the evolution of impact basins: cooling, subsidence, and thermal

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 90, NO. B14, PAGES 12,415-12,433, DECEMBER 10, 1985

The Evolution of Impact Basins' Cooling, Subsidence, and Thermal Stress

STEVEN R. BRATT ! AND SEAN C. SOLOMON

Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technolo•7y, Cambrid•7e

JAMES W. HEAD

Department of Geolo•7ical Sciences, Brown University, Providence, Rhode Island

Potentially important contributors to the topography and tectonics of multi-ring impact basins are the thermal contraction and thermal stress that accompany the loss of heat emplaced during basin formation. Heat converted from impact kinetic energy and contributed from the uplift of isotherms during cavity collapse are important components in the energy budget of a newly-formed basin. That the subsequent cooling may have been an important factor in the tectonic evolution of the Orientale basin is suggested by the deep central depression and by a surrounding region of extensive fissuring. To test these concepts, we develop models for the anomalous temperature distribution immediately following basin formation, and we calculate the resulting elastic displacement and stress fields that then would accompany cooling of the basin region. All models predict subsidence of the basin floor and a near-surface stress field consistent with fissuring. In addition, the rates of cooling and of accumulation of thermal stress are in agreement with the inferred timing of fissure formation in Orientale. The sensitivity of the predicted displacements and stresses to the initial temperature field allows us to place bounds on the quantity and distribution of impact heat emplaced during basin formation. In order to be consistent with the observed topography and the distribution of fissures in the Orientale basin, the buried heat deposited during the basin-forming event was between 1032 and 1033 erg. It is likely that most of this heat was concentrated within a distance of 100-200 km from the point of impact.

INTRODUCTION

Multi-ring impact basins on the Moon exhibit wide variations in their present geometry and structure [Hartmann and Wood, 1971; Wilhelrns, 1973; Wood and Head, 1976]. Some of the variations may be related to differences in the properties of the lithosphere or impacting projectile at the time of basin formation [e.g., Melosh and McKinnon, 1978; Holsapple and Schmidt, 1982]. Many of the observed variations likely reflect different degrees of modification of initial basin geometry and structure on time scales long compared to those for cavity excavation and ring formation. The subdued topographic relief of basins formed early in lunar history when the lithosphere was relatively warm is probably a consequence of lateral flow of crustal material over times scales ranging up to millions of years [Solomon et al., 1982; Bratt et al., 1985a]. The infilling of impact basins with mare basalt, on a somewhat greater time scale, led to loading of the lunar lithosphere and consequent subsidence and flexurally-induced tectonic ac- tivity [Solomon and Head, 1979, 1980; Comer et al., 1979].

Thermal contraction and thermal stress accompanying the loss of heat emplaced during basin formation are two additional and potentially important contributors to the long- term modification of an impact basin [Bratt et al., 1981]. During impact a significant fraction of the projectile kinetic energy is converted to buried heat [O'Keefe and Ahrens, 1976, 1977]. Further, the uplift of lower crustal and upper mantle material during collapse of the excavated cavity and formation

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of the multi-ring basin [Melosh and McKinnon, 1978] results in a corresponding uplift of the crustal and mantle isotherms, an additional source of heat beneath newly formed basins. Conduction of this anomalous heat to the surface gives rise to lithospheric thermal contraction and stress.

In this paper we assess the contribution of thermal contraction and thermal stress to the topography and tectonics of large lunar impact basins. Exploratory models are developed for the temperature structure following basin formation, for the subsequent cooling of the basin region, and for the resulting thermal displacements and stresses as functions of time. The subsidence and stress at the surface are compared with topography and tectonic features in the comparatively well-preserved Orientale basin [Head, 1974; Church et al., 1982]. On the basis of these comparisons we derive approximate constraints on the quantity and distribution of heat implanted during the basin-formation process.

GEOLOGICAL OBSERVATIONS: THE ORIENTALE BASIN

The Orientale basin (Figure 1), the youngest and best preserved of all lunar impact basins [Head, 1974; Moore et al., 1974], is an important source of information about the formation and modification of impact basins on all the terrestrial planets. Only the centralmost 220 km of the 900-km-diameter topographic depression is extensively covered by mare basalt [Head, i974], leaving exposed many geologic units and tectonic features that are presumably hidden beneath mare units in other nearside basins. Because Orientale is the youngest major basin on the Moon [Wilhelms, 1979], it has been left relatively undisturbed by ejecta deposits from other large impact events. Orientale is thus a nearly ideal location to look for tectonic and topographic expressions of basin cooling.

A careful documentation of the principal structural features and morphological units within the Orientale basin has been made by Church et al. [1982]; see Figure 2. The plains and

12,415

12,416 BRATT ET AL.' THERMAL STRESS NEAR COOLING IMPACT BASINS

Fig. 1. The Orientale basin. The basin center is located near 20øS, 95øW ' the diameter of the outer ring (Cordillera Mountains) is about 900 kin. Lunar Orbiter photograph LO IV-194M.

corrugated facies, which surround and probably underlie the central mare, are interpreted as cooled impact-melt material [Head, 1974]. Head [1974] and Church et al. [1982] suggest that the pitted and cracked texture of the corrugated facies resulted from cooling and internal thermal contractions acting during the time interval of melt-sheet cooling, about 10 3 to 10 '• yr [Onorato et al., 1978]. The outer Rook Mountains, believed to represent the rim of the original impact cavity, bound the outer edge of the corrugated and plains facies at a

radial distance of about 310 km from the basin center [Head, 1974, 1977].

Mare ridges and arcuate rilles (graben) constitute the innermost and outermost tectonic features, respectively (Figure 2). Both types of features postdate the emplacement of the central mare units and are consistent with the pattern of stresses produced by lithospheric loading and flexure [Comer et al., 1979; Solomon and Head, 1980]. No graben such as those found in Orientale and other mascon mare basins have been identified

BRATT ET AL.' THERMAL STRESS NEAR COOLING IMPACT BASINS 12,417

-2

i_ GRABEN •1

=MARE RIDGES_• PITTED AND CRACKED

INNER MARE.d_. CORRUGATED AND PLAINS

OUTER MARE

STRUCTURAL FEATURES

I FACIES

_L. DOMICAL • RADIAL T '"T-

I I I

6 CENTRAL ORIENTALE 0 I00 200 300 400 500 600

r, krn

Fig. 2. Generalized topographic profile of Orientale, after Head et al. [-1981]. The z = 0 datum corresponds to the typical level of surrounding terrain. Also shown are the locations of ring structures, stratigraphic facies, and tectonic features, after Church et al. [1982].

in the vicinity of farside basins not filled by mare units. It is therefore unlikely that the rille systems were formed in response to basin cooling, a process which would be expected to affect basins of a given age and size to a similar extent.

Tectonic features not obviously related to basin formation, melt-sheet cooling, or mare loading are the narrow fissures that occur within the corrugated and plains facies of Orientale (Figure 3). Most fissures are generally concentric to the basin and are concentrated in an annular band between 150 and 230

km from the basin center (Figure 4). In the southern quadrant of the basin, portions of some of the innermost fissures are buried by basalts of the inner mare (Figure 3), so that fissuring may have extended somewhat inward of 150 km distance prior to emplacement of the central mare units. In other quadrants, however, the corrugated and plains facies are exposed inward of 150 km radial distance and lack extensive fissuring. Neither vertical nor horizontal offsets are resolvable across individual

fissures, though some component of dip-slip or strike-slip motion cannot be excluded for some of these features. The

fissures differ in both geometry and distribution from the graben. Whereas graben have fiat floors up to 4 km in width, fissures have narrow V-shaped cross sections. The graben tend to occur outward of the zone of most intense fissuring, but the innermost graben and outermost fissures overlap in radial range (Figure 4).

The fissures are most readily interpreted as extension fractures formed under tensional horizontal stress at low confining pressure. An understanding of such a stress regime may be obtained from a Mohr failure envelope. In Figure 5 is shown the failure envelope derived by Brace [1964] from laboratory measurements of the strength of Frederick diabase under axial extension and compression. At high confining pressures, the failure envelope is defined by straight lines, signifying a constant angle of shear failure. At low confining pressure, however, the envelope bends to intersect the normal-stress axis at a right angle [Gri•t•ts and Handin, 1960; Muehlberger, 1961] to be consistent with the formation of extension fractures orient-

ed normal to the direction of greatest extensional stress. Ex- tension fractures will be favored over shear failure when the

Mohr circle approaches the failure envelope only along the normal stress axis. Consider, for example, the state of stress characterized by Mohr circle a (Figure 5) with diameter a• - 0.3, where 0.• and 0' 3 are the maximum and minimum prin-

cipal stresses and stress is positive in compression. Mohr circle a indicates that extensional fractures develop in diabase when 0'3 reaches -0.4 kbar and when 0'• is less than 1.5 kbar. Because most fissures in Orientale are approximately concentric to the basin (Figure 4), 0'3 must have been generally horizontal and radial to the basin center at the time of fissure

formation. Further, the magnitude of this radial stress in the vicinity of these fissures must have generally exceeded the extensional strength, approximately 0.2 to 0.4 kbar for com- petent igneous rocks at low confining pressure [Brace, 1964]. The direction of 0'• may have been either vertical or horizontal (i.e., azimuthal).

