The algorithm is designed to work with two arbitrary elliptical orbits.
Due to its nature some classes of problem (i.e. objects in secular,
mean-motion and Kozai resonances, including extreme cases of
(*e*, )-coupling) may be inappropriate for this approach. The
basic assumptions involved are:

- 1.
- each orbit has fixed semi-major-axis (
*a*), eccentricity (*e*) and inclination (*i*); - 2.
- the rate of variation of the argument of perihelion ()is greater than that of any of the other elements;
- 3.
- the variations of the respective arguments of perihelia are not correlated.

Let us suppose that each orbit has semi-major axis, eccentricity and
inclination (*a _{1}*,

(1) |

Figure 1:
Intersection of orbital planes, illustrating the mutual
inclination, , and the definition of the argument of
perihelion, , of one orbit measