floyd herbert -- primordial electrical induction heating

8/10/2019 Floyd Herbert -- Primordial Electrical Induction Heating

1/9

ICARUS 78,

4 0 2 - 4 1 0 ( 1 9 8 9 )

Pr im ord ia l E lec t r ica l Indu c t ion Hea t ing o f s te ro ids

F L O Y D

H E R B E R T

L u n a r a n d P l a n e t a r y L a b o r a t o D , , U n i v e r s i t y ~ [ A r i z o n a , T u c s o n , A r i z o n a 8 .5 72 1

R e c e i v e d M a y 2 0 , 1 9 88 ; r e v i s e d A u g u s t I l , 1 9 88

R e c e n t l y o b s e r v e d s y s t e m a t i c t r en d s o f a s t e r o id a l c o m p o s i t i o n w i t h r e sp e c t t o s i z e a n d

h e l i o c e n t r i c d i s t a n c e a r e i n t e r p r e t e d a c c o r d i n g t o t h e h y p o t h e s i s t h a t p r i m o r d i a l a s te r o i -

d a l t h e r m a l e v o l u t i o n w a s d r i v e n b y e l e c t r o m a g n e t i c i n d u c t i o n i n a d e n s e p r e - m a i n -

s e q u e n c e s o l a r w i n d . I t i s f o u n d t h a t t h e s p a t i a l d i s t r i b u t i o n o f i n d u c t i o n h e a t i n g c o r r e -

s p o n d s w e l l w i t h t h a t i n f e r r e d f r o m t h e p r e s e n t - d a y d i s t r i b u t i o n o f a s t e r o i d a l

c o m p o s i t i o n a l t y p e s w h e n r e a s o n a b l e p r o t o - S o l a r - S y s t e m c h a r a c t e r i s t i c s a r e a s s u m e d .

F r o m t h is m a t c h i s d e d u c e d a r o u g h c o n s t r a in t o n t h e i n d u c t i o n h e a t i n g e p o c h ' s d u r a t i o n

( ~ 1 0 s y e a r s ) a n d t h e t e m p e r a t u r e - d e p e n d e n t e l e c tr i c a l c o n d u c t i v it y o f t h e p r o t o a st e r -

o i da l m a ter i a l ( a b o ut th e s a m e a s tha t o f the C 2 m ete o r i t e M ur ch es o n) . ~ 1989 Academ ic

Press , Inc.

1 . I N T R O D U C T I O N

I t h a s b e e n k n o w n f o r s o m e t i m e t h a t

t h e r e a r e s y s t e m a t i c t r e n d s i n t h e s p a t i a l

d i s t r i b u t io n o f a s t e r o id a l t a x o n o m i c t y p e s ,

w h i c h a r e u s u a l l y i n t e r p r e t e d a s c o m p o s i -

t i o n a l v a r i a t i o n s ( C h a p m a n e t a l 1975,

Z e l l n e r a n d B o w e l l 1 9 77 , B o w e l l

e t a l

1 97 8) . T h e v i g o r o u s p r o g r a m s o f a s t e ro i d

o b s e r v a t i o n s o v e r t h e p as t d e c a d e h a v e r e-

s u l t e d i n t h e s i g n i fi c a nt i m p r o v e m e n t o f t h e

c h a r a c t e r i z a t i o n o f a s t e r o id a l c o m p o s i t i o n

a s a f u n c t i o n o f si z e a n d d i s t a n c e f r o m t h e

s u n ( e. g . , G r a d i e a n d T e d e s c o 1 98 2, T h o -

l e n , 1 9 8 4 , Z e l l n e r

e t a l

1985 , Be l l 1986 ,

M a t s o n 1 98 6). F i g u r e 1 s h o w s a n a p p r o x i -

m a t i o n o f th e d i s t r i b u t i o n o f a s t e r o i d a l

c o m p o s i t i o n t y p e s w i t h r e s p e c t t o h e li o c e n -

t r ic d i s t a n c e o f t h e m a j o r a s t e r o i d a l t a x o -

n o m i c t y p e s i n t h e m a i n b e lt . T h e s e c u r v e s

a re h a n d - s m o o t h e d a p p r o x i m a t i o n s o f u n-

p u b l i s h e d p r e l i m i n a r y b i a s - c o r r e c t e d o b s e r -

v a t i o n al d a t a (C . R . C h a p m a n , p r i v a t e c o m -

m u n i c a t i o n ) . O f t h o s e t y p e s s h o w n t h e

m o s t p r i m i t iv e a r e p r o b a b l y c l a s s e s P a n d

D , w h i c h a r e b e l i e v e d t o b e c a r b o n a c e o u s

b u t u n s a m p l e d b y a n y w e l l - k n o w n c l a s s o f

m e t e o r i t e . T y p e C a n d s i m i la r t y p e s a r e

g r o u p e d t o g e t h e r a s an i n t e r m e d i a t e c l a s s

0019-1035 /89 3 . 00

Copyright et') 1989 by Academ ic P ress, Inc.

