that that than

A T E C H N I Q U E O F E V A L U A T I N G F U E L LOSSES DUE T O M E T E O R O I D P U N C T U R E A N D

S O M E T I M E L Y E X A M P L E S¹ 2 A n d r e w H . J a z w i n s k i

M a r t i n C o m p a n y , B a l t i m o r e 3 , M a r y l a n d A B S T R A C T

A n i m p o r t a n t h a z a r d to which s p a c e v e h i c l e s a r e e x p o s e d is l o s s of fuel due to m e t e o r o i d puncture of fuel t a n k s . T h e m e t e o r o i d s p e c t r u m of s i g n i f i c a n c e h e r e is p o o r l y d e f i n e d , s i n c e s a t e l l i t e m e a s u r e m e n t s - - t o d a t e - - h a v e dealt with m e t e - o r o i d s of c o n s i d e r a b l y s m a l l e r m a s s than those of i n t e r e s t . V i s u a l , photographic and r a d i o - r a d a r data do not y i e l d m a s s d i r e c t l y . A c o n s e r v a t i v e m e t e o r o i d e n v i r o n m e n t is d e f i n e d . W i t h the aid of a h y p e r v e l o c i t y p e n e t r a t i o n r e l a t i o n ( 1 8 ) 5 a hole p r o d u c t i o n m o d e l is i n t r o d u c e d - - a n d methods a r e p r e - sented f o r c a l c u l a t i n g fuel l o s s e s as a function of t i m e and p r o b a b i l i t y . T w o s p a c e m i s s i o n s a r e a n a l y z e d as e x a m p l e s : 1) E a r t h - M o o n t r a n s f e r of a l i q u i d h y d r o g e n fuel tank to be used f o r E a r t h - r e t u r n , and 2) s t o r a g e of a s i m i l a r fuel tank in an E a r t h - p a r k i n g o r b i t f o r l a t e r r e n d e z v o u s . Tank skins of s t e e l and aluminum a r e c o n s i d e r e d , with equivalent tank skin t h i c k n e s s as a p a r a m e t e r . It is found that fuel l o s s e s m a y i n d e e d be significant and that m e a n s should be taken to l i m i t o r e l i m i n a t e t h e m by s h i e l d i n g the fuel tanks.

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

The advent of space e x p l o r a t i o n has s t i m u l a t e d i n t e r e s t in the m e t e o r o i d e n v i r o n m e n t and i t s e f f e c t s on s p a c e v e h i c l e s . K o r n h a u s e r ( 1 ) c a l c u l a t e d h o l e a r e a p r o d u c e d and p r e s s u r e

P r e s e n t e d at the A R S L u n a r M i s s i o n s M e e t i n g , C l e v e l a n d , Ohio, July 17-19, 1962.

^ R e s e a r c h conducted at G e n e r a l D y n a m i c s / A s t r o n a u t i c s , San D i e g o , C a l i f o r n i a .

2

Senior E n g i n e e r , A e r o - S p a c e M e c h a n i c s D e p a r t m e n t . 3

N u m b e r s in p a r e n t h e s e s i n d i c a t e R e f e r e n c e s at end of p a p e r .

l o s s e s v s t i m e at v a r i o u s p r o b a b i l i t y l e v e l s , a s s u m i n g an a v e r a g e c o n s e r v a t i v e situation. B j o r k ( 2 ) e s t i m a t e d s t e e l and aluminum a r m o r w e i g h t s r e q u i r e d to p r o t e c t a v e h i c l e at any confidence l e v e l . T h e author ( 3 - 5 ) e s t i m a t e d c o n s e r v a t i v e fuel l o s s e s on the Centaur v e h i c l e , using at f i r s t an a p p r o x i - m a t e p r o b a b i l i s t i c a p p r o a c h . M o r e r e c e n t l y E d m i s t o n (6) d e v e l o p e d r e l a t i o n s for the m e t e o r o i d h o l e a r e a p r o d u c e d as a function of t i m e and p r o b a b i l i t y , based on t w o p e n e t r a t i o n m o d e l s .

F o u r distinct p r o b l e m a r e a s e x i s t in evaluating fuel l o s s e s due to m e t e o r o i d p u n c t u r e . T h e y a r e 1) p r o p e r t i e s of the m e t e o r o i d e n v i r o n m e n t which the v e h i c l e t r a v e r s e s , 2) the p e n e t r a t i o n p r o c e s s w h e r e b y h o l e s a r e p r o d u c e d in the v e h i c l e , 3) flow r e l a t i o n s f o r the e s c a p i n g f u e l , and 4) p r o b a b i l i s t i c e v a l u a t i o n of the fuel l o s s e s . T h e s e p r o b l e m s a r e d i s c u s s e d in the s e q u e l .

T h e v e h i c l e d e s i g n e r w i s h e s to know the a v e r a g e o r m e a n effect of the m e t e o r o i d e n v i r o n m e n t on his v e h i c l e , as w e l l as to have s o m e m e a s u r e of the d i s p e r s i o n f r o m this m e a n - - i . e . , that e f f e c t which w i l l not be e x c e e d e d at s o m e high p r o b a b i l i t y . Both e f f e c t s a r e p r e s e n t e d h e r e . T h e a n a l y s i s is a p p l i e d to two s p e c i f i c c a s e s : 1) A fuel tank t r a n s p o r t e d to the M o o n to be used f o r E a r t h - r e t u r n , and 2) a fuel s t o r a g e tank in an E a r t h - p a r k i n g o r b i t . F u e l l o s s e s a r e studied as a function of t a n k - s k i n thickness for aluminum and s t e e l s k i n s .

M E T E O R O I D E N V I R O N M E N T

C u r r e n t k n o w l e d g e of the m e t e o r o i d e n v i r o n m e n t is v e r y l i m i t e d . Its p r o p e r t i e s of i n t e r e s t in r e l a t i o n to the p r e s e n t p r o b l e m include m e t e o r o i d d e n s i t y , m a s s - f l u x , m a s s - v e l o c i t y and s p a t i a l d i s t r i b u t i o n s . I d e a l l y , it would be d e s i r a b l e to know the e n v i r o n m e n t a l p a r a m e t e r s as a function of t i m e and p o s i t i o n in s p a c e . T h i s is v i r t u a l l y i m p o s s i b l e , because of the e x i s t e n c e of m e t e o r o i d s h o w e r s ( 7 ) , as w e l l as m e t e o r s h o w e r s ( 8 ) - - i n addition to the s p o r a d i c b a c k g r o u n d . T h e s e have been known to e x c e e d the s p o r a d i c background^by s e v e r a l o r d e r s of m a g n i t u d e . O n l y the s p o r a d i c m e t e o r o i d flux can be c o n s i d e r e d h e r e .

