HAL Id: jpa-00224320
https://hal.archives-ouvertes.fr/jpa-00224320
Submitted on 1 Jan 1984
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
STICKING EXPERIMENTS AND
NON-GRAVITATIONAL COMPONENT OF THE MECHANISM OF THE GROWTH OF PLANETS
J. Leliwa-Kopystyński
To cite this version:
J. Leliwa-Kopystyński. STICKING EXPERIMENTS AND NON-GRAVITATIONAL COMPONENT OF THE MECHANISM OF THE GROWTH OF PLANETS. Journal de Physique Colloques, 1984, 45 (C8), pp.C8-109-C8-112. �10.1051/jphyscol:1984821�. �jpa-00224320�
JOURNAL DE PHYSIQUE
Colloque C8, supplement au n ° i l , Tome 45, novembre 198* page C8-109
STICKING EXPERIMENTS AND NON-GRAVITATIONAL COMPONENT OF THE MECHANISM OF THE GROWTH OF PLANETS
J . L e l i w a - K o p y s t y n s k i
Institute of Geophysics, Warsaw University, ul. Pasteura 7, 02-093 Warszawa, Poland
Abstract - The experiments concerning collisional sticking of metallic bodies (Pb, Sn, Fe ) have been performed. The impact velocity range was 50 to 650 m/s. The mapping of the results of collisions (rebound, sticking, partial sticking) in the impact angle - impact velocity plane was made. The application of this results to planetary accretion problems is presented.
I - COLLISIONAL STICKING OF SOLID GRAINS
However the problem of formation and growth of planets during the early stage of evolution of the Solar System have been intensively studied theoretically during the last decades, any experimental foun- dation is very esential. In particuliar the collisional experiments concerning growth of solid grains are of special interest. Three main mechanism of mass increase as a result of collision should be taken into account: Van der Waals interaction for micron-size bodies, gra- vitational interaction for kilometer-size and biger bodies, and ther- momechanical sticking for intermediate sizes. Such a classification beeing rather roughly can be improved taking into account two impor- tant factors: collisional velocity and composition of colliding bodies.
This work deals with the collisional experiments concerning ductile materials. For convenience we call one of colliding bodies the target and the other the projectile. However such a definition has no special meaning for the solid grains orbiting in the space, it is usefull for description of laboratory experiments. Let us define two parameters:
1. Mass change parameter, being a parameter of individual collision f _ mass of target after collision - mass of target before collision
mass of projectile
2. Sticking coefficient, being a statistical parameter number of collisions with sticking Y " total number of collisions
We introduce the classification: S = 1 perfect sticking, 0< <S <1 par- tial sticking, 5 = 0 rebound, and <$ < 0 mass losses by droplets or solid fragments production. The parameter <$ depends on a lot of vari- ables, mainly: material constants, geometrical parameters, kinemati- cal parameters, and ambient conditions especially pressure and tempe- rature. For collision in vacuum of spheres made from the identical materials there is
Résumé - Des expériences sur la jonction métal-métal produite par des collisions de moyenne vitesse (de 50 à 650 m/s) ont été effectuées. Les résultats des collisions sont présentés dans le plan de coordonnées : angle de collision - vitesse de collision.
L'application de ces résultats aux problèmes d'accrétion en Planétologie est étudiée.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984821
C8-110 JOURNAL DE PHYSIQUE
8 = 6 ( m a t e r i a l p a r a m e t e r s , r l , r 2 , V , ly, T o )
w h e r e r , , r a r e t h e r a d i u s , V i s t h e i m p a c t v e l o c i t y , Y. i s t h e i m p a c t a n g l e a n d T~ i s t h e t e m p e r a t u r e o f s p h e r e s . F o r f i x e d n a t e r i a l , s i z e a n d t e t i a e r a ' ? u r e t h e p a r a f i e t e r 8 i e p e n d s o n it an; y o n l y . T h e s e a r c h o f 6 = ( V , w ) i s t h e m a i n e x p e r i m e n t a l p u r p o s e o f t h i s o o r k . T a k i n g i n t o a c c o u n t t h e v a l u e o f d t h e s t i c k i n g c o e f f i c i e n t 1 c a n b e r e d e f i - n e d m o r e p r e c i s e l y a s f o l l o n
nuniber o f c o l l i s i o n s v r i t h CS = 1
7 4 = t o t a l n u m b e r o f c o l l i s i o n s
T h e c a l c u l a t i o n o f 1, c a n b e d o n e o n t h e b a s e o f n a p o f 6, a n d i f t h e s i z e a n d v e l o c i t y d i s t r i b u t i o n s o f c o l l i d i n g g r a i n s a r e k n o w n .