All rights of reproduction in any form reserved.

a n d a r e b e l i e v e d t o b e s i m i l a r t o c a r b o n a -

c e o u s c h o n d r i t i c m e t e o r i t e s , w h i c h in t u r n

a p p e a r t o b e t h e r e su l t o f a q u e o u s a l t e r a t io n

a n d t h u s i n t e r m e d i a t e i n t h e r m a l p r o c e s s i n g

a s w e ll . B o t h t h e S a n d M t y p e s a r e b e -

l i e v e d t o b e i g n e o u s d i f f e r e n t i a t e s ( G a f f e y

1 9 8 4 ) ; t h e M t y p e s a p p e a r t o b e m e t a l l i c

i ro n a n d t h e S t y p e s a p p e a r t o e x p o s e b o t h

i r o n a n d i g n e o u s r o c k . T h e E t y p e s a r e a

m i n o r c l a s s , b u t m a y w e l l b e i g n e o u s a l s o .

T h i s o u t w a r d s u c c e s s i o n o f t y p e s t h u s p r e -

s u m a b l y r e p r e s e n t s a s e q u e n c e o f m e t a -

m o r p h i c t y p e s f r o m i g n e o u s l y d i f f e r e n t i -

a t e d t h r o u g h a q u e o u s l y m e t a m o r p h o s e d t o

p r i m i t i v e . T h e m o s t n a t u r a l i n f e r e n c e t o b e

d r a w n f r o m t h is i s t h a t a h i g h ly h e l i o c e n -

t r i c - d i s t a n c e - d e p e n d e n t h e a t s o u r c e w a s re -

s p o n s i b l e f o r m e t a m o r p h o s i n g t h e a s t e r o id

be l t (Be l l 1986).

F i g u r e 2 ( a l s o d e r i v e d f r o m u n p u b l i s h e d

p r e l i m i n a r y b i a s - c o r r e c t e d d a t a f r o m C . R ,

C h a p m a n ) s h o w s t h e d i s t ri b u t io n o f g r o u p s

o f a s t e r o i d a l c o m p o s i t i o n a l t y p e s w i t h r e-

s p e c t t o s i z e i n t h e m a i n b e l t ( o u t t o z o n e

I l i a ) . A s i n F ig . I t h e r e i s a n e v i d e n t t r e n d :

h e r e t h e i g n e o u s c l a s s e s a p p e a r t o c l u s t e r at

i n t e r m e d i a t e s i z e s . W h e t h e r t h e a p p a r e n t

r e d u c t i o n i n S o b j e c t s a t t h e s m a l l e s t d i a m -

e t e r s i s c o n t i n u e d b e l o w t h e l im i t o f t h e

4 2


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404 FLOYD HERBERT

rates as of the time of solidification of the

feldspars investigated. Interest in the 26A1

heating hypothesis revived when Le e e t a l .

(1977) and Hutcheon e t a l . 1 9 7 8 ) found evi-

dence implying primordial 26AI/AI values of

about 50 x 10 -6 in Ca/Al-rich inclusions

(CAls) from the C3 (i.e., unmelted) me-

teorite Allende. This concentration when

present in bulk material is easily enough to

melt an asteroid larger than about 10 km,

although the average inferred primordial

Allende 26A1 conce ntrat ion is orde rs of

magnitude less. Since then, however, more

evidence for thermally significant primor-

dial 26AI/AI has only been found in one mi-

croscopic CAI in one sample of the H3

chondrite Dhajala (Hinton and Bischoff

1984), despite searches in other parts of AI-

lende (Lee e t a l . 1979, Hinton and Bischoff

1984), Dhajala, or other H3 or LL3 chon-

drites (Hinton and Bischoff 1984).

Moreover the CAIs are not representa-

tive of the fabric of chondr ites, being inclu-

sions which appear to have formed inde-

pendently and incorporated later when the

chondritic parent bodies formed. More-

over, the bulk material of the ordinary and

carbonaceous chondrites, including the

only two with the 26Al-daughter CAIs, is

unlikely ever to have approached the melt-

ing point. By contrast, a recent search for

indications of primordial 26AI in a meteori te

(a mesosiderite) which actually w a s the

result of melting has also been negative

(Papanastassiou

e t a l .

1987). In addition to

the lack of visible role for 26A1 in the

thermal metamorphosis of any known me-

teoritic sample, the variation of composi-

tional type with respect to size and helio-

centric distance discussed earlier is hard to

explain as a result of a heat source as non-

selective

as 26Al.

