray" and "between-ray" craters, this is still not large enough to account for the observed excess of a factor of 15. A second possibility is that at least some of the lunar primary impacts have produced exceptionally large numbers of secondary craters. Perhaps some of the recent primary
~02
ALLAN H. MARCUS
,o,J
'l
•
"% %; % °° %. *% ",. .. • "o
i0 II,
• • I 0 i,
~ ~ ' ° . o ~I=: I0', v
0. I0°. X z IO".
I
i~
,'0"~
I'0•
I0'
I O "3
crater
diameter,
x, km
Fro. 7. Upper points are "within ray" cumulative number density N (.x) oa Mare (3og[fitt~n, lower points are "between ray" cumulative number density on Mare Cognitum, after Shoemaker (1965). Solid curve is predicted from primary craters, dashed curve is predicted from primary and secondary craters with ~ = 3, M0c2.~ = 0.1, dotted curve is predicted from primary and secondary craters with ~ = 3, M0d .~5 = 0.4. projectiles impacted on the Moon with exceptionally high velocity, tile violent explosion producing unusually large quantities of ejeeta. Although m u c h of this ejecta would escape from the Moon, it is possible t h a t an unusually large q u a n t i t y would be retained, creating a locally high density of secondary craters. This possibility is further developed ill a letter b y Roberts which is quoted in the Appendix. Finally, the surplus of small craters m a y be due to the fact t h a t m a n y of t h e m are, in
one sense or another, of internal origin. T h e y m a y be caused by degassing (Kuiper and H a r t m a n n , 1964; Miyamoto, 1964; Fielder, 1965) or b y collapse (subsidence) as has been argued b y Kopal (1965a). The two sources of uncertainty in our analysis n m s t always be kept in mind. First of all, the formation of secondary impact craters is not really well understood in a quantitative sense and the model functions we have been using nlay be quite inaccurate. Secondly, the crater-density estimates are based on small numbers of craters and it is
STOCHASTIC MODEL OF LUNAR CRATERS V.
603
,o.t ,o'-[ I 0 ~' .~
Id E 2¢
L
10'
~.
~--Io" X
z
io" i 0 -L
i 0 "s '
i0"i. .6
I0
i .I
!
I0
/O °
I 0 "~
crater
I
IO'
d~am¢ter,
x,km.
hG. 8. Points are smoothed cumulative number density N(x) after Kopal (1965b) based on Shoemaker (1965). Lower curve is predicted from primary craters, upper curve is predicted from primary and seconda~craters with o = 3, Moc2.2~= 0.1. thus possible that these estimates are inaccurate b y a factor of 2 or so. Detailed empirical s t u d y of crater counts from the Ranger 7 photographs is still needed to clear tip this last difficulty.
to be at least as complete as those reported b y others and yield about the same diameter distribution. Crater distributions from the Ranger 8 and 9 have recently become available (Jet
Note added in proof. In a letter to the
Propulsion Lab., Calif. Inst. Technol. Tech. Rept. 32-800, Pt. II). The diameter distribu-
author (May 22, 1966), W. H. Osborn has reported a revision of the crater counts in his Scope paper, based on improved scale and altitude determinations from the Ranger 7 photographs. The revised counts appear
tion is similar in shape to that found in Mare Cognitum, with similar density in Alphonsus but rather higher density in Mare Tranquillitatis. This suggests that somewhat less crater-filling activity has occurred in
604
ALLAN H. MARCUS [] - 3 8 0 + -38! A - 372 Z - 379 O - 373 • - 359 4~- 3f7
+ Q
0 - 24~
IO ~ .I
+
I O*
i 0 "~.
vt
E
Q. x
z \ \ 0
i0 "b.
16"
to-'
:o. crater
diameter,
x.
,'o'
km
Fro. 9. Points are cumulutive number density N(x) in Mare Cognitum from indicated Ranger 7 photographs, after Osborn (1065). Lower curve is predicted from primary craters, upper curve is predicted from primary and secondary craters with oJ - 3, M0c2-~ -- 0.1. Mare Tranquillitatis than in the other two regions. A detailed analysis of these data will be published separately. APPENDIX
Excerpts from a letter by W. A. Roberts (September 12, 1966): "Any counts of secondary craters for modelling studies, such as Tulalip, will be incomplete due to the discrete boundaries of the test pad. Incomplete counts for Sedan are related to available time, both from a
contractual point of view and because the e]ecta blanket was radioactive, requiring that field work be limited and performed iu haste. "In your conclusions, you suggest that exceptionally high velocity impacts may have produced unusually large quantities of ejecta. Such large quantities of ejecta would have been expelled from an appropriately large crater volume. On the moon, the lack of atmosphere and the low gravity will probably result in higher impact (terminal)
STOCHASTIC MODEL OF LUNAR CRATERS V.