I1 - EXPERIMENTS
T h e r e i s known a l o t o f e x p e r i m e n t s c o n n e c t e d w i t h p l a n e t o l o g y d e a l i n , g w i t h c o l l i s i o n a l b e h a v i o u r o f b r i t t l e m a t e r i a l s ( c e n t i m e t e r o r d e c i m e - t e r s i z e d i-oclc-rock c o l l i s i o n s ) . A d e c r e a s i n o o f t a r g e t a n d p r o j e c t i l e m a s s i s o b s e r v e d a s a r u l e . T h e e x p e r i m e n t s l e a d i n g t o i n c r e a s i n g o f t a r g e t m a s s c a n b e e x p e c t e d f o r d u c t i l e m a t e r i a l s , t h e r e f o r e f o r m e t a l - m e t a l c o l l i s i o n s a n d e v e n f o r r o c k - r o c k c o l l i s i o n s , b u t i n e l e v a t e d t e m p e r a t u r e , i n t h e b r i t t l e - d u c t i l e t r a n s i t i o n r e g i o n n e a r a m e l t i n g p o i n t .
mls, veiocity V m/s, velocity V b )
Vcos Y = 90 m I s
d I I I I I I S V L I 1 I
lo0 20° 30" 40" 50" 60" 10" 20" 30"
impact angle Y impact angle V
F i g . 1 - T h e m a p s o f c o l l i s i o n a l r e s u l t s a s o b s e r v e d f o r P b - P b i m p a c t . E a c h r e c t a n g l e c o r r e s p o n d t o o n e e x p e r i m e n t . T h e s i z e o f r e c t a n g l e c o r r e s p o n d t o a n e x p e r i m e n t a l e r r o r . T h e n u m b e r n e x t t o r e c t a n g l e d e - n o t e s t h e m a s s c h a n g e p a r a m e t e r 6 . T h e c o n t i n u o u s o r d a s h e d l i n e s a r e
t h e l i m i t s o r u n c e r t a i n l i m i t s o f d i f f e r e n t - t y p e m a t e r i a l r e s p o n c e f o r i m p a c t . ( a ) S p h e r i c a l p r o j e c t i l e a n d s t i f f m o u n t e d " i n f i n i t e " t a r - g e t . (b) H e m i s p h e r i c a l p r o j e c t i l e a n d f r e e h a n g e d s p h e r i c a l t a r g e t .
