Apse-line

We note that the scattering angle x an<3 the polar angle at t —> oo are related according to % + 23(oo) = n. The dipole moment fi(R) is parallel to the intermolecular axis. Introducing a Cartesian frame with its x-axis along the apse line and the y-axis in the collision plane as illustrated in Fig. 5.7, we get from the definition of for i = 1,... 

In this section we consider a scattering event, viewed in a frame where the origin of the relative vector r is fixed (i.e., equivalent to the center of mass frame), as sketched in Fig. 2.3. The line joining the two molecules when at the points of closest approach, rmim is called the apse-line. This apse-line passes through O, the intersection of the two asymptotes g2i and g 21. The unit vector k of the apse-line bisects the angle between —g2i and g2i, as the orbit of the second molecule relative to the first is symmetrical about the apse-line. This symmetry is a consequence of the conservation of angular momentum L = pgh = pg h [61] [35]. 

For example, in the particular case when the molecules are rigid elastic spheres the apse-line becomes identical with the line of center at collision. In this case the distance d 2 between the centers of the spheres at collision is connected with their diameters d, d/2 by the relation [77] ... 

In the analysis of a binary collision between molecules with velocities Ci and C2 the direction of the line of centers at collision can be specified by the unit vector k along the apse-line (i.e, in the particular case of rigid spherical molecules, the apse-line corresponds to the line joining the centers of the two molecules at the instant of contact), Fig. 2.7. This unit vector is precisely characterized by the polar angles (V ,<( )-... 

To deduce the formula for the dense gas collision frequency a modified relation for the volume of the collision cylinder is required. As mentioned in chap 2, it is customary to consider the motion of particles 2 relative to the center of particles 1 (see Fig 2.2). For a binary molecular collision to occur the center of particle 2 must lie on the sphere of influence with radius di2 about the center of particle 1, see Fig 2.7. The radius of the sphere of influence is defined by (2.152). Besides, since the solid angle dk centered about the apse line k is conveniently used in these calculations in which the billiard ball molecular model is adopted, it is also necessary to specify the direction of the line connecting the centers of the two particles at the instant of contact [86]. The two angles 6 and 4> are required for this purpose. Moreover, when the direction of the apse line lies in the range of 0, 4> and 6 - - dO, 4> + d4>, at the instant of collision, the center of particle 2 must lie on the small rectangle da cut out on the sphere of influence of particle 1 by the angles dO and d< >. The area of this rectangle is ... 

In the discussion of dilute gases in sect 2.4.2 the corresponding surface area element is determined by the product da = bdbd(f>, as illustrated in Fig 2.10. For the billiard ball molecular model the link between the two surface element formulas when centered about the apse line is defined analogous to (2.159). 

It is required that the component of the relative velocity perpendicular to the apse line should be unchanged in a collision, thus the impulse J12 must act entirely in the k direction. On this demand, J12 can be determined from (4.21), (4.22) and (2.123). The impulse of the force exerted by particle 1 on particle 2 is given by ... 

It will be useful below 10 consider another way to determine vi and vi in terms of Vi and V2. We note that since the forces are central, the orbit of particle 2 in the relative coordinate frame is symmetrical about the apse line. This is the line from the origin to the center of particle 2, at the point of closest approach. ... 

To see that the orbit must be symmetrical about the apse line, consider the time-reversed motion in the relative frame. By rotating the coordinate system about the origin, one should be able to make the time-reversed trajectory coincide with that of the forward trajectory. 

To construct Enskog s extension of the Boltzmann equation to higher densities, we consider only the change in /(r, v, /) with time due to collisions, since only this term is affected by the density of the gas. Let us first consider the change in /(r, v, /) due to the direct collisions. For a dilute gas we have argued that the number of direct collisions taking place in 8t in time 8t between molecules with velocity v and molecules with velocity Vi, with apse line in direction dk about k is... 

Fig. 16. Direct (a) and restituting (b) collisions for a collision of two hard spheres. We consider a relative coordinate system centered on particle 1 with z axis in the direction of g=Vi—V. In (a) the apse line is in the direction k, and in (b), the apse line is in the direction —fc. The action sphere of radius a about particle 1 is indicated by the circle.

Figure 4.2 The collision trajectory in the c.m. system. The solid curve represents a trajectory with initial velocity v, impact parameter b, and mass fi. The relative separation R(t) is uniquely defined in terms of the distance /land the orientation angle if. The trajectory is symmetric about the apse line, which passes from the origin through Rq where Rq is the distance of closest approach. The final deflection angle is X =jt — 2 0 where ifo is the value of f at the mid-point of the trajectory.

As for dilute gases, corresponding to any direct collision specified by the variables c, Cl, k there is an analogous inverse collision in which c, ci are the velocities of the molecules after the collision, while -k is the direction of the apse-line. In such a collision the center of the second molecule is at r - - di2k, while the point of contact is at r -I- idi2k. The correction function is still determined at the point of contact between the two particles, hence the collision frequency for the inverse collision is approximated as ... 

The relative velocities of the centers of the spheres immediately before and after a collision are still given by (2.111). For these inelastic particle collisions it is required that the relative particle velocity component normal to the plane of contact, g2i k (before collision) and k (after collision) satisfy the empirical relation (2.110) [68], If the restitution coefficient therein is equal to one, the collision is elastic, which means that there is no energy loss during collision. Otherwise the collision is inelastic, which means that there is energy dissipation during collision. It is required that the component of the relative velocity perpendicular to the apse line should be unchanged in a collision, thus the impulse J12 must act entirely in the k direction. On this demand, J12 can be determined from (4.47), (4.48) and (2.110). 

Inhnitesimal solid angle centered around the apse-line (steradians, sr)... 

---