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Large impact craters and the moon's orientation
Earth and Planetary Science Letters, 26 (1975) 353-360 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
LARGE IMPACT CRATERS AND THE MOON'S ORIENTATION* H.J. MELOSH Division o f Geological and Planetary Sciences, California Institute of Technology, Pasadena, Calif. (USA} Received October 29, 1974 Revised versions received February 10, and April 29, 1975
This paper investigates the idea that large impact events have caused the moon to change its orientation in space. It is found that the very largest impact events, such as thOse which formed Imbrium and Orientale, probably did reorient the moon. This reorientation is primarily due to the change in the moon's moments of inertia consequent upon crater formation. The impulse delivered by the impact can at most unlock the moon's synchronous rotation for a few thousand years, and is thus not of major importance. The moon will attain its new orientation in less than a few times 104 years as a result of tidal friction. Since the large craters eventually are f'flled by isostatic rebound and extrusive igneous activity, the moon may eventually regain its original orientation unless other phenomena cause new changes in the distribution of mass on its surface.
1. Introduction The purpose of this paper is to explore the possibility that large impact cratering events have significantly altered the moon's orientation. As we shall show, the impulse delivered to the m o o n by an impact has only a transient effect on its orientation; however, the change in the moon's m o m e n t of inertia caused b y the crater may, in a few cases, be sufficiently large to interchange maximum and intermediate principal moments. If the moon was rotating synchronously in orbit about the earth at the time that this interchange t o o k place, tidal damping would result in a complete reorientation of the moon on a geologically short time scale. Briefly, what happens when a sufficiently large crater is formed on the moon is the following: (1) The impulse delivered by the impact may cause the moon to rotate non-synchronously for a few thousand years before tidal friction again brings the moon into synchronous rotation. (2) The moon reorients in a few thousand to a few * Contribution No. 2551, Division of Geological and Planetary Sciences, California Institute of Technology.
tens of thousands of years, with the new maximum m o m e n t o f inertia perpendicular to the orbital plane [ 1 ], the intermediate m o m e n t tangent to the orbit, and the minimum m o m e n t on the e a r t h - m o o n line. (3) The crater slowly f'dls, either by isostatic uprise of underlying material or by extrusive igneous activity. (4) As the crater is filled, its effect on the moon's moment of inertia declines sufficiently for the old principal axes to reassert themselves, and a second reorientation takes place, bringing the moon back to approximately its original orientation (in the absence of other changes in the moon's mass distribution). The details of the story outlined above will be presented as follows: We shall first discuss the impulse delivered by the impact, and changes in the m o m e n t of inertia of the moon brought about by craters, establishing criteria for when this change may be considered likely to reorient the moon. We shall also discuss the dynamic evolution of the moon shortly after the cratering event, and compute the time required for tidal friction to bring the moon into its new orientation. The question of the stability of large craters will then be addressed, and we shall see how long a large crater may survive unfdled on the surface of the moon.
354 2. Reorientation of the moon by impacts
2.1. Impulse delivered by the impact A large impact event on the moon can affect its rotation and orientation in two ways: First, the impulse delivered by the impact itself can change the angular momentum of the moon, and second, the cra ter formed as a result of the impact can change the moment of inertia of the moon, thus leading to a reorientation. The impulse delivered by even a very large impact is extremely small, that of Imbrium being no larger than about 3% of the moon's present rotational angular momentum [2]. Despite this small value, however, such an impulse can have an important effect on the moon's orientation. Thus, if the moon had one face locked to the earth before the impact occurred, it is possible for the impact to deliver enough rotational energy to the moon to unlock it and give it a slow non-synchronous rotation until tidal damping again brings it into synchronization. In the absence of changes in the moon's moment of inertia the final orientation will, of course, be the same as before the impact except for a possible 180 ° rotation about the polar axis. We can obtain a lower limit on the crater size associated with an unlocking event if we compare the energy required to unlock the moon today [3], 2.1 × 1027 erg, with the energy derived from the maximum possible impulse [2] (i.e. from a tangential impact on the moon's equator by a body moving at the moon's escape velocity), AE = LAL/C = 1.23 X 106r 3 erg, where r is the crater radius in centimeters, L is the moon's present total rotational angular momentum, AL is the impulse delivered by the impact, and C is the moon's moment of inertia about its polar axis. These two energies are equal for craters of 120 km radius, thus fixing a strict lower limit on the size of a crater which could have been associated with an unlocking event. There are only ca. 23 craters visible on the moon with radii greater than this lower limit. Hence, we can conclude that only the very largest impact events may have been associated with the unlocking of the moon's synchronous rotation. Events such as those which produced Imbrium, Orientale, Nectaris, and Serenitatis are the prime candidates. Although in a few cases the impulse delivered by the impact may have had a significant effect on the moon, this effect is generally short-lived. Since the im-
pulse is much smaller than the original angular momentum of the moon, the moon's axis of rotation is nearly unchanged. All that happens, even for the largest impacts, is that the moon's rotation becomes non-synchronous by a small amount. Tidal friction will rapidly bring the moon back into synchronous rotation, and the only permanent effect this episode could have on the moon's orientation would be a 180 ° rotation about the polar axis, thus interchanging near and far sides.
2.2. Moment of inertia change caused by impacts The moon's principal moments of inertia, denoted C > B ~ A, are normally oriented as shown in Fig. 1. The axis of the greatest principal moment is perpendicular to the moon's orbital plane, the intermediate axis is tangent to the orbit, and the minimum axis is along the line connecting the centers of the earth and moon. The reason for this orientation is that it is the lowest energy configuration for a synchronously rotating moon. If the moon's moments of inertia were significantly changed by the production of a large crater, the principal axes would suddenly be rotated into new positions [4]. These new positions would not, in general, coincide with the minimum energy orientation shown in Fig. 1. The moon's new principal axes would oscillate about their minimum energy configuration in much the same way that a pendulum oscillates about its minimum energy configuration. As this oscillatory motion is damped by tidal friction, the new principal axes are eventually brought into the minimum energy config-
H Earth
0.,-~/~'o'I plo,.'po Y Fig. 1. The coordinate system used for discussing the moon's orientation is shown above. This is a rotating coordinate system in which both the earth's and the moon's apparent position is fixed. Also shown is the lowest energy orientation of the axes of the moon's principal moments of inertia C > B > A.