Photogeologic observations provide important constraints on the timing of fissure formation [Church et al., 1982]. Be- cause mare basalts flood portions of fissures, the fissures must predate at least the most recent episode of mare volcanism within the central basin. In addition, Church et al. [1982] concluded that the fissures postdate the cooling of the corrugated and plains facies (impact melt sheet) on which they formed. One reason for this view is that portions of the corrugated and plains facies are not cut by fissures, indicating that fissuring is the result of processes other than cooling and contraction of the melt sheet. On the basis of these relative age relations and estimates for the ages of the Orientale impact event and of the central mare deposits [Greeley, 1976], Church et al. [1982] concluded that the deformation that led to fissure formation occurred during a 100 to 200 m.y. time period following basin formation. They further suggested, on the basis of preliminary thermal evolution models for impact basins [Bratt et al., 1981], that the fissures were principally the result of thermal stress.

It has also been proposed that a portion of the topographic relief of the Orientale basin (Figure 2) may be the result of thermal contraction [Bratt et al., 1981; Church et al., 1982]. The total relief from the central depression to the Cordillera

12,418 BRATT ET AL.: THERMAL STRESS NEAR COOLING IMPACT BASINS

Fig. 3. A portion of the Orientale basin, showing (top to bottom) inner mare, the corrugated and plains facies, and the domical facies of Head [1974] and Church et al. [1982]. A number of fissures are evident on the corrugated and plains units. Lunar Orbiter photograph LO IV-195H1; width of image is 270 km.

Mountains exceeds 9 km [Head et al., 1981]. The strongest signature of thermal contraction postdating basin formation is likely to be concentrated in the central basin region. The topographic relief between the basin center and the foot of the

inner Rook Mountains about 180 km distant is nearly 4 km. In addition, the central basin region is floored by mare basalts up to perhaps 1 km in thickness [Head, 1974]. We regard the 5 km of relief from the base of central mare units to the foot of


Fig. 4.

ORIENTALE STRUCTURE

e e e ß ß

• ß

i•NNE'R ; ß

ß

ß

ß ß

-.

ß ß

ß

•[- •"• CORDILLERA MTNS.

•\• OUTER ROOK MTNS.

t

/

!

Sketch map of Orientale, showing basin rings, fissures (thin lines) and graben (double hatched lines). Features were mapped from Lunar Orbiter photograph LO IV-194M (Figure 1), a slightly oblique view.

the inner Rook Mountains as an upper bound on the subsidence experienced by Orientale as a result of cooling and thermal contraction.

Below we present models for the thermal contraction and thermal stress that accompany cooling of an Orientale-size basin. We use as primary constraints the subsidence of the central basin region by as much as 5 km and the formation of fissures as times less than 200 m.y. after basin formation and at distances of 150 to 230 km from the basin center.

BASIN THERMAL EVOLUTION

We describe the thermal evolution of an impact basin region with a simple analytical model obeying cylindrical symmetry. Let r, 0, and z be the cylindrical coordinates radius, azimuth, and depth. Because of linearity we need consider only the anomalous temperature field beneath a basin, i.e., the temperature in excess of the pre-impact thermal gradient. It is sufficient to solve the problem of the thermal evolution of a halfspace in which the initial anomalous temperature is uni- formly T O within a buried vertical cylinder, extending from r -- 0 to r -- a and from z -- h• to z - h2, and the temperature is zero outside the cylinder. The solution for any cylindrically symmetric initial distribution of temperature may then be ap-

proximated by a sum of suitably weighted solutions for uniform cylinders. This geometry is illustrated in Figure 6.

The solution for the temperature field in a halfspace due to cooling of a buried cylinder initially at temperature To begins with the solution for the temperature in an infinite medium due to an instantaneous temperature change To within a unit volume of material located at the origin [Carslaw and Jaeger, 1959]:

To e_tr2+=2•/c2 (1) T(r, z' t) = •3/2c3 where t is time, c = 2(kt) •/2 and k is thermal diffusivity. We next apply the method of images to satisfy the boundary condition T - 0 at the free surface, and we integrate over a cylindrical volume of initial temperature T 0. The solution for a buried cylinder of uniform initial temperature is then

•o • • 2'c: /2rr'\ To e-,.:/c: -,/ Iot•--7-; dr' r(r, z' t) = •7 r'e

.Ierf(h27Z)-erf(h'7Z)-erf(h2c+Z)+erf(h• (2)


or, kbar

Fig. 5. Mohr failure envelope for Frederick diabase derived from the laboratory experiments of Brace [1964]. The quantities a and ß are normal and shear stress, respectively; a• and a 3 are the greatest and least compressive principal stresses. Circle a represents a stress field capable of producing extensional fissures trending at an angle 0 = 90 ø from a 3. The stress field corresponding to circle b leads to shear failure along faults trending at 0 = _ 66 ø from a3.

where err is the error function [Gautschi, 1964] and I o is a modified Bessel function [Olver, 1964]. The integral in equation (2) is identical to the P function described by Masters [1955]. A thermal evolution model for an impact basin region follows from (2) and from a specification of an initial distribution of anomalous temperature. We defer discussion of the temperature field immediately following basin formation to a later section.

THERMAL DISPLACEMENT AND STRESS

Differential cooling of an elastic medium will lead to spa- tially variable thermal contraction and to thermal stress. To calculate these quantities from a thermal evolution model, we use the method of thermoelastic displacement potentials [Goodier, 1937]. During a time interval At, there will be a non-uniform change in the temperature field given by

AT= T(r, z ; t) - T(r, z ; t - At) (3)

where the thermoelastic displacement potentials are

c)i = • dr' dO' dz' (6) v

(]•2 = • r' dr' dO' dz' (7)

are integrals over volume V in the half-space, weighted by the inverse of the distance scalars

R•'- [(r - r') 2 - 2rr' cos (0 - 0') + (z -- j,)211/2 (8)

R2'= [(r - r') 2 -- 2rr' cos (0 -- 0') + (z + z')2] 1/2 (9)

The vector operator applied to &2 is

V2&2 = (3- 4v)V&2 + 2V[z(O•)]-4(1-v)V2(z&2)e• The response of an elastic whole-space to such a temperature (10) change can be represented by a distribution of centers of contraction (or dilatation) of magnitude:

0•A T(1 + v) ,8 = 12•(1 - v) (4)

where 0• is the volumetric coefficient of thermal expansion and v is Poisson's ratio [Goodier, 1937]. The solution for thermal stress in a halfspace must satisfy the additional condition that the surface be free of shear and normal tractions. To meet

these boundary conditions, Mindlin and Cheng [1950] modified the formulation of Goodier [1937] by imposing a center of contraction, a double force, and a doublet, each of appropriate strength, at the image point (r, -z) corresponding to each center of contraction (r, z) in the half-space. As a result, the thermal displacement field u = (u, v, w) in cylindrical coordinates is given by

r

AT:O

U --' -- Vlpl -- V2{P2 (5)

Fig. 6. Geometry of the thermoelastic stress problem for a cooling impact basin. Over each time interval At, cylinders of constant temperature change AT can be superposed to form any axisymmetric distribution of temperature change.


where V 2 is the scalar Laplacian and ek is the unit vector in the z direction.

By symmetry, we can solve for the displacement field by approximating the temperature change field AT(r, z; t) by a set of cylinders of uniform temperature change, computing the displacements due to the cooling of individual cylinders, and then summing the resulting displacement solutions. The calcu- lation of displacements from (5) reduces to differentiation if the potential of the distribution of temperature changes is known. Unfortunately, the solution for the potential surrounding a cylinder does not condense to a simple equation. An expression for the potential of an infinitely thin disk of radius a has been derived by C. H. Thurber (personal com- munication, 1980) and is given by

•b,(r, z) = 2n/• •o (4k)! F 2k k= 24n(k !)2(2k)!