2. ELECTRICAL INDUCTION HEATING

Another of the possible heat sources re-

sponsible for the primordial melt events in-

ferred from asteroidal spectra is electrical

induction heating driven by a hypothesized

dense solar-wind-like plasma outflow (often

loosely termed a T Tauri solar wind )

from the pre-main-sequence Sun. This in-

duction phenomenon, analogous to the

induced currents driven through Io's

ionosphere by Jupiter's corotating magne-

tospheric plasma, could have efficiently

coupled the kinetic energy flux of the hypo-

thetical primordial solar wind to the deep

interiors of proto-asteroids. Although the

assumed solar wind energy flux is small

(comparable to that of present-day sun-

light), its coupling to the deep interior

would have generated large interior temper-

atures in order to develop the thermal gra-

dient necessary to drive enough outwardly

diffusing heat. In this respec t the phenome-

non is analogous to the greenhouse effect.

By contrast with the case of

26AI,

there

are size- and distance-dependent variations

in the efficiency of inductive heating

(Sonett e t a l . 1968, 1970, Herbert and

Sonett 1978, 1979, 1980, Lebofsky e t a l .

1988). Naturally, induced fields are strong-

est near the Sun, and the temperature de-

pendence of the electrical conductivity of

the asteroidal material allows the surface

temperature of the body (controlled largely

by insolation and thus by heliocentric dis-

tance) to modulate the induced current.

Moreover, the magnetic deflection of

plasma by induced currents also reduces

heating in larger body sizes, while of cour se

thermal conduction limits the temperature

rise in smaller bodies. Conseq uently, in any

protosolar plasma environment there is

some body size at which maximum temper-

ature increase occurs (Herbert and Sonett

1978, 1979, 1980). This size and distance

dependence of heating was originally sug-

gested (Herbert and Sonett 1978, 1979,

1980) as being the cause of the difference

between the apparently igneous surface of

4 Vesta, usually believed to be the eucrite

parent body (McCord and Gaffey 1974,

Drake 1979), and the unmelted surfaces of

2 Pallas and 1 Ceres (Chapman e t a l . 1974,

Lebofsky 1978, Larson and Veeder 1979).

The present work explores this sugges-

tion further, in application to the now


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I N D U C T I O N H E A T I N G O F A S T E R O I D S 4 05

g r e a t l y e x p a n d e d o b s e r v a t i o n a l d a t a b a s e

s u m m a r i z e d i n F i g s . 1 a n d 2 . T h u s t h e in -

f e r r e d m a x i m u m p r i m o r d i a l h e a t i n p u t a t i n-

t e r m e d i a t e a s t e r o i d a l s i z e s a n d a t t h e i n n e r

e d g e o f t h e b e l t i s c o m p a r e d w i t h t h e r e s u l t s

o f m o d e l c a l c u l a t i o n s o f i n d u c t i o n h e a t i n g .

T h e e x a c t e l e c t r i c a l h e a t i n g p a t t e r n d e -

p e n d s o n th e c o n d i t i o n s a s s u m e d , h o w e v e r .

F o r e x a m p l e , i f t h e s o l a r m a s s l o s t is s m a ll

b u t c o n f i n e d t o a s h o r t t im e i n t e r v a l , s u c h

as 10 4 y e a r s , t h e s i z e a t w h i c h m a x i m u m

h e a t i n g o c c u r s i s a r o u n d 1 0 k m . T h i s c o n -

t r a s t s w i t h t h e 1 0 6 - y e ar i n t e r v a l c a s e w i t h

l a r g e s o l a r m a s s l o s s a s s u m e d b y H e r b e r t

a n d S o n e t t ( 1 9 7 8 , 1 9 7 9 , 1 9 8 0 ) , a m o d e l

w h i c h y i e ld e d a m a x i m u m t e m p e r a t u r e a t

a n a s t e r o i d d i a m e t e r o f a b o u t 2 0 0 k m .

M o d e l s w h i c h a s s u m e t h e a s s e m b l y o f

l a rg e r a s t e r o id s f r o m a l r e a d y - m e l t e d p r o t o -

a s t e r o i d s l e a d to a n o t h e r i n t e r e s t i n g s u it e o f

p o s s i b il i ti e s a n a l o g o u s t o s c e n a r i o s i n v e s t i -

g a t e d b y W o o d ( 1 9 7 9 ) . I n p a r t i c u l a r t h i s

p o s s i b i l i t y m a y b e r e l e v a n t t o t h e c a s e o f

t h e p a r e n t b o d y o f t h e b a s al t i c a c h o n d r i t e s

w h e r e d i f f e r e n t m e l t re g i o n s o c c u r r i n g a t

w i d e l y s e p a r a t e d t i m e s ( u p t o 1 08 y e a r s

a p a r t ) h a v e b e e n i n f e r r e d b y M i t t l e f e h l d t

( 197 9) .