355
uration. The ultimate result of this process is to reorient the moon. Since the new lunar pole can be virtually anywhere on the moon's surface (depending on the size and location of the impact crater), we see that this process may be very important in determining which portion of the moon shall face the earth. In order to estimate the importance of this process, we must have some idea o f how large a crater must be before it can cause a significant change in the moon's moment of inertia. We shall presume that a "significant change" is one which alters the differences of the principal moments by amounts equal to or greater than their differences at the present time [5]. Given this criterion, we shall use two extreme models to calculate the moments of inertia of a crater. In this way we hope to bracket the most probable value for the difference of the principal moments caused by crater formation. Crater model A (see Fig. 2a) represents the crater as a simple circular hole of depth h and radius aO, where a is the radius of the moon and 0 is the angle (in radians) between the center of the hole and its edge as measured from the center of the moon. The model A crater has no rim, thus representing an impact in which all of the throwout escaped the moon. In the case that h ~ a, but where the curvature of the moon cannot be neglected, we have: I I ( A ) =/2 (A) 7T
a4hp (1 - cos 0)(4 + cos 0 + cos20) (la) 3 2rr I3(A) = - - - a4hp (1 - cos 0)2(2 + cos 0) (lb) 3 where 13 is the crater's moment of inertia about an axis passing through the center of the moon and the center of the circular crater, 11 and 12 are the crater's moments about axes perpendicular to the axis of I3, and O is the density of the moon's crust. The impor-
Fig. 2. (a) Crater m o d e l A. Simple rimless hole o f depth h and radius aO. Co) Crater m o d e l B. Same as A with an ejecta blanket extending to two crater radii from center. Mass in blanket equals mass removed from crater.
tant quantity for our purposes is the difference I3(A) -I,(A): I3(A) - I I ( A ) = 7ra4hp cos 0 ( 1 - cos20)
(2)
Thus, the net effect of this crater is to increase the relative moment o f inertia about the axis passing through the centers of the moon and the crater. Crater model B (see Fig. 2b) is the same as model A except that it includes a rim. In this model we assume that all of the material excavated from the crater is spread uniformly between one and two radii from the crater's center. This blanket of excavated material has a thickness t: h t =
1 + 2cos0
(3)
In the limit where h ,~ a, we compute for model B:
I3(B) - I, (B) = 2 [/3(.4) - I1 (.4)] (1 - cos 0) × (1 + 2 cos 0)
(4)
As might be expected, the presence of a rim equal in mass to the material excavated from the crater greatly reduces the moment of inertia difference. This reduction is greater for smaller craters, but even for a crater the size of Imbrium it is a factor 5. The differences 13 - 11 according to both models are plotted in Fig. 3 for various crater depths and radii. Craters the size of Imbrium or Orientale would have to be 1 0 - 3 0 km deep to significantly affect the moon's orientation. Note that these are the depths of a cylindrical hole whose volume equals that of the real crater. Since real craters are cup-shaped, the depth of the center of the crater below the pre-impact surface, h*, must be larger than the corresponding h in our models. In observed craters such as Arizona Meteor Crater, or the Jangle U nuclear explosion crater [6] we find h* 3h. For these craters h* is approximately 1/3 of their radius, so h ~ 30 km is reasonable for craters with radii of 300 km or larger (i.e. h ~ r/lO where r is the crater radius). Both models demonstrate that only the largest craters could have significantly changed the moon's moments of inertia. Furthermore, model B is likely to be the more realistic representation of a crater. Current work indicates that very little of the material from the crater would be ejected from the moon [7]. Thus, it seems that only Imbrium, and perhaps Orientale, could have had any chance of reorienting the moon.
356 104°[
i
i
i
MODEL A
i
MODEL B
/ krn
~
+
I039
/
+
km
i
C-A
E d
o c
B-A
B-A
km
I1)
LS 103a 0
÷
c
oE 1037
+
3-
.'2_
z,3 1
0
0
100
3
+
c ~
ea ~
-
Y,Y"
6
_E
6 ~ 200 500 400 500 Crater Radius, km
600
700 0
100
200 3o0 400 500 Crater Radius, km
600
700
Fig. 3. Moment of inertia differences as a function of crater radius and depth for two crater models. Model A represents the crater as a rimless circular hole in the moon's surface. Model B includes a rim of mass equal to that excavated from the crater. Both models neglect the depth of the crater relative to the moon's radius, but they include the effect of the moon's curvature. The horizontal dotted lines on the graphs are the differences between the moon's moments of inertia at the present time. where G is the gravitational constant, M® the earth's mass, R the earth-moon distance, and:
2.3. Behaviour o f the moon after a large impact Shortly after the impact, the moon's orientation will be a very complex function o f time. Aside from the probable unlocking o f its synchronous rotation, the moon will suffer large librations o f its new principal axes about their minimum energy positions. All of these motions will be damped by tidal friction, which will eventually bring the moon into synchronous rotation and orient it as in Fig. 1. An estimate o f the time required for tidal friction to brake the m o o n ' s non-synchronous rotation is already available in the literature. Goldreich and Sorer [8] have shown that the earth's tidal torque on the moon's spin is: 9 GM~a s N =
4 Q'R 6
(5)
a ' = O (1
387ra4/al
+ 3~-~m~!
