[PDF] - Application of Granular Kinetics to Ring Processes Jim Jenkins PDF Document

SLIDE 1

Application of Granular Kinetics to Ring Processes Jim Jenkins Cornell University with Volker Simon, Brian Lawney, and Joe Burns Dilute, three-dimensional ring: repeat and extend Goldreich and Tremaine’s calculation of the relationship between optical depth and coefficient of restitution. Dense, two-dimensional ring: introduce and interpret a simple numerical simulation of the flow around a moonlet in the absence of gravitational interactions between the moonlet and the disk particles and between the disk particles.

SLIDE 2

Velocity distribution function: f ( ; ,t)d d c x c x number density: f ( ; , n( , t)d t)  c c c x x Averages: 1 fd n   

c

c mean velocity: ( ,t)   u u x c velocity fluctuation: = ( ,t)   C c u C x second moment:   K C C

 

T tr  K , 1 ˆ T 3   K K 1 third moment:    Q C C C

 

ijk ipp jk jpp ki kpp ij

1 Q Q Q Q 5       ,

i ipp

1 q Q 2  

SLIDE 3

Balance equations mass:

s

mn     ,

3

d n / 6   

 

t       u  linear momentum

   

3

GM t            u R u u K R   second moment

         

T

t                   K u K + K u + K C C u Q      Explicit form

 

1 1/2

1 f (c; ,t) ex K 2 8 n p



          K C x C ,

 

K det  K

SLIDE 4

Collisions

1 2

and c c pre-collisional velocities, 1 and  

2

c c post-collisional velocities, unit vector k directed from 1 to 2, coefficient of restitution e, relative velocity

1 2

  g C C , unit vector j in the plane of g and k, perpendicular to k.

 

e     g k g k

 

1 1

1 e 2      c c g k k

 

2 2

1 e 2      c c g k k

Total change of second moment

  

1 1 2 2 1 1 2 2

1 (1 e)( ) 2 (1 e)( ) ( )                            g k g k k k g j k C C C C C C C j j k C 

SLIDE 5

Collisional production of second moment

1 2 1 2 3/2 3/2

1 [ ] m f f d( )d d d 2 6 T (1 e) d

 

        

g k

C C g k k c c    ˆ (1 e) 2   A B  

3/2

( / T) d

 

  

g k

A k k k Kk k

 

1/2

ˆ ( / T) ( / T)d

 

     

g k

B k i i k k Kk k Ki k

SLIDE 6

Second moment

 

1 2

K K / 2T    ,

 

1 2

K K / 2T 1         Cylindrical polar: r, , z 

r 1

cos   e e 1 cos2 sin 2 = sin 2 1 cos2 1 T 2                         K Nearly homogeneous , and constant; T T(z), (z), u u u(r)



        

SLIDE 7

Balance equations at lowest order

1/2 3

GM u(r) (r)r, (r) r          

 

r rr zrr

4 K Q z



       

 

r z

K Q z

  

      

 

zz zzz

Q z      

   

rr r zr

1 K 4K Q 2 z

  

       

SLIDE 8

Eigenvector basis

   

2 11 zrr 2 zr z

3 T 1 sin 2 cos Q 2 z sin 2 Q sin Q z z

 

                      

   

2 22 zrr 2 zr z

3 T 1 sin 2 sin Q 2 z sin 2 Q cos Q z z

 

                     

 

33 zzz

Q z      

   

zrr z zr

1 1 T 5 3 1 cos2 sin 2 Q 2 2 z Q cos2 Q z z

 

                           

SLIDE 9

Integrate the last over z:

 

5 cos2 3 1     Integrate the isotropic part over z:

 

1/2 2 2 3/2 3/2

6 3 sin tr( ) 2 T dz 1 e T dz d

   

      

 

A Using this, the 33 component is

33

ˆ tr( (1 e ) )     A With the last two, the difference between the 22 and 11 components is

   

33 22 11

3 1 ˆ       

SLIDE 10

Approximate

4 2 2 4 2 2 2 3

2 3 9 tr( ) 70 7 21 2 2 35 4 2 4                        A 

2 2 3 2 22 11

8 2 2 (1 e) 7 2 35 6 2 11 11                         

4 2 2 4 2 2 3 2 33

4 6 15 ˆ (1 e) 42 2 6 3 105 11 11 11                             

Solve the last two balance equations for α and β in terms of 1 e   

   