W h i p p l e ( 9 ) r e p o r t s that m o r e than 90% of a l l photographic 3

m e t e o r s have d e n s i t i e s as l o w as 0.05 g m / c m . R a d i o - r a d a r m e t e o r s m a y have s i m i l a r d e n s i t i e s . T h i s a p p l i e s to m e t e o r s

-4

of m a s s r o u g h l y g r e a t e r than 10 g m , c o m m o n l y r e f e r r e d to as dust b a l l s . ' T h e s e dust b a l l s m a y b r e a k up, h o w e v e r , p r o - ducing s m a l l e r p a r t i c l e s with m o r e c o n v e n t i o n a l d e n s i t i e s ,

414

p e r h a p s m o r e l i k e those of s i l i c a t e r o c k s .

The a s t e r o i d s m a y be the s o u r c e of a l a r g e f r a c t i o n of the -4

m e t e o r o i d s p e c t r u m of s m a l l e r m a s s e s ( < 1 0 g m ) . B r o w n (10, 11) c o r r e l a t e s the m a s s - f r e q u e n c y d i s t r i b u t i o n of m e t e - o r i t e s with that of the a s t e r o i d s . T h e c o r r e l a t i o n is g o o d .

3

M e t e o r i t e s r a n g e in d e n s i t y f r o m 2. 7 to 7. 9 g m / c m ( s t o n e s - i r o n s ) , the r e l a t i v e abundance being about 16 t o 1, r e s p e c - t i v e l y . T h i s would p l a c e the a v e r a g e d e n s i t y of a s t e r o i d a l

3 m e t e o r o i d s at about 3 g m / c m .

In v i e w of the uncertainty that is p r e s e n t , the fact that -4

m e t e o r o i d s of m a s s l e s s than 10 g m m a y p l a y a c r u c i a l r o l e in contributing to fuel l o s s e s (as w i l l be s e e n l a t e r ) and of the d e s i r e f o r a c o n s e r v a t i v e e s t i m a t e of fuel l o s s e s , a density of 3 g m / c m w i l l be a s s u m e d for a l l m e t e o r o i d s . 3

G e o c e n t r i c v e l o c i t i e s of m e t e o r s have been m e a s u r e d . W h i p p l e ( 9 ) c a l c u l a t e s the a v e r a g e v e l o c i t y of photographic m e t e o r s to be 28 k m / s e c . A r a n g e of 11-72 k m / s e c is p o s - s i b l e , s i n c e ( 9 ) m e t e o r s a r e p e r m a n e n t m e m b e r s of the s o l a r s y s t e m . One would e x p e c t s m a l l e r m e t e o r o i d s t o have l o w e r v e l o c i t i e s . On this b a s i s , W h i p p l e ( 9 ) c o n s t r u c t e d a table of v e l o c i t y as a function of m a s s . T h i s is plotted in F i g . 1 and r e p r e s e n t s the m o s t r e a s o n a b l e v e l o c i t y d i s t r i b u t i o n which p r e s e n t l y can be deducted. A c o n s e r v a t i v e f i g u r e of 28 k m / s e c m a y be used for a l l m e t e o r o i d s .

Data as to the flux of m e t e o r o i d s have been obtained f r o m v i s u a l , photographic and r a d i o - r a d a r o b s e r v a t i o n s - - a n d of late f r o m counting d e v i c e s on r o c k e t s and s a t e l l i t e s . Data in the f i r s t c a t e g o r y ( i . e . , v i s u a l , photographic and r a d i o - r a d a r ) y i e l d m e t e o r flux as a function of m a g n i t u d e . It is then n e c - e s s a r y to r e l a t e magnitude to the m a s s of the p a r t i c l e . T o d a t e , no r e l i a b l e r e l a t i o n e x i s t s . W h i p p l e ( 9 ) a s s i g n e d a v a l u e of 25 g m to a z e r o magnitude m e t e o r , w h i l e W a t s o n (8) c h o s e the v a l u e 0. 25 g m . On t h e s e b a s e s t w o c u r v e s a r e obtained of the flux of m e t e o r o i d s of m a s s equal to and g r e a t e r than m a s s m , as a function of m a s s . T h e s e have been e x t r a - polated to p a r t i c l e s of s m a l l e r m a s s ( 8 , 9) and a r e shown in F i g . 2. T h e t w o e s t i m a t e s c a n , p e r h a p s , be taken as bounds on the s p o r a d i c m e t e o r o i d e n v i r o n m e n t in the m a s s r a n g e

-4

f r o m 10 to 1 g m .

A r a t h e r c o m p l e t e l i s t of in situ r o c k e t and s a t e l l i t e m e a s - u r e m e n t s has been c o m p i l e d "By the author (5) and is

t h o r o u g h l y r e f e r e n c e d . T h e m o s t s i g n i f i c a n t m e a s u r e m e n t s a r e p r e s e n t e d in F i g . 2. T h e y a r e l i s t e d in o r d e r of d e c r e a s - ing s i g n i f i c a n c e , the s o l i d s y m b o l s r e p r e s e n t i n g the m o s t s i g n i f i c a n t m e a s u r e m e n t s . T h e points plotted a r e based on an a s s u m e d v e l o c i t y of 15 k m / s e c , s i n c e s a t e l l i t e s e n s o r s

4

r e s p o n d to m o m e n t u m . T h e data have not been c o r r e c t e d for E a r t h - s h i e l d i n g , as is s o m e t i m e s the c u s t o m ( 1 2 ) , s i n c e , w h i l e in E a r t h¹ s v i c i n i t y , a v e h i c l e w i l l s e e the s h i e l d e d flux and t h e r e a r e t h e o r e t i c a l indications ( 1 3 - 1 5 ) that flux d e c r e a s e s with i n c r e a s i n g a l t i t u d e .