(6)
where Q is the quality factor for the whole moon, # the m o o n ' s rigidity, and m is the moon's mass. For small planets, such as the moon, the second term in eq. 6 dominates the first since most o f the tidally induced distortion energy is elastic rather than gravitational. The rate at which the non-synchronous angular velocity, w, is dissipated is thus: dw dt
N -
C
45 GMe 2 a a -
8
R6
mQ'
(7)
where we have put C ~. 2/5 ma 2. Goldreich and Soter [8] have argued that 10 < Q < 100 for the moon, so that if we take Q ~ 100 we should get an upper limit
357
to the tidal damping time. If the impact gave the moon a non-synchronous rotation with an angular frequency equal to 3% of its present angular frequency (corresponding to the maximum possible impulse which the Imbrium event could have delivered [2]), then at the present earth-moon distance this rotation would be damped out in less than 7,000 years (where we have used/a = 10 x1 dyne/cm 2 to make our estimate). Since the damping time depends upon R 6, a smaller earthmoon distance at the time of the impact would cut this estimate considerably. Having established that the non-synchronous rotation of the moon would be damped extremely rapidly, it remains to establish a similar result for the periodic librations of the moon about its final orientation. We shall simplify the situation somewhat by supposing the librations to be simple, pendulum-like oscillations of frequency 0%. Let 0m be the moon's maximum angular excursion during one oscillation (measured from the equilibrium point). The energy lost per cycle due to tidal friction is four times that lost in moving from the equilibrium point to the maximum excursion: /m AE = -4
NdO = -4NOra
(8)
0
assuming that N is independent of the moon's angular excursion O. Thus: dE dt
-
Wo 21r
zXE = -
2~Oo 7r
NO m
(9)
The rate of change of the energy involved in the periodic motion is/~ = 6oo2C0m0m , so that: dOm _ dt
N lrwoC
(lO)
The time required to reduce 0 m from 1 radian to zero is: 1
Tdamp = [dOm]dt[
(11)
To estimate Tdamp, we need wo. A representative value is obtained by considering the energy required to twist the moon about its polar axis as a function of angle. (By choosing the polar axis rather than some other axis, we avoid complications due to the moon's angular momentum. These complications are not essential for the estimate of the damping time, and our result is only
changed slightly by their inclusion.) It is a straightforward problem to compute the interaction energy of the moon's quadrupole moment with the tidal potential of the earth; the resulting equation of motion for the moon is: 20x + COo2 sin 2 0 x = 0
(12)
where 0 x is the angle of rotation about the x-axis and: ]/ 3GMcB - A Wo =
R3
C
(13)
At the present time coo = 9.8 X 10 -s sec -~, which yields a free libration period of about two years. Using the same values for N and C as in our previous estimate of the rotational damping time, we find that Tdamp 3 X 104 yr for Q -- 100. Thus, the librations of the moon are damped very rapidly, even at its present distance from the earth. The energy dissipated in the moon by this process is negligibly small, amounting to only ca. 3 tacal/g of the moon, and so cannot have any noticeable effect on the thermal state of the moon's interior. We can conclude that the complex motions of the moon's orientation immediately following a major impact event will be quickly damped. Both the nonsynchronous rotation and the librational oscillations will cease within a few tens of thousands of years after the event. If the crater produced by the impact was sufficiently large, the moon's final orientation may be quite different from its initial orientation, even apart from a possible 180 ° rotation about the polar axis due to the unlocking and recovery of synchronous rotation. These estimates of the damping time of the moon's librational motions are very strict upper limits. If the moon was closer to the earth at the time of the impacts, or if its Q was lower, the damping timewould be correspondingly decreased. Moreover, we have been treating the moon as a cold elastic body, thus neglecting the possibility of internal damping. In fact, if the moon had a fluid core at the time of the major impacts, much of the orientational energy could be dissipated by the viscous coupling of core and crust. It thus seems likely that the moon (which even now could have a small fluid core [9]) also dissipated some of its orientational energy in this way. If, for example, this process were characterized by a Q of 100, the damping time would be only 200 years. Note that this
358 reduction in damping time applies only to the oscillatory part of the moon's motion; the non-synchronous rotation must be damped tidally since it requires a net loss of angular momentum. Having established that the moon would probably be reoriented by major impact events, and that it would relax into its new orientation very shortly after such an event, it is reasonable to ask how long it retains its new orientation. It takes a large, deep crater to change the moon's moment of inertia enough to cause reorientation. Such deep craters are not present on the moon's surface today. If such craters ever existed, they must have been frilled since their formation, and it is important to know how long it took for this to happen. 3. The stability of very large craters There seems to be little doubt that the initial cavity excavated by a large impact has a depth h* of at least 1/3 of its radius [10,11]. The crucial question from the point of view of reorientation is whether a sufficiently deep cavity can remain unfilled for more than the 10 4 years required for reorientation to occur. Craters on the moon are modified by at least three processes [ 12] discussed in the following subsections.
3.1. Mechanical collapse This is the most serious modification because it occurs almost immediately after the crater is formed. If mechanical collapse of the crater succeeds in replacing enough of the excavated material, the equivalent depth h may be too small for the change in the moment of inertia induced by the crater to reorient the moon. However, there are several reasons for believing that mechanical collapse did not result in sufficient Idling to prevent Imbrium from reorienting the moon. First, it has been shown [13] that large impact craters are stable against deep seated slumping in the absence of processes which can relieve the overburden pressure. Modifications of the crater are limited to relatively shallow slumping in regions near the rim where the slope of the initial cavity exceeds the angle of repose of pulverized rock: 3 0 - 4 0 ° . There is no mechanical reason why a depression 60 km deep and 1,000 km in diameter cannot be supported on the
moon, as long as the marginal slopes have been reduced to 3 0 - 4 0 ° or less by slumping. Altitude differences of 10 km have been similarly supported on the earth, with its 6 times larger gravity, for periods exceeding 10 m.y. Such static analyses, however, may not apply during the few minutes immediately after the impact event, when large seismic disturbances and possibly tensional stress waves are propagating through the rocks adjoining the crater. Even if large slumps are then possible, it seems unreasonable that they should extend further from the impact point than twice the radius of the initial cavity. Existing data [14], however, indicate that only about 50% of the excavated material lies within two radii from the impact point. Thus, even such widespread slumping as this could at most halffill the crater. Such a collapse would still not prevent Imbrium from reorienting the moon.
3.2. Extrusive igneous activity It is not known to what extent igneous activity can frill the basins of large craters. Basalt floods have certainly played some role in filling all large basins, but they are probably not more than a few kilometers thick [15]. The fact that the last basalt flows in Imbrium occurred ca. 7 × 108 yr after excavation of the basin [16] provides an upper limit on both the time required to fill the basin by basalt flows, and the time required to fill the basin by any process. If any appreciable uplift of Imbrium's floorhad occurred after the latest basalt flows, it would have resulted in updoming of the floor and fracture systems which are not observed. We thus conclude that the Imbrium basin was frilled to nearly its present level, and that the reorientation sequence was completed less than 7 × l0 s yr after the impact event.
