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Global fracture patterns of a despun planet: Application to Mercury
ICARUS38, 243-250 (1979)
Global Fracture Patterns of a Despun Planet: Application to Mercury J. B U R T P E C H M A N N
AND H. J. M E L O S H
California Institute of Technology, Pasadena, California 911~5
Received October 16, 1978; revised December 19, 1978 We have determined the global fracture patterns resulting from combinations of stresses due to tidal despinning and contraction or expansion. We find that Mercury's lineament pattern is consistent with a history of despinning and contraction. According to our model, the observed tectonic pattern implies that the despinning process reached completion before the planet ceased contracting. Our model predicts a stress due to contraction which is up to 1.8 times the maximum despinning stress on Mercury. The maximum contractional stress could be as large as 4 times the maximum despinning stress if the oldest fractures on the planet are N-S thrust faults in the equatorial region. INTRODUCTION Evidence for the tectonic a c t i v i t y on M e r c u r y can be presented in t e r m s of a global fracture pattern. Burns (1976) a n d Melosh (1977) consider the possibility t h a t the fracture p a t t e r n on M e r c u r y m a y be a consequence of stresses caused b y tidal despinning a n d contraction of the planet. Since these studies were made, more work has been done with Mariner 10 photographs. Besides the planimetrically arcuate escarpm e n t s recognized early b y S t r o m et al. (1975), the fracture p a t t e r n consists of older N E a n d N W trending lineament systems (Dzurisin, 1978), an E - W trending lineament s y s t e m near the poles (Strom and Malin, in preparation), a n d a poorly developed N - S trending lineament s y s t e m (Dzurisin, 1978). T h e scarps a n d the N E and N W lineaments a p p e a r to h a v e a global distribution. However, the N E a n d N W trends become more nearly N - S in the polar regions. T h e a r c u a t e scarps are commonly interpreted as t h r u s t or high-angle reverse faults (Strom et al., 1975; Dzurisin, 1978). Melosh and Dzurisin (1978) believe t h a t the scarps h a v e a rough N - S trend
due to the influence of despinning stresses and t h a t the N E and N W lineaments are strike-slip faults or joints caused b y despinning stresses. U n d e t e c t e d systems of lineaments and scarps m a y exist since illumination effects a n d incomplete coverage of the planet m a k e interpretation of the Mariner 10 p h o t o g r a p h s difficult. We examine the hypothesis t h a t the fracture p a t t e r n on M e r c u r y is due to stresses generated b y despinning and contraction b y v a r y i n g the relative strength and time dependence of these stresses a n d determining the resultant global fracture patterns. Our calculations cover all episodes of faulting including those which occur after the entire lithosphere has fractured. I n this case the lithosphere is t r e a t e d as a plastic medium. This work is based on calculations of fracture p a t t e r n s b y Melosh (1977). I n his work the relative time dependence of the stresses was not varied, and the t y p e of faulting occurring in the plastic lithosphere was not determined. Thus, we present a more complete picture of the global fracture p a t t e r n s resulting f r o m despinning a n d contraction or expansion. 243
0019-1035/79/050243-08502.00/0 Copyright O 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
244
PEGHMANN AND MELOSH
Our results m a y be used in interpreting lineament systems on any despun planet or satellite.
5 a~" -
X\5
We model the planet's lithosphere as a thin shell over a fluid interior. T h e equatorial bulge relaxes as the planet is tidally despun, and stresses build up in the lithosphere. A thin lithosphere cannot support the bulge nonhydrostatically; it deforms to follow the hydrostatic shape t h r o u g h o u t the despinning episode. If the planet despins without contracting or expanding, the stress relaxation is b y the development of N E and N W strike-slip faults which extend from the equator to midlatitudes and b y E - W normal faults which extend from midlatitudes to the poles (Melosh, 1977; Burns, 1976). At the other extreme, pure contraction produces randomly oriented thrust faults which cover the planet. Pure expansion produces randomly oriented normal faults which cover the planet (Solomon, 1978). Various combinations of these processes lead to more complicated fracture patterns. T h e stresses at any latitude due to despinning and contraction or expansion are computed from the elastic equations of Vening-Meinesz (1947) until the maximum stress difference at the latitude reaches the yield stress. After the yield stress is reached, the stress is given b y the plastic equations of Melosh (1977). T h e maximum stress difference is held at the yield stress. At the b o u n d a r y between the elastic and plastic regions of the planet a0e is required to be continuous, a~r need not be continuous. T h e elastic equations are: 5 -
[too - ~ ( t ) ] ~ ,
12 x \5---~/(5
- 3 cos 2x)
+ 2.
(t), \1 --a/
[mo - m ( t ) ] u
(1 -{-#~ (1 + 9cos2X)
METHOD
o-,eo
12
r
(t)
+o'/
( 1 + o"~ Ar 4- 2#
(t), kl --a/
o,,,, °
= O.