I n d u c t i o n h e a t i n g i s q u i t e d e p e n d e n t o n

t h e e l e c t r i c a l c o n d u c t i v i t y o f t h e m a t e r i a l t o

b e h e a t e d . M a n y c a n d i d a t e m a t e r ia l s ' c o n -

d u c t i v i ti e s h a v e b e e n m e a s u r e d , b u t s in c e

t h e n a t u r e o f t h e p r i m o r d i a l m i x o f m a t e -

r ia l s t h a t o r i g i n a l l y a c c r e t e d i n t o t h e a s t e r -

o i d s i s u n k n o w n , i t s e l e c t r i c a l c h a r a c t e r i s -

t i c s a r e s t i l l m y s t e r i o u s . W h a t e v e r t h e

p r o t o a s t e r o i d a l c o m p o s i t i o n w a s , i t i s l i k e ly

t o h a v e c o n s i s t e d o f a l a rg e v a r i e t y o f c o m -

p o u n d s , i n c l u d i n g t r a c e s o f u s u a l l y v o l a t i l e

i o n ic c o m p o u n d s , a s i tu a t io n t h a t f a v o r s

c o n d u c t i v i t i e s g r e a t e r t h a n t h a t t y p i c a l f o r

i g n e o u s m i n e r a l s a n d t h u s f a v o r s t h e l i k e -

l i h o o d o f s i g n i f i c a n t e l e c t r i c a l c u r r e n t s .

E x a m p l e s o f c a n d i d a t e m a t e r ia l s w h o s e

c o n d u c t i v i t i e s h a v e b e e n m e a s u r e d a r e c a r -

b o n a c e o u s c h o n d r i t e s ( S c h w e r e r e t a l .

1 9 7 1 , B r e c h e r

e t a l .

1 9 7 5 , D u b a

e t a l .

197 3 ,

D u b a

e t a l .

1 9 74 , D u b a a n d B o l a n d 1 98 4)

a n d o r d i n a r y c h o n d r i t e s ( E v e r n d e n a n d

V e r h o o g e n 1 9 56 , S c h w e r e r

e t a l .

1971,

B r e c h e r 1 9 7 3 , B r e c h e r

e t a l .

1 9 7 5 ) . H o w -

e v e r , i n a s m u c h a s th e c a r b o n a c e o u s c h o n -

d r i t e s a r e b e l i e v e d t o h a v e b e e n s u b j e c t e d

t o a q u e o u s a l t e ra t i on , e v e n t h e s e m o s t

p r i m i t i v e m e t e o r i t i c s a m p l e s a r e n o t p r i m i -

t i r e e n o u g h ( a l t h o u g h t h e y a r e t h e m o s t

s u i t a b l e t h a t h a v e h a d e l e c t r i c a l c o n d u c t i v -

i ty m e a s u r e m e n t s p e r f o r m e d ) . T h e s e p o s -

s ib l e a n a l o g s o r s u r v i v i n g s a m p l e s o f pr i-

m o r d i a l m a t e r i a l h a v e r o o m t e m p e r a t u r e

c o n d u c t i v i t i e s v a r y i n g f r o m 1 0 - t to 1 S / m .

E l e c t r i c a l c o n d u c t i v i t y i s s t r o n g l y d e p e n -

d e n t i n m o s t c a s e s o n t e m p e r a t u r e , o x y g e n

f u g a c i t y , a n d t h e p r e s e n c e o f i m p u r i ti e s

s u c h a s w a t e r o r c a r b o n ( P a r k h o m e n k o

1 9 67 , D u b a a n d B o l a n d 1 98 4). T h e p r e s e n c e

o f a d e e p ( w i t h th i c k n e s s m o r e t h a n a f e w

p e r c e n t o f b o d y s iz e ) re g o l it h c a n f u r t h e r

c o m p l i c a t e t h e s c e n a r i o ; l i m i t a ti o n o f e l e c -

t r i c a l c u r r e n t b y t h e s e r i e s r e s i s t a n c e a s s o -

c i a t e d w i t h p o r o s i ty r e d u c e s h e a t i n g b u t th e

t h e r m a l b l a n k e t i n g e f f e c t c a n h e l p r e t a i n

h e a t .

M o s t w o r k o n p r i m o r d i a l i n d u c t iv e h e a t -

i n g ( S o n e t t e t a l . 1 9 6 8 , 1 9 7 0 , H e r b e r t a n d

S o n e t t 1 9 7 8 , 1 9 79 , 1 98 0 ) h a s a s s u m e d a so -

l a r w i n d p l a s m a f l u x c o m p a r a b l e t o t h e

l a r g e H f l u x e s ( u p t o n e a r l y 1 0 - 6 M G / y e a r )

s e e n i n T T a u r i s ta r s ( K u h i 1 9 64 ). E v e n

l a r g e r o u t f l o w s o c c u r a t F U O r i o n i s s t a r s ,

w h i c h a p p e a r t o b e T T a u r i s t a r s i n a n o u t -

b u r s t p h a s e ( H e r b i g 1 9 7 7 ) . V e r y m a s s i v e

o u t f lo w s a p p e a r t o o c c u r f r e q u e n t l y in w h a t

i s b e l i e v e d t o b e t h e p r e - T - T a u r i s t a g e o f

s o l a r - m a s s s t a r s , i n v o l v i n g l a r g e e l e c t r o n

d e n s i t i e s (1 08 t o 1 0 II c m - 3 ) , m a s s l o s s e s i n

t h e r a n g e 10 -7 to 10 -6

M o / y e a r

a n d o u t f l o w

v e l o c i t i e s a r o u n d 1 00 k m / s e c ( L a d a 1 9 85 ,

1 9 8 7 ) . E d w a r d s

e t a l .