3. 3. lsostatic rebound High-temperature creep of the rocks beneath the moon's surface is certain to lead to isostatic rebound of any large, uncompensated, depression. The major uncertainty in the rate of this process is the value of the moon's effective viscosity. In view of the uncertainties involved, a crude order of magnitude estimate of the isostatic rebound rate will suffice to illustrate the problem. We Shall represent
359 the crater as a surface load pgu(t)Jo(otr/d) where g is the acceleration of gravity at the moon's surface, u(t) is the depth of the center of the relaxing crater below the pre-impact surface (u(t = O) = h*), r is distance from the crater's center, d is the crater radius, and t~ is some constant of order unity. Let this load be applied to a half space of uniform Newtonian viscosity r~. We find that u(t) = h* exp(--t/w) where relaxation time: "c = 2a~/pgd
(14)
Thus, the larger the crater, the faster it relaxes. The real problem in estimating the relaxation rate of large craters is to determine the effective r/. Estimates in the literature range from 2 X 102s poise for craters less than 30 km in diameter [17] to 1026 or 1027 poise required to support the mascous for 3 X 109 yr [18]. Accepting these values for r~ yields relaxation times ~- ranging from 100 m.y. to 5,000 m.y. for a crater 500 km in radius. Unfortunately, such numbers should not be taken too seriously. We do not know what the detailed thermal structure of the moon was when the mare-forming impacts took place. Since r/may be an exponential function of the temperature, this lack of knowledge is very significant. If at the time o f the impacts r~ was as low as the value estimated for the earth's asthenosphere, 1022 poise, then we would find Z --~ 70,000 years. In this last case the moon would barely have time to reorient before the crater was t~flied. In spite of all the above uncertainties, however, it does not seem improbable that a large impact event, such as the one which excavated Imbrium, could have reoriented the moon for a period exceeding several million years. Ultimately, of course, some combination of isostatic rebound and extrusive igneous activity will fill the crater. In the case of Imbrium this took less than 700 m.y. As the crater t'dls, its effect on the moon's moments of inertia declines and the moon can return to its old orientation, barring complications (such as another large impact).
can cause even more profound reorientation. Very large craters, such as Imbrium and larger, may be expected to migrate to a position near one of the moon's poles in a few tens of thousands of years after their formation. This position cannot be retained indefinitely, however, for the crater is eventually Idled by a combination of collapse, isostatic rebound, and extrusive igneous activity. As the crater is filled, the differences of its moments of inertia decline at a corresponding rate, and the moon's orientation undergoes new alterations. It is thus probable that the moon has not always had its present orientation. The terrain near Imbrium has probably spent some time in the lunar polar regions, as well as near the moon's equator at the limb (this progression occurred as Imbrium first marked the maximum principal axis, then the intermediate, and finally the minimum; these changes are due to fdling, then overfdling Imbrium to form a mascon). At present there is no observational evidence for the reality of these reorientations. This lack is not surprising, since we do not expect the moon's orientation to have a strong effect on its surface. Nevertheless, we hope to have demonstrated the probable existence of an episode of reorientation in the moon's past and thus sketched out another aspect of lunar history. The idea that the moon's orientation can be greatly altered by large impacts may be useful in circumstances of which we are not presently aware.
Acknowledgment I would like to thank G.J. Wasserburg, whose question sparked this investigation, and with whom I have had several enlightening discussions. P. Goldreich also aided in clarifying my ideas on the importance of tidal effects on the moon's figure.
N,~tes and: references 4. Conclusion We have seen that large impact events have a strong influence on the moon's orientation. The impulse delivered by such events is capable of unlocking the moon's synchronous rotation. The change in the moon's moment of inertia due to crater formation
1 We shall always work in the approximation that the moon's orbit is circular and that its polar axis is perpendicular to the orbital plane. We shall also neglect the tides on the moon raised by the sun. Since tides on the moon due to the sun are only 11180 the height of those due to the earth, this is an excellent approximation. 2 This estimate is arrived at following W.M. Kaula, Phys.
360 Earth. Planet. Inter. 2 (1969) 123. We put the maximum mass which could have formed a crater of diameter r cm at: M = 6.54 X 108 (ra/v 2) g where v is the velocity of the infaUing object in cm/sec. The maximum angular momentum AL which M could deliver to the moon is obtained for a tangential impact at the moon's equator:
5
AL = Mva = 6.54 X 108 (r3a/v) g cm2/sec where a is the moon's radius in cm. The smallest possible v is 2.4 km/sec, the moon's escape velocity. Thus, for Imbrium with r = 550 km:
6 7
AL = 7.9 X 1034 g cm2/sec which is only about 3% of the moon's present rotational angular momentum, 2.3 X 1036 g cm2/see. 3 The energy required to interchange the axes of the intermediate principal moment, B, with that of the minimum principal moment, A, is given by:
E -
3 GM. R~- (B - A ) 2
where G is the gravitational constant, M . is the mass of the earth, and R is the earth-moon distance. At the present timeB - A = 2.01 X 1038 g cm 2, so: E = 2.1 X 1027 erg 4 The new principal axes of the moon and crater together
8 9 10 11 12 13 14 15 16 17 18
are found by summing their inertia tensors and then diagonalizing the result. It makes no difference whether we perform the diagonalization about the center of mass of the system or about the center of mass of the moon alone. The difference between the two results is of the order of the mass moved out of file crater divided by the mass of the moon, and so is entirely negligible. At the present time C - A = 5.60 × 1038 g cm 2 and B - A = 2.01 X 1038 g cm 2. These values are based on C = 8.88 X 1041 g cm 2 along with the values for # and ",/given by P.L. Bender et al. in Science 182 (1973) 229. Obtained from data presented by E.M. Shoemaker, in: The Solar System, 4 (Univ. of Chicago Press, 1961) 301. D.E. Gault, E.M. Shoemaker, H.J. More, Spray ejected from the lunar surface by meteoroid impact, NASA Technical Note D-1767. P. Goldreich and S. Soter, Icarus 5 (1966) 375. U. Nakamura et al., Geophys. Res. Lett. 1 (1974) 137-140. R.J. Pike, Geophys. Res. Lett. 1 (1974) 291. D.E. Gault et al., in: Shock Metamorphism of Natural Materials (lVlono Book Corp., 1968) 87-99. W.L. Quaide et al., Annals, N.Y. Acad. Sci. 123 (1965) 563. B. Dent, AGU Abstract T40, EOS 54 (1973) 1207. H.J. Melosh, Unpublished (1974). R.H. Carlson and G.D. Jones, JGR 70 (1965) 1897. K.A. Howard, D.E. Wilhelms and D.H. Scott, Rev. Geophys. Space Phys. 12 (1974) 309. F. Tera et al., Lunar Science IV (1973) 723. A.W.G. Kunze, Phys. Earth Planet. Inter. 8 (1974) 375 J. Arkani-Hamed, The Moon 6 (1973) 100.