(2)
r (3)
a00o, a , e , and a,r" are the elastic meridional, azimuthal, and radial stresses, respectively. We use a rigidity, g, of 6.5 X 1011 d y n / c m ~ and a Poisson's ratio, a, of 0.5. Using # = 0.25 would have only a small effect on the results (Melosh, 1977). = latitude, re(t) = the centripetal acceleration at the equator divided b y the gravitational acceleration ( = o ~ r 3 / G M , where oJ = rotation rate, r = radius, M = mass of planet, and G = gravitational constant). m0 is the initial value of re(t). ( h r / r ) ( t ) = fractional change in the planet's radius. T h e method just described gives only an approximate solution to the full P r a n d t l Reuss equations of plastic flow. It guarantees, however, t h a t both the stress equilibrium equations and the associated flow law are satisfied (Kachanov, 1974). These equations lead to a plane stress elasticplastic problem with a Tresca failure criterion. In the strike-slip region, the principal deviatoric plastic strains in the plane, 1 ~'s0 = ~( 0e -- e~) and ~'~ = ½(%~ -- ~e0), are equal and opposite, as are the deviatoric stresses, thus satisfying the associated flow law ~'e0/#'** = g ~ / a ' , ~ . In the thrust fault region the meridional strain, ese, but not the stress, ace, is purely elastic, and only ~'~ and ~',, are affected b y plastic yielding. Here the associated flow law merely determines g , . Similarly, in the normal fault region ~ is purely elastic and gee and g,, are related b y the flow law. Thus, this method satisfies two of the main requirements for an exact solution. It fails, however, to correctly relate the areal strain, ~x = ~e0 + ~¢~, to the areal
GLOBAL FRACTURE PATTERNS stress, aA = a00 + ~**. Normally, this is assumed to be given b y an elastic law, eA ---- (1/2 ~)[(1 a ) / ( 1 + a)]aA, where a is Poisson's ratio. T h e approximation used in this paper overestimates cA, resulting in slightly incorrect meridional displacements, u0, of the elastic lithosphere. These errors, however, are seldom more t h a n 20% for realistic models. An interesting, if academic, result is t h a t if Poisson's ratio equals 1, the m e t h o d used here is exact. Since the true failure law of the lithosphere is not well known, the effect of performing an exact solution for m a n y different cases does not seem worthwhile. T h e model presented here gives sufficiently accurate results for qualitarD re
245
tive comparisons with planetary fracture systems. Since the actual time dependence of the elastic stresses given b y (1), (2), and (3) is unknown, a change proportional to ( 1 - e -tl') was assumed. Thus m o re(t) = m0(1 -- e-t/'D) and Ar(t)/r = (hr/r)f (1 -- e-t/"o), where rD = time constant for despinning, ro = time constant for contraction or expansion, and (hr/r)f = asy m p t o t i c value for change in planet's radius. This time dependence requires t h a t at t = 0, m = m0, while as t--* oo, m--+0. There are three variable parameters in our model calculations. These are given below.
time constant for despinning (4) time constant for contraction or expansion '
2~E(1 + a ) / ( 1 - a ) J ( h r / r ) , B=
(5/12)p[-(1 q- a ) / ( 5 q- a)Jl0m0 asymptotic azimuthal or meridional stress due to contraction or expansion ,
(5)
asymptotic azimuthal despinning stress at ~ = 0 ° (5/12)m0~[(1 -F ~)/(5 -F ~)-]10 5' --
Y asymptotic azimuthal despinning stress at ), = 0 ° (6)
yield stress stress. An initial rotation rate between 10 and 40 hr (1.25 X 10 - 8 _ < m 0 < 0 . 2 ) is assumed, implying t h a t the planet was initially rotating rapidly with a period similar to t h a t of all the other planets except Venus. T h e planet's radius is allowed to change by a maximum of 0.1% or a b o u t aoo/Y = -- ~ (5 -- 3 cos 2~) 5"(1 - e-tl")) 2.5 km. This value is reasonable in light of W/3 5"(1 -- e--"t/'D), (7) the estimation of 1-2 k m of contraction inferred from the arcuate scarps b y Strom a , ~ / Y = ~o(1 -F 9 cos 2h) 5" (1 - e- ' / ' " ) -t-/~ 5" (1 e-a~/'D). (8) et al. (1975). Possible causes of contraction are cooling of the core (Strom et al., 1975) T h e ranges for the parameters /~ and 5" and cooling of the lithosphere (Solomon, are determined b y the values assumed for 1976). Expansion of the planet is considered the initial rotation rate, the fractional in our calculation for completeness. The change in the planet's radius, and the yield yield stress in the lithosphere is given a
Y is the yield stress. T h e previous calculations done b y Melosh are equivalent to a = 1. B is negative for expansion. Using these parameters and the explicit time dependence given above, the equations for ae0 and a ~ become:
-
-
246
PECHMANN AND MELOSH 4.0-
3.0-
2.0-
[ "L,
1.0-
1 oil thrust
2..~.~-thrust; earlier strike-slip 3"~- thrush earlier strike-slip " ~ °ndn°rm°l
stri~ 1
I
and L
normol; eorlier thrust
¢
o5
strike-slip and normal
-1.0-
__]
6 all normal
l J
-2.0 -
-3.0-
-4.0 ,
0.1
,
0.2
,
0.4 0.5
,
i
0.75 1.0 a
,
i
2.0
,
4D 5D
,
7.5 10.0
FIG. 1. Global fracture patterns just after the last latitude to remain elastic has failed. See text for details. ~, = 11. v e r y large range of 10 to 500 bars (Chinnery, 1964; Jefferys, 1952). T h e resulting range for ~ is [/~1< 4.2. T h e m i n i m u m value for is 1.85 ; the m a x i m u m value is ~ . ~ = 7.39 X 104mo . . . .
(9)
where
to decrease the largest stress difference. T h e stress increments are c o m p u t e d elastically, added to the plastic stress field, and the stresses adjusted to satisfy the yield criteria. Thus, if a e e = So, a,~ -- S~, and arr = 0 at time t, such t h a t the maxim u m stress difference is Y, then at time t2 = tz +
(Ar/r)max
5.28 X 10-3
I~1
181
m0 m,~ = 5.28
At:
,~.(t,) = S , +
daoo e
~
at,
(10)
t, At"
(11)
t]
[8] >---0.0264, do" ~ ~,e
= 0.2,
]8[ ~ 0.0264.