The m o s t s i g n i f i c a n t m e a s u r e m e n t s have r e p o r t e d l y (12) been taken on the E x p l o r e r V I I I s a t e l l i t e , w h e r e m e a s u r e m e n t s w e r e made in t h r e e m o m e n t u m r a n g e s . T h e t h r e e E x p l o r e r V I I I points fall on a good s t r a i g h t l i n e , m a r k e d " D i r e c t " in F i g . 2 , which a l s o fits the o t h e r s a t e l l i t e data r a t h e r w e l l . T h i s line m a y be t a k e n , at p r e s e n t , as the best r e p r e s e n t a - tion of the s p o r a d i c m e t e o r o i d e n v i r o n m e n t in the m a s s r a n g e

-9 -6

f r o m 10 to 10 g m . A c l o s e r i n s p e c t i o n of the s a t e l l i t e data ( 5 , 12) i n d i c a t e s that no altitude dependence of the flux is d i s c e r n i b l e . Î h e mean altitudes of the s a t e l l i t e orbits vary, at m o s t , by 800 k m ( 5 ) - - a n d the rocket data cannot be c o n s i d e r e d significant.

A t the present t i m e , then, no altitude dependence can be deduced e x p e r i m e n t a l l y and, t h e r e f o r e , it w i l l be assumed that m e t e o r o i d flux is Earth-distance independent in cislunar s p a c e .

It can be seen that no m e a s u r e m e n t s of m e t e o r o i d flux a r e -6 -4 -3 available in the m a s s range f r o m 10 to 10 o r 10 g m . T h i s , as w i l l be seen, is unfortunately the most important m a s s r e g i m e as far as fuel l o s s e s due to puncture a r e con- cerned. Sensor surfaces have been v e r y s m a l l in the past, because of the weight limitation. Hence, they have had to be of high sensitivity to r e c o r d a significant number of hits.

Consequently, a c o n s e r v a t i v e m a s s - f l u x r e l a t i o n has to be estimated. It is bel i e v e d that W h i p p l e ' s c u r v e ( F i g . 2) r e p r e s e n t s such a relation. M e t e o r o i d flux, t h e r e f o r e , w i l l be assumed to follow the law

ι ^m i n - 1 2 . 2 - 1 . 0 r η

Φ * 10 m [ l j w h e r e m is the m e t e o r o i d m a s s in g r a m s and φ is the number

3

R u s s i a n s e n s o r s r e p o r t e d l y r e s p o n d to the k i n e t i c e n e r g y of the i m p a c t i n g p a r t i c l e .

416

of p a r t i c l e s of m a s s g r e a t e r than and equal to m p e r m - s e c . A m o r e c o m p l e t e d i s c u s s i o n of m e t e o r o i d flux is g i v e n in R e f s . 5 and 3.

H O L E A R E A D U E T O M E T E O R O I D P U N C T U R E

M e t e o r o i d puncture of a v e h i c l e tank skin i n v o l v e s a h y p e r - v e l o c i t y i m p a c t p r o c e s s which of late has r e c e i v e d c o n s i d e r - able attention. A n e x c e l l e n t c o m p i l a t i o n and a n a l y s i s of e x - p e r i m e n t a l and t h e o r e t i c a l w o r k is p r e s e n t e d by H e r r m a n n and Jones ( 1 6 ) . U n t i l r e c e n t l y , v e l o c i t i e s attained in the l a b - o r a t o r y have not e x c e e d e d 6 k m / s e c . S o m e shots at v e l o c i t i e s up to 12 k m / s e c have been obtained ( 1 7 ) . A s can be s e e n , m e t e o r o i d v e l o c i t i e s have b a r e l y been a p p r o a c h e d in the l a b - o r a t o r y . E x t r a p o l a t i o n of t h e s e e x p e r i m e n t a l data taken at low v e l o c i t i e s to the much h i g h e r m e t e o r o i d v e l o c i t i e s is s t r i c t l y i n v a l i d ( 5 , 1 6 ) , s i n c e the p h y s i c a l p r o c e s s i n v o l v e d is d i f f e r - ent. A t h y p e r v e l o c i t i e s , s t r e n g t h e f f e c t s a r e n e g l i g i b l e ( 1 6 , 18) during p a r t of the i m p a c t p r o c e s s , b e c a u s e the p r e s s u r e s i n v o l v e d far e x c e e d the m a t e r i a l s t r e n g t h .

T h e m o s t a p p l i c a b l e t h e o r e t i c a l w o r k a p p e a r s to be that of Β j o r k ( 1 8 ) , who s o l v e d the equations for a h y d r o d y n a m i c i m - pact of i r o n on i r o n and aluminum on aluminum n u m e r i c a l l y in the i n v i s c i d , adiabatic a p p r o x i m a t i o n . He obtained the r e - lations

A l on A l : ρ * 1. 09 ( m v )^{1 f 3} [2]

1 /3 F e on F e : ρ « 0. 606 ( m v ) ⁷

w h e r e ρ is the depth of p e n e t r a t i o n in a s e m i - i n f i n i t e t a r g e t in c m , m is the m e t e o r o i d m a s s in g m , and ν is the m e t e o r o i d v e l o c i t y in k m / s e c .

B j o r k (19) r e c e n t l y s u g g e s t e d that the p e n e t r a t i o n in a g i v e n t a r g e t m a t e r i a l by p r o j e c t i l e s of d i f f e r e n t m a t e r i a l s having the s a m e m a s s and v e l o c i t y is p r o p o r t i o n a l to the i n i - t i a l i n t e r f a c e v e l o c i t y on i m p a c t . H e r r m a n n and Jones (16) show that i n i t i a l i n t e r f a c e v e l o c i t i e s v * for i m p a c t s of d i s - s i m i l a r m a t e r i a l s a r e g i v e n with a m a x i m u m e r r o r of about 20% for m o s t m a t e r i a l s by

v * - ( p^p/ p^t)^{1 / 3} v / 2 [ 3 ]

for p r o j e c t i l e v e l o c i t i e s ν a b o v e 3 k m / s e c , w h e r e ρ and p.

a r e p r o j e c t i l e and t a r g e t d e n s i t i e s , r e s p e c t i v e l y .