•0 a r•2k + 1 it,2 + r2(2_ Z,)232k+ 1/2 dr' (11) The integration in (11) can be solved by parts for any given k; for example, the k - 0 term reduces to

2•r/•[(a 2 + r 2 + (z - z')2) •/2 - (r 2 + (z - z')2) •/2] (12)

which, for r- 0, is simply the potential of a thin disk at a point on the axis [Moulton, 1914, p. 113]. The number of terms necessary to evaluate (11) is a function of the location of the observation point, (r, z- z') or (r, z + z'), relative to the disk.

To obtain the potential of a cylinder of finite thickness and constant temperature change, we numerically integrate equation (11) over z'. The displacement potential •b2 may similarly be computed by substituting the quantity (z + z') for (z - z') in (11) prior to integration over z'. The displacement field follows directly from centered-difference numerical differentiation of •b x and •b2 as prescribed by (5) and (10).

The accuracy of this method was tested by approximating a buried sphere by a stack of buried cylinders. We compared the potentials for the two problems, and we also compared the displacement field produced by the cooling of the cylinder stack with the displacements at the free surface resulting from the cooling of a buried sphere [Mindlin and Cheng, 1950]. The difference between our approximate solution and the exact solution can be made to vanish by using suitably thin cylinders. We are confident, therefore, that the approximations we have adopted can adequately represent the displacement field due to the cooling and contraction associated with any axisymmetric initial temperature structure.

In the models that follow, we examine the displacements and stresses only at the lunar surface. Thus, the normal stress azz, the shear stresses %z and a0z, and the local temperature change are all zero. The shear stress aro is also zero by symmetry. The thermal stresses at the free surface are calculated from the displacement field in (5) using standard formulae [Timoshenko and Goodier, 1970, p. 456]:

*" = -(1 + v)(1 - 2v) [(1 - v)a. + v(aoo + azz)] (13)

O'00 •- (1 + v)(1 - 2v)

[(1 - v)%o + v(e,, + ezz)] (14)

where E is Young's modulus, stress is positive in compression, strain is positive in extension, and the strain components are given by

• = Ou/Or (15)

%o = u/r (16)

•zz = Ow/Oz (17)

Differentiation in (15) and (17) is accomplished numerically using the centered difference method.

The stresses given by (13) and (14) are relative to the ambient state of stress at t = 0, immediately following basin formation. While we concentrate below on the time-dependent thermal stress produced during basin cooling, the possible modi- fying influence of a non-zero prestress is also discussed.

ANELASTIC EFFECTS

The equations presented above provide a simple procedure to compute the thermoelastic displacements and stresses for an axisymmetric thermal evolution model. Several uncertainties in the underlying assumptions are noteworthy. Per- haps least certain is the assumption that the moon behaves as a conductive, elastic half-space during basin thermal evolution. The thermal model will be valid only if convective transport of heat in the shallow sub-basin region is unimportant, and the stress field will be correct only if the material under- going temperature change behaves completely elastically. It is lik. ely, however, that at high temperatures stress will be rapidly relieved by ductile flow. While equations (4)-(10) may still give a reasonable approximation to the overall displacement field for the purpose of estimating subsidence of the surface, equations (13)-(17) will not give a correct representation of the thermal stress field in the presence of strongly anelastic behavior. From the depth distribution of intraplate earthquakes, Chen and Molnar [1983] estimate that the earth's crust and upper mantle display brittle behavior only at temperatures less than 250ø-450øC and 600ø-800øC, respectively. From flex- ural studies, McNutt and Menard [1982] conclude that the base of the elastic lithosphere beneath ocean basins is defined approximately by the 550øC isotherm. While the lunar thermal profile is highly uncertain, particularly near the time of major basin formation, the thermal history model of Solomon and Head [1979] indicates that only the upper 25 km of lunar crust were at temperatures less than 250øC and that little or none of the mantle was cooler than 500øC at the time of

formation of Orientale (about 3.8 b.y. ago). The anomalous heat emplaced during basin formation further reduces the volume of subsurface material capable of elastically sustaining stress. Some caution, of course, should be exercised in transfer- ring to the moon information on the depth extent of elastic behavior on the earth. Neither the average time-dependent lunar temperature profile nor the rheology of the lunar lithosphere is well known. Material near the lunar surface un- doubtedly cooled rapidly after basin formation, however, even if mass flow did not contribute significantly to heat transport. It is thus likely that the upper lunar crust began to accumulate elastic stress (and episodically to undergo brittle deformation) shortly after basin formation as a result of material cooling and contracting at greater depth.

In the simple models considered here, it is not possible to examine completely the effects of depth- and temperature- dependent rheology on near-surface displacements and stresses. To simulate some of the effects of rapid stress relaxation in regions of high temperature, however, we may postu- late that material does not contribute to the thermal stress

field until it has cooled below some elastic "blocking temperature" Te I-Turcotte, 1974, 1983' Bratt et al., 1985b]. This pos-


PRE- IMPACT

CRUST

IMPACT HEATING

CAVITY EXCAVATION • • m I I • ß I I I I I I/

',,,,,,, ,,,,,,i j?

', .... •,, ,) ," ' -,-- .',', ,•.-• -,.-. -..• t"

CAVITY COLLAPSE

ISOTHERM UPLIFT

Fig. 7. Schematic view of the two principal sources of deep heating that accompany basin formation.

tulate can be readily implemented into the procedure detailed above by stipulating that if either T(r, z; t) or T(r, z; t -- At) in (3) exceeds Te, then that temperature is set equal to Te. The deepest nonzero contour of AT that contributes to thermal stress at the surface would then follow closely the T = Te isotherm, and Te can be regarded as approximating the temperature at the base of the mechanically strong elastic lithosphere of the moon. We include an elastic blocking temperature in a subset of the stress models discussed below. The

blocking temperature hypothesis is used only for the calcula- tion of thermal stress. We apply equations (3) and (5) in their unmodified form to predict basin subsidence in response to fooling.

INITIAL THERMAL STRUCTURE

The impact of a basin-forming projectile with a planet in- volves a •ransfer of projectile kinetic energy primarily to heating of the target area and secondarily to ejection of some portion of heated target material [e.g., O'Keefe and Ahrens, 1976, 1977]. The deep structure of the youngest nearside basins [Bratt et al., 1985a] suggests that the excavated cavities for basins the size of Orientale penetrated to the lower crust or upper mantle and that the volume of ejected material is on the order of 10 7 km 3. Collapse of the excavated cavity was accomplished at least partly by uplift of the underlying crust and upper mantle [Head, 1974; Melosh and McKinnon, 1978]; the uplifted material would be hotter than surrounding material even in the absence of other effects of the impact. Thus immediately after the basin-forming event, the basin region was subjected to two primary sources of deep heating (Figure 7): (1) conversion of impact kinetic energy to buried heat (hereinafter referred to as impact heating), and (2) uplift of crustal and mantle isotherms during collapse of the excavated cavity (hereinafter termed isotherm uplift). We consider both

sources of heat below, and we derive simple relationships for the anomalous temperature beneath a newly formed basin. We then apply these temperature distributions to investigate the contribution of cooling to the topography and tectonics of Orientale.

Impact Heating

The quantity and distribution of heat emplaced during a hypervelocity impact, as well as the size of the basin remaining after excavation and short-term modification, are complex functions of the mass, volume, and velocity of the projectile and of the gravity, density, and strength near the planet surface '[e.g., Holsapple and Schmidt, 1982]. Given only the observed basin morphology, it is difficult to estimate the original kinetic energy (Es:) of the impacting body. Many of our present conceptions about the energetics and mechanics of basin formation follow from scaling of smaller craters formed by ancient impacts on the earth [Shoemaker, 1960; Grieve et al., 1977-], nuclear and chemical explosions in the field [e.g., Vaile, 1961; Nordyke, 1961, 1977], and impact experiments in the laboratory [e.g., Gault and Wedekind, 1977; Schmidt, 1977]. Valuable information on cratering has also been derived from analytical and numerical simulations of crater formation [e.g., Gault and Heitowit, 1963; Maxwell, 1977; O'Keefe and Ahrens, 1976, 1977]. Estimates for the total kinetic energy (Es:) required to form an Orientale-size basin on the moon, however, can vary by several orders of magnitude depending on the scaling law applied. For instance, assuming that the crater produced by the Teapot-Ess explosion (yield = 5 x 10 TM erg, diameter = 89 m [Nordyke, 1961]) is a good terrestrial analog of larger impact craters [Shoemaker, 1960], the kinetic energy required to form Orientale ranges from about 10 3• erg if Es: •' D 3 to 1035 erg if Es: .,. D 4 [Holsapple and Schmidt, 1982], where D is the crater diameter.