Calculations were done for 0.1 < a < 10. After the m a x i m u m stress difference has reached the yield stress and faulting has occurred at all latitudes, we suppose t h a t the lithosphere is in a plastic state. T h e m a x i m u m stress difference is held at the yield stress. At this point the problem becomes highly nonlinear. I n order to extend our model past the time of full failure, we assume t h a t changes in the plastic state of the lithosphere m a y occur a t some latitudes if the stresses a t these latitudes due to the ongoing contraction a n d despinning tend
a , , ( t , ) = S , -}- ~-~
T h e superscript e denotes stresses given b y the elastic expressions (1) a n d (2). If the incremental t e r m s are such t h a t the new m a x i m u m stress difference is greater t h a n Y, abe and a~¢ are set equal to their original plastic values, Se and S¢. If, however, the m a x i m u m stress difference is less t h a n Y, aes and arC are allowed to change in this incremental fashion until a new mode of faulting occurs. We actually only determine whether a change is possible, i.e., whether the incremental t e r m s ever
GLOBAL FRACTURE
N° 6 ~ 90 R
wO°ti ~ 3 0 ° ~
1-s
E
-90° S FZG. 2. T h e f r a c t u r e p a t t e r n w h e n t h e last l a t i t u d e to r e m a i n elastic h a s j u s t failed for a = 0.15, /9 = 3.2, "r = 11. T h i s p a t t e r n is r e p r e s e n t a t i v e of t h o s e in area 2 in Fig. 1. T h e b o u n d a r i e s will be at different l a t i t u d e s for different v a l u e s of t h e p a r a m e t e r s , a, /~, a n d -y. T h e m o v e m e n t on t h e strike-slip f a u l t s is c o n s i s t e n t w i t h E - W compression.
change sign and, hence, act to decrease the largest stress difference. DISCUSSION T h e global fracture patterns at the time at which the last latitude to remain elastic has just failed are shown in Fig. 1 as a function of a and ~ with ~/ = 11, a typical intermediate value. In the discussion the fractures are referred to as faults. However, the global stress field m a y also control the orientation of joints with no displacement. At each labeled a calculations were done at 17 or more values of ~. In Fig. 1 the positions of the boundaries are generally correct, but the details of the boundaries reflect the coarse grid size. In area 1 in Fig. 1 thrust faults due to dominant contraction stresses cover the planet. If the despinning stresses are strong enough to control the orientation of the local principal stress axes, they will cause the preferred strike of the thrust faults to be N-S. In area 2, N E and N W strike-slip faulting occurs
247
PATTERNS
first at low or midlatitudes in a limited latitude range. T h e entire planet, including this region of strike-slip faults, is eventually covered b y thrust faults. An illustration of the fracture pattern in area 2 is shown in Fig. 2. Area 3 is similar to area 2 except that, besides early strike-slip faults, E - W normal faults occur first in limited regions near the poles. In areas 4 and 5, N E and N W trending strike-slip faults occur from equator to midlatitudes and E - W normal faults occur from midlatitudes to the poles. Area 4 differs from area 5 in t h a t thrust faults trending N - S occur in area 4 before the strike-slip and normal faults. These thrust faults are confined to a limited latitude range in low or midlatitudes. Figure 3 is an illustration of the fracture p a t t e r n in area 4. In area 6 normal faults due to dominant expansion cover the planet. T h e y tend E - W if the despinning stresses are strong enough to control their orientation. T h e global fracture patterns for ~/mln and "~ma~ are shown in Figs. 4 and 5. In areas labeled "incomplete" in Fig. 4 there are latitude ranges in which the yield stress is
N 90°
-90" S FIG. 3. T h e f r a c t u r e p a t t e r n w h e n t h e last l a t i t u d e to r e m a i n elastic h a s j u s t failed for a = 0.1, = 2.6, ~ = 11. T h i s p a t t e r n is r e p r e s e n t a t i v e of t h o s e in area 4 in Fig. 1. T h e b o u n d a r i e s will be a t different l a t i t u d e s for different v a l u e s of a,/~, a n d 3'.
248
PECHMANN AND MELOSH 4`0" 1 all thrust
3.0" 2"~rust;
2.0.
earlier strike-slip
eanv ss ] 1,0- 2;.incomplet~rnrush
I
t' incomplete thrust
2'
I
5' incomplete strike-slip and normal
/9 o-
5 strike-slip end normal
-1.0-
-2.0-
I
I
I 6 all normal
-3.0 -
-4.0-
I
0'.40~ 0.75 1.0
0.2
0.1
210
4:0 5.0
715 10.0
a
FIG. 4. Global fracture patterns just after the last latitude to remain elastic has failed. = ~=in = 1.85. I n areas labeled "incomplete" there is a range of latitudes at which the yield stress is never reached.
Global p a t t e r n s including changes which occur after the lithosphere becomes fully plastic are shown for ~ = 11 in Fig. 6. T h e t y p e of faulting which occurs after full failure of the lithosphere is shown superimposed on the areas outlined in Fig. 1. T h e
never reached. F r o m comparison of Figs. 1, 4, and 5 it can be seen t h a t as long as the yield stress is exceeded at all latitudes, the p a r a m e t e r ~ has little effect on these p a t terns except in areas 2 a n d 3. These areas shrink and area 4 enlarges as ~ is increased.
'°1 I
3.0
I all
thrust
4 strike-slip and normal; earlier thrust
2.0
I 1.o-
I
,8 o-
I
5 strike-sllp and normal
I
-1.0 -
6 oil normal
-2.0-
-3.0 -4.0 0.1
0:2
0'4 0.5 0.+5 do cl
210
41O5'`0 7'.51o`0
Fro. 5. Global fracture patterns just after the last latitude to remain elastic has failed.
= ~(~).