Equation 2 , t h e r e f o r e , can be m o d i f i e d f o r d i f f e r e n t p r o - j e c t i l e m a t e r i a l s to y i e l d

ρ « 1.09 ( P^p/ P^{A 1})^{1/ 3} ( m v )^{1 / 3} [ 4 ]

for i m p a c t s on aluminum t a r g e t s , and

ρ * 0.606 ( p^p/ p^{F e})^{1 / 3} ( m v )^{1 / 3} [ δ ]

for i m p a c t s on i r o n t a r g e t s .

It has been o b s e r v e d (16) that, if a p r o j e c t i l e w i l l p e n e t r a t e a s e m i - i n f i n i t e t a r g e t to a depth of p , it w i l l puncture a thin plate of t h i c k n e s s 1,5 p . T h e r e f o r e , in t e r m s of plate t h i c k - n e s s , Eqs,4 and 5 b e c o m e

t^{A 1} « 1.64 ( P^p/ P^A1)^{1/ 3} ( m v )^{1 / 3} [ 6 ]

t^{F e} « 0 . 9 0 8 ( p^p/ p^{F e})^{1 / 3}( m v )^{1 /3}. [ ? ]

T h e s e r e l a t i o n s w i l l be used in further a n a l y s i s .

Equations 6 and 7 m a y be c o m b i n e d with E q . 1 to y i e l d the flux of m e t e o r o i d s which w i l l puncture a thin v e h i c l e skin of t h i c k n e s s t. T h i s w i l l be c a l l e d , a f t e r B j o r k ( 2 ) , the p e n e - t r a t i n g flux ψ. G i v e n a skin t h i c k n e s s t , and taking v e l o c i t i e s f r o m F i g . 1, o r a s s u m i n g a m e t e o r o i d v e l o c i t y of 28 k m / s e c , E q s . 6 and 7 y i e l d the s m a l l e s t m a s s m e t e o r o i d s which w i l l puncture the s k i n . T h i s w i l l be c a l l e d the t h r e s h o l d m a s s m^.

A l l m e t e o r o i d s of m a s s g r e a t e r than m^ w i l l l i k e w i s e punc- t u r e the skin. U s i n g v e l o c i t i e s f r o m F i g . 1 and m e t e o r o i d

3

d e n s i t y = * 3 g m / c m , was c a l c u l a t e d as a function of skin t h i c k n e s s f o r both s t e e l and aluminum skins and a p p e a r s

— 6 in F i g . 3. It i s seen that m e t e o r o i d s of m a s s as l o w as 10 g m m a y be i m p o r t a n t in contributing to fuel l o s s e s .

It w i l l be a s s u m e d that the h o l e p r o d u c e d by a m e t e o r o i d i m p a c t i s of constant d i a m e t e r throughout i t s depth, the d i a m - e t e r b e i n g equal to the e n t r a n c e d i a m e t e r p r o d u c e d by the m e t e o r o i d . T h e depth of p e n e t r a t i o n p, g i v e n in E q s . 4 and 5, i s a l s o the r a d i u s of the h o l e . T h e r e f o r e , the a r e a of a h o l e p r o d u c e d by a m e t e o r o i d of m a s s m i s g i v e n by

418

a ( m ) = π ρ² « π C² < P^m/ p^{A 1 Q r F e}> ^{2 / 3} [mv(m3^{2 / 3}

^

w h e r e C * 1.09 f o r aluminum t a r g e t s , C * 0.606 f o r s t e e l t a r g e t s , and p ^m is the m e t e o r o i d d e n s i t y . F o r * 3, E q . 8 is quite a c c u r a t e f o r m e t e o r o i d s of m a s s n e a r the t h r e s h o l d m a s s , s i n c e the r a t i o of skin t h i c k n e s s to d i a m e t e r of t h r e s - hold m e t e o r o i d ( a s s u m i n g a s p h e r i c a l m e t e o r o i d ) i s 6 and 2. 33 f o r aluminum and s t e e l t a r g e t s , r e s p e c t i v e l y .

F o r m e t e o r o i d s of m a s s g r e a t e r than the t h r e s h o l d m a s s , Eq.8 w i l l o v e r e s t i m a t e the actual a r e a - - t h e o v e r e s t i m a t e i n - c r e a s i n g as m a s s i n c r e a s e s . T h i s i s b e c a u s e (16) the radius of a c r a t e r p r o d u c e d by a g i v e n p r o j e c t i l e in a s e m i - i n f i n i t e t a r g e t is g r e a t e r than the r a d i u s of a h o l e p r o d u c e d by the s a m e p r o j e c t i l e in a thin p l a t e . Since the s i g n i f i c a n c e ( i . e . , n u m b e r ) of m e t e o r o i d s d e c r e a s e s with i n c r e a s i n g m a s s , a b o v e the t h r e s h o l d m a s s , the e r r o r introduced by using E q . 8 f o r a l l m e t e o r o i d s w i l l not be substantial. A t any r a t e , E q . 8 g i v e s a c o n s e r v a t i v e e s t i m a t e of a r e a .

H o l e a r e a v e r s u s m e t e o r o i d m a s s f o r both aluminum and s t e e l is g i v e n in F i g . 4.