BRATT ET AL.: THERMAL STRESS NEAR COOLING IMPACT BASINS 12,423

TABLE 1. Models of Thermal Stress for the Orientale Basin

Model Number

Buried

Impact Decay Blocking Uplift Heat E s, Constant Temperature Heating Figure 1032 erg s, km T e, øC Included? Number(s)

A ......... yes 10, 11 B 1 25 .-. no 12, 14 C 1 90 ... no 13, 15 D 7 50 ... no 16, 17 E 7 50 800 yes 18

Numerical models of small impacts (Eg • 1016 erg) per- formed by O'Keefe and Ahrens [1976] suggest that more than 90% of Eg goes into heating the projectile and target; let us call this quantity En. While most of En goes toward heating basin ejecta, the results of O'Keefe and Ahrens [1976] indicate that perhaps 25% of this heat is implanted in non-ejected target material and an additional unknown fraction of heated ejecta returns to rest within the basin. It is possible that less heat is ejected from the target region during a basin-size impact because excavation of heated target material may have been impeded by an abrupt increase in strength at the lunar Moho [Bratt et al., 1985a].

The spatial distribution of anomalous temperatures produced by shock heating under a large basin is largely un- constrained. Though the distribution of impact-related energy density within the target region can be derived from cratering models [e.g., Gault and Heitowit, 1963; O'Keefe and Ahrens, 1975], little consideration has been given to recovering the temperature field from subsurface energy density. For simplicity, we assume that the implanted heat per unit volume decreases exponentially with distance within a hemispherical volume centered at the point of impact on the pre-impact surface [Kaula, 1979]. This geometry is illustrated in Figure 7. An exponential decay of impact heat density is partially supported by the geologic observation that shock metamorphism and melting are concentrated at shallow depths below terrestrial craters [e.g., Dence, 1971; Grieve and Cintala, 1981] and by the roughly exponential relationship between the internal energy density and depth in finite-difference models of the formation of the Imbrium basin (Figure 5 of O'Keefe and Ahrens [1975]).

We compute the spatial distribution of impact-generated temperature as a function of slant-range distance q- (r 2 + z2) 1/2 from the center of symmetry on the pre-impact sur-

face using the relation

T(q) = 6e-q/s/pCv (18)

TABLE 2. Adopted Values of Parameters Used in Thermal Stress Models

Variable Description Value Source

Pc

Cp

k

E

crustal density 2.9 g/cm 3 Solomon [1975] upper mantle 3.4 g/cm3 Solomon [ 1975]

density specific heat 1.2 x 107 Solomon and

erg/g øC Longhi [1977] thermal 0.01 cm2/s Schatz and

diffusivity Simmons [1972] volumetric thermal 2.7 x 10- 5 øC- • Baldridge and

expfinsion Simmons [1971]; coefficient Skinner [1966]

Young's modulus 7 x 10 TM dyn/cm 2 Mizutani and Osako [1974]; Simmons and Brace [1965]

where s is a decay constant (the distance from the center of symmetry at which heating falls to 1/e of its peak value), p is density (either crustal or mantle), and Cp is specific heat. The latent heat of phase changes is ignored. The constant E is an energy density obtained from the equation

or•Ee-•/s27tq 2 dq = E n (19) where L is the value of q at which the temperature due to impact heating drops below some arbitrarily small value.

A significant fraction of Ex acts to heat material ejected during basin excavation. To estimate the amount of heat in material thrown beyond the basin rim, we utilize a model for nearside crustal structure derived from gravity and topographic data [Bratt et al., 1985a]. The difference between the assumed pre-impact crustal thickness and the thickness of non- mare crust beneath the youngest basins such as Orientale pro- vides a lower bound on the depth from which material was permanently excavated from the basin. We remove from the top of the target region a plug of heated material of thickness equal to the apparent depth of excavation (Figure 7). Left behind is a thickness of crust equivalent to that inferred at present and a spherical cap of heated crust and upper mantle. Though the return of heated ejecta probably contributed to the post-impact thermal structure of basins, we assume here that the contribution of returned ejecta to buried heat is small in comparison to that of shock heating of non-excavated target material. Let E• be the total quantity of impact heat left beneath the final basin. Scaling arguments cited above suggest that En may lie in the range 103• to 1035 erg for Orientale. The quantity En and the decay constant s are treated as free parameters in the description of impact heating in the thermal models that follow.

Isotherm Uplift

Uplift of ambient crustal and mantle isotherms during cavity collapse and basin formation may be treated as a source of heat independent of impact heating. The extent of uplift heating may be estimated for young basins from the preserved relief of the lunar Moho [Bratt et al., 1985a], the basin age, and a temperature profile obtained from a global thermal history model. For simplicity, the uplift of isotherms during basin formation is assumed to follow a vertical trajec- tory, with the maximum uplift occurring in the central region of the basin. The shape of the uplifted volume (Figure 7) is assumed to be' that of a truncated cone with upper and lower radii taken from the crustal structure modei of Bratt et al. [1985a]. The pre-impact temperature distribution is assumed to be constant below 100 km depth. The anomalous temperature distribution is calculated from the difference between the

uplifted temperature profile and the pre-impact temperature profile. The temperature change contributed by isotherm uplift is maximum at the surface and at •he basin center and, by assumption, is zero at 100 km and greater depth.

TEMPERATURE AND THERMAL STRESS

MODELS FOR ORIENTALE

Following the above guidelines, we present models for the contributions of isotherm uplift and impact heating to the temperature distribution beneath the newly formed Orientale basin. A summary of the models presented below and the associated values of free parameters is given in Table 1. Adopted values of physical parameters common to all models are given in Table 2.


_20 •00

r, km

200 0 200 400 I I I I I I I

60O I

2O

E

6O

8O

IO0

I ORIENTALE I

CRUST

MANTLE

Fig. 8. Structure of the crust and upper mantle beneath Orientale determined from an inversion of gravity and topographic data over the lunar nearside [Bratt et al., 1985a]. The dashed line at the base of the crust delineates the Moho as computed in the inversion along a profile from 30øS, 100øW to 5øS, 85øW. The solid line represents an azimuthally averaged Moho profile used in the estimate of isotherm uplift in this paper.

Isotherm Uplift

The shape of the Moho beneath Orientale (Figure 8) pro- vides a measure of the extent of uplift of lower crust and upper mantle during basin formation. This measure is strictly only a lower bound, since the newly formed basin may have been modified by such processes as long-term viscous relaxation. Because Orientale is the youngest lunar basin and preserves a large amount of topographic relief, however, the effects of long-term modification processes are thought to be minor. The uplifted mantle (Figure 8) may be approximated by a truncated cone with an upper radius of 50 km, a lower radius of 310 km, and a height of 55 km. From the estimated age of the basin (•-3.8 b.y.), the ambient temperature profile taken from a global thermal history model [Solomon and Head, 1979], and the Moho relief shown in Figure 8, the anomalous temperature field resulting solely from isotherm uplift can be estimated. Figure 9 shows the pre-impact and post-uplift temperature profiles beneath the center of Orientale as well as the anomalous temperature distribution, equal to the difference AT between these two curves. At the lunar surface AT is

nearly 600øC. The initial anomalous temperature field produced by isotherm uplift is shown as a function of r and z in Figure 10. The total anomalous heat contributed by isotherm uplift beneath Orientale is 1.4 x 1032 erg.

A thermal history model for the basin (model A) in which uplift heating is the sole contribution to the anomalous temperature field is illustrated in Figure 10. By t = 10 m.y. (Figure 10b), much of the heat in the upper 20 km of model A has left the basin region. By 100 m.y. (Figure 10c) only about 30% of the initial heat remains. By 500 m.y. (Figure 10d) less than 1% of the heat is left. Thus most of the thermal contraction and

stress contributed by isotherm uplift will take place within 100 m.y. of basin formation for this model.

The major uncertainties in the contribution of isotherm uplift to basin thermal evolution are the adopted pre-impact temperature profile and the extent and distribution of uplift. It is difficult ,to assess error in the adopted global temperature

profile, but an estimate for the error in the amount of uplift beneath Orientale follows from the uncertainty of about + 10 km in the crustal thickness beneath the basin center [Bratt et al., 1985a]. We have computed the basin thermal histories subsequent to isotherm uplift by amounts 10 km greater and less than for model A; temperatures differ by less than 10% from those shown in Figure 10.

The surface displacements and thermal stresses predicted by model A at several times after basin formation are shown in

Figure 11. The center of the basin subsides (Figure 11a) about

T, øC AT, øC 0 250 500 750 0 250 500

0 / • ' '• 'a ' ' I b 20 -

POST-

_ PRE-

.