LARGE IMPACT CRATERS AND THE MOON'S ORIENTATION* H.J. MELOSH Division o f Geological and Planetary Sciences, California Institute of Technology, Pasadena, Calif. (USA} Received October 29, 1974 Revised versions received February 10, and April 29, 1975
This paper investigates the idea that large impact events have caused the moon to change its orientation in space. It is found that the very largest impact events, such as thOse which formed Imbrium and Orientale, probably did reorient the moon. This reorientation is primarily due to the change in the moon's moments of inertia consequent upon crater formation. The impulse delivered by the impact can at most unlock the moon's synchronous rotation for a few thousand years, and is thus not of major importance. The moon will attain its new orientation in less than a few times 104 years as a result of tidal friction. Since the large craters eventually are f'flled by isostatic rebound and extrusive igneous activity, the moon may eventually regain its original orientation unless other phenomena cause new changes in the distribution of mass on its surface.
1. Introduction The purpose of this paper is to explore the possibility that large impact cratering events have significantly altered the moon's orientation. As we shall show, the impulse delivered to the m o o n by an impact has only a transient effect on its orientation; however, the change in the moon's m o m e n t of inertia caused b y the crater may, in a few cases, be sufficiently large to interchange maximum and intermediate principal moments. If the moon was rotating synchronously in orbit about the earth at the time that this interchange t o o k place, tidal damping would result in a complete reorientation of the moon on a geologically short time scale. Briefly, what happens when a sufficiently large crater is formed on the moon is the following: (1) The impulse delivered by the impact may cause the moon to rotate non-synchronously for a few thousand years before tidal friction again brings the moon into synchronous rotation. (2) The moon reorients in a few thousand to a few * Contribution No. 2551, Division of Geological and Planetary Sciences, California Institute of Technology.
tens of thousands of years, with the new maximum m o m e n t o f inertia perpendicular to the orbital plane [ 1 ], the intermediate m o m e n t tangent to the orbit, and the minimum m o m e n t on the e a r t h - m o o n line. (3) The crater slowly f'dls, either by isostatic uprise of underlying material or by extrusive igneous activity. (4) As the crater is filled, its effect on the moon's moment of inertia declines sufficiently for the old principal axes to reassert themselves, and a second reorientation takes place, bringing the moon back to approximately its original orientation (in the absence of other changes in the moon's mass distribution). The details of the story outlined above will be presented as follows: We shall first discuss the impulse delivered by the impact, and changes in the m o m e n t of inertia of the moon brought about by craters, establishing criteria for when this change may be considered likely to reorient the moon. We shall also discuss the dynamic evolution of the moon shortly after the cratering event, and compute the time required for tidal friction to bring the moon into its new orientation. The question of the stability of large craters will then be addressed, and we shall see how long a large crater may survive unfdled on the surface of the moon.
354 2. Reorientation of the moon by impacts
2.1. Impulse delivered by the impact A large impact event on the moon can affect its rotation and orientation in two ways: First, the impulse delivered by the impact itself can change the angular momentum of the moon, and second, the cra ter formed as a result of the impact can change the moment of inertia of the moon, thus leading to a reorientation. The impulse delivered by even a very large impact is extremely small, that of Imbrium being no larger than about 3% of the moon's present rotational angular momentum [2]. Despite this small value, however, such an impulse can have an important effect on the moon's orientation. Thus, if the moon had one face locked to the earth before the impact occurred, it is possible for the impact to deliver enough rotational energy to the moon to unlock it and give it a slow non-synchronous rotation until tidal damping again brings it into synchronization. In the absence of changes in the moon's moment of inertia the final orientation will, of course, be the same as before the impact except for a possible 180 ° rotation about the polar axis. We can obtain a lower limit on the crater size associated with an unlocking event if we compare the energy required to unlock the moon today [3], 2.1 × 1027 erg, with the energy derived from the maximum possible impulse [2] (i.e. from a tangential impact on the moon's equator by a body moving at the moon's escape velocity), AE = LAL/C = 1.23 X 106r 3 erg, where r is the crater radius in centimeters, L is the moon's present total rotational angular momentum, AL is the impulse delivered by the impact, and C is the moon's moment of inertia about its polar axis. These two energies are equal for craters of 120 km radius, thus fixing a strict lower limit on the size of a crater which could have been associated with an unlocking event. There are only ca. 23 craters visible on the moon with radii greater than this lower limit. Hence, we can conclude that only the very largest impact events may have been associated with the unlocking of the moon's synchronous rotation. Events such as those which produced Imbrium, Orientale, Nectaris, and Serenitatis are the prime candidates. Although in a few cases the impulse delivered by the impact may have had a significant effect on the moon, this effect is generally short-lived. Since the im-
pulse is much smaller than the original angular momentum of the moon, the moon's axis of rotation is nearly unchanged. All that happens, even for the largest impacts, is that the moon's rotation becomes non-synchronous by a small amount. Tidal friction will rapidly bring the moon back into synchronous rotation, and the only permanent effect this episode could have on the moon's orientation would be a 180 ° rotation about the polar axis, thus interchanging near and far sides.
2.2. Moment of inertia change caused by impacts The moon's principal moments of inertia, denoted C > B ~ A, are normally oriented as shown in Fig. 1. The axis of the greatest principal moment is perpendicular to the moon's orbital plane, the intermediate axis is tangent to the orbit, and the minimum axis is along the line connecting the centers of the earth and moon. The reason for this orientation is that it is the lowest energy configuration for a synchronously rotating moon. If the moon's moments of inertia were significantly changed by the production of a large crater, the principal axes would suddenly be rotated into new positions [4]. These new positions would not, in general, coincide with the minimum energy orientation shown in Fig. 1. The moon's new principal axes would oscillate about their minimum energy configuration in much the same way that a pendulum oscillates about its minimum energy configuration. As this oscillatory motion is damped by tidal friction, the new principal axes are eventually brought into the minimum energy config-
H Earth
0.,-~/~'o'I plo,.'po Y Fig. 1. The coordinate system used for discussing the moon's orientation is shown above. This is a rotating coordinate system in which both the earth's and the moon's apparent position is fixed. Also shown is the lowest energy orientation of the axes of the moon's principal moments of inertia C > B > A.