GLOBAL FRACTURE PATTERNS
249
4.0-
no change
strike-slip fallowed by normal spread from poles no farther than 48 °
3.0-
no change
4 ~ 2,0-
1.0-
thrust '
~d Ifrom
J
I
normal spread from poles no farther .,~...;on 48°
equator I
tshprr~Sa~"from I e_.quotor
,B o-1.0-
normal spread from poles
change "•'1--no 6
~
I
-2.0 -
]
-3.0 -
no change
no change
-4.0 0.1
o:2
o:4 o:~
o.~5 f.o
i.o
4'.0 5'.0
7:5 to~
a
FIG. 6. Modifications to the global fracture patterns in Fig. 1 which occur after the whole lithosphere has fractured. ~, = 11.
extent in latitude of these new faults is variable since it depends on the model parameters a, B, and ~. In area 5 when ~ > 0 and < 1, younger thrust faults are developed in regions covered by strike-slip and normal faults. These thrust faults could eventually cover the planet if the contraction were large enough. T h e orientation of these thrust faults is N - S if despinning stresses are strong enough to control their orientation. This pattern of older strike-slip and normal faults and younger thrust faults which cover the planet is consistent with M e r c u r y ' s tectonic pattern. However, the systematic shift of the N E and N W lineaments towards N - S in the polar regions is not explained. This shift m a y be due in part to enhancement differences caused by changes in illumination with latitude (Dzurisin, 1978). Since area 4 is identical to area 5 except for thrust faults at low latitudes which are older t h a n the strikeslip faults, the discovery on Mercury of N - S lineaments near the equator older t h a n the N E and N W lineaments would increase the allowable range of f~. These ancient
N - S lineaments may be associated with the poorly developed N - S lineament system of Dzurisin (1978). However, the observed N - S lineaments do not appear to be confined to the equatorial region. CONCLUSION
These theoretical calculations and the new observational evidence of E - W lineaments near the pole have significantly strengthened the case for the importance of tidal despinning in Mercurian tectonics. T h e observed p a t t e r n falls within area 5 with a < 1 and B > 0 in Fig. 4 even if the arcuate scarps have no N - S alignment. If the observed p a t t e r n is, in fact, a consequence of despinning and contraction, some limits on the range of parameters characterizing these processes may be defined. T h e requirement a < 1 implies that the time scale for despinning is shorter t h a n the time scale for contraction. According to Goldreich and Soter (1966) the time required to despin M e r c u r y from a rotation period of 20 hr to the present period of
250
PECHMANN AND MELOSH
56.6 d a y s is 0.2 t o 2.0 B.Y. d e p e n d i n g on t h e v a l u e of t h e p l a n e t a r y d i s s i p a t i o n c o n s t a n t , Q. T h e l a r g e s t v a l u e for/~ in a r e a 5 is a b o u t 1.8, i m p l y i n g t h a t t h e m a x i m u m s t r e s s d u e to c o n t r a c t i o n m u s t b e less t h a n 1.8 t i m e s t h e m a x i m u m s t r e s s d u e to d e s p i n n i n g . H o w e v e r , if a set of a n c i e n t N - S l i n e a m e n t s were d i s c o v e r e d in t h e e q u a t o r i a l region, a m a x i m u m c o n t r a c t i o n a l stress u p to 4 t i m e s t h e m a x i m u m stress d u e to d e s p i n n i n g w o u l d b e a l l o w e d . I n t h i s case t h e y o u n g e r t h r u s t f a u l t s s h o u l d s h o w l i t t l e or no N - S alignment. O u r c a l c u l a t i o n s m a y b e a p p l i e d to o t h e r d e s p u n p l a n e t s since t h e p a r a m e t e r i z e d elastic s t r e s s e q u a t i o n s c o n t a i n no t e r m s explicitly dependent on Mercurian properties. T h e r a n g e s a l l o w e d for t h e t h r e e p a r a m e t e r s m a y b e d i f f e r e n t for o t h e r p l a n e t s , b u t o u r c a l c u l a t i o n s a r e e a s i l y e x t e n d e d to l a r g e r ranges. Specifically, t h e p a t t e r n s we h a v e f o u n d m a y b e useful in i n t e r p r e t i n g s u r f a c e f e a t u r e s of t h e G a l i l e a n s a t e l l i t e s of J u p i t e r .
REFERENCES BURNS, J. A. (1976). Consequences of the tidal slowing of Mercury. Icarus 28, 453-458. CHINNERY, M. A. (1964). The strength of the Earth's crust under horizontal shear stress. J. Geophys. Res. 59, 2085-2089. DZURISIN, D. (1978). The tectonic and volcanic history of Mercury as inferred from studies of scarps, ridges, troughs, and other lineaments. J. Geophys. Res. 83, 4883-4906. GOLDREICH P., AND SOTER. S. (1966). Q in the solar system. Icarus 5, 375-389. JEFFREYS, H. (1952). The Earth, 3rd ed. Cambridge Univ. Press, London/New York. KACI-IANOV, L. M. (1974). Fundamentals of tha Theory of Plasticity, p. 246. MIR, Moscow. MELOSH, J. (1977). Global tectonics of a despun planet. Icarus 31, 221-243. MELOSH, J., AND DZURISIN, D. (1978). Mercurian global tectonics: A consequence of tidal despinning? Icarus 35, 227-236. SOLOMON,S. C. (1976). Some aspects of core formation in Mercury. Icarus 28, 509-522. SOLOMOn, S. C. (1978). On volcanism and thermal tectonics on one-plate planets. Geophys. Res. Lett. 5, 461-463. STROM, R. G., ANDMALIN,M. (1978). Geologic map of the Bach quadrangle (H-15), Mercury. USGS Miscellaneous Investigations Map. In preparation.
STROM, R. G., TRASK, N. J., AND GUEST, J. E. ACKNOWLEDGMENT This work was supported in part by a National Science Foundation Graduate Research Fellowship.
(1975). Tectonism and volcanism on Mercury. J. Geophys. Res. 80, 2378-2507. VENING-MEINESZ, F. A. (1947). Shear patterns of the Earth's crust. Tra~s. Amer. Geophys. Union 28, 1 61.