(1 9 80 ) h a v e o b s e r v e d

S + a n d m a s s l o s s e s o f 10 -9 t o 10 -7 M G / y e a r

w i t h v e l o c i t i e s u p t o 2 0 0 k m / s e c i n s p e c t r a

o f T T a u r i s t a r s; t h e y i n fe r t h e p r e s e n c e o f

o p a q u e p r o t o p l a n e t a r y a c c r e t i o n d i s k s .

I f su c h f lu x e s c o n t i n u e o v e r t i m e s c a l e s

o f o r d e r 106 y e a r s , a l a rg e f r a c t i o n o f a s o l a r

m a s s c a n b e l o s t . F o r t h e S u n t o h a v e l o s t

m o r e t h a n a b o u t h a l f o f it s o ri g in a l m a s s is

u n l i k e l y a n d t h e l i m i t m a y w e l l b e m o r e


5/9

4 06 F L O Y D H E R B E R T

s t r i n g e n t t h a n t h a t ( W e i d e n s c h i l l i n g 1 97 8) ,

b u t i n t e n s e s o l a r w i n d f l u x e s s u c h a s t h o s e

m e n t i o n e d a b o v e b u t o f s h o r t e r d u r a ti o n

a r e n o t r u l e d o u t . S h o r t e r t i m e s c a l e s p r e f -

e r e n t i a l l y f a v o r i n d u c t i o n h e a t i n g i n s m a l l e r

b o d i e s . T h e a c t u a l o u t f l o w t i m e i s n o t w e l l

k n o w n ; s o m e t h e o r e ti c a l t r e a t m e n t s o f t h e

s o u r c e o f t h e f l o w (e . g . , S h u e t a l . 1987)

s u g g e s t t h a t t h e f l o w c o u l d b e d r i v e n b y i n -

f l ow i n g p r o t o p l a n e t a r y d i s k m a t e r i a l an d

t h u s m i g h t b e e p i s o d i c . I n su c h a c a s e t h e r e

c o u l d b e s i g n if i c an t e n e r g y f lu x a t a n u m b e r

o f t im e s c a l e s .

3. MODEL DESCRIPTION

T h e t i m e - i n d e p e n d e n t t r a n s v e r s e m a g -

n e t i c ( u n i p o l ar ) i n d u c t i o n m o d e l u s e d in t h e

p r e s e n t w o r k h a s b e e n d e s c r i b e d m o s t c o m -

p l e te l y b y S o n e t t e t a l . ( 1 9 7 0 ) . D e p e n d e n c e

o f i n d u c t i o n o n f l u c t u a t i o n s i n t h e f i el d s

( s u c h a s th e t r a n s v e r s e e l e c t r i c m o d e a n d

s k i n d e p t h e f f e c t s ) a r e m i n i m a l i n b o d i e s a s

s m a l l a s a s t e r o i d s i f m o s t o f th e p o w e r l ie s

b e l o w a f e w m H z i n f r e q u e n c y a n d t h e c o n -

d u c t i v i t y i s l e s s t h a n 1 0 -~ S / m . T h u s V - J -

0 o r V 2q5 - - - V l o g o - V ~b i s s o l v e d f o r

a s p h e r i c a l ly s y m m e t r i c a s t e r o i d a l m o d e l

w i t h a c o n s t a n t e l e c t r i c f ie ld a t th e b o u n d -

a r y ( l o n g w a v e l e n g t h l i m i t) . H e r e J is t h e

c u r r e n t d e n s i t y , ~b i s t h e e l e c t r i c a l p o t e n t i a l ,

a n d cr i s t h e e l e c t r ic a l c o n d u c t i v i t y , w h i c h

v a r i e s w i t h t e m p e r a t u r e a n d c o m p o s i t i o n .

T h e m o t i o n a l e l e c t r i c f ie ld is g i v e n b y t h e

p r o d u c t o f t h e p r im o r d i a l s o l a r w i n d s p e e d

( a s s u m e d i n d e p e n d e n t o f h e l i o ce n t r ic d i s -

t a n c e d ) a n d t r a n s v e r s e m a g n e t i c fi el d ( a s -

s u m e d ~ d i); b o t h a s s u m p t i o n s a r e sa t is -

f ie d f o r t h e p r e s e n t - d a y s o l a r w i n d . T h e

e l e c t r i c f ie l d a t t h e a s t e r o i d ' s s u r f a c e i s a s -

s u m e d t o b e d i m i n is h e d f r o m t h e v a l u e o f

t h e m o t i o n a i f ie ld b y t h e m a g n e t i c f l u x d e -

f l e c ti o n f a c t o r d e r i v e d b y S o n e t t e t a l .