355
uration. The ultimate result of this process is to reorient the moon. Since the new lunar pole can be virtually anywhere on the moon's surface (depending on the size and location of the impact crater), we see that this process may be very important in determining which portion of the moon shall face the earth. In order to estimate the importance of this process, we must have some idea o f how large a crater must be before it can cause a significant change in the moon's moment of inertia. We shall presume that a "significant change" is one which alters the differences of the principal moments by amounts equal to or greater than their differences at the present time [5]. Given this criterion, we shall use two extreme models to calculate the moments of inertia of a crater. In this way we hope to bracket the most probable value for the difference of the principal moments caused by crater formation. Crater model A (see Fig. 2a) represents the crater as a simple circular hole of depth h and radius aO, where a is the radius of the moon and 0 is the angle (in radians) between the center of the hole and its edge as measured from the center of the moon. The model A crater has no rim, thus representing an impact in which all of the throwout escaped the moon. In the case that h ~ a, but where the curvature of the moon cannot be neglected, we have: I I ( A ) =/2 (A) 7T
a4hp (1 - cos 0)(4 + cos 0 + cos20) (la) 3 2rr I3(A) = - - - a4hp (1 - cos 0)2(2 + cos 0) (lb) 3 where 13 is the crater's moment of inertia about an axis passing through the center of the moon and the center of the circular crater, 11 and 12 are the crater's moments about axes perpendicular to the axis of I3, and O is the density of the moon's crust. The impor-
Fig. 2. (a) Crater m o d e l A. Simple rimless hole o f depth h and radius aO. Co) Crater m o d e l B. Same as A with an ejecta blanket extending to two crater radii from center. Mass in blanket equals mass removed from crater.
tant quantity for our purposes is the difference I3(A) -I,(A): I3(A) - I I ( A ) = 7ra4hp cos 0 ( 1 - cos20)
(2)
Thus, the net effect of this crater is to increase the relative moment o f inertia about the axis passing through the centers of the moon and the crater. Crater model B (see Fig. 2b) is the same as model A except that it includes a rim. In this model we assume that all of the material excavated from the crater is spread uniformly between one and two radii from the crater's center. This blanket of excavated material has a thickness t: h t =
1 + 2cos0
(3)
In the limit where h ,~ a, we compute for model B:
I3(B) - I, (B) = 2 [/3(.4) - I1 (.4)] (1 - cos 0) × (1 + 2 cos 0)
(4)
As might be expected, the presence of a rim equal in mass to the material excavated from the crater greatly reduces the moment of inertia difference. This reduction is greater for smaller craters, but even for a crater the size of Imbrium it is a factor 5. The differences 13 - 11 according to both models are plotted in Fig. 3 for various crater depths and radii. Craters the size of Imbrium or Orientale would have to be 1 0 - 3 0 km deep to significantly affect the moon's orientation. Note that these are the depths of a cylindrical hole whose volume equals that of the real crater. Since real craters are cup-shaped, the depth of the center of the crater below the pre-impact surface, h*, must be larger than the corresponding h in our models. In observed craters such as Arizona Meteor Crater, or the Jangle U nuclear explosion crater [6] we find h* 3h. For these craters h* is approximately 1/3 of their radius, so h ~ 30 km is reasonable for craters with radii of 300 km or larger (i.e. h ~ r/lO where r is the crater radius). Both models demonstrate that only the largest craters could have significantly changed the moon's moments of inertia. Furthermore, model B is likely to be the more realistic representation of a crater. Current work indicates that very little of the material from the crater would be ejected from the moon [7]. Thus, it seems that only Imbrium, and perhaps Orientale, could have had any chance of reorienting the moon.
356 104°[
i
i
i
MODEL A
i
MODEL B
/ krn
~
+
I039
/
+
km
i
C-A
E d
o c
B-A
B-A
km
I1)
LS 103a 0
÷
c
oE 1037
+
3-
.'2_
z,3 1
0
0
100
3
+
c ~
ea ~
-
Y,Y"
6
_E
6 ~ 200 500 400 500 Crater Radius, km
600
700 0
100
200 3o0 400 500 Crater Radius, km
600
700
Fig. 3. Moment of inertia differences as a function of crater radius and depth for two crater models. Model A represents the crater as a rimless circular hole in the moon's surface. Model B includes a rim of mass equal to that excavated from the crater. Both models neglect the depth of the crater relative to the moon's radius, but they include the effect of the moon's curvature. The horizontal dotted lines on the graphs are the differences between the moon's moments of inertia at the present time. where G is the gravitational constant, M® the earth's mass, R the earth-moon distance, and:
2.3. Behaviour o f the moon after a large impact Shortly after the impact, the moon's orientation will be a very complex function o f time. Aside from the probable unlocking o f its synchronous rotation, the moon will suffer large librations o f its new principal axes about their minimum energy positions. All of these motions will be damped by tidal friction, which will eventually bring the moon into synchronous rotation and orient it as in Fig. 1. An estimate o f the time required for tidal friction to brake the m o o n ' s non-synchronous rotation is already available in the literature. Goldreich and Sorer [8] have shown that the earth's tidal torque on the moon's spin is: 9 GM~a s N =
4 Q'R 6
(5)
a ' = O (1
387ra4/al
+ 3~-~m~!