Global Fracture Patterns of a Despun Planet: Application to Mercury J. B U R T P E C H M A N N
AND H. J. M E L O S H
California Institute of Technology, Pasadena, California 911~5
Received October 16, 1978; revised December 19, 1978 We have determined the global fracture patterns resulting from combinations of stresses due to tidal despinning and contraction or expansion. We find that Mercury's lineament pattern is consistent with a history of despinning and contraction. According to our model, the observed tectonic pattern implies that the despinning process reached completion before the planet ceased contracting. Our model predicts a stress due to contraction which is up to 1.8 times the maximum despinning stress on Mercury. The maximum contractional stress could be as large as 4 times the maximum despinning stress if the oldest fractures on the planet are N-S thrust faults in the equatorial region. INTRODUCTION Evidence for the tectonic a c t i v i t y on M e r c u r y can be presented in t e r m s of a global fracture pattern. Burns (1976) a n d Melosh (1977) consider the possibility t h a t the fracture p a t t e r n on M e r c u r y m a y be a consequence of stresses caused b y tidal despinning a n d contraction of the planet. Since these studies were made, more work has been done with Mariner 10 photographs. Besides the planimetrically arcuate escarpm e n t s recognized early b y S t r o m et al. (1975), the fracture p a t t e r n consists of older N E a n d N W trending lineament systems (Dzurisin, 1978), an E - W trending lineament s y s t e m near the poles (Strom and Malin, in preparation), a n d a poorly developed N - S trending lineament s y s t e m (Dzurisin, 1978). T h e scarps a n d the N E and N W lineaments a p p e a r to h a v e a global distribution. However, the N E a n d N W trends become more nearly N - S in the polar regions. T h e a r c u a t e scarps are commonly interpreted as t h r u s t or high-angle reverse faults (Strom et al., 1975; Dzurisin, 1978). Melosh and Dzurisin (1978) believe t h a t the scarps h a v e a rough N - S trend
due to the influence of despinning stresses and t h a t the N E and N W lineaments are strike-slip faults or joints caused b y despinning stresses. U n d e t e c t e d systems of lineaments and scarps m a y exist since illumination effects a n d incomplete coverage of the planet m a k e interpretation of the Mariner 10 p h o t o g r a p h s difficult. We examine the hypothesis t h a t the fracture p a t t e r n on M e r c u r y is due to stresses generated b y despinning and contraction b y v a r y i n g the relative strength and time dependence of these stresses a n d determining the resultant global fracture patterns. Our calculations cover all episodes of faulting including those which occur after the entire lithosphere has fractured. I n this case the lithosphere is t r e a t e d as a plastic medium. This work is based on calculations of fracture p a t t e r n s b y Melosh (1977). I n his work the relative time dependence of the stresses was not varied, and the t y p e of faulting occurring in the plastic lithosphere was not determined. Thus, we present a more complete picture of the global fracture p a t t e r n s resulting f r o m despinning a n d contraction or expansion. 243
0019-1035/79/050243-08502.00/0 Copyright O 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
244
PEGHMANN AND MELOSH
Our results m a y be used in interpreting lineament systems on any despun planet or satellite.
5 a~" -
X\5
We model the planet's lithosphere as a thin shell over a fluid interior. T h e equatorial bulge relaxes as the planet is tidally despun, and stresses build up in the lithosphere. A thin lithosphere cannot support the bulge nonhydrostatically; it deforms to follow the hydrostatic shape t h r o u g h o u t the despinning episode. If the planet despins without contracting or expanding, the stress relaxation is b y the development of N E and N W strike-slip faults which extend from the equator to midlatitudes and b y E - W normal faults which extend from midlatitudes to the poles (Melosh, 1977; Burns, 1976). At the other extreme, pure contraction produces randomly oriented thrust faults which cover the planet. Pure expansion produces randomly oriented normal faults which cover the planet (Solomon, 1978). Various combinations of these processes lead to more complicated fracture patterns. T h e stresses at any latitude due to despinning and contraction or expansion are computed from the elastic equations of Vening-Meinesz (1947) until the maximum stress difference at the latitude reaches the yield stress. After the yield stress is reached, the stress is given b y the plastic equations of Melosh (1977). T h e maximum stress difference is held at the yield stress. At the b o u n d a r y between the elastic and plastic regions of the planet a0e is required to be continuous, a~r need not be continuous. T h e elastic equations are: 5 -
[too - ~ ( t ) ] ~ ,
12 x \5---~/(5
- 3 cos 2x)
+ 2.
(t), \1 --a/
[mo - m ( t ) ] u
(1 -{-#~ (1 + 9cos2X)
METHOD
o-,eo
12
r
(t)
+o'/
( 1 + o"~ Ar 4- 2#
(t), kl --a/
o,,,, °
= O.