1/2 2 2

8 3 2772 2960 9393 8 3 10 12 11 4851 2960 9393 2960 9393                       

Lowest order in :

 

1/2

and 5 /14 5 / 2      

2 2

7 1013 2 396     

SLIDE 11

Limitations

 

5 cos2 3 1     with

 

1/2

5 5 and 14 2      

implies that 0.3688  

r e

0.6312 (0.6270) 

SLIDE 12

Isothermal

 

3

z (1 2 ) T GM z r        

2 0 exp

2(1 2 )             

1/2 z

T   

 

1/2 2 2 3/2

6 tr( )T 3 sin 2 dz 1 e dz d

   

      

 

A

 

2 3/2 1/2

6 3 sin 2 1 e t T d r( ) 2        A

Optical depth

1/2 1/ 2 1/2

3(1 ) 3 T (z)dz d d 2



             



 

2 2 1/2

2 tr( ) 3 sin 2 1 (1 ) e        A

SLIDE 13

and (z) ) T(z 

 

2

T z z (1 2 )         

3/2 1/2 z

d 6 (2 ) 3 T sin 2 T tr( ) 2 d d q z           A

1/2 1/2 z 2 2 1 2

5 d (1 2 )(5 4 ) dT q T 4(2 ) d 2d T 4d T dz              

 

2 2

ˆ 9 tr( ) d 33 49 (1 ) 28 T       K

1

4 (6 13) d 5 T   

2 2

4 (6 13) d 35 T    

SLIDE 14

Differential Equations

2

T / T   F /   

1/2

z / T    F F 2 (1 2 )                 

 

2 2 1 2 2 2 3

C C F S F S         

1 1

C C ( )  

2 2

C C ( )  

2 3 2 1/2 2 1 0

C F d d S T C F d

 

       

 

Initial Conditions

(0) 1   (0)   

 

F 0 1 

SLIDE 15 i : ヨ

C , ' 2 ･ ) I

一針2

3 ･号F ㊥2 ,

1 1 4

w h e r e S i s t h e n

n

d i m e n s i

n

a l s h e a r r a t e ,

S =

f l ( r ) d C l …3 αs i n 2 X ,

C 2 = C 3 =

6 8 * ( 2

e

* )

q 3 / 2 t r ( A ) ,

5 汀l / 2 ( I

2

β) ( 5

4

β) ( 2

6

* ) M l ヱ

c

s

2 X

5

α

3 ( 1 十β) ' . ∨享訪

a =

2 ' β-

1 壁

1 4 ' a n d

w i t h

a n d S I M O N A N D J E N K I N S

( 5 8 ) ( 5 9 ) ( 6 )

t r ( A , 諸( 7 + 7 α2 . 2 J I P 2

2

α恒i ) I ( 6 1 ,

U p

n

i n t e g r a t i n g ( 5 4 )

v

e r t h e r i n g t h i c k n e s s , w e

b

t a i n

S

許

J

)

. S F O 2 d f ( ) . D F 2 3 d E T l

w h e r e

( 6 2 ) ( 6 3 )

W i t h ( 6 3 ) , w e m a y e m p l

y

E q . ( 3 6 ) t

s
l

v e f

r

S i n t e r m s

f

J a n d t

w

r i t e ( 5 4 ) a s

㊥〝=

2

( ㊥′) 2

β≡

+ B J F ( J F ㊥

I

) , C 弓

C 2 C 3

( 6 4 ) ( 6 5 )

w h e r e

F I G . L D i m e n s i

n

l e s s t e m p e r a t u r e d i s t n ' b u t i

n

v e r s u s n

n

d i n e n

s

i

n

a l a x I ' a l d t ' s t a n c e g f

r

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

f

E * . A s e ' → . 3 6 8 8 ,

⑳→1 . i s p

s

i t i v e f わr a l l v a l u e s

r

g 串. T h e i n t r

d

u c t i

n
r

t h e i n t e g r a l p a r a m e t e r J p e r m i t s u s t

e

a s i 一y i m p

s

e t h e c

n

d i

t

i

n

t h a t t h e h e a t h x v a n i s h e s a t l a r g e i a n d a l S

r

e m

v

e s

t h e q u a n t i t y S f r

m

t h e e q u a t i

n

s . T h i s i s a n a d v a n t a g e

b e c a u s e J r e m a i n s 爪n i t e f

r

a J I v a l u e s

f

S * , W h i l e S i n ･

c r e a s e s w i t h

u

t b

u

n d a s C * a p p r

a

c h e s t h e v a l u e a t

w h i c h s i n 2 X

.