F U E L F L O W

F l u i d flow out of s m a l l o r i f i c e s is c u r r e n t l y not w e l l u n d e r - s t o o d . B e r n o u l l i flow f r o m a r e s e r v o i r at z e r o - g into a v a c - ι uum w i l l be u s e d . T h e exhaust v e l o c i t y is t h e r e f o r e g i v e n by

v^e = ( 2 P / p^f)^{1 / 2} [ 9 ]

w h e r e Ρ is the tank p r e s s u r e and p^ the fuel d e n s i t y . T h e flow r a t e is t h e r e f o r e

Q « r a ( 2 P / p^f)^{1 / 2} [ίο]

w h e r e r is an o r i f i c e c o e f f i c i e n t and a the a r e a . T h e m a s s flow r a t e i s

Q^m « r a ( 2 P p^f)^{1 / 2}. [ l l ]

W h e n d e a l i n g with c r y o g e n i c l i q u i d s , the question m a y be r a i s e d w h e t h e r the liquid m i g h t not f r e e z e w h i l e expanding through the h o l e , thus t e m p o r a r i l y p l u g g i n g the hole and d e - c r e a s i n g the e f f e c t i v e flow r a t e . E a c h situation must be

e x a m i n e d independently f o r this e f f e c t . M E A N F U E L LOSSES

A v e h i c l e with tank a r e a A , e x p o s e d to a p e n e t r a t i n g flux ψ for a t i m e / w i l l suffer an a v e r a g e n u m b e r of punctures λ * ψ Α τ , T h e p r o b a b i l i t y that a m e t e o r o i d in the m a s s r a n g e

dm w i l l puncture the tank skin is

-αφ/ψ [12]

w h e r e -d<\> is the n u m b e r of m e t e o r o i d s in d m . N o w f r o m E q . l

-d<\> * ( K / m²) dm [13]

-12 2

w h e r e Κ * 10 . T h e r e f o r e ,

-αΦ/ψ ^ mt/ m2j d m Q4J

Of c o u r s e , the t o t a l p r o b a b i l i t y is 0

J -αΦ/ψ * J ( m^t/ m²) d m * 1 Qo]

ψ m^t

T h e a r e a of the a v e r a g e puncture is s i m p l y

Ae * ^ ( a ( m ) m^t/ m²) d m « π C² m^t (p^/p^{Ai or 23 /}

( [ m v ( m ) ]^{2 / 3}/ m²) dm [ΐβ]

mt

with the aid of E q . 8. Κ the m e t e o r o i d v e l o c i t y is a c o m p l i - c a t e d function of m a s s , the i n t e g r a l in E q . 16 is r a t h e r d i f f i - c u l t . A s s u m i n g a constant v e l o c i t y , v ( m ) « V q (28 k m / s e c ) , E q . 16 is e a s i l y i n t e g r a t e d

« ^ 2/3 2/3 Α * 3 τ τ Γ fp / ρ^Λ1 _Λ Γ ι η , ν Ί

e r m KA 1 o r F e ) [_ t oj

3 at [17]

w h e r e a^ is the a r e a p r o d u c e d by the m e t e o r o i d of t h r e s h o l d m a s s .

420

In the a v e r a g e s i t u a t i o n , t h e n , the t o t a l h o l e a r e a p r o d u c e d a f t e r t i m e τ is Α λ * Α Ψ AT , and the fuel l o s s e s at the end

e e of the m i s s i o n a r e

LA V *^{( Q}^^{a )} J A ^e ψ Α T d r « ( Q / a ) A^e ψ A =

( Q / a ) A^e λ τ^τ/ 2 [18]

Equation 18 w i l l be used subsequently in c a l c u l a t i n g a v e r a g e o r m e a n fuel l o s s e s . T h e a v e r a g e a r e a has been used in the past by the author ( 3 , 4 ) in c a l c u l a t i n g the a p p r o x i m a t e fuel l o s s e s at h i g h e r p r o b a b i l i t i e s .

M e t e o r o i d s i m p a c t i n g a s p a c e v e h i c l e constitute a r a n d o m p r o c e s s d e v e l o p i n g in t i m e ( a s s u m i n g that m e t e o r o i d flux is i t s e l f r a n d o m ) and, t h e r e f o r e , one would e x p e c t it to o b e y a P o i s s o n d i s t r i b u t i o n in the p a r a m e t e r λ . L a m b d a is a l s o the m e a n of the P o i s s o n d i s t r i b u t i o n . Since the d i s t r i b u t i o n is an a s y m m e t r i c o n e , m e a n fuel l o s s e s a r e not the m o s t p r o b a b l e losses*, n e i t h e r a r e they l o s s e s which w i l l not be e x c e e d e d half the t i m e - - a s is the c a s e on a n o r m a l d i s t r i b u t i o n . T h e P o i s s o n d i s t r i b u t i o n a p p r o x i m a t e s a n o r m a l d i s t r i b u t i o n for l a r g e λ ( 2 0 ) , h o w e v e r , w h i l e d e p a r t i n g f r o m it r a d i c a l l y f o r s m a l l λ .

F U E L LOSSES A T H I G H E R P R O B A B I L I T I E S

In t h e o r y , an infinite n u m b e r of e v e n t s m a y o c c u r w h e r e b y the t o t a l a r e a due to m e t e o r o i d punctures w i l l not e x c e e d a^

(as a,p-> oo). T h e r e m a y be no p u n c t u r e s . T h e r e m a y be one puncture w h o s e a r e a d o e s not e x c e e d a ^ . In g e n e r a l , t h e r e m a y be k punctures w h o s e c o m b i n e d a r e a does not e x c e e d a,p, p r o v i d e d k is l e s s than o r equal t o the i n t e g r a l p a r t of a ^ / a ^ . If P^ is the p r o b a b i l i t y of obtaining k p u n c t u r e s , and p'k is the c o n d i t i o n a l p r o b a b i l i t y that, g i v e n k p u n c t u r e s , t h e i r c o m b i n e d a r e a w i l l not e x c e e d a^T, then the p r o b a b i l i t y that the t o t a l hole a r e a due t o m e t e o r o i d punctures w i l l not e x c e e d

_N

p ( aT > "

1

A ^

w h e r e Ν is the i n t e g r a l p a r t of a^/a^.

a T i s

A s discussed p r e v i o u s l y , it is reasonable to assume that the m e t e o r o i d flux obeys a P o i s s o n distribution, i . e . ,

P^k « e ^{_ X} ( X^k/ k « ) [20]

O b v i o u s l y ,

m ( a^T) a^T

p'j « J

( 3 / 2 ) a

J

with the aid of E q . 8, assuming v ( m ) * V q * constant, and

• . ^ - ( k - l ) a ^ Ρ ^aT ~^at ~^a3 ~^β ' ' "^ak

P k " ) ^{f ( a}k^{) d a}k ' - - J ^{f ( a}2^{) d a}2

at ^at

aT ~^a2^_' ' ' ^{_ a}k

f i a j i d a j [22]

I

at

3/2 -5 /2

w h e r e f(a) * ( 3 / 2 ) a^ ,a . T h e s e integrals b e c o m e p r o - g r e s s i v e l y m o r e difficult as k i n c r e a s e s .

