## Theory

Dealing with a directional meteoroid distribution (like that described by Divine) as it encounters a gravitating body such as the Earth is a daunting mathematical task. However, by invoking the often-taught but seldom-used Liouville's theorem, it is possible to greatly simplify the problem. Liouville's theorem states that an ensemble of particles moving under the influence of conservative forces (such as gravity) maintains its density in phase space (see ). For an ensemble of N particles traveling from point "A" to point "B" in a conservative field,

A3xa A3va A3xb A3vb '

where the A terms represent the dimensions of the six-dimensional "box". No matter how the six-dimensional phase space "box" changes in shape due to forces, its volume containing the ensemble remains constant. Defining the functions

where N is the number of particles, N(x) the spatial density, and N(x,v) the phase space density, we can rewrite equation 4 in its differential form where £2 refers to the direction of the velocity, xt /—'—\ xt ,—"—\ N. (xA, v.) d3v, NR(xR, vj d3v„ NA(xA,vA) = NB(xB,vB) or 2 J ' = 2° ' ■ (6)

The numerators in equation 6 are spatial densities, but defined for a narrow range of velocities. We now define the flux magnitude at a point per unit speed per unit solid angle direction <5,

A new approach to applying interplanetary meteoroidflux models _ ~ v N(x, v) d3v

dv dQ Equation 6 now becomes

Unlike in equation 1, the full directional information is here preserved, and a velocity distribution is used rather than assuming a single velocity.

In order to obtain the meteoroid flux within a gravitational field (point "B") per unit speed and unit solid angle from a particular direction, simply trace the hyperbolic orbit of the meteoroids that came from that particular direction at that speed backwards out of the gravitational field to a point in interplanetary space (point "A") and define the flux per unit speed per unit solid angle from the appropriate direction there. Note that if the orbit projected back out of the gravitational field does not have sufficient energy to leave the gravitational field, then the interplanetary component is zero from that direction. Likewise, if the orbit traced back encounters the body of the gravitating planet or moon, then that flux direction is "shadowed" by the bulk of the body and the flux contribution is zero. Figure 1. The reference frame for interplanetary flux calculations uses r, the range from the Sun, and X, the ecliptic latitude. The y-axis represents the direction of increasing ecliptic longitude and the X-axis of increasing ecliptic latitude. That meteoroids arrive with dimensionless speed b from the direction defined by 0 and (p.

Figure 1. The reference frame for interplanetary flux calculations uses r, the range from the Sun, and X, the ecliptic latitude. The y-axis represents the direction of increasing ecliptic longitude and the X-axis of increasing ecliptic latitude. That meteoroids arrive with dimensionless speed b from the direction defined by 0 and (p.

This method also works in transforming between reference frames, such as between a spacecraft frame and an Earth-centered frame ("SC" is the spacecraft frame, "E" the Earth-centered inertial frame, and "IS" the heliocentric interplanetary space inertial frame):

vsc~3 ^>sc(xscvsc) = ve~3 <&e(xe,ve)= vis~3 \$IS(xIS,vIS). (9)

The next step is to compute <I>is using the reference frame in figure 1 with the definitions b = r,=——e = Jb2 sin2 9 (b2 - 2) + 1 , and cosi = -cos<p cosX, (10)

V M* 2-b where p. is the solar gravitational constant. The Jacobian can be constructed for these parameters,

M.J. Mainey

(Z-D'je sini and the spatial density in equation 3 can now be written as

M.J. Mainey

db d0 dip

This can be written in terms of the true velocity and solid angle

This can be written in terms of the true velocity and solid angle 4n3 r, e (2 - b2) -^/r - rt ^(1 + e)r, - (1 - e)r (sin i) \ V