I n t h e c

u

r s e

f

s

l

v i n g f

r

t h e d e n s i t y

a n d t e m p e r a t u r e p r

f

i l e s , J i s d e t e r m i n e d i t e r a t i v e l y .

T h e b

u

n d a r y c

n

d i t i

n

s a t t h e m i d p l a n e a r e ㊥( )

1

, ' ( )

,

a n d F ( )

)

. ( 6 6 ) E q u a t i

n

s ( 5 3 ) a n d ( 6 4 ) h a v e b e e n s

l

v e d s u b j e c t t

t

h e

b

u

n d a r y c

n

d i t i

n

s ( 6 6 ) u s i n g a R u n g e

K

u t t a m e t h

d
b

e g l n n l n g w i t h a n i n i t i a l g u e s s f

r

J . O n c e t h e p r

f

i l e i s

d e t e r m i n e d , ∫ i s r e c a 】c u l a t e d u s i n g ( 6 3 ) a n d t h e i n t e g r a

t

i

n

s

f

( 5 3 ) a n d ( 6 4 ) a r e r e p e a t e d . T h i s p r

c

e d u r e r e q u i r e s

n

l y a f e w i L e r a t i

n

s t

c
n

v e r g e .

T y p i c a l p r

f

i l e s f

r

㊨ a n d F a r e s h

w

n i n F i g s . 1 a n d

2 f

r

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

f

8 * ･ I n a l l c a s e s , t h e t e m p e r a t u r e

i n c r e a s e s w i t h d i s t a n c e f r

m

t h e m i d p ] a n e , r e a c h i n g a

c

n

s t a n t v a l u e a s i b e c

m

e s v e r y I a r g e . T h e l a r g e s t s u c h

t e m p e r a t u r e

c

c u r s i n t h e m a t h e m a t i c a l , b u t n

t

p h y s i c a l ,

l i m i t

f

p e r f e c t l y e l a s t i c c

u

i s i

n

s ･ , i t I ' s a b

u

t 1 5 % l a r g e r

t h a n t h e I . e m p e r a t u r e

n

t h e m i d p l a n e . F

r

8 8

.

3 6 8 8 , t h e t e m p e r a t u r e i s c

n

s t a n t t h r

u

g h

u

t . F r

m

F i g . 2 i t c a n b e s e e n t h a t t h e d e n s i t y p r

m

e s d

n
t

v a r y a p p r e c i a

b

l y f

r

d i f f e r e n t e * a n d a r e c )

s

e t

t

h e i s

t

h e r m a l . p r

触:

F *

e

x p L

2 ( 1

2

P )

( 6 7 )

SLIDE 16 i コ

O

N T H E V E R T I C A L S T R U C T U R E O F D I L U T E P L A N E T A R Y R I N G S 1 2 3 E 4

F I G . 2 . D i m e n s i

n

l c s s d e n s i t y d i s t r i b u t i

n

F v e r s u s n

n

d i m e n s i

n

a J

a x i a 一 d i s t a E t C e f f

r

d i f f e 作n 【 v a l u e s

f

e * . T h e j s

t

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

蝕

c

n

. e s p

n

d s t

s

事- ･ 3 6 8 8 .

F i n a l l y , F i g . 3 s h

w

s a d e t a i l e d e n e r g y b a l a n c e b r S *

.

I , w i t h p r

d

u c t i

n

( P ) , d i s s i p a t i

n

( D ) , a n d c

n
d

u c t i

n

( C ) d e 負n e d b y

P

B

J F O 2 , D

B

J 2 F 2 3 , a n d

C

㊥2

〝 + 2 ( ㊥' ) 2 . ( 6 8 )

i t i s s e e n t h a t p r

d

u c t i

n

a s w e l ] a s d j s s J P a t j

n
f

k j n e l j c

e

n e r g y i s h i g h e s t a t t h e m i d p l a n e . T h e d i s s i p a t i

n
f

e n ･ e r g y n e a r t h e c e n t r a l p l a n e d

m

i n a t p s t h e p r

d

u c t i

n

,

w h e r e a s f

r

l a r 砦e a x i a l d i s t a n c e s m

r

e e n e r g y i s g e n e r a t e d

I

t h a n c a n b e d i s s i p a t e d ; t h e r e f

r

e , h e a t i s c

n

d u c t e d f r

m

l a r g e a x i a l d i s t a n t e s t

t

h e c e n t r a l p l a n e . . 6 . 5 . 4 . 3 . 2 . 1 .