Equation 22 has been d e r i v e d by Edmiston ( 6 ) . Although his penetration m o d e l was different, the functional dependence of a r e a on m a s s was the s a m e . In addition, he assumed a s i m i l a r flux-mass relationship. His e x p r e s s i o n for p£ is t h e r e f o r e exactly the s a m e as E q . 22, although a^ is obtained differently.

Equation 21 can be integrated r e a d i l y , yielding

p\ * 1 - ( a^t/ a^T)^{3 / 2} [23]

Edmiston (6) d e r i v e d a l o w e r bound formula for p£ ( a ^ / a ^ ) which results in a c o n s e r v a t i v e estimate of a^ at any p r o b - ability. This formula is

Pk ^{( a}T^{/ a}t⁾ - [ ρ Ϊ ^{( a}T ^{/ k a}t^} , a^T/ a^t > k [24]

422

W i t h the aid of this e x p r e s s i o n , E q . 19 m a y be w r i t t e n , f o r a^T - n a^t

P( n a J *e~^X

n-1 J 1 + y ( ^^k/ k ! ) [l - ( k / n )^{3 / 2}J ) , η - 1. 2 , . . .

- J [25]

F o r any v a l u e of Ρ (na^), E q . 25 m a y be s o l v e d f o r a^ * na^ as a function of X. T h i s has been done f o r Ρ (na^) * 0. 9986 ( 3 σ ) and f o r Ρ (na^t> * 0. 9772 ( 2 σ ) and a p p e a r s in F i g . 5. T h e s e c u r v e s m a y be fitted by p o l y n o m i a l s in X , y i e l d i n g

a^T - a^t f(X) » a^t ί ( ψ Α τ ) [26]

The fuel l o s s e s , then, at the g i v e n p r o b a b i l i t y Ρ (na^) a r e g i v e n by

TT

LP( n a^t) " (^{Q a}t ^/a) J^T ^ A r ) d T [27]

Ο

w h e r e the l o w e r l i m i t T ^q is the t i m e c o r r e s p o n d i n g t o the λ v a l u e at which a^ b e g i n s t o a s s u m e p o s i t i v e v a l u e s as g i v e n in F i g . 5.

E A R T H " M O O N T R A N S F E R O F A F U E L T A N K 2

A c y l i n d r i c a l fuel tank with 25 m c y l i n d r i c a l e x p o s e d a r e a m i g h t be used to t r a n s p o r t l i q u i d h y d r o g e n to the m o o n to be used f o r E a r t h - r e t u r n . Such a tank might hold a p p r o x i -

5

m a t e l y 2500 lb of fuel under p r e s s u r e of 23 psia. T h e den- s i t y of l i q u i d h y d r o g e n i s = 4. 1 l b / f t . A n o r i f i c e c o e f f i - cient of 0. 6 i s a p p r o p r i a t e in t h i s c a s e . T r a n s f e r t i m e w i l l be taken t o be 66 h o u r s .

5 H i g h e r tank p r e s s u r e s might be used to p r e v e n t boiloff.

T h e question of w h e t h e r p l u g g i n g m a y o c c u r under t h e s e conditions m i g h t be a s k e d . T h e author b e l i e v e s that p l u g g i n g w i l l not o c c u r . T h e l i q u i d w i l l be out of the h o l e b e f o r e it has a chance t o f r e e z e , in v i e w of the v e r y high exhaust v e -

4

l o c i t i e s i n v o l v e d (~10 c m / s e c ) . Some pertinent e x p e r i m e n t s 7

a r e now b e i n g conducted , including the m e a s u r e m e n t of o r i - f i c e c o e f f i c i e n t s .

M e a n , 2σ⁹ and 3σ fuel l o s s e s w e r e c a l c u l a t e d as a function of skin thickness f o r both s t e e l and aluminum s k i n s . T h e y a p p e a r in F i g s . 6 and 7. It was a s s u m e d that the tank is an infinite r e s e r v o i r f o r f u e l ; thus, l o s s e s a r e s e e n to e x c e e d its c a p a c i t y , which is a p p r o x i m a t e l y i n d i c a t e d .

A n i n s p e c t i o n of t h e s e f i g u r e s r e v e a l s that 2σ and 3σ fuel l o s s e s b e c o m e l e s s than the m e a n l o s s e s f o r s u f f i c i e n t l y thick skin. A t f i r s t , this m a y be s u r p r i s i n g . It m e a n s , h o w e v e r , that punctures b e c o m e r a r e indeed and can o c c u r only late in the m i s s i o n , at a g i v e n p r o b a b i l i t y . When t h e y do o c c u r , the hole a r e a must be l a r g e - - b e c a u s e the skin is t h i c k . T h e r e - f o r e , they m a k e a l a r g e c o n t r i b u t i o n to m e a n l o s s e s , w h e r e - - on the a v e r a g e - - p u n c t u r e s m a y be thought of as o c c u r r i n g in the m i d d l e of the m i s s i o n .

M o r e o v e r , 2σ and 3σ l o s s e s g o t o z e r o w h i l e m e a n l o s s e s r e m a i n f i n i t e . T h i s is b e c a u s e , at the g i v e n p r o b a b i l i t y , t h e r e a r e no p u n c t u r e s , w h i l e s o m e s t i l l c o n t r i b u t e to the m e a n . T h e rarjid d e c r e a s e of 2σ and 3σ l o s s e s is a s s o c i a t e d with the "jump ¹ in a r e a f r o m z e r o to a^ o b s e r v e d in F i g . 5.

T h e r e can be no hole unless its a r e a is at l e a s t as l a r g e as that p r o d u c e d by a m e t e o r o i d of t h r e s h o l d m a s s .