( 19 7 0) . T h e j o u l e h e a t i n g r a t e i s g i v e n b y

lV, l 2 T h e s o l a r w i n d p l a s m a ( H + e ) d e n -

s i t y i s a s s u m e d p r o p o r t i o n a l b o t h t o d ~-

a n d t h e n e t s o l a r m a s s l o s s d i v i d e d b y t h e

i n d u c t i o n e p o c h d u r a t i o n . T h e m a s s lo s s is

a s s u m e d i s o t r o p i c ; s i n c e t h e r e l e v a n t p a -

r a m e t e r is t h e s o l a r w i n d d e n s i t y , i f t h e

m a s s l o ss i s p r e d o m i n a n t l y e q u a t o r i a l t h e

n e t m a s s l o ss is c o r r e s p o n d i n g l y r e d u c e d .

T h e e l e c t r i c a l c o n d u c t i v i t y f u n c t i o n o f

t h e p r o t o - a s t e r o i d a l b u l k m a t e r i a l i s a n i m -

p o r t a n t u n k n o w n q u a n t i t y . F o r th e p u r-

p o s e s o f m o d e l i n g a fu n c t i o n o f th e f o r m

m i n { o - 0e~7, 0.1 S / m } i s u s e d . A t h i g h t e m -

p e r a t u r e s t h e e l e c t ri c a l c o n d u c t i v i t y o f r e a l

c a n d i d a t e m a t e r i a l s o f t e n s t o p s i n c r e a s i n g

w i th t e m p e r a t u r e , a s t h o u g h a c o n d u c t i v e

m i n e r a l w e r e b e i n g v o l a t i z e d o r d e s t r o y e d

b y c h e m i c a l r e a c t i o n s ( c f. D u b a a n d B o l a n d

1984) .

A l t h o u g h a m o r e f a m i l ia r t e m p e r a t u r e d e -

p e n d e n c e f o r t h e e le c t r i c a l c o n d u c t i v i t y o f

r o c k y s e m i c o n d u c t o r s is e -E /k r, w it h E t h e

a c t i v a t i o n e n e r g y o f t h e c o n d u c t i o n m o d e ,

m o s t r e a l m i n e ra l s h a v e s e v e r a l c o n d u c t i o n

m o d e s w i t h d i f f e r i n g a c t i v a t i o n e n e r g i e s

w h i c h o f t e n p r o d u c e a n o v e r a l l c o n d u c t i v -

i t y p r o f i l e w i t h 9 l o g

o / O T o n l y

w e a k l y v a r i -

a b l e a n d t h u s a p p r o x i m a t a b l e b y a c o n s t a n t

( ~ a ) a s a s s u m e d h e r e . E x a m p l e s o f t hi s

b e h a v i o r a r e s h o w n i n F ig . 3, in w h i c h m e a -

s u r e m e n t s o f o -( T) m a d e b y B r e c h e r

e t a l .

( 1 97 5 ) a n d D u b a a n d B o l a n d ( 1 9 8 4 ) a r e

c o m p a r e d w i th t h e c o n s t a n t 0 l og

( r / O T

f o r m . G i v e n t h a t t h e n a t u r e o f t h e t r u e p r i-

m o r d i a l c o n d u c t i v i t y f u n c t i o n i s s o p o o r l y

k n o w n , p a r a m e t e r i z a t i o n o f t h e m o d e l c o n -

d u c t i v i t y f u n c t i o n b y m o r e t h a n t w o v a r i -

a b l e s s e e m s u n j u s ti f ie d a t t h e p r e s e n t l e v el

o f a p p r o x i m a t i o n .

E a c h c a s e c o n s i d e r e d is a c t u a l l y a g r id o f

a s t e r o i d t h e r m a l h i s t o r y m o d e l s , w i t h a l l

p a r a m e t e r s h e l d c o n s t a n t e x c e p t h e l i o c e n -

t r ic d i s t a n c e a n d a s t e r o i d a l s i z e. T h e r e s u l t s

w il l b e s u m m a r i z e d b y a c o n t o u r p l ot o f t h e

m a x i m u m i n te r io r t e m p e r a t u r e r e a c h e d i n

e a c h a s t e r o i d m o d e l p l o t t e d a s a f u n c t i o n o f

t h e s i z e a n d d i s t a n c e o f t h a t a s t e r o i d .

4. RESULTS

I n o r d e r t o e x p l o r e t h e c h a r a c t e r i s t i c s o f

t h e i n d u c t i o n h e a t i n g m o d e l a n d t o f in d pa -

r a m e t e r r a n g e s t h a t r e su l t in e v o l u t i o n a r y

m o d e l s th a t h a v e o u t c o m e s c o n s i s t e n t w i th

o b s e r v a t i o n s o f a s t e r o id a l c o m p o s i t i o n a l

t y p e s a s s u m m a r i z e d in F i g s. 1 a n d 2 , a


6/9

IN D U C T IO N H E A T IN G O F A S T E R O ID S 4 0 7

.e,

o _=

_

De~ees (K)

t000 600 400 370 200

,, , , J i

0 ~ 3b0 c 160 c b*c

/

-t - s_ k .