60 -

70 -

80 -

90 -

I00

Fig. 9. (a) Pre-impact thermal profile 3.8 b.y. ago [Solomon and Head, 1979] and temperature distribution beneath the center of the newly formed Orientale basin due only to isotherm uplift. (b) Anoma- lous temperature profile contributed by isotherm uplift beneath the basin center.


r, km I MODEL A I

200

3OO

t--O

O• [ I I I i [

IOO

IOøC 200 -

_

c 100 m.y. 300 i I i

0 I00 200 300

_,ooo

b I0 m.y.

Fig. 10. Basin thermal evolution for model A. The initial field of anomalous temperature (t = 0) is that produced only by isotherm uplift. Also shown are the anomalous temperature fields at 10, 100, and 500 m.y. after basin formation.

0.2 km in the first 10 m.y. and an additional 0.2 krn during the next 90 m.y. The horizontal distribution of subsidence is controlled by the horizontal extent of uplifted material beneath the basin (Figure 8). By 500 m.y., a region out to 200 km radius has experienced at least 100 m of subsidence.

The amount of subsidence predicted at the center of the basin for model A (.--0.4 km) is about an order of magnitude less than the observed relief of the central basin depression. Even if upper mantle isotherms were raised to the surface during basin formation, the total accumulated subsidence would not exceed 1 km. Thus, if a large portion of the relief associated with the central depression is a consequence of thermal subsidence, an additional source of initial heat is required.

The radial displacement u (Figure ,1 l c) is toward the center of the basin. The maximum value occurs between 150 and 200 ,

km radial distance and reaches 200 m by 500 m.y. This value is about half that of the subsidence of the basin center over the same time interval.

In the central basin region both horizontal stresses (Figures l lb and 11d) are compressional and similar in magnitude. Because azz and all shear stresses are zero at the surface, thrust faulting should be the dominant mode of stress release near the center of the basin. Both stress components accumulate most rapidly in the first 10 m.y. and reach 1.6 kbar by 100 m.y. With increasing radial distance r, aoo approaches zero. In contrast, a, becomes extensional at r greater than about 200 km (Figure l lb). The zone of maximum extensional a, is located between 250 and 350 km radial distance. By 100 m.y., a, reaches -0.4 kbar at r = 290 km.

As discussed above, fissuring most likely occurred within 100 to 200 m.y. after basin formation and in a stress regime where a, was more negative than -0.2 to -0.4 kbar. In

model A, the region of the basin where a, satisfies this criterion at 100 m.y. is from r = 230 to 400 km. Thus while the magnitudes and signs of principal stresses for this model are consistent with fissure formation, the predicted fissures would be at radial distances significantly greater than observed. If fissures originated by thermal stress, we conclude that the anomalous heat contributed by conversion of impact kinetic energy must have been at least comparable in magnitude to that contributed by isotherm uplift. Further, the effects of impact heating were probably concentrated at lesser distance from the basin center than were those of isotherm uplift.

Impact Heating

As discussed above, the magnitude and distribution of impact heating are parameterized by equations (18) and (19) plus a correction for the quantity of impact heat carried away from the excavated cavity by heated ejecta. The crustal structure beneath Orientale (Figure 8) suggests that at least 55 km thickness of shock-heated crust was removed during the excavation of the central portion of the cavity. We have assumed that the uppermost 55 km of the hemispherical distribution of impact heat (equation (18)) was transported outside the basin as hot ejecta. The decay constant s and the net quantity of buried heat Ea remaining beneath the newly formed basin are taken to be free parameters. Following our earlier discussion, we begin by assuming that E a = 1032 erg and we show later the effect of varying this poorly known quantity.

In Figures 12 and 13 are shown the initial distributions of anomalous temperature and the cooling histories predicted by impact heating models for Orientale with Ea = 1032 erg and with s = 25 km (model B) and 90 km (model C). Most of the initial heat in model B is concentrated within a small volume

near the basin surface. The initial anomalous temperature im-

E

0.1

0.2

0.3

0.4 /

0.5

r, km I00 200 $00 400

I I I , I - : : ß - - - I ' - m .•..__, __•_.______

I••//// _ _ _ _

// SUBSIDENCE /

/500 m.y.

.-..... •.500 m.y.

1.5 •

I.O \

0.5

t=lOm. _

I I I I I I

\ \

i i I

RADIAL STRESS

E

r, km 0 I00 200 300 400

I i

-O.O5

\ //// \\\ IO• -0.20 - • /

500 m.y. RADIAL DISPLACEMENT

-O. 25

2.0

1.5

- ---- • 500 m y. d

- IO•• - • •,•,•% \ HOOP STRESS _

- I Fig. 11. Displacements and thermal stresses at the basin surface for model A (Figure 10). Curves shown represent

accumulated values at 10, 100, and 500 m.y. after basin formation. (a) Subsidence w; (b) radial stress a,,, (c) radial displacement u, and (d) azimuthal or hoop stress aoo.


IO0

200

I:: 3oo

• o

IO0

2OO

I00 200

500 ø

•• i00 o

300 I I

r, km

o IOO 200

? / I00ø /

I MODEL B I 300

I I

I0 m.y.

i i

300

IOøC

50 o

I00 m.y. I

3oc

5 o

_

I I,

500 m.y. I I

Fig. 12. Basin thermal evolution for mod61 B. The initial anomalous temperature field is due to implanted impact kinetic energy' the total heat E B buried beneath the basin is 1032 erg and the decay constant s is 25 km (see equation (19)). Also shown are the anomalous temperature fields at 10, 100, and 500 m.y. after basin formation.

mediately beneath the central basin region exceeds 1000øC. By 100 m.y. after basin formation, only 27% of the initial energy remains beneath the basin. The initial heat is distributed over

a larger volume in model C relative to model B, and therefore

IOO

200

3OO

o

IO0

200

300

I00 200 300

/i ø i / 'i /1 ' I / i '/

00 o • ?50 / j

o

- f=O

,., ,':ø, .I , ,

i I I I I I

I • i

the initial temperatures are considerably reduced. For the •ame reason, heat leaves the basin more slowly. About 60% of the initial energy remains buried in the subsurface after 100 m.y. The thermoelastic effects of cooling are illustrated for

r, km MODEL o IOO 200 300

, , ,

OiC I0 :.y.

• 5o C 8 ø

500 m.y. , I I, , I, , I, ,,I

Fig. 13. Basin thermal evolution for model C. The model includes impact heating with Ea - 10 32 erg and s -- 90 km. See Figure 12 for further explanation.


Fig. 14.

r, km 0 I00 200 300 400

0.5 /'•-I00 m.y. IMODEL BI i.o 500 m.y. _

SUBSIDENCE

1.5

2.0 r I I I I I I I I

81 21.6{I • • I • • • • • • b_

61 •5.,• d• _

4 -

RADIAL STRESS -

......... , t •--• .......... ! = I0 m.y• ••••a•-• •• - • 500 m.y. -

-4 L I I I I I I I I

Surface subsidence w and radial thermal stress a,, for basin thermal model B (Figure 12). Curves shown represent accumulated values at 10, 100, and 500 m.y. after basin formation.

models B and C in Figures 14 and 15, respectively. Shown in each figure are w and a,, at the lunar surface versus r and t. The shapes of the curves for u and aoo and their relationships to w and a, are similar to those for model A (Figure 11).

Comparison of Figures 14 and 15 shows that the patterns of accumulated displacement and stress reflect the distribution of initial heat (defined by s) in each model. Most of the subsidence in model B is confined to the first 100 m.y. and to radial distances less than 150 km. Subsidence near r = 0 (Figure 14a) is 1.8 km by 100 m.y. Subsidence beneath the basin center for model C (Figure 14a) is lower in magnitude (only •0.2 km by 100 m.y.) but takes place over a broader region. Also, because heat is lost more slowly in model C, a significant proportion of the total subsidence takes place between 100 and 500 m.y. Accumulated radial stress (Figures 14b and 15b) in the center of both models is compressional and exceeds 1 kbar. Maxi- mum extensional stress in model B occurs between r = 90 and 130 km and accumulates to about -3 kbar. The distribution

of a, in model C contains a relatively broader region of extension; maximum accumulated extensional stress is -0.2 kbar at 100 m.y.