(2)
r (3)
a00o, a , e , and a,r" are the elastic meridional, azimuthal, and radial stresses, respectively. We use a rigidity, g, of 6.5 X 1011 d y n / c m ~ and a Poisson's ratio, a, of 0.5. Using # = 0.25 would have only a small effect on the results (Melosh, 1977). = latitude, re(t) = the centripetal acceleration at the equator divided b y the gravitational acceleration ( = o ~ r 3 / G M , where oJ = rotation rate, r = radius, M = mass of planet, and G = gravitational constant). m0 is the initial value of re(t). ( h r / r ) ( t ) = fractional change in the planet's radius. T h e method just described gives only an approximate solution to the full P r a n d t l Reuss equations of plastic flow. It guarantees, however, t h a t both the stress equilibrium equations and the associated flow law are satisfied (Kachanov, 1974). These equations lead to a plane stress elasticplastic problem with a Tresca failure criterion. In the strike-slip region, the principal deviatoric plastic strains in the plane, 1 ~'s0 = ~( 0e -- e~) and ~'~ = ½(%~ -- ~e0), are equal and opposite, as are the deviatoric stresses, thus satisfying the associated flow law ~'e0/#'** = g ~ / a ' , ~ . In the thrust fault region the meridional strain, ese, but not the stress, ace, is purely elastic, and only ~'~ and ~',, are affected b y plastic yielding. Here the associated flow law merely determines g , . Similarly, in the normal fault region ~ is purely elastic and gee and g,, are related b y the flow law. Thus, this method satisfies two of the main requirements for an exact solution. It fails, however, to correctly relate the areal strain, ~x = ~e0 + ~¢~, to the areal
GLOBAL FRACTURE PATTERNS stress, aA = a00 + ~**. Normally, this is assumed to be given b y an elastic law, eA ---- (1/2 ~)[(1 a ) / ( 1 + a)]aA, where a is Poisson's ratio. T h e approximation used in this paper overestimates cA, resulting in slightly incorrect meridional displacements, u0, of the elastic lithosphere. These errors, however, are seldom more t h a n 20% for realistic models. An interesting, if academic, result is t h a t if Poisson's ratio equals 1, the m e t h o d used here is exact. Since the true failure law of the lithosphere is not well known, the effect of performing an exact solution for m a n y different cases does not seem worthwhile. T h e model presented here gives sufficiently accurate results for qualitarD re
245
tive comparisons with planetary fracture systems. Since the actual time dependence of the elastic stresses given b y (1), (2), and (3) is unknown, a change proportional to ( 1 - e -tl') was assumed. Thus m o re(t) = m0(1 -- e-t/'D) and Ar(t)/r = (hr/r)f (1 -- e-t/"o), where rD = time constant for despinning, ro = time constant for contraction or expansion, and (hr/r)f = asy m p t o t i c value for change in planet's radius. This time dependence requires t h a t at t = 0, m = m0, while as t--* oo, m--+0. There are three variable parameters in our model calculations. These are given below.
time constant for despinning (4) time constant for contraction or expansion '
2~E(1 + a ) / ( 1 - a ) J ( h r / r ) , B=
(5/12)p[-(1 q- a ) / ( 5 q- a)Jl0m0 asymptotic azimuthal or meridional stress due to contraction or expansion ,
(5)
asymptotic azimuthal despinning stress at ~ = 0 ° (5/12)m0~[(1 -F ~)/(5 -F ~)-]10 5' --
Y asymptotic azimuthal despinning stress at ), = 0 ° (6)
yield stress stress. An initial rotation rate between 10 and 40 hr (1.25 X 10 - 8 _ < m 0 < 0 . 2 ) is assumed, implying t h a t the planet was initially rotating rapidly with a period similar to t h a t of all the other planets except Venus. T h e planet's radius is allowed to change by a maximum of 0.1% or a b o u t aoo/Y = -- ~ (5 -- 3 cos 2~) 5"(1 - e-tl")) 2.5 km. This value is reasonable in light of W/3 5"(1 -- e--"t/'D), (7) the estimation of 1-2 k m of contraction inferred from the arcuate scarps b y Strom a , ~ / Y = ~o(1 -F 9 cos 2h) 5" (1 - e- ' / ' " ) -t-/~ 5" (1 e-a~/'D). (8) et al. (1975). Possible causes of contraction are cooling of the core (Strom et al., 1975) T h e ranges for the parameters /~ and 5" and cooling of the lithosphere (Solomon, are determined b y the values assumed for 1976). Expansion of the planet is considered the initial rotation rate, the fractional in our calculation for completeness. The change in the planet's radius, and the yield yield stress in the lithosphere is given a
Y is the yield stress. T h e previous calculations done b y Melosh are equivalent to a = 1. B is negative for expansion. Using these parameters and the explicit time dependence given above, the equations for ae0 and a ~ become:
-
-
246
PECHMANN AND MELOSH 4.0-
3.0-
2.0-
[ "L,
1.0-
1 oil thrust
2..~.~-thrust; earlier strike-slip 3"~- thrush earlier strike-slip " ~ °ndn°rm°l
stri~ 1
I
and L
normol; eorlier thrust
¢
o5
strike-slip and normal
-1.0-
__]
6 all normal
l J
-2.0 -
-3.0-
-4.0 ,
0.1
,
0.2
,
0.4 0.5
,
i
0.75 1.0 a
,
i
2.0
,
4D 5D
,
7.5 10.0
FIG. 1. Global fracture patterns just after the last latitude to remain elastic has failed. See text for details. ~, = 11. v e r y large range of 10 to 500 bars (Chinnery, 1964; Jefferys, 1952). T h e resulting range for ~ is [/~1< 4.2. T h e m i n i m u m value for is 1.85 ; the m a x i m u m value is ~ . ~ = 7.39 X 104mo . . . .
(9)
where
to decrease the largest stress difference. T h e stress increments are c o m p u t e d elastically, added to the plastic stress field, and the stresses adjusted to satisfy the yield criteria. Thus, if a e e = So, a,~ -- S~, and arr = 0 at time t, such t h a t the maxim u m stress difference is Y, then at time t2 = tz +
(Ar/r)max
5.28 X 10-3
I~1
181
m0 m,~ = 5.28
At:
,~.(t,) = S , +
daoo e
~
at,
(10)
t, At"
(11)
t]
[8] >---0.0264, do" ~ ~,e
= 0.2,
]8[ ~ 0.0264.