-2 ~ ,-3e'~sT

~....10- 4cO ~O3T

-6 lO-=e ~ . . . . ~ . . . , . . . . ~ -

tO00/T (K t

FIG. 3. Electrical conductivity as a function of tem-

perature. The measurements of Brecher

e t a l

(1975) at

low temperature and Duba and Boland (1984) at high

temperature of the conductivities of samples of the

meteorites Allende and Murcheson are shown. The

solid-dot curves are for Allende and the open-circle

curves are for Murcheson. Since different samples

were used, the different determinations are not neces-

sarily consistent (e.g., see the curves for Allende).

Also shown are some of the analytic conductivity

functions used in the models of the present work la-

beled by their analytic form.

TABLE II

PARAMETERS THAT VARY BETW EEN MODELS

Figure number 4a 4b 4c 4d 5

Duration of induction

epoch (year) 107 107 104 104 10~

Solar mass loss (Mu) 0.30 0.03 0. 02 5 0.03 0.10

a (K -~) 0.025 0.003 0.0 25 0.003 0.025

~r0 (S/m) 10 7 10-5 10 -s 10 ~ 10 x

h e l i o c e n t r i c d i s t a n c e , a n d a r e o t p r o f i le s o f

t e m p e r a t u r e v s r a d i u s w i t h i n a s i n g l e b o d y .

T h e h i g h e s t c o n t o u r l e v e l i n e a c h c a s e a p -

p r o x i m a t e l y i n d i c a t e s t h e s i z e a n d d i s t a n c e

r a n g e i n w h i c h s o m e p o r t i o n ( g e n e r a l l y

m o s t o f t h e i n t e ri o r ) o f a m o d e l a s t e r o i d

a t t a in s m e l t i n g t e m p e r a t u r e . F i g u r e s 4 a a n d

4 b w e r e c o m p u t e d a s s u m i n g a 1 0 7-y e ar d u -

r a t i o n f o r t h e i n d u c t i o n h e a t i n g e r a , w h i l e

F i g s . 4 c a n d 4 d a r e f o r a 1 0 4 - y e ar d u r a t i o n .

F i g u r e s 4 a a n d 4 c r e s u l t e d f r o m ~ = 0 . 0 2 5

( e l e c t r i c a l c o n d u c t i v i t y h i g h l y t e m p e r a t u r e

d e p e n d e n t ) w h i l e F i g s . 4 b a n d 4 d a r e f o r

= 0 .0 0 3 ( w e a k l y d e p e n d e n t ) .

T h e w e a k l y t e m p e r a t u r e - d e p e n d e n t

c a s e s s h o w l it tl e v a r i a t i o n w i t h h e l i o c e n t r i c

d i s t a n c e , w h i l e t h e s t r o n g l y t e m p e r a t u r e -

d e p e n d e n t c a s e s h a v e s i g n i f i c a n t m e l t i n g

n u m b e r o f m o d e l s w e r e c o m p u t e d . T h e s e

m o d e l s i n v o l v e th e a s s u m p t i o n o f p a r a m e -

t e r s w h i c h a r e s u m m a r i z e d in T a b l e s 1 a n d

I I .

F i g u r e 4 c o n s i s t s o f fo u r c o n t o u r p l o ts o f

m a x i m u m t e m p e r a t u r e (C ) r e a c h e d b y

e a c h o f a g r id o f a s t e r o i d m o d e l s s p a n n in g

t h e i n d i c a t e d r a n g e s o f d i a m e t e r a n d h e l io -

c e n t r i c d i s t a n c e . N o t e t h a t e a c h c o n t o u r

p l o t s u m m a r i z e s t h e r e s u l t s o f a s u i te o f

m o d e l a s t e r o id s o f d if fe r in g d i a m e t e r a n d

TABLE

PARAMETERS USED IN ALL MODELS

Asteroid density

Specific heat

Solar wind speed

Solar surface magnetic field

Solar rotational velocity

Insolation tempera ture at I AU

2500 kg/ m 3

733 J/K/kg

400 km/sec

100G

10 -4 ra d/s ec

0oc

~- 5I~I ~ 1

iNNf 1

.2 ~.5 2.6 3.~ 31 2.2 2.8 3

Heliocentric Distance (A.U.)

FIG. 4. Contour plots of maximum temperature at-

tained by each asteroid within each of four suites of

models. The effects of induction epoch duration and

tempera ture depe ndence of electrical conductivity are

investigated by variation of these parameters. Figures

4a and 4b are for l07 years and Figs. 4c and 4d are for

104 years . Figure s 4a and 4c were ge nerate d u sing ~ =

0.025OK 1 and Fig s. 4b and 4d ar e for 0.003K ~.