.

I 1 2 3 E 4

F I G . 3 . E n e r g y b a l a n c e w i t h p r

d

u c t i

n

( P ) , d i s s i p a t i

r

t ( D ) , a n d c

n

d u c t i

n

( C ) f

r

e +

.

1 v e r s u s n

n

d i m e n s i

n

a l a x i a l d i s t a n c e f .

1 1 5 O f c

u

r s e , t h e v a r i a t i

n

s

f

t h e v

l

u m e f r a c t i

n

a n d

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

n

l y r e l a t i v e t

t

h e i r v a l u e s

n

t h e m i d p t a n e

f

t h e r l n g . I n

r

d e r t

d

e t e r

■

m i n e t h e i r m i d p l a n e v a l u e s , a d d i t i

n

a l i n f

r

m a t i

n

i s r e

q

u i r e d . I n t h e n e x t s e c t i

n

, w e i n t r

d

u c e a d d i t i

n

a l i n f

r
m

a t i

n

a n d s

l

v e f

r

t h e s e m i d p l a n e v a l u e s a n d t h e

p a r t i c l e d i a m e t e r .

S A T U R N ' s A ･ R I N G

W e n

w

g l V e r O u g h e s t i m a t e s

f

t h e g r a n u l a r t e m p e r a

■

t u r e a n d t h e s

l

i d v

l

u m e f r a c t i

n
n

t h e m i d p l a n e a n d

t h e p a r t i c l e d i a m e t e r i n S a t u m ' s A

r

i n g ･ A s a r e s u l t

f

t h e 6 / 7 L i n d b l a d r e s

n

a n c e , t h e r e i s a t

r

q u e T 6 / 7 e x e r t e d a t t h e

u

t e r e d g e

f

t h e A

r

i n g . W e R r s t e q u a t e t h i s t

r

q u e

t

t

h ¢ m

m

e n t

f

t h e i n t e g r a t e d s h e a r s t r e s s ,

･ 6 ′7

4

w R i l

G

p K , p d z ･ ( 6 9 )

w h e r e R A i s t h e

u

t e r r a d i u s

f

t h e A

r

i n g , R A

l

･ 3 7 × l

g

m , w e d e 触e t h e s u r f a c e m a s s d e n s i t y ∑ b y

∑

2

( . T p d z ･ ( 7 )

a n d u s e t h e f a c t t h a t ∑主3 k g m ~ 2 a n d T 6 / 7 / ∑主1 . 1 3

× l

l

l m 4 s e c ~ 2 ( c u z z i e t a Z . , 1 9 8 4 ) . T h e n , u p

n

w r i t i n g K , p ≡ T α s i n 2 x a n d u s i n g t h e v a r i a b l e s

f

t h e p r e v i

u

s s e c t i

n

i n ( 6 9 ) a n d ( 7 ) , w e h a v e

t h a t t h e m i d p J a n e t e m p e r a t u r e a n d t h e m i d p J a n e v

l

u m e

f r a c t i

n

a r e g J V e n b y

T
a

n d

㌔/ 7 1

2 打∑R 2 ^ α s i n 2 X ∑f l

l

n

F d f ( ( . e F ◎2 d f )

.

( 7 1 ,

( ( . I F d f )

l

･ ( 7 2 ,

R e c a J 】i n g t h e d e h i t j

z

l ( 5 5 ) , t h e p a r t i c l e d i a m e t e r i s g i v e n

i n t e r m s

f

t h e s e b y

d

f

l '

( 7 3 ) w h e r e ∫ i s g i v e n i n t e m s

r

J b y ( 6 2 ) . W h e n t h e s e q u a n t i

t

i e s a r e e v a l u a t e d

n

t h e n u m e r i c a l s

l

u t i

n

s f

r

F a n d , t h e i r v a l u e s a r e d e t e r m i n e d a s f u n c t i

n

s

f

e * .