F i g u r e 1 p r e s e n t s t h e maps o f mass change p a r a m e t e r f o r Pb-Pb c o l l i s i o ~ ~ s a s o b t a i n - ed i n two d i f f e r e n t s e r i e s o f e x p e r i m e n t s , a c c o r d i n g t o ( 1 ) and t o t h e e x p e r i m e n t s done by K . K a n i , J . Leliwa-KopysCytiski and T. M a t s u i . Two a c c e l e r a t i n g guns have been used : t h e gun of Tokyo I n s t i t u t e o f Technology (TIT, r e f e r e n c e 1 ) and t h o s e of Okayama U n i v e r s i t y (OU). F o r TIT gun t h e s p h e r i c a l m e t a l l i c p r o j e c t i l e , r a d i u s 4 mm, have been a t t a c h e d t o c y l i n d r i c a l p l a s t i c o r aluminium s a b o t and a c c e l e r a t e d by powder s y s t e m ( f o r v e l o c i t i e s V < 200 m / s ) . The c o n i c a l s a b o t c a t c h e r mounted between gun muzzle and t a r g e t have s e p a r a t e d a p r o j e c t i l e from s a b o t . P r o j e c t i l e impacted a n " i n f i n i t e " t a r g e t ( t a r g e t m a s s / p r o j e c t i l e mass > 300) ; impacted s u r - f a c e was i n o b l i q u e p o s i t i o n r e l a t i v e l y t o t h e t r a j e c t o r y of p r o j e c t i l e . By means of TIT gun t h e c o l l i s i o n s Pb-Pb, Sn-Sn and Fe-Fe was i n v e s t i g a t e d . The OU powder gun a c c e l e r a t e d a h e m i s p h e r i c a l p r o j e c t i l e , r a d i u s 7 mm, a t t a c h e d t o a p l a s t i c s a b o t w i t h a r a d i u s 7 . 5 m. No s a b o t c a t c h e r have been u s e d s o t h a t a s p h e r i c a l Pb t a r g e t , r a d i u s 15 mm, was impacted by p r o j e c t i l e and s a b o t f l y i n g t o g e t h e r .
However t h e maps p r e s e n t e d on F i g . l a and l b a r e found f o r d i f f e r e n t g e o m e t r i c a l c o n d i t i o n s , t h e common f e a t u r e s c a n be d i s t i n g u i s e d , m a i n l y a n e x i s t a n c e o f c e r t a i n a r e a s : 1 - rebound o f p r o j e c t i l e w i t h p o s s i b l e p l a s t i c d e f o r m a t i o n , & = 0 ; 2 - p e r - f e c t s t i c k i n g w i t h c r a t e r i n g by p l a s t i c d e f o r m a t i o n and by s u r f a c e m e l t i n g , 6 = 1 ; 3 - p a r t i a l s t i c k i n g w i t h s t i c k i n g e f f e c t g r e a t e r t h a n t h e p r o d u c t i o n of e j e c t a , O<6<1 ; 4 - mass l o s s e s a r e a i n which t h e p r o d u c t i o n of m o l t e n d r o p l e t s i s g r e a t e r t h a n t h e s t i c k i n g e f f e c t , 6<0 ; 5 - d r o p l e t - l i k e rebound of p r o j e c t i l e . The c o n t i - nous l i n e s d e n o t e t h e 1-2 and 2-3 b o r d e r y which c a n be c l e a r l y d i s t i n g u i s h e d i n t h e F i g . l a . The dashed l i n e s a r e t h o s e which a r e n o t found v e r y e x a c t e l y , o r even t h e y a r e t h e h y p o t h e t i c a l l i m i t s o f o t h e r a r e a s .
The r e s u l t s f o r Pb-Pb c o l l . i s i o n s s u g g e s t t h a t t h e r e i s a s h a r p b o r d e r l i n e between t h e a r e a o f p l a s t i c rebound and t h e a r e a of p e r f e c t t h e r m a l s t i c k i n g ( w e l d i n g ) . I t c a n be f i t t e d t o c o n s t a n t normal v e l o c i t y l i n e .
V c o s = V n , s t ( m a t e r i a l p a r a m e t e r s , t e m p e r a t u r e , p r o j e c t i l e t o t a r g e t r a d i u s r a t i o )
Vn s t i s t h e minimum i n c i d e n t v e l o c i t y e n o u t h f o r p r o d u c i n g t h e s t i c k i n g i n t h e caBe of normal i m p a c t .
I n t h e T a b l e 1 t h e e x p e r i m e n t a l r e s u l t s f o r t h e c o l l i s i o n s o f s p h e r i c a l p r o j e c t i l e w i t h t h e f l a t " i n f i n i t e " t a r g e t f o r t h r e e m e t a l s a r e p r e s e n t e d .