7/9

4 0 8 F L O Y D H E R B E R T

500 ~ ~ ~

20O

~ 5o

N

6

2

22 2.5 ~B 3 1

4

HefiocentricOislonce A.U)

F~G. 5. Contour plots of maximum temperature at-

tained by each asteroid within a suite of models repre-

senting an app rox imat e fit to the data in Figs. I and 2.

c o n f i n e d t o t h e i n n e r e d g e o f t h e b e l t . G i v e n

t h e o b s e r v e d d i s t r i b u t i o n o f a s t e r o i d a l

t y p e s ( c f . F i g . 1 ) , t h e l a t t e r c a s e s s e e m

m o r e r e a l i s ti c . T h e v a l u e s o f o -, a n d o~ in

t h e s e c a s e s c o r r e s p o n d r o u g h l y t o v a l u e s

m e a s u r e d b y D u b a a n d B o l a n d ( 19 8 4) f o r

t he C 2 c a r b o n a c e o u s c h o n d r i t e M u r c h e s o n .

A l t h o u g h a s m e n t i o n e d a b o v e t h e c a r b o n a -

c e o u s c h o n d r i t e s a r e p r o b a b l y n o t u n a l -

t e r e d s a m p l e s o f th e o r i g in a l a s t e r o i d a l b u l k

m a t e r i a l , t h e s e m e t e o r i t e s a r e p r o b a b l y a s

c l o s e t o t h o s e m a t e r i a l s a s c a n p r e s e n t l y b e

o b t a i n e d .

T h e d u r a t i o n o f t h e i n d u c t i o n h e a t i n g e p i-

s o d e a f f e c t s t h e s i z e r a n g e o f a s t e r o i d

m o d e l s i n w h i c h m e l t i n g o c c u r s . A 1 0 4 -y e ar

i n t e r v a l i s m o s t l i k e l y t o m e l t a s t e r o i d s w i t h

d i a m e t e r s b e t w e e n I a n d 2 0 t o 5 0 k i n, w h i l e

a 1 0 Y -y ea r d u r a t i o n f a v o r s d i a m e t e r s b e -

t w e e n 1 0 a n d 2 0 0 to 50 0 k m . A c o m p a r i s o n

w i t h F i g . 2 s u g g e s t s t h a t p e r h a p s a n i n t e r -

m e d i a t e t i m e i n t e r v a l , s u c h a s 105 y e a r s i s

o p t i m u m . C o m p a r i s o n w i th F ig . 1 i n d i c a t e s

t h a t l a r g e a i s p r e f e r r e d . T h u s a fi n al m o d e l

( F i g . 5 ) s h o w s a c a s e w i t h t h e s e p r e v i -

s i o n a l l y d e t e r m i n e d p a r a m e t e r s .

5. DISCUSSION

A s a r e s u lt o f th e m o d e l i n g w o r k in t h e

p r e v i o u s s e c t io n s , a n u m b e r o f n e w t h in g s

h a v e b e e n l e a rn e d . O n e o f t h e m o s t i m p o r -

t a n t i s t h e l o w v a l u e o f A M ~ ( 0 .0 3 t o 0 .1

M


8/9

INDUCTION HEATING OF ASTEROIDS 409

obse rvat iona l data) indicate that the pre-

ferr ed size range for melt i ng is less than 100

km, is accret ion from bodies which were

nearly or s t i l l par t ly mol ten. The t ime con-

stant for loss of heat fr om a bod y with a 10-

km solid crust is on the order of 106 years

and for a body wi th a 30-km crust i s about

107 years . Con seq uen t ly i f Vesta accre ted

in 106 to 107 years from 100-km bodies

which were s ti ll par t ly mol te n, the resul t i ng

body would l ikely show a complex his tory

of part ial melt events. With a 100-km or

thicker crus t the in ter ior could s tay mol t en

for up to 108 years, while at the same t ime

material could have accreted at the surface

that could show evid ence for comple te so-

l id i f icat ion imme diate ly af ter the e nding of

the solar wind induc t ion heat ing epoch. Ac-

cret ion of such bodies in to a larger as teroid

would p roduce a paren t body fo r meteor i t es

showing a very complex thermal his tory .

This scenario is one which could poss ibly

fit with the intri guing result of Mitt lefehl dt

(1979) that the basal t ic achondri te parent

body seems to show e v idence fo r ep i sodes

of di fferent iat ion separated spat ial ly and

also chron olog ical ly by as mu ch as 108

years . The spat ial gradient ins tabi l i ty of in-

duct ion heat ing (Herbert and Sonet t 1979)

could wel l account for the spat ial i solat ion

of mel t regions wi thin the basal t ic achon-

dr i t e paren t body . Al though the p resen t

specu la t ion concern i ng accre t ion o f a l ready

mel ted proto-as teroids does not fu l ly solve

the pr oble m of the 108-year separa t ion of

crys tal l iza t ion t imes , i t may serve as a

s t a r t ing po in t fo r a more comprehens ive

model .

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