Β j o r k ( 2 ) c o n c l u d e d that, weight f o r w e i g h t , aluminum skins a r e s u p e r i o r to s t e e l skins b e c a u s e the p e n e t r a t i n g flux, ψ , is l o w e r . F i g u r e s 6 and 7 (as w e l l as 8 and 9) exhibit t h i s , in that the l o s s e s g o to z e r o in a g r e e m e n t with

Note added in p r o o f : P r e l i m i n a r y v i s u a l o b s e r v a t i o n s at the G e n e r a l D y n a m i c s / A s t r o n a u t i c s A e r o p h y s i c s L a b o r a - t o r y s e e m to indicate that no p l u g g i n g o c c u r s under the c o n - ditions d e s c r i b e d . High s p e e d m o t i o n p i c t u r e s r e v e a l e d that i n t e r m i t t e n t p l u g g i n g a c t u a l l y does take p l a c e . It is so r a p i d , h o w e v e r , that the o r i f i c e c o e f f i c i e n t is not a p p r e c i a b l y d e - c r e a s e d unless the hole is e x t r e m e l y s m a l l . T h e value of 0.6, used h e r e , a p p e a r s to be r e a s o n a b l y good f o r the s i z e h o l e s under c o n s i d e r a t i o n . T h e r e a r e s o m e indications that the o r i f i c e c o e f f i c i e n t m a y be as high as 0.8 f o r the l a r g e r h o l e s .

7G e n e r a l D y n a m i c s / A s t r o n a u t i c s A e r o p h y s i c s L a b o r a t o r y . 424

such a r e l a t i o n . When h o l e s a r e a l l o w e d , h o w e v e r , the i n - v e r s e is t r u e . T h i s is b e c a u s e , w h i l e w e i g h t f o r w e i g h t , the p e n e t r a t i n g flux in aluminum skins is l o w e r than in s t e e l s k i n s , the h o l e s p r o d u c e d in aluminum a r e l a r g e r - - t h e o v e r a l l r e s u l t being h i g h e r l o s s e s f r o m an aluminum tank.

S T O R A G E T A N K I N A N E A R T H P A R K I N G O R B I T

A liquid h y d r o g e n fuel tank m i g h t be p l a c e d in an e a r t h - p a r k i n g o r b i t for l a t e r r e n d e z v o u s . F o r the p u r p o s e of a n u m e r i c a l e x a m p l e , it w i l l be a s s u m e d that the tank has an

2

e x p o s e d c y l i n d r i c a l a r e a of 50 m , c a p a b l e of holding a p p r o x - i m a t e l y 7000 lb of fuel. It is r e q u i r e d that the tank r e m a i n in o r b i t f o r 30 days b e f o r e r e n d e z v o u s . P r e s s u r e , d e n s i t y and o r i f i c e c o e f f i c i e n t w i l l be taken to be the s a m e as in the p r e v i o u s e x a m p l e . M e a n , 2σ and 3σ fuel l o s s e s a r e g i v e n in F i g s . 8 and 9 as a function o f skin t h i c k n e s s f o r both s t e e l and aluminum s k i n s . A g a i n , an infinite r e s e r v o i r of fuel is a s s u m e d . H e r e , fuel l o s s e s a r e s u b s t a n t i a l l y h i g h e r than in the p r e v i o u s e x a m p l e , b e c a u s e m i s s i o n t i m e is c o n s i d e r a b l y l o n g e r . R e m a r k s s i m i l a r t o t h o s e in the p r e v i o u s e x a m p l e apply h e r e .

C O N C L U S I O N S

A r e a s o n a b l y c o n s e r v a t i v e e s t i m a t e of the m e t e o r o i d e n v i r o n m e n t was m a d e and m e t h o d s p r e s e n t e d f o r the

e v a l u a t i o n of fuel l o s s e s due to puncture of v e h i c l e fuel t a n k s , r e s u l t i n g in c o n s e r v a t i v e e s t i m a t e s of fuel l o s s e s . F u e l l o s s e s w e r e studied as a function of tank skin thickness f o r s t e e l and aluminum skins for t w o s p a c e m i s s i o n s . It was found that, w e i g h t f o r w e i g h t , a s t e e l skin is s u p e r i o r t o an aluminum skin f o r such t h i c k n e s s e s as a l l o w punctures at a g i v e n p r o b a b i l i t y . If such t h i c k n e s s e s a r e used as to e x - clude a l l punctures at a g i v e n p r o b a b i l i t y , the i n v e r s e is t r u e . S t e e l skins a r e a l w a y s s u p e r i o r in t e r m s of m e a n l o s s e s . The s i g n i f i c a n c e of m e a n l o s s e s is l i m i t e d , h o w e v e r , when the a v e r a g e number of punctures is v e r y s m a l l .

In g e n e r a l , fuel l o s s e s due to m e t e o r o i d puncture w e r e found to be s i g n i f i c a n t for s i n g l e - s k i n t a n k s . M e t e o r o i d s h i e l d s should be c o n s i d e r e d as a m e a n s of r e d u c i n g p o s s i b l e fuel l o s s e s . W e i g h t f o r w e i g h t , t h i n , s p a c e d skins have been found to be s u p e r i o r to s i n g l e skins in r e d u c i n g p e n e t r a t i o n by p r o j e c t i l e s ( 1 6 ) . I f s h i e l d s a r e u s e d , the t h i c k n e s s e s d i s c u s s e d h e r e m a y be thought of as " e f f e c t i v e " skin t h i c k - n e s s e s .

A c o m p a r i s o n of fuel l o s s e s on the r e l a t i v e l y s h o r t t r a n s - f e r m i s s i o n ( F i g s . 6 and 7) with those in the r e l a t i v e l y long parking o r b i t ( F i g s . 8 and 9) r e v e a l s that, although a t r a d e - off m a y e x i s t b e t w e e n fuel lost and added w e i g h t of s h i e l d i n g for s h o r t m i s s i o n s , in g e n e r a l no such t r a d e o f f e x i s t s f o r long o n e s . A puncture t o w a r d the beginning of a long m i s s i o n m a y c a u s e the l o s s of a l l fuel. T h e r e f o r e , any m e t e o r o i d puncture during a long m i s s i o n must be c o n s i d e r e d a " k i l l ,^{1 1} and the v e h i c l e must be s h i e l d e d t o e l i m i n a t e a l l punctures at the d e s i r e d p r o b a b i l i t y .