I t i s , p e r h a p s , m

r

e c

n

v e t l i e n t t

h

a v e t h e s e q t l a n t i l i e s

g J V e n a S f u n c t i

n

s

f

t h e

p

t i c a l d e p t h T d e 丘n e d b y

･
S

I . # y d z = 言! . x F d f

宅J

I . * F d f ･ ( 7 4 ,

SLIDE 17

6/7 Lindblad resonance

2 11 4

2

6/7 rf

T 4 R ρK dz 1.13×10 m s



  



Mass density

2

2 dz 300 kg m

 

   



6/7 2 2

Fd 1 T 2 R sin 2 F d T

 

       

 

1/2 s

1 2 T Fd



     



2 3 1/2 2 2 1 0

C F d T d C F d

 

      

 

SLIDE 18

S I M O N A N D J E N K I N S

( 2 3 T 4

F I G . 4 . T h e q u a n t i t i e s ヽ塙( m s e c l t ) , v

,

a ( m ) , a n d e * v e r s u s T .

B e c a u s e ( 7 4 ) r e l a t e s T a n d S *

n

s

l

u t i

n

s , t h i s i s e a s y t

d
.

T h e g r a p h s

f

T

,

u

,

d , a n d e * v e r s u s ナa r e s h

w

n

i n F i g . 4 . T h e r e l a t i

n

s h i p b e t w e e n S * a n d T t h a t r e s u l t s i s e s s e n

t

i a l l y i d e n t i c a l L

t

h e r e l a t i

n

s h i p s b e t w e e n S * a n d T d e t e r ･

m i n e d b y G

l

d r e i c h a n d T r e m a i n e ( 1 9 7 8 ) a n d A r a k i ( 1 9 9 1 ) . T h a t i s , t h e v a r i a t i

n
f

t e m p e r a t u r e n

r

m a l t

t

h e p l a n e

f

t h e r i n g d

e

s n

t

h a v e a d i s c e m i b l e e f f e c t

n

t h e r e l a ･

t

i

n

s h i p b e t w e e n t h e

p

t i c a l d e p t h a n d t h e c

e

f G c i e n t

f

r e s t i t u t i

n

. W e c a n t a k e a d v a n t a g e

f

t h i s l a c k

f

s e n s i t i v ･

l t y t O t h e v a r i a t i

n
f

t h e g r a n u l a r t e m p e r a t u r e n

r

m a l t

t

h e r I ' n g t

b

t a i n s i m p l e a p p r

x

. i m a t e f

r

m s

f

( 7 1 )

(

7 3 ) t h a t a r e m

r

e a m e n a b l e t

p

h y s I C a l i n t e r p r e t a t i

n

.

t f w e t a k e t h e t e m p e r a t u r e t

b

e u n i f

r

m a n d u s e t h e

i s

t

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

n

( 6 7 ) i n t h e i n t e g r a l s ( 7 1 ) a n d ( . 7 2 ) , t h e r e l a t i

n

F

r

r e s p

n

d i n g t

(

7 ) ) i s c l e a r l y a

d e t e r m i n a t i

n
f

t h e m i d p l a n e v a l u e

f

K , P t h a t h a s b e e n

s

l

v e d f

r

T

t

.

T

㌔/

7 1

2 打∑R 2 A α S i n 2 X

(

7 5 )

T h e q u a n t i t y α s i n 2 x i s t h e k i n e m a l i c v i s c

s

i t y i n t r

d

u c e d

b y G

l

d r e i c h a n d T r e m a i n e ( 1 9 7 8 ) , n

r

m a l i z e d b y 2 T

/

3 f l . A s t h e y s h

w

, t h i s n

r

m a l i z e d k i n e m a t i c v i s c

s

i t y

丘r s t i n c r e a s e s w i t h T a n d t h e n d e c r e a s e s . T h e m i n i m u m

i n t h e c u r v e

f

T

v

e r s u s T i n F i g ･ 4 i s a c

n

s e q u e n c e

f

t h i s . l n t h e s a m e l i m i t , ( 7 2 ) b e c

m

e s a r e l a t i

n

b e t w e e n

t h e m i d p l a n e v a l u e

f

K z z a n d I ,

:

∑f l 1

Z /

=

p , ∨註v T . ( I I 2 β)

(

7 6 ) B e c a u s e β i s a m

n
t
n

e i n c r e a s i n g f u n c t i

n
f

S * , t h e

i = 1

m a x i m u m i n t h e c u r v e

f

I ,

v

e r s u s T i s i n h e r L ' t e d f r

m

t h e m i n i m u m i n t h e c u r v e

f

T D V e r s u s T . T h e p a r t i c l e d i a m e t e r

t h e n f

l

l

w

s f r

m

( 7 3 ) w i t h J = ヽ巧, a n d t h e

p

t i c a l d e p t h

i s

3 q 2 α s i n

丁 =

2 X

2 ㌔( 2

6

* ) t r ( A )V

r 二両. ( 7 7 )

U p

n

e m p l

y

i n g ( 5 9 )

(

6 1 ) , t h e s e 叩a y b e e x p r e s s e d i n t e r m s

f

8 * a l

n

e .