T a b l e 1 m a t e r i a l
Pb - Pb Sn - Sn Fe - Fe
For t h e p r o j e c t i l e t o t a r g e t mass r a t i o e q u e l t o one t h e v a l u e s o f V a r e e s t i - mated t o b e h i g h e r a b o u t 40%. Taking i n t o a c c o u n t t h e h e a t c o p a c i t y gfs!he c o l l i - d i n g b o d i e s i t c a n be e s t i m a t e d t h a t i f t e m p e r a t u r e i n c r e a s e s t h e V n , s t d e c r e a s e s . Near t h e m e l t i n g p o i n t t h e V i s p r o b a b l y e q u a l t o a h a l f of i t s room tempera-
n,st
t u r e v a l u e . Such a n e x t r a p o l a t i v n c a n b e done a l s o f o r t h e o t h e r m a t e r i a l s , e . q . s i l i c a t e s f o r t h e t e m p e r a t u r e s above b r i t t l e - d u c t i l e t r a n s i t i o n .
From t h e F i g . 1 i t c a n be e a s l y deduced t h a t t h e a s s u m p t i o n t h a t q=1 ( f r e q u e n t l y a d o p t e d i n t h e a c c r e t i o n t h e o r i e s ) i s n o t t r u e .
I11 -'APPLICATION FOR THE PROTOPLU'ETARY CO1,LISIOXS.
The c o - p l a n a r b e l t o f o r b i t i n g b o d i e s ( g r a i n s , p 1 a n e t e s i m a l s ) w i l l . be c o n s i d e r e d . L e t u s assume t h a t t h e m a j o r s e m i a x e s o f a l l o f t h e e l l i p t i c a l o r b i t s o f t h i s b o d i e s i s a = c o n s t ; t h e e x c e n t r i c i t i e s a r e e (O<e<e < < I ) . I t can be c a l c u l a t e d t h a t ap- p r o a c h v e l o c i - t y V of two o r b i t i n g b o d i e s inm%e moment of c o l l i s i o n i s g i v e n by a f o r m u l a
' n , s t
8 0 - 100 m/s
138 - 168 m/s 450 - 543 m/s
mean p r e s s u r e f o r p e r p e n d i c u l a r impact w i t h v e l o c i t y V
n , s t 0.1 - 0 . 2 GPa
0 . 2 - 2 GPa
2 - 1 0 GPa
JOURNAL DE PHYSIQUE
where
is the a @ =.
unlt .
mean orbital velocity.
1,s x 10"m is the astronomical a denotes the angle between the long axes of the orbits. The mean value of a square of approach veloci- ty, with respect to different excent- ricities and different a is
2 2 2 2
<v > = ? -
"0 max
It can be found that the unelastic energy trapped inside of the sticking colliding spherical bodies (ml, m2 and r1, r2 denote their mass and ra- dius respectively) is
where )I is an impact angle and A < 5 1 7 is a factor depending on the moments of inertia and on rotational energy.
On the other hand the gravitational energy of mutual attraction in the moment of
collision is mlm2
= G -
Egrav r +r 1 2
For the case the most interesting for planetary accretion is ml>>m2 and r >>r
1 2 '
where the index one denotes the growing embryo of a planet. In such the case ~t can be estimates that
< AEorb > L max
-- E i 1.06 x 1018
(2 4
grav
t l . Pl .
where p, is the density of growing planet, in kg.~n-~, and r, is in meters. This relationlis presented on the Fig. 2 for the growin planets hade from iron, pl = 8 . 6 ~ 1 0 ~ kg.~n-~ or from silicates, pl = 3.0~10 kg.~n-~. 5 Below the respective llne the collisional energy prevails the gravitational energy of planet formation.
In conclusion we find that the inner planets during their formation have grown and trapped the energy mainly due to orbital collisions ; the outer ones have grown mainly due to embryo-grains gravitational at~raction.
Reference :
J. LELIWA-KOPYSTY~SKI, T. TANIGUCI-11, K. KONDO and A. SAWAUKA, Sticking in Moderate Velocity Oblique Impact Application to Planetology; Icarus, 57, 280-293 (1984).