Neither model B nor model C satisfactorily predicts the location of fissures within Orientale or matches the full mag-

__

nitude of relief of the central depression. However, these

models indicate the sensitivity of the displacement and stress fields to the values of the parameters Ea and s. The magnitudes of displacement and thermal stress scale linearly with Ea, but the distributions of those quantities depend on s. Therefore, if some portion of the relief of the central depression and the band of fissures are products of thermal stress and if the elastic half-space model adequately represents the response of the moon during the time of basin formation and modification, we may constrain the quantity and distribution of heat implanted during basin formation by varying s so as to match the horizontal extent of the central depression and the locus of fissuring and varying Ea so as to match a given fraction of the relief of the central depression. Of course, it is important to keep in mind that the effects of isotherm uplift (e.g., model A) will have to be added to the effects of impact heating to estimate the total thermoelastic response of the moon to the basin formation event.

A model for impact heating following the above guidelines, model D, is shown in Figure 16a. For this model Ea = 7 X 1032 erg and s = 50 km. Significant impact heating extends

to radial distances and depths comparable to the radius of Orientale (310 km). Near-surface temperatures within 150 km of basin center exceed the liquidus temperatures of most igneous rocks; the rate of cooling in this model is underesti-


E

•o.:> /

/500 m.y. 0.3

I

r, km 0 I00 200 300 400

ß .-'- '"" '""= cI

I•/ /•• _ SUBSIDENCE

1.5

1.0

I I I I I I I I

'"• b • ,X500 m.y.

- IO•', • X'N•\, RADIAL STRESS _

Fig. 15. Surface subsidence w and radial stress a. for basin thermal model C (Figure 13).

mated for the first ,--10 m.y. because equation (2) does not account for convective heat transport. By 100 m.y., 20% of the initial energy has been lost; by 500 m.y. about 20% of E s remains in the target region.

The surface subsidence and radial stress for model D are

shown in Figure 17. The center of the basin subsides by 4.6 km after 500 m.y. This subsidence is in addition to the 0.4 km of subsidence contributed by the loss of heat from isotherm

IOO

2OO

3OO

o

IOO

2OO

300

I00 200 300

' /I / I/ I /

3000ø00// / - • ••.5o ø 50 o i I ••1 i t:øl t/

_ 250 ø / /

1 i•• Io o c ,

r, km I MODEL D I o IOO 200 300

I•1 I 10 ø (3

25 ø

50 ø

500 m.y. I

Fig. 16. Basin thermal evolution for model D. The model includes impact heating with E B - 7 x 10 TM erg and s = 50 km. See Figure 12 for further explanation.


-2

-4

r, km 0 I00 200 :300

I i _ i- - : -_ ;_ - -' -' i •

2 •.yd/// I / / $UaS•OENCE

4r.. //

5 I I I I I I I

4OO I

34.2 • b • -

31.9 •', - 19..5 . •\ -

• _ RADIAL STRESS

t = IOm.y. .....----- --

I00 m.y..• .• 500 m.y. _

Fig. 17. Surface subsidence w and radial stress a, for model D (Figure 16).

uplift (model A). The sum of the subsidence from models A and D accounts for essentially all of the present relief from the base of the central mare to the foot of the inner Rook Moun-

tains. Also, that most of the subsidence occurs at radial distances less than 200 km is in agreement with the observed topographic profile (Figure 2). Since some of the relief of the Orientale central depression may not be the result of thermal contraction, the amounts of cooling and of subsidence in model D should be regarded as upper bounds.

By 100 m.y. after basin formation in model D, accumulated values of a, reach nearly final values (Figure 17). This is in agreement with the inferred timing of fissuring in the corrugated and plains facies of Orientale. Near the center of the basin, a, and %o are compressional and exceed the compressive strength even of unfractured igneous rock at low confining pressure, typically 2-5 kbar [Brace, 1964]. Radial stress is most extensional between about 100 and 250 km radial dis-

tance. Between 10 and 100 m.y. after basin formation, the predicted location of maximum accumulated extensional stress moves from r = 140 to r = 180 km. The greatest extensional stress reaches -2 kbar by 10 m.y. and -3 kbar by 100 m.y. Thus even by 10 m.y., a,, exceeds the extensional strength of unfractured rock at low confining pressure [Brace, 1964]. Fis- suring is likely within 10 m.y. of basin formation and would be predicted to occur earlier if convective heat transport were included in the thermal model. The location of maximum ex-

tensional a•r in Figure 17 is consistent with the location of the band of fissures (150-230 km radial distance) within Orientale.

Effect of an Elastic Blocking Temperature

As discussed above, material at sufficiently elevated temperatures will not likely contribute significantly to the thermal stress field. This effect has been parameterized with an elastic blocking temperature [Turcotte, 1974, 1983] in thermal stress model E depicted in Figure 18. In model E, the anomalous temperature field contains contributions from both isotherm uplift (model A) and impact heating. To determine whether a parcel of material is above or below the elastic blocking temperature Te, the ambient thermal gradient (Figure 9) must be added to the anomalous temperature. We use a blocking temperature of 800øC, corresponding to the temperature at the greatest depth of earthquakes in terrestrial intraplate settings [Chen and Molnar, 1983]. The use of a blocking temperature does not affect the calculated subsidence, which should reflect the combined solutions to the full thermal contraction problem for the isotherm uplift and impact heating cases.

Model E (Figure 18) combines isotherm uplift from model A and impact heating from model D. Immediately following basin formation, temperatures beneath the center of the basin exceed T e at all depths. During early cooling, thermal stress thus accumulates only in the shallow crust exterior to the regions most extensively heated by the basin formation pro-


r, km

:50 I00 200 $00 I i I I I I 400

I

• • • • • ,500 rn. y.

I00 m.y.

/. ,• • RADIAL STRESS t =10 rn.y.

[MODEL E l

I I I I

Fig. 18. Surface radial stress a,, for model E, a combination of uplift heating from model A (Figure 10) and impact heating from model D (œa = 7 x i032 erg and s = 50 km, Figure 16). An elastic blocking temperature T e of 800øC has been assumed.

cess. This effect is reflected in the shape of the a,, distribution at 10 m.y. after basin formation. The zone of contraction con- tributing most to thermal stress during this time interval occurs not beneath the basin center but near r ~ 150 km, where temperatures are below 800øC. The cooled, elastic surface layer is pulled toward this annular region, thus producing zones of mild extension near r--0 and r- 300 km. By 100 m.y., most of the crust beneath the basin has cooled below Te and, as a result, has begun to contribute to and accumulate thermal stress. By this time, the distribution of a, begins to resemble models without a blocking temperature. The magnitude of a,, however, is everywhere less than in previous models. This is especially evident near the basin center, where a, in model D (Figure 17) exceeds 30 kbar at 500 m.y. while a, in model E is an order of magnitude less. The position of the surface zone experiencing maximum radial extension is strongly controlled by both the value of Te and the radial extent of isotherm uplift beneath the basin. The result is a region of extensional stress that is broader, smaller in the magnitude of stress, and located at a greater radial distance from the basin center compared to the same model without a blocking temperature. In thermal stress models with lower adopted values for Te, these effects are more pronounced. After 100 m.y. in model E, a, exceeds the extensional strength of igneous rock only for r > 250 km. Even though model D provided a good fit to the topography and the location of fissuring within Orientale among models not including the effects of a blocking temperature, model E demonstrates that with the inclusion of a blocking temperature a value of s less than 50 km is necessary to match the location of fissuring in Orientale. If a blocking temperature of 800øC is appropriate, thermal stress calculations for models otherwise similar to E

indicate that s should be about 20 km to predict fissuring at the distance range observed.

DISCUSSION

The models presented above suggest that the emplacement of heat during the formation of an impact basin and the subsequent loss of that heat were important contributors to the topography and tectonics of lunar impact basins. Beyond this qualitative result, we may use the results of these models to place approximate constraints on the quantity and distribution of impact heat implanted during the formation of the Orientale basin. These estimates are based on the assumption that the observed fissuring is a product of thermal stress

[Church et al., 1982] and that the distribution of anomalous temperatures resulting from isotherm uplift is relatively well known. Given these assumptions, we note first that isotherm uplift alone predicts poorly the location of fissuring within Orientale. It follows that heat converted from impact kinetic energy must have been at least as important to the early thermal budget of the basin. Expressed differently, EB is probably comparable to or greater than 1032 erg, the total amount of anomalous heat contributed by isotherm uplift.