T h e fuel l o s s e s p r e s e n t e d h e r e a r e c o n s e r v a t i v e in another s e n s e . A s fuel is l o s t , the r e m a i n i n g fuel w i l l a s s u m e a z e r o - g r a v i t y c o n f i g u r a t i o n . S o m e of the tank a r e a w i l l then be a d - jacent t o e m p t y s p a c e o r , in the c a s e of liquid h y d r o g e n , t o a g a s u l l a g e . T h e flow r a t e f o r h y d r o g e n g a s , under the c o n - ditions d i s c u s s e d , is about one-tenth of that f o r liquid h y d r o g e n . R E F E R E N C E S

1 K o r n h a u s e r , Μ . , " S a t e l l i t e p r e s s u r e l o s s e s caused by m e t e o r o i d i m p a c t s , " A R S J. 30, 475-479 ( 1 9 6 0 ) .

2 B j o r k , R . L . , " M e t e o r o i d s v e r s u s .space v e h i c l e s , "

A R S J. 3 1 , 803-807 ( 1 9 6 1 ) .

3 J a z w i n s k i , Α . Η . , " F u e l l o s s e s on a t y p i c a l 6. 2 hour Centaur m i s s i o n due to m e t e o r o i d p u n c t u r e , G e n e r a l D y n a m i c s / A s t r o n a u t i c s R e p t . A E 6 1 - 1 0 4 2 , A u g u s t 1961.

4 J a z w i n s k i , Α . Η . , " F u e l l o s s e s on a t y p i c a l 6. 2 hour centaur m i s s i o n due to m e t e o r o i d p u n c t u r e , A d d e n d u m :

" V a r i a b l e F l i g h t T i m e , " G e n e r a l D y n a m i c s / A s t r o n a u t i c s R e p t . A E 6 1 - 1 0 4 2 A d d e n d u m , D e c e m b e r 12, 1961.

5 J a z w i n s k i , Α . Η . , " M e t e o r o i d puncture of s p a c e v e - h i c l e s with a p p l i c a t i o n to fuel l o s s e s on the c e n t a u r , G e n e r a l D y n a m i c s / A s t r o n a u t i c s R e p t . A E 6 2 - 0 4 5 3 , M a y 29, 1962.

6 E d m i s t o n , R . Μ . , " T h e p r o d u c t i o n of a m e t e o r o i d hole a r e a in a s p a c e v e h i c l e n e a r the E a r t h , " Inst. A e r o - s p a c e S c i e n c e s P a p e r N o . 6 2 - 2 9 , January 1962.

7 Dubin, Μ . , A l e x a n d e r , W . Μ . , and B e r g , Ο . Ε . ,

" C o s m i c dust s h o w e r s by d i r e c t m e a s u r e m e n t s , " S y m p o s i u m on the A s t r o n o m y and P h y s i c s of M e t e o r s ( 1 9 6 1 ) .

426

8 W a t s o n , F . G . , B e t w e e n the P l a n e t s , ( H a r v a r d U n i v e r - s i t y P r e s s , C a m b r i d g e , M a s s . ) ( 1 9 5 6 ) , Chap. 7,

9 W h i p p l e , F . L . , " T h e m e t e o r i t i c r i s k t o s p a c e v e h i c l e s , "

V i s t a s in A s t r o n a u t i c s , ( 1 9 5 8 ) .

10 B r o w n , H a r r i s o n , " T h e d e n s i t y of m a s s d i s t r i b u t i o n of m e t e o r i t i c b o d i e s in the n e i g h b o r h o o d o f the e a r t h¹ s o r b i t , "

J. G e o p h y s . R e s e a r c h 65, 1679 (June 1960).

11 B r o w n , H a r r i s o n , " A d d e n d u m : T h e d e n s i t y and m a s s d i s t r i b u t i o n o f m e t e o r i t i c b o d i e s in the neighborhood of the e a r t h ' s o r b i t , " J. G e o p h y s . R e s e a r c h 66, 1316 ( 1 9 6 1 ) .

12 M c C r a c k e n , C . W . , and A l e x a n d e r , W . Μ . , " T h e d i s t r i b u t i o n of s m a l l i n t e r p l a n e t a r y dust p a r t i c l e s in the v i c i n i t y of e a r t h , " S y m p o s i u m on the A s t r o n o m y and P h y s i c s of M e t e o r s ( 1 9 6 1 ) .

13 B e a r d , D . Β . , " i n t e r p l a n e t a r y dust d i s t r i b u t i o n , "

A s t r o p h y s . J o u r n a l , 192, 496-506 (1959).

14 S i n g e r , S. F . , " i n t e r p l a n e t a r y dust n e a r the e a r t h , "

N a t u r e 192, 321-323 ( 1 9 6 1 ) .

15 W h i p p l e , F . L . , " P a r t i c u l a t e contents of s p a c e , "

M e d i c a l and B i o l o g i c a l A s p e c t s of the E n e r g i e s of S p a c e , C o l u m b i a U n i v e r s i t y P r e s s , N e w Y o r k ( 1 9 6 1 ) , Chap. 3.

16 H e r r m a n n , W . , and J o n e s , Α . Η . , " S u r v e y of h y p e r - v e l o c i t y i m p a c t i n f o r m a t i o n , " M a s s . Inst. T e c h . , A S R L R e p t . N o . 9 9 - 1 , S e p t e m b e r 1961.

17 E i c h e l b e r g e r , R . L . , and G e h r i n g , J. W . , " E f f e c t s of m e t e o r o i d i m p a c t on s p a c e v e h i c l e s , " B R L R e p t . N o . 1155, D e c e m b e r 1961.

18 B j o r k , R . L . , " E f f e c t s o f a m e t e o r o i d i m p a c t on

s t e e l and a l u m i n u m in s p a c e , " X t h I n t e r n a t i o n a l A s t r o n a u t i c a l C o n g r e s s , London (1959),

19 F i f t h H y p e r v e l o c i t y I m p a c t S y m p o s i u m , D e n v e r (1961).

20 F e l l e r , W . , P r o b a b i l i t y T h e o r y and Its A p p l i c a t i o n s , John W i l e y & S o n s , I n c . , L o n d o n ( 1 9 5 0 ) , Chap. 7.

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