(6)
where Q is the quality factor for the whole moon, # the m o o n ' s rigidity, and m is the moon's mass. For small planets, such as the moon, the second term in eq. 6 dominates the first since most o f the tidally induced distortion energy is elastic rather than gravitational. The rate at which the non-synchronous angular velocity, w, is dissipated is thus: dw dt
N -
C
45 GMe 2 a a -
8
R6
mQ'
(7)
where we have put C ~. 2/5 ma 2. Goldreich and Soter [8] have argued that 10 < Q < 100 for the moon, so that if we take Q ~ 100 we should get an upper limit
357
to the tidal damping time. If the impact gave the moon a non-synchronous rotation with an angular frequency equal to 3% of its present angular frequency (corresponding to the maximum possible impulse which the Imbrium event could have delivered [2]), then at the present earth-moon distance this rotation would be damped out in less than 7,000 years (where we have used/a = 10 x1 dyne/cm 2 to make our estimate). Since the damping time depends upon R 6, a smaller earthmoon distance at the time of the impact would cut this estimate considerably. Having established that the non-synchronous rotation of the moon would be damped extremely rapidly, it remains to establish a similar result for the periodic librations of the moon about its final orientation. We shall simplify the situation somewhat by supposing the librations to be simple, pendulum-like oscillations of frequency 0%. Let 0m be the moon's maximum angular excursion during one oscillation (measured from the equilibrium point). The energy lost per cycle due to tidal friction is four times that lost in moving from the equilibrium point to the maximum excursion: /m AE = -4
NdO = -4NOra
(8)
0
assuming that N is independent of the moon's angular excursion O. Thus: dE dt
-
Wo 21r
zXE = -
2~Oo 7r
NO m
(9)
The rate of change of the energy involved in the periodic motion is/~ = 6oo2C0m0m , so that: dOm _ dt
N lrwoC
(lO)
The time required to reduce 0 m from 1 radian to zero is: 1
Tdamp = [dOm]dt[
(11)
To estimate Tdamp, we need wo. A representative value is obtained by considering the energy required to twist the moon about its polar axis as a function of angle. (By choosing the polar axis rather than some other axis, we avoid complications due to the moon's angular momentum. These complications are not essential for the estimate of the damping time, and our result is only
changed slightly by their inclusion.) It is a straightforward problem to compute the interaction energy of the moon's quadrupole moment with the tidal potential of the earth; the resulting equation of motion for the moon is: 20x + COo2 sin 2 0 x = 0
(12)
where 0 x is the angle of rotation about the x-axis and: ]/ 3GMcB - A Wo =
R3
C
(13)
At the present time coo = 9.8 X 10 -s sec -~, which yields a free libration period of about two years. Using the same values for N and C as in our previous estimate of the rotational damping time, we find that Tdamp 3 X 104 yr for Q -- 100. Thus, the librations of the moon are damped very rapidly, even at its present distance from the earth. The energy dissipated in the moon by this process is negligibly small, amounting to only ca. 3 tacal/g of the moon, and so cannot have any noticeable effect on the thermal state of the moon's interior. We can conclude that the complex motions of the moon's orientation immediately following a major impact event will be quickly damped. Both the nonsynchronous rotation and the librational oscillations will cease within a few tens of thousands of years after the event. If the crater produced by the impact was sufficiently large, the moon's final orientation may be quite different from its initial orientation, even apart from a possible 180 ° rotation about the polar axis due to the unlocking and recovery of synchronous rotation. These estimates of the damping time of the moon's librational motions are very strict upper limits. If the moon was closer to the earth at the time of the impacts, or if its Q was lower, the damping timewould be correspondingly decreased. Moreover, we have been treating the moon as a cold elastic body, thus neglecting the possibility of internal damping. In fact, if the moon had a fluid core at the time of the major impacts, much of the orientational energy could be dissipated by the viscous coupling of core and crust. It thus seems likely that the moon (which even now could have a small fluid core [9]) also dissipated some of its orientational energy in this way. If, for example, this process were characterized by a Q of 100, the damping time would be only 200 years. Note that this
358 reduction in damping time applies only to the oscillatory part of the moon's motion; the non-synchronous rotation must be damped tidally since it requires a net loss of angular momentum. Having established that the moon would probably be reoriented by major impact events, and that it would relax into its new orientation very shortly after such an event, it is reasonable to ask how long it retains its new orientation. It takes a large, deep crater to change the moon's moment of inertia enough to cause reorientation. Such deep craters are not present on the moon's surface today. If such craters ever existed, they must have been frilled since their formation, and it is important to know how long it took for this to happen. 3. The stability of very large craters There seems to be little doubt that the initial cavity excavated by a large impact has a depth h* of at least 1/3 of its radius [10,11]. The crucial question from the point of view of reorientation is whether a sufficiently deep cavity can remain unfilled for more than the 10 4 years required for reorientation to occur. Craters on the moon are modified by at least three processes [ 12] discussed in the following subsections.
3.1. Mechanical collapse This is the most serious modification because it occurs almost immediately after the crater is formed. If mechanical collapse of the crater succeeds in replacing enough of the excavated material, the equivalent depth h may be too small for the change in the moment of inertia induced by the crater to reorient the moon. However, there are several reasons for believing that mechanical collapse did not result in sufficient Idling to prevent Imbrium from reorienting the moon. First, it has been shown [13] that large impact craters are stable against deep seated slumping in the absence of processes which can relieve the overburden pressure. Modifications of the crater are limited to relatively shallow slumping in regions near the rim where the slope of the initial cavity exceeds the angle of repose of pulverized rock: 3 0 - 4 0 ° . There is no mechanical reason why a depression 60 km deep and 1,000 km in diameter cannot be supported on the
moon, as long as the marginal slopes have been reduced to 3 0 - 4 0 ° or less by slumping. Altitude differences of 10 km have been similarly supported on the earth, with its 6 times larger gravity, for periods exceeding 10 m.y. Such static analyses, however, may not apply during the few minutes immediately after the impact event, when large seismic disturbances and possibly tensional stress waves are propagating through the rocks adjoining the crater. Even if large slumps are then possible, it seems unreasonable that they should extend further from the impact point than twice the radius of the initial cavity. Existing data [14], however, indicate that only about 50% of the excavated material lies within two radii from the impact point. Thus, even such widespread slumping as this could at most halffill the crater. Such a collapse would still not prevent Imbrium from reorienting the moon.
3.2. Extrusive igneous activity It is not known to what extent igneous activity can frill the basins of large craters. Basalt floods have certainly played some role in filling all large basins, but they are probably not more than a few kilometers thick [15]. The fact that the last basalt flows in Imbrium occurred ca. 7 × 108 yr after excavation of the basin [16] provides an upper limit on both the time required to fill the basin by basalt flows, and the time required to fill the basin by any process. If any appreciable uplift of Imbrium's floorhad occurred after the latest basalt flows, it would have resulted in updoming of the floor and fracture systems which are not observed. We thus conclude that the Imbrium basin was frilled to nearly its present level, and that the reorientation sequence was completed less than 7 × l0 s yr after the impact event.