velocities and longer trajectories for the missies. The higher impact velocities will result in correspondingly wider secondary craters for given conditions of missle and target since the diameter would be a function of the kinetic energy or about V 2/a. This could m e a n that, for a given lower limit of resolution, a missle of a given size which would not produce a visible secondary crater from a p r i m a r y on earth could produce a resolvable structure on the moon tending to increase the n u m b e r of secondaries above some diameter limit. Most of the large secondaries seen adjacent to Sedan a p p e a r to have been formed b y composite missies. One such missle, completely winnowed on a longer trajectory, m a y have resulted in eleven small craters within a radius of about 75 feet west of Sedan. On the moon, the longer trajectories of the composite missies will allow more time for the composite missles to disperse. Although there is no atmospheric drag to break up these missies, small differences in ejection angle and velocity in these masses (composite missies, or unconsolidated masses) would result in more dispersion for the longer flight times and larger numbers of secondaries. " B o t h Sedan and Scooter were deeply buried charges, and the ejection velocity is obviously a function of scaled burst depth. Missle velocities near Sedan were of the order of 0.1 km. Missle velocities of these masses resulting in secondary craters on the moon could easily be an order of magnitude higher from the v e r y shallow scaled burst depths of the p r i m a r y craters. This would yield craters (given missle mass) a b o u t 5 X larger. I do not have enough d a t a to resolve the problem of the ejecta blanket when the parameters scaled depth of burst, yield, explosion medium, and explosive type are considered . . . . "
605
V. G. (1963). Crater frequency and the interpretation of lunar history. Icarus 2, 466-480. FrsLVER, G. (1965). "Lunar Geology." Lutterworth Press, London. HAIrr~ANN, W. K. (1965). Secular changes in meteoritic flux through the history of the solar system. Icarus 4, 207-213. KoPAL, Z. (1965a). The nature of secondary craters photographed by Ranger VII. Math. Note 430. Boeing Scientific Research Lab., Seattle, Washington. KOPAL, Z. (1965b). Topography of the Moon. Space Sci. Rev. 4, 737-855. KvlPER, G, P., A~D HARTMA~r, W. K. (1965). Interpretation of Ranger VII Records. Jet Propulsion Lab., Cali]. Inst. Technol. Tech. Rept.
32-700, Pt. II, Chapter 3, 32-74. Maacvs, A. H. (1964). A stochastic model of the formation and survival of lunar craters. I. Distribution of diameter of clean craters. Icarus 3, 460-472. MARCUS, A. H. (1966a). A stochastic model of the formation and survival of lunar craters. II. Approximate diameter distribution of all observable craters. Icarus 5, 165-177. MARCUS, A. H. (1966b). A stochastic model of the formation and survival of lunar craters. III. Filling and disappearance of craters. Icarus 5, 178189. MILLER, B. P. (1965). Distribution of small lunar craters based on Ranger VII photographs. J. Geophys. Res. 70, 2265-2266. MIYAMOTO,S. (1064). Morphological aspects of the lunar crust. Icarus 3, 486-490. MOORE, H. J., KAC~-ADOORL~,R., A~D WmSHI~E, H. G. (1964). A preliminary study of craters produced by missle impacts. U. S. Geol. Survey, Astrogeol. Studies Ann. Progr. Rept., July 1, 1963 to July 1, 196/~, Pt. B, 58-92. OSBORN, W. H. (1965). Lunar crater counts. Scope (J. Fed. Univ. Astron. Soc.) 7, 63-76. ROBERTS, W. A. (1964). Secondary craters. Icarus
3, 348--364. ROBERTS, W. A. (1966). Personal communication.
SHOEMAKEr, E. M. (1962). Interpretation of lunar craters. In "Physics and Astronomy of the Moon" (Z. Kopal, ed.), pp. 283-359. Academic Press, New York. SHORMArd':R,E. M. (1965). Preliminary analysis of REFERENCES the fine structure of the lunar surface in Mare BRINKMANN,R. T. (1966). Lunar crater distribuCognitum. Jet Propulsion Lab., Call]. Inst. Techtions from the Ranger 7 photographs. J. Geonol. Tech. Rept. 32-700, Pt. II, Chap. 4, 75-134. phys. Res. 71, 340--342. VAILR, R. B. (1961). Pacific craters and scaling DODD, R. W., JR., SALISBURY, J. W., AND SMALLEY, laws. J. Geophys. Res. 66, 3413--3438.