SHELL OF THICKNESS �R
( 4 4) Since these definitions do not vary appreciably
(Wetherill, 1980) , the latter will be used here for ease of
comparison with previous work. By substituting (44) into
(4 3) , the minimum deflection in an encounter (xd) can now be calculated.
3. 4 THE MEAN AND ROOT-MEAN -SQUARE DEFLECTIONS IN AN ENCOUNTER
For an encounter where the planetary mass is large and the impact parameter is small, the deflection x will be large; a small relative velocity would enhance the deflection
(c. f. equation 4 1 ) . In such an encounter all memory of the
original traj ectory will be lost , excepting that the Jacobi constant (Tisserand invariant) will be conserved (e. g. Kresak , 197 9) .
However , most passages are distant , so that the angular deflection is small and the new particle orbit is not
substantially different from the original orbit. In a phase
space described by some form of heliocentric angular coordinates, multiple encounters by the particle with different planets
40.
deflection. Multiple encounters thus result in a
diffusion of the orbit until it no longer resembles the original particle orbit . The expectation value of the
square of the ultimate deflection (x ) is just the sum of the u
squares of the individual deflections; i. e . , for i encounters,
I I
Opik (1966 b ) required that an accumulated Xu = 90° should occur for a complete ' loss of memory ' by the particle of its original orbit .
The above indicates that the mean and/or root -mean square deflection per encounter would be of use in describing the diffusion of a particle orbit .
these two parameters.
In this section I derive
The Rutherford Scattering differential cross-section for deviation x is:
do ( 4 5 )
(Landau and Lifshitz, 19 76; p . 53 ).
For a particular encounter the bracket is a constant,
and the trigonometric part can be integrated (formula
322 , p . 568, CRC Handbook ) to give a total cross -section as:
Equation (4 6 ) can be used to determine an alternative expression to (40 ) for the collision cross -section , as follows .
For an impact upon the planet, the minimum deflection required
is Xmin = Xg , given by ( 4 2 ) The maximum deflection
the collision cross -section is:
[ (Gm1 )2
] [ _2
ac = TI
-;;; sin (
Xg/2) ( 4 7 )
Substitution of the relevant planetary parameters and any required relative velocity shows that this renders
41.
the same result as found from (40) . In all numerical work
I have used (40) since it requires less computation.
A non -obvious error in Weidenschilling (1975a) should
be pointed out here . Clearly his equations (12) and (13) do
not lead to his equation (1 7) , a factor of TI being missing. Similarly , the TI is missing from his equation (18) . H owever,
this is correct since his normalised units are equivalent
to having a unit of time of (l /2TI ) times the planet 's orbital period , and a unit of area (1 / TI ) . This follows from taking the unit of length as being the planet 's semi -maj or axis , and the unit of velocity the planet 's orbital velocity. Weiden schilling 's equation (18 ) therefore gives an answer which is numerically correct , and for his units all is right if the factor of n is deleted from his equation (12) .
Equations (45) and (46) can now be used to calculate the probability of obtaining any angular deflection x within
an interval d x. This depends only upon the relative velocity
V for any particular planetary encounter since xd is a function
of V. Since xmin = xd and xmax = 180° this probability is
P (x l v) d x = dn 0 - 2
sin (xd/2) -1 ( 4 8)
Note that this is the probability of such a deflection per encounter , and the definition of an encounter depends upon the definition of d (equation 44 ) . By accepting a larger
value of d , and hence a smal ler minimum deflection Xd ' the probabil ity o f a certain large deflection per encounter i s decreased markedly . The probabi lity o f such a deflection per unit time remains constant .
The mean deflect ion ( xBAR) per encounter can now be found . This is
X g
J X do XBAR 0nc
Xd
where one is the non-collisional cross-section: or
o = nd 2 - o
nc c
4 2 .
s ince any X � x results in a collision . This cross-section g can be used in equation ( 7 ) to find P0 , the probability of a de flection .
After cancel lation of the constants , X BAR = [ sin -2 ( X d / 2 )
( 5 0 )
which can be integrated by parts (formu la 5 , p . 5 4 2 ;
formu la 3 0 8 , p . 5 6 7 ; formu la 3 2 2 , p . 5 68 , CRC Handbook ) to give :
( 5 1 )
in a
Next the root-mean -square deflection XRMs is derived similar way. Since
1
rg
2 x2 do
XRMS 0nc
Xd
4 3.
an equation for X�s is derived which is identical to (50 )
except that the x inside of the integral is replaced by x2 .
The equation can be integrated as before ( using additionally
formula 293 , p. 566 , CRC H andbook ) to get :
1-xd2 sin- 2 (xd/2 )
+ 4xd cot (xd/2 )
x; sin-2 (Xg/2 )
1
- 4 x g cot (Xg/2 )
[ +B £n (sin (Xg/2 ) ) - 8 £n (sin (Xd/2 ) )J (52 )
In Weidenschilling (19 75 a ) the equivalent to this (his equation 2 7 ) contained an error .
A typical value of xRMS � (2 . 6 to 2 . B )xd is derived.
3. 5 PROBABILITY OF EJECTION
Using the deflection range xd to xg it is now possible
to calculate the probability of an encounter resulting in the particle being ejected from the solar system.
The planet is considered to be a scattering centre which is momentarily on a circular orbit of radius
R ,
so that the circular velocity is:At this solar distance the critical heliocentric velocity for ejection from the solar system is
so that V2 = 2V2
C 0
As discussed previously, in the encounter it is
( 5 4 )
44.
assumed that VRC ' the relative velocity of the particle and the planet taken to be on a circular orbit, is conserved in magnitude but not in direction. In Figure 4a,
v
62 is the particle velocity and 0 is the encounter angle, given by equation (3 7) with y1 = 0 since a circular orbit is assumed. Thuscos 8 = cos y2 cos (a1 ±a2)
and VRC is given by
v
2 =v
2 +v
2 -�v v
cose
RC 2 o � 2 o( 5 5 )
( 5 6 )
Figure 4b shows the geometry after the encounter.
A deflection x results in VRC being conserved with its
orientation changed from E to E', and thus its heliocentric
velocity is changed to v�2 . This new velocity will not in
general be in the same plane as v62 , and it is assumed that all orientations of the new relative velocity vector VRC are equally likely for a given E' (i. e. the scattering is rotationally symmetric) .
This is more clearly shown in Figure 5. A deflection x results in a new direction of VRC which may be anywhere on the circle ABCD.
VI >- V
B2 r c
45 .