The distribution of impact heating has been assumed in this paper to follow an exponential decay with distance characterized by a fixed decay constant s (equation (18)). When no blocking temperature is considered, s must be about 50 km (model D, Figures 16-17) to predict correctly the occurrence of fissuring in the distance range 150 to 230 km. With the inclusion of a blocking temperature (model E, Figure 18), an even greater concentration of impact heat near the point of impact is required to match the fissure positions. We therefore suggest that the decay of impact heat density with distance from the point of impact for an Orientale-size event must be rapid and that for the exponential parameterization assumed in this paper s must be less than or equal to 50 km. For comparison, the energy density distribution shown in Figure 5 of O'Keefe and Ahrens [1975] for their numerical model of the formation of the Imbrium basin falls off approximately exponentially with distance with a decay constant of about 20 km. As a measure of the parameterization used here, with s = 50 km about half the buried impact heat lies inward of r - 100 km and 90% of the heat lies inward of r - 200 km.

If 5 km can be regarded as an upper bound on the thermal subsidence that has occurred within the central region of the Orientale basin, then the calculations of this paper also permit an estimate of an upper bound on EB. A superposition of model A (isotherm uplift) and model D (Ea = 7 x 1032 erg and s = 50 km) accounts for the entire relief of the central depression. Further, if s is less than 50 km, as suggested by the thermal stress models that incorporate an elastic blocking temperature, subsidence at the center of the basin increases for a given value of E• (compare models B and C, Figures 14 and 15). On these grounds 7 x 1032 erg is an upper bound on Ea.

It should be recalled that these bounds on Ea have been estimated without regard to sources of stress other than thermal stress. The state of stress immediately following basin formation is unknown. Residual stresses may have remained after shock release and cavity collapse, but such stresses


should have largely relaxed beneath the central basin region because of the elevated temperatures. A persistent source of stress that did not relax is that associated with basin topographic relief and its compensatio n by lateral variation in density (e.g., crustal thickness) at depth. If the pre-mare basin was in a state of nearly complete isostatic compensation, then the basin relief would give rise to a horizontal stress of order pgh, where p is the density, g is the gravitational acceleration, and h is the variation in topography [e.g., deffreys, 1970, pp. 249- 268]. The Orientale basin inward of the Cordillera Mountains lies below the level of surrounding terrain (Figure 2), which would add a horizontal compressive stress at shallow depths in the basin interior. The band of extensive fissuring occurs in terrain 2 to 4 km below the local datum, suggesting that 100- 200 bars of horizontal compressive stress should be expected from topography. This additional stress would have little effect on the development of fissures predicted by the stress models except to delay slightly the time at which a, first satisfied the criterion for extensional failure.

The calculations of subsidence and the inferred constraints

on E• and s were obtained without consideration of the specific volume change that accompanies freezing. Several of the basin thermal models have initial temperatures well in excess of that necessary to induce melting (Figures 12 and 16). The temperatures immediately beneath the central basin region are probably unrealistically high in these models, a consequence of neglecting the heat of fusion and of the simplistic exponential relation for the distribution of impact heat. Geologi- cal arguments and scaling from melt volumes in terrestrial craters suggest that the Orientale melt sheet has an average thickness inward of the Outer Rook Mountains of about 1 km

[Head, 1974]. The additional subsidence contributed by freezing of this melt sheet should not exceed a few hundred meters. In the basin thermal models, of course, it is the integrated heat rather than any given value of initial temperature that is important for the subsidence problem.

If we accept that Ea represents about 25% [O'Keefe and Ahrens, 1976] of the original kinetic energy E s of the impacting projectile and we use the bounds on E• suggested above, then Es for the Orientale event was in the range 4 x 1032 erg to 3 x 1033 erg. These values for Es may be useful for estimating scaling relations of the form Es •- D n for large impact craters; scaling Orientale from Teapot-Ess, for instance, would favor n _• 3.4-3.6, values similar to that derived by Valle [1961] from small terrestrial craters. The estimate of impact kinetic energy derived here for Orientale may also provide constraints on models of planetary accretion calling for the impact of large planetesimals and for models of the early thermal histories of planets in which the fractional conversion of impact kinetic energy to heat and the spatial distribution of that heat are important parameters [e.g., Kaula, 1979].

CONCLUSIONS

We have explored the hypothesis that thermal stress has contributed significantly to the topography and tectonics of lunar multi-ringed basins. Thermal models have been calculated for a variety of assumptions about initial basin heating contributed by impact kinetic energy and uplift of isotherms during cavity collapse and basin formation. Thermal stresses and displacements have been calculated from the time- dependent thermal models using analytic expressions for the response of an elastic halfspace. Some stress models have included the effects of an elastic blocking temperature [Turcone, 1974, 1983] to account approximately for high-temperature

anelastic effects. These solutions have been compared with the topographic relief [Head et al., 1981], the location of extensional fissures, and the timing of fissure formation [Church et al., 1982] in the relatively well-preserved Orientale basin.

For all basin models considered, basin cooling and accumulation of thermal stress is most rapid within 100 m.y. after basin formation, in agreement with the inferred timing of ils- suring within Orientale. The predicted state of stress in the region of fissuring (a, extensional, aoo compressional and smaller in magnitude) predicts well the form and orientation of fissures if these features are the product of extensional failure.

On the basis of the thermal stress models, the topographic relief of the central basin depression and the range of radial distances from basin center over which extensive fissuring occurred constrain the magnitude E• and distribution of kinetic energy that was converted to buried heat beneath the newly formed Orientale basin. E• must be comparable to or greater than 1032 erg because the contribution of impact heating to thermal stress must be at least comparable to that of isotherm uplift. E• must be less than or equal to 7 x 1032 erg in order to be consistent with the topography of the central basin depression. The impact heat was concentrated within 100-200 km of the point of impact.

It is important to emphasize that there is an untested ele- ment of uncertainty in the ability of our models to represent the earliest portions of basin thermal history and the anelastic response of material at high temperature to cooling. The models presented here nonetheless suggest that cooling and thermal stress contributed significantly to the topography and tectonics of multi-ringed basins and that constraints on the quantity and distribution of impact heat emplaced during basin formation may be derived from geological observations of the youngest basins on the moon and on other planets and satellites.

Acknowledgments. We thank W. F. Brace and C. H. Thurber for helpful discussions, Jay Melosh and an anonymous reviewer and as- sociate editor for constructive criticism of an earlier draft, and Jan Nattier-Barbaro for assistance with manuscript preparation. This research was supported by the NASA Planetary Geology and Geophys- ics Program under grants NSG-7081 and NSG-7297.

REFERENCES

Baldridge, W. S., and G. Simmons, Thermal expansion of lunar rocks, Proc. Lunar Planet. Sci. Conf 2nd, 2317-2321, 1971.

Brace, W. F., Brittle fracture of rocks, in State of Stress in the Earth's Crust, edited by W. R. Judd, pp. 110-178, Elsevier, New York, 1964.

Bratt, S. R., S.C. Solomon, and J. W. Head, The evolution of multi- ringed basins: Cooling, subsidence, and thermal stress (abstract), Lunar Planet. Sci., 12, 109-111, 1981.

Bratt, S. R., S.C. Solomon, J. W. Head, and C. H. Thurber, The deep structure of lunar basins: Implications for basin formation and modification, J. Geophys. Res., 90, 3049-3064, 1985a.

Bratt, S. R., E. A. Bergman, and S.C. Solomon, Thermoelastic stress: How important as a cause of earthquakes in young oceanic lithosphere 9., j. Geophys. Res., 90, 10,249-10,260, 1985b.

Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., 510 pp., Oxford University Press, New York, 1959.

Chen, W.-P., and P. Molnar, Focal depths of intracontinental and intraplate earthquakes and their implications for the thermal and mechanical properties of the lithosphere, J. Geophys. Res., 88, 4183- 4214, 1983.

Church, S., J. W. Head, and S.C. Solomon, Multi-ringed basin in- teriors: Structure and early evolution of Orientale (abstract), Lunar Planet. Sci., 13, 98-99, 1982.

Comer, R. P., S.C. Solomon, and J. W. Head, Elastic lithosphere thickness on the moon from mare tectonic features: A formal inver-

sion, Proc. Lunar Planet. Sci. Conf loth, 2441-2463, 1979.

BRATT ET AL.: THERMAL STRESS NEA• COOLING IMPACT BASINS 12,433

Dence, M. R., Impact melts, J. Geophys. Res., 76, 5552-5565, 1971. Gault, D. E., and E. D. Heitowit, The partitioning of energy for

hypervelocity impact craters formed in rock, in Proceedings of the Sixth Hypervelocity Impact Symposium, vol. 2, pp. 419-456, Nation- al Technical Information Service, Springfield, Va., 1963.

Gault, D. E., and J. A. Wedekind, Experimental hypervelocity impact into quartz sand, II, Effects of gravitational acceleration, in Impact and Explosion Cratering, edited by D. J. Roddy, R. O. Pepin, and R. B. Merrill, pp. 1231-1260, Pergamon, New York, 1977.