■

l t i s i n t e r e s t l n g t h a t t h e i n t r

d

u c t i

n
f

s

m
d

e s t a n a m

u

n t

f

i n f

r

m a t i

n

p e r m i t s s u c h e x p l i c i t p r e d i c t i

n

s t

b

e m a d e

f

t h e m i d p l a n e v

)

u m e f r a c t i

n

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

n

t e x t

f

t h i s s i m p l e T n O d e J

f

t h e r l n g .

■

A C K N O W L E D G M E N T S

T h e a u t h

r

s h a v e b e m e 飢t e d f r

m

c

n

v e r s a t i

n

s w i t h J . A . B u r n s .

T h e y a r e a l s

i

n d e b t e d t

C

. Z h n g f

r

h i s a s s i s t a n c e w i t h t h e e t l e r g y

n u X , F i n a l l y , t h e y a r e g r a t e f u l t

S

. A r a k i f わr s u g g e s t i

n

s t h a t l e d t

t

h e i m p r

v

e m e n t

f

t h e m a n u s c n p t . T h i s w

r

k w a s b e g u n w h e n V

l

k e r S i n

n

w a s a n e x c h a n g e s t u d e n t a t C

r

n e t ) U n i v e r s i t y . H e i s p l e a s e d t

t

h a n k F . G . K

l

l m a n n a n d W . J l . S a c h s e , w h

i

n i t i a t e d t h i s e x c h a n g e

p r

g

r a m a n d t h e r e b y l a i d t h e f

u

r l d a t i

n

f

r

t h e c u r r e n t c

l

l a b

r

a l i

n

･ H e i s a l s

a

p p r e c i a t i v e

f

t h e h

s

p i t a ) i t y s h

w

n t

h

i m d u r i n g v i s i t s t

t

h e D e p a r t m e n t

†

T h e

r

e t i c a l a n d A p p l i e d M e c h a n i c s , C

m

e l L U n i v e r

s

l l y , I ' n t h e s u m m e r s

f

1 9 8 8 a n d 1 9 8 9 . T h e s e v i s i t s w e r e s u p p

r

t e d b y

t h e U . S . A r m y R e s e a r c h O f G c e t h r

u

g h t h e M a t h e m a t i c a l S c i e n c e s

I n s t i t u t e a t C

r

n e H U n i v e r s i t y .

R E F E R E N C E S

A R A K l , S . , A N D S . T R E M A J P J E 1 9 8 6 . T h e d y n a m i c s

f

d e n s e p a r t i c l e

d i s k s . J c a r t L S 6 5 , 8 3

I

9 .

A R A K I , S . 1 9 8 8 . T h e d y n a m i c s

f

p a r t i c l e d i s k s . I l . E f f e c t s

f

s p l n

d e g r e e s

f

f r e e d

m

. I c a r L J S 7 6 , 王8 2

I

9 8 .

A R A K l , S . 1 9 9 1 . T h e d y n a m i c s

f

p a r t i c l e d i s k s . I l l . D e n s e a n d s p i n n i r L g

p a r t i c l e d i s k s . J c L l r 〃S 9 , 1 3 9

1

7 1 . C u Z Z I , ∫. N . , ∫. ∫. L I S S A U E R , し. W . E s p

s

I T O , ∫. a . H

L

B E R G , 巳. A . M A R O U F , G . し. T Y L E R , A N D A . B

I

S C T I O T 1 9 糾. S a t u r r l l s r i n g s : P r

p

･ e r t i e s a n d p r

c

e s s e s . 1 n p l a n e t a r y L h ' T 7 g S ( 良. J . G r e e n b e r g a n d A ･

B r a h i c , E d s ) . p p . 7 3

2

.