3. 3. lsostatic rebound High-temperature creep of the rocks beneath the moon's surface is certain to lead to isostatic rebound of any large, uncompensated, depression. The major uncertainty in the rate of this process is the value of the moon's effective viscosity. In view of the uncertainties involved, a crude order of magnitude estimate of the isostatic rebound rate will suffice to illustrate the problem. We Shall represent
359 the crater as a surface load pgu(t)Jo(otr/d) where g is the acceleration of gravity at the moon's surface, u(t) is the depth of the center of the relaxing crater below the pre-impact surface (u(t = O) = h*), r is distance from the crater's center, d is the crater radius, and t~ is some constant of order unity. Let this load be applied to a half space of uniform Newtonian viscosity r~. We find that u(t) = h* exp(--t/w) where relaxation time: "c = 2a~/pgd
(14)
Thus, the larger the crater, the faster it relaxes. The real problem in estimating the relaxation rate of large craters is to determine the effective r/. Estimates in the literature range from 2 X 102s poise for craters less than 30 km in diameter [17] to 1026 or 1027 poise required to support the mascous for 3 X 109 yr [18]. Accepting these values for r~ yields relaxation times ~- ranging from 100 m.y. to 5,000 m.y. for a crater 500 km in radius. Unfortunately, such numbers should not be taken too seriously. We do not know what the detailed thermal structure of the moon was when the mare-forming impacts took place. Since r/may be an exponential function of the temperature, this lack of knowledge is very significant. If at the time o f the impacts r~ was as low as the value estimated for the earth's asthenosphere, 1022 poise, then we would find Z --~ 70,000 years. In this last case the moon would barely have time to reorient before the crater was t~flied. In spite of all the above uncertainties, however, it does not seem improbable that a large impact event, such as the one which excavated Imbrium, could have reoriented the moon for a period exceeding several million years. Ultimately, of course, some combination of isostatic rebound and extrusive igneous activity will fill the crater. In the case of Imbrium this took less than 700 m.y. As the crater t'dls, its effect on the moon's moments of inertia declines and the moon can return to its old orientation, barring complications (such as another large impact).
can cause even more profound reorientation. Very large craters, such as Imbrium and larger, may be expected to migrate to a position near one of the moon's poles in a few tens of thousands of years after their formation. This position cannot be retained indefinitely, however, for the crater is eventually Idled by a combination of collapse, isostatic rebound, and extrusive igneous activity. As the crater is filled, the differences of its moments of inertia decline at a corresponding rate, and the moon's orientation undergoes new alterations. It is thus probable that the moon has not always had its present orientation. The terrain near Imbrium has probably spent some time in the lunar polar regions, as well as near the moon's equator at the limb (this progression occurred as Imbrium first marked the maximum principal axis, then the intermediate, and finally the minimum; these changes are due to fdling, then overfdling Imbrium to form a mascon). At present there is no observational evidence for the reality of these reorientations. This lack is not surprising, since we do not expect the moon's orientation to have a strong effect on its surface. Nevertheless, we hope to have demonstrated the probable existence of an episode of reorientation in the moon's past and thus sketched out another aspect of lunar history. The idea that the moon's orientation can be greatly altered by large impacts may be useful in circumstances of which we are not presently aware.
Acknowledgment I would like to thank G.J. Wasserburg, whose question sparked this investigation, and with whom I have had several enlightening discussions. P. Goldreich also aided in clarifying my ideas on the importance of tidal effects on the moon's figure.
N,~tes and: references 4. Conclusion We have seen that large impact events have a strong influence on the moon's orientation. The impulse delivered by such events is capable of unlocking the moon's synchronous rotation. The change in the moon's moment of inertia due to crater formation
1 We shall always work in the approximation that the moon's orbit is circular and that its polar axis is perpendicular to the orbital plane. We shall also neglect the tides on the moon raised by the sun. Since tides on the moon due to the sun are only 11180 the height of those due to the earth, this is an excellent approximation. 2 This estimate is arrived at following W.M. Kaula, Phys.
360 Earth. Planet. Inter. 2 (1969) 123. We put the maximum mass which could have formed a crater of diameter r cm at: M = 6.54 X 108 (ra/v 2) g where v is the velocity of the infaUing object in cm/sec. The maximum angular momentum AL which M could deliver to the moon is obtained for a tangential impact at the moon's equator:
5
AL = Mva = 6.54 X 108 (r3a/v) g cm2/sec where a is the moon's radius in cm. The smallest possible v is 2.4 km/sec, the moon's escape velocity. Thus, for Imbrium with r = 550 km:
6 7
AL = 7.9 X 1034 g cm2/sec which is only about 3% of the moon's present rotational angular momentum, 2.3 X 1036 g cm2/see. 3 The energy required to interchange the axes of the intermediate principal moment, B, with that of the minimum principal moment, A, is given by:
E -
3 GM. R~- (B - A ) 2
where G is the gravitational constant, M . is the mass of the earth, and R is the earth-moon distance. At the present timeB - A = 2.01 X 1038 g cm 2, so: E = 2.1 X 1027 erg 4 The new principal axes of the moon and crater together
8 9 10 11 12 13 14 15 16 17 18
are found by summing their inertia tensors and then diagonalizing the result. It makes no difference whether we perform the diagonalization about the center of mass of the system or about the center of mass of the moon alone. The difference between the two results is of the order of the mass moved out of file crater divided by the mass of the moon, and so is entirely negligible. At the present time C - A = 5.60 × 1038 g cm 2 and B - A = 2.01 X 1038 g cm 2. These values are based on C = 8.88 X 1041 g cm 2 along with the values for # and ",/given by P.L. Bender et al. in Science 182 (1973) 229. Obtained from data presented by E.M. Shoemaker, in: The Solar System, 4 (Univ. of Chicago Press, 1961) 301. D.E. Gault, E.M. Shoemaker, H.J. More, Spray ejected from the lunar surface by meteoroid impact, NASA Technical Note D-1767. P. Goldreich and S. Soter, Icarus 5 (1966) 375. U. Nakamura et al., Geophys. Res. Lett. 1 (1974) 137-140. R.J. Pike, Geophys. Res. Lett. 1 (1974) 291. D.E. Gault et al., in: Shock Metamorphism of Natural Materials (lVlono Book Corp., 1968) 87-99. W.L. Quaide et al., Annals, N.Y. Acad. Sci. 123 (1965) 563. B. Dent, AGU Abstract T40, EOS 54 (1973) 1207. H.J. Melosh, Unpublished (1974). R.H. Carlson and G.D. Jones, JGR 70 (1965) 1897. K.A. Howard, D.E. Wilhelms and D.H. Scott, Rev. Geophys. Space Phys. 12 (1974) 309. F. Tera et al., Lunar Science IV (1973) 723. A.W.G. Kunze, Phys. Earth Planet. Inter. 8 (1974) 375 J. Arkani-Hamed, The Moon 6 (1973) 100.