Calculations were done for 0.1 < a < 10. After the m a x i m u m stress difference has reached the yield stress and faulting has occurred at all latitudes, we suppose t h a t the lithosphere is in a plastic state. T h e m a x i m u m stress difference is held at the yield stress. At this point the problem becomes highly nonlinear. I n order to extend our model past the time of full failure, we assume t h a t changes in the plastic state of the lithosphere m a y occur a t some latitudes if the stresses a t these latitudes due to the ongoing contraction a n d despinning tend
a , , ( t , ) = S , -}- ~-~
T h e superscript e denotes stresses given b y the elastic expressions (1) a n d (2). If the incremental t e r m s are such t h a t the new m a x i m u m stress difference is greater t h a n Y, abe and a~¢ are set equal to their original plastic values, Se and S¢. If, however, the m a x i m u m stress difference is less t h a n Y, aes and arC are allowed to change in this incremental fashion until a new mode of faulting occurs. We actually only determine whether a change is possible, i.e., whether the incremental t e r m s ever
GLOBAL FRACTURE
N° 6 ~ 90 R
wO°ti ~ 3 0 ° ~
1-s
E
-90° S FZG. 2. T h e f r a c t u r e p a t t e r n w h e n t h e last l a t i t u d e to r e m a i n elastic h a s j u s t failed for a = 0.15, /9 = 3.2, "r = 11. T h i s p a t t e r n is r e p r e s e n t a t i v e of t h o s e in area 2 in Fig. 1. T h e b o u n d a r i e s will be at different l a t i t u d e s for different v a l u e s of t h e p a r a m e t e r s , a, /~, a n d -y. T h e m o v e m e n t on t h e strike-slip f a u l t s is c o n s i s t e n t w i t h E - W compression.
change sign and, hence, act to decrease the largest stress difference. DISCUSSION T h e global fracture patterns at the time at which the last latitude to remain elastic has just failed are shown in Fig. 1 as a function of a and ~ with ~/ = 11, a typical intermediate value. In the discussion the fractures are referred to as faults. However, the global stress field m a y also control the orientation of joints with no displacement. At each labeled a calculations were done at 17 or more values of ~. In Fig. 1 the positions of the boundaries are generally correct, but the details of the boundaries reflect the coarse grid size. In area 1 in Fig. 1 thrust faults due to dominant contraction stresses cover the planet. If the despinning stresses are strong enough to control the orientation of the local principal stress axes, they will cause the preferred strike of the thrust faults to be N-S. In area 2, N E and N W strike-slip faulting occurs
247
PATTERNS
first at low or midlatitudes in a limited latitude range. T h e entire planet, including this region of strike-slip faults, is eventually covered b y thrust faults. An illustration of the fracture pattern in area 2 is shown in Fig. 2. Area 3 is similar to area 2 except that, besides early strike-slip faults, E - W normal faults occur first in limited regions near the poles. In areas 4 and 5, N E and N W trending strike-slip faults occur from equator to midlatitudes and E - W normal faults occur from midlatitudes to the poles. Area 4 differs from area 5 in t h a t thrust faults trending N - S occur in area 4 before the strike-slip and normal faults. These thrust faults are confined to a limited latitude range in low or midlatitudes. Figure 3 is an illustration of the fracture p a t t e r n in area 4. In area 6 normal faults due to dominant expansion cover the planet. T h e y tend E - W if the despinning stresses are strong enough to control their orientation. T h e global fracture patterns for ~/mln and "~ma~ are shown in Figs. 4 and 5. In areas labeled "incomplete" in Fig. 4 there are latitude ranges in which the yield stress is
N 90°
-90" S FIG. 3. T h e f r a c t u r e p a t t e r n w h e n t h e last l a t i t u d e to r e m a i n elastic h a s j u s t failed for a = 0.1, = 2.6, ~ = 11. T h i s p a t t e r n is r e p r e s e n t a t i v e of t h o s e in area 4 in Fig. 1. T h e b o u n d a r i e s will be a t different l a t i t u d e s for different v a l u e s of a,/~, a n d 3'.
248
PECHMANN AND MELOSH 4`0" 1 all thrust
3.0" 2"~rust;
2.0.
earlier strike-slip
eanv ss ] 1,0- 2;.incomplet~rnrush
I
t' incomplete thrust
2'
I
5' incomplete strike-slip and normal
/9 o-
5 strike-slip end normal
-1.0-
-2.0-
I
I
I 6 all normal
-3.0 -
-4.0-
I
0'.40~ 0.75 1.0
0.2
0.1
210
4:0 5.0
715 10.0
a
FIG. 4. Global fracture patterns just after the last latitude to remain elastic has failed. = ~=in = 1.85. I n areas labeled "incomplete" there is a range of latitudes at which the yield stress is never reached.
Global p a t t e r n s including changes which occur after the lithosphere becomes fully plastic are shown for ~ = 11 in Fig. 6. T h e t y p e of faulting which occurs after full failure of the lithosphere is shown superimposed on the areas outlined in Fig. 1. T h e
never reached. F r o m comparison of Figs. 1, 4, and 5 it can be seen t h a t as long as the yield stress is exceeded at all latitudes, the p a r a m e t e r ~ has little effect on these p a t terns except in areas 2 a n d 3. These areas shrink and area 4 enlarges as ~ is increased.
'°1 I
3.0
I all
thrust
4 strike-slip and normal; earlier thrust
2.0
I 1.o-
I
,8 o-
I
5 strike-sllp and normal
I
-1.0 -
6 oil normal
-2.0-
-3.0 -4.0 0.1
0:2
0'4 0.5 0.+5 do cl
210
41O5'`0 7'.51o`0
Fro. 5. Global fracture patterns just after the last latitude to remain elastic has failed.
= ~(~).