G

L

D R E t C H , P . , A N D S . T R E M A l N E 1 9 7 8 . T h e v e 一o c i t y d i s p e r s i

n

i n S a t u r n ' s n n g s . t c E z r n L ･ 3 4 , 2 2 7

2

3 9 .

J

E N K J N S , ∫. T . , A N D M . W . R I C H M A N 1 9 8 8 . P l a n e s i m p l e s h e a r

f

s m

t

h

i r l e J a s t i c c i r c u l a r d i s k s : T h e a n i s

t

r

p

y

f

t h e s e c

n

d m

m

e n t i n t h e

d i l u t e a n d d e n s e 一i m i t s . J . F t L I f d M e c h . 1 9 之, 3 1 ト3 2 8 .

R l c H M A N , M . W . I 9 8 9 . T h e s

u

r c e

f

s e c

n

d m

m

e n t i l l d i l u t e g r a n u l a r n

w

s

f

h i g h l y i n e ) a s t i c s p h e r e s . ) . R F l e O l . 3 3 , I 2 9 3

1

3 6 .

S H U H K M A N , G . 1 9 糾. C

H

i s i

n

a l d y n a m i c s

r

p a r t i c 一e s i n S a t u r n

s

r i n g s . A s t r

n

. Z h . 6 1 . 9 8 5

1

4 : t r a n s l a t e d i n S

t

l . A s r r

l

7 . 2 8 ,

5 4 7

5

8 5 . s T E W A ふT , G . A . . D . N . C . L I ‖, A N D P . B

D

E N H E I M E . R , 1 9 8 4 . C

l

l i s i

n

i r l d u c e d t r a n s p

r

t p r

c

e s s e s i n p l a n e t a r y r l n g S . l n P l

1

7 e t L T r T ･ L u n g s ( R ･ J ･ G r e e r l b e r g a n d A ･ B r a h i c , E d s . ) , p p ･ / 4 4 7

5

I 2 . Z H A N G , C . 1 9 9 3 . K J ' n Q t L ' c T h e

r

y f

r

R a p L ' d G r a f 7 L L l a r F l

w

s , P h . D . D i s

s

e r t a t i

n

, C

r

r t e l l U n i v e r s i t y , I t h a c a , N Y .

′

SLIDE 19

Shock Waves around a Moonlet in a Planar Ring Steady, homogeneous energy balance:

r

P



     

2

4a(1 e) T d   

1/2

2 d a G     (1 e) 2   

2

(16 7 ) G 16(1 )       ˆ P (p a) 2 D

  

    

 

p 1 2 G T    

 

1/2 1/2 1

G 3 2 1 a dT 2 8 (5 3 )G               

2 2 3 2 2 2

T 1 1 G G(3 2G) 3G 1 d 16(1 ) 4 G (5 3 )                       

SLIDE 20

Dimensionless ring temperature

 

*

T T / d    versus area fraction 

SLIDE 21

Isentropic Sound Speed

4 3 2 2 4 s

p 9 32 24 128 a T 64(1 )                   0.2, 0.8  

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 5 10 15 20 25 30 35

Area Fraction (ν) Normalized Sound Speed (a/T1/2)

Non−dilute Dilute

SLIDE 22

Mach Number

u y   

 

2 2 3 2 2 2 3

64G 5 8 3 u y T d G (20G ) (3 2 G 12G) 3 G                              

2 4 3 2 4

a 9 32 24 128 T 64(1 )         

Mach Number M

2 2 2

u / T M a / T 

SLIDE 23

Mach number M u / a 

, with

 

2 s

a p /    , normalized by dimensionless vertical displacement,

*

y y / d  , versus sound speed

SLIDE 24

Simulations Two-dimensional flow of identical, frictionless, circular disks Event-driven simulations of hard-particles Homogeneous Hill equations

2 2 2

d y dx 2 3 y dt dt     

2 2

d x dy 2 dt dt   

Time made dimensionless by Ω, lengths by d moonlet diameter D Parameters: D/d, e, ν

SLIDE 25

Particle velocity

D/d = 25, ν = 0.5, e = 0.3

260 270 280 290 300 310 320 330 75 80 85 90 95 100 105 110 115 120 125

Azimuthal Direction (Particle Diameters) Radial Direction (Particle Diameters)

SLIDE 26

D / d 30, 25, 15, 10  with e 0.3  and 0.5  

SLIDE 27

e 0.3, 0.5, 0.6, 0.8  with D / d 25  and 0.5  

SLIDE 28