Gautschi, W., Error function and Fresnel integrals, in Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun, pp. 295-329, National Bureau of Standards, Washington, D.C., 1964.

Goodier, J. N., On the integration of the thermoelastic 'equations, Philos. Mag., 7, 1017-1032, 1937.

Greeley, R., Modes of emplacement of basalt terrains and an analysis of mare volcanism in the Orientale basin, Proc. Lunar Planet. Sci. Conf. 7th, 2747-2759, 1976.

Grieve, R. A. F., and M. J. Cintala, A method for estimating the initial impact conditions of terrestrial cratering events, exemplified by its application to Brent crater, Ontario, Proc. Lunar Planet. Sci. Conf. 12th, 1609-1621, 1981.

Grieve, R. A. F., M. R. Dence, and P. B. Robertson, Cratering processes: As interpreted from the occurrence of impact melts, in Impact and Explosion Cratering, edited by D. J. Roddy, R. O. Pepin, and R. B. Merrill, pp. 791-814, Pergamon, New York, 1977.

Griggs, D. T., and J. W. Handin, Observations on fracture and a hypothesis of earthquakes, Mem. Geol. $oc. Am., 79, 347-364, 1960.

Hartmann, W. K., and C. A. Wood, Moon: Origin and evolution of multi-ring basins, Moon, 3, 3-78, 1971.

Head, J. W., Orientale multi-ringed basin interior and implications for the petrogenesis of lunar highland samples, Moon, 11, 327-356, 1974.

Head, J. W., Origin of outer rings in lunar multi-ringed basins: Evi- dence from morphology and ring spacing, in Impact and Explosion Cratering, edited by D. J. Roddy, R. O. Pepln, and R. B. Merrill,

_

pp. 563-573, Pergamon, New York, 1977. Head, J. W., E. Robinson, and R. Phillips, Topography of the Orien-

tale basin (abstract), Lunar Planet. Sci., 12, 421-423, 1981. Holsapple, K. A., and R. M. Schmidt, On the scaling of crater dimen-

sions, 2, Impact processes, J. Geophys. Res., 87, 1849-1870, 1982. Jeffreys, H., The Earth, 5th ed., 525 pp., Cambridge University Press,

New York, 1970. Kaula, W. M., Thermal evolution of earth and moon growing by

planetesimal impacts, J. Geophys. Res., 84, 999-1008, 1979. Masters, J. I., Some applications in physics of the P function, J. Chem.

Phys., 23, 1865-1874, 1955. Maxwell, D. E., Simple Z model of cratering, ejection and overturned

flap, in Impact and Explosion Cratering, edited by D. J. Roddy, R. O. Pepin, and R. B. Merrill, pp. 1003-1008, Pergamon, New York, 1977.

McNutt, M. K., and H. W. Menard, Constraints on yield strength in the oceanic lithosphere derived from observations of flexure, Geo- phys. J. R. Astron. Soc., 71, 363-394, 1982.

Melosh, H. J., and W. B. McKinnon, The mechanics of ringed basin formation, Geophys. Res. Lett., 5, 985-988, 1978.

Mindlin, R. D., and D. H. Cheng, Thermoelastic stress in the semi- infinite solid, J. Appl. Phys., 21,931-933, 1950.

Mizutani, H., and M. Osako, Elastic-wave velocities and thermal dif- fusivities of Apollo 17 rocks and their geophysical implications, Proc. Lunar Sci. Conf. 5th, 2891-2901, 1974.

Moore, H. J., C. A. Hodges, and D. H. Scott, Multiringed basins-- Illustrated by Orientale and associated features, Proc. Lunar Sci. Conf. 5th, 71-100, 1974.

Moulton, F. R., Celestial Mechanics, 2nd ed., 437 pp., Macmillan, New York, 1914.

Muehlberger, W. R., Conjugate joint sets at small dihedral angle, J. Geol., 69, 211-219, 1961.

Nordyke, M.D., Nuclear craters and preliminary theory of mechanics of explosive crater formation, J. Geophys. Res., 66, 3439-3459, 1961.

Nordyke, M.D., Nuclear cratering experiments: United States and Soviet Union, in Impact and Explosion Cratering, edited by D. J. Roddy, R. O. Pepin, and R. B. Merrill, pp. 103-124, Pergamon, New York, 1977.

O'Keefe, J. D., and T. J. Ahrens, Shock effects from a large impact on the moon, Proc. Lunar Sci. Conf. 6th, 2831-2844, 1975.

O'Keefe, J. D., and T. J. Ahrens, Impact ejecta on the moon, Proc. Lunar Sci. Conf. 7th, 3007-3025, 1976.

O'Keefe, J. D., and T. J. Ahrens, Impact-induced energy partitioning, melting, and vaporization on terrestrial planets, Proc. Lunar Sci. Conf 8th, 3357-3374, 1977.

Olver, F. W. J., Bessel functions of integer order, in Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun, pp. 355-434, National Bureau of Standards, Washington, D.C., 1964.

Onorato, P. I. K., D. R. Uhlmann, and C. H. Simonds, The thermal history of the Manicouagan impact melt sheet, Quebec, d. Geophys. Res., 83, 2789-2798, 1978.

Schatz, J. F., and G. Simmons, Thermal conductivity of earth materi- als at high temperatures, J. Geophys. Res., 77, 6966-6983, 1972.

Schmidt, R. M., A centrifuge cratering experiment: Development of a gravity-scaled yield parameter, in Impact and Explosion Cratering, edited by D. J. Roddy, R. O. Pepin, and R. B. Merrill, pp. 1261- 1278, Pergamon, New York, 1977.

Shoemaker, E. M., Penetration mechanics of high velocity meteorites, illustrated by Meteor crater, Arizona, Int. Geol. Congr., 21st, Proc. Sect. 18, 418-434, 1960.

Simmons, G., and W. F. Brace, Comparison of static and dynamic measurements of compressibility of rocks, d. Geophys. Res., 70, 5649-5656, 1965.

Skinner, B. J., Thermal expansion, in Handbook of Physical Constants, edited by S. P. Clark, Jr., pp. 75-96, Geological Society of America, Boulder, Colo., 1966.

Solomon, S.C., Mare •olcanism and lunar crustal structure, Proc. Lunar Sci. Conf 6th, 1021-1042, 1975.

Solomon, S.C., and J. W. Head, Vertical movement in mare basins: Relation to mare emplacement, basin tectonics, and lunar thermal history, d. Geophys. Res., 84, 1667-1682, 1979.

Solomon, S.C., and J. W. Head, Lunar mascon basins: Lava filling, tectonics, and evolution of the lithosphere, Rev. Geophys., 18, 107- 141, 1980.

Solomon, S.C., and J. Longhi, Magma oceanography, 1, Thermal evolution, Proc. Lunar Sci. Conf. 8th, 583-599, 1977.

Solomon, S.C., R. P. Comer, and J. W. Head, The evolution of impact basins: Viscous relaxation of topographic relief, J. Geophys. Res., 87, 3975-3992, 1982.

Timoshenko, S. P., and J. N. Goodier, Theory of Elasticity, 3rd ed., 567 pp., McGraw-Hill, New York, 1970.

Turcotte, D. L., Are transform faults thermal contraction cracks?, J. Geophys. Res., 79, 2573-2577, 1974.

Turcotte, D. L., Thermal stress in planetary elastic lithospheres, Proc. Lunar Planet. Sci. Conf. 13th, Part 2, d. Geophys. Res., 88, suppl., A585-A587, 1983.

Vaile, R. B., Jr., Pacific craters and scaling laws, d. Geophys. Res., 66, 3413-3438, 1961.

Wilhelms, D. E., Comparison of Martian and lunar multiringed circu- lar basins, d. Geophys. Res., 78, 4084-4095, 1973.

Wilhelms, D. E., Relative ages of lunar basins (abstract), Reports of Planetary Geology Program, 1978-1979, NASA Tech. Memo., 80339, 135-137, 1979.

Wood, C. A., and J. W. Head, Comparison of impact basins on Mer- cury, Mars and the moon, Proc. Lunar Sci. Conf., 7th, 3629-3651, 1976.

S. R. Bratt, Science Applications International Corporation, 10210 Campus Point Drive, San Diego, CA 92121.

J. W. Head, Department of Geological Sciences, Brown University, Providence, RI 02912.

S. C. Solomon, Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139.

(Received January 28, 1985; revised August 19, 1985;

accepted September 9, 1985.)

the evolution of impact basins: cooling, subsidence, and thermal

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