GLOBAL FRACTURE PATTERNS
249
4.0-
no change
strike-slip fallowed by normal spread from poles no farther than 48 °
3.0-
no change
4 ~ 2,0-
1.0-
thrust '
~d Ifrom
J
I
normal spread from poles no farther .,~...;on 48°
equator I
tshprr~Sa~"from I e_.quotor
,B o-1.0-
normal spread from poles
change "•'1--no 6
~
I
-2.0 -
]
-3.0 -
no change
no change
-4.0 0.1
o:2
o:4 o:~
o.~5 f.o
i.o
4'.0 5'.0
7:5 to~
a
FIG. 6. Modifications to the global fracture patterns in Fig. 1 which occur after the whole lithosphere has fractured. ~, = 11.
extent in latitude of these new faults is variable since it depends on the model parameters a, B, and ~. In area 5 when ~ > 0 and < 1, younger thrust faults are developed in regions covered by strike-slip and normal faults. These thrust faults could eventually cover the planet if the contraction were large enough. T h e orientation of these thrust faults is N - S if despinning stresses are strong enough to control their orientation. This pattern of older strike-slip and normal faults and younger thrust faults which cover the planet is consistent with M e r c u r y ' s tectonic pattern. However, the systematic shift of the N E and N W lineaments towards N - S in the polar regions is not explained. This shift m a y be due in part to enhancement differences caused by changes in illumination with latitude (Dzurisin, 1978). Since area 4 is identical to area 5 except for thrust faults at low latitudes which are older t h a n the strikeslip faults, the discovery on Mercury of N - S lineaments near the equator older t h a n the N E and N W lineaments would increase the allowable range of f~. These ancient
N - S lineaments may be associated with the poorly developed N - S lineament system of Dzurisin (1978). However, the observed N - S lineaments do not appear to be confined to the equatorial region. CONCLUSION
These theoretical calculations and the new observational evidence of E - W lineaments near the pole have significantly strengthened the case for the importance of tidal despinning in Mercurian tectonics. T h e observed p a t t e r n falls within area 5 with a < 1 and B > 0 in Fig. 4 even if the arcuate scarps have no N - S alignment. If the observed p a t t e r n is, in fact, a consequence of despinning and contraction, some limits on the range of parameters characterizing these processes may be defined. T h e requirement a < 1 implies that the time scale for despinning is shorter t h a n the time scale for contraction. According to Goldreich and Soter (1966) the time required to despin M e r c u r y from a rotation period of 20 hr to the present period of
250
PECHMANN AND MELOSH
56.6 d a y s is 0.2 t o 2.0 B.Y. d e p e n d i n g on t h e v a l u e of t h e p l a n e t a r y d i s s i p a t i o n c o n s t a n t , Q. T h e l a r g e s t v a l u e for/~ in a r e a 5 is a b o u t 1.8, i m p l y i n g t h a t t h e m a x i m u m s t r e s s d u e to c o n t r a c t i o n m u s t b e less t h a n 1.8 t i m e s t h e m a x i m u m s t r e s s d u e to d e s p i n n i n g . H o w e v e r , if a set of a n c i e n t N - S l i n e a m e n t s were d i s c o v e r e d in t h e e q u a t o r i a l region, a m a x i m u m c o n t r a c t i o n a l stress u p to 4 t i m e s t h e m a x i m u m stress d u e to d e s p i n n i n g w o u l d b e a l l o w e d . I n t h i s case t h e y o u n g e r t h r u s t f a u l t s s h o u l d s h o w l i t t l e or no N - S alignment. O u r c a l c u l a t i o n s m a y b e a p p l i e d to o t h e r d e s p u n p l a n e t s since t h e p a r a m e t e r i z e d elastic s t r e s s e q u a t i o n s c o n t a i n no t e r m s explicitly dependent on Mercurian properties. T h e r a n g e s a l l o w e d for t h e t h r e e p a r a m e t e r s m a y b e d i f f e r e n t for o t h e r p l a n e t s , b u t o u r c a l c u l a t i o n s a r e e a s i l y e x t e n d e d to l a r g e r ranges. Specifically, t h e p a t t e r n s we h a v e f o u n d m a y b e useful in i n t e r p r e t i n g s u r f a c e f e a t u r e s of t h e G a l i l e a n s a t e l l i t e s of J u p i t e r .
REFERENCES BURNS, J. A. (1976). Consequences of the tidal slowing of Mercury. Icarus 28, 453-458. CHINNERY, M. A. (1964). The strength of the Earth's crust under horizontal shear stress. J. Geophys. Res. 59, 2085-2089. DZURISIN, D. (1978). The tectonic and volcanic history of Mercury as inferred from studies of scarps, ridges, troughs, and other lineaments. J. Geophys. Res. 83, 4883-4906. GOLDREICH P., AND SOTER. S. (1966). Q in the solar system. Icarus 5, 375-389. JEFFREYS, H. (1952). The Earth, 3rd ed. Cambridge Univ. Press, London/New York. KACI-IANOV, L. M. (1974). Fundamentals of tha Theory of Plasticity, p. 246. MIR, Moscow. MELOSH, J. (1977). Global tectonics of a despun planet. Icarus 31, 221-243. MELOSH, J., AND DZURISIN, D. (1978). Mercurian global tectonics: A consequence of tidal despinning? Icarus 35, 227-236. SOLOMON,S. C. (1976). Some aspects of core formation in Mercury. Icarus 28, 509-522. SOLOMOn, S. C. (1978). On volcanism and thermal tectonics on one-plate planets. Geophys. Res. Lett. 5, 461-463. STROM, R. G., ANDMALIN,M. (1978). Geologic map of the Bach quadrangle (H-15), Mercury. USGS Miscellaneous Investigations Map. In preparation.
STROM, R. G., TRASK, N. J., AND GUEST, J. E. ACKNOWLEDGMENT This work was supported in part by a National Science Foundation Graduate Research Fellowship.
(1975). Tectonism and volcanism on Mercury. J. Geophys. Res. 80, 2378-2507. VENING-MEINESZ, F. A. (1947). Shear patterns of the Earth's crust. Tra~s. Amer. Geophys. Union 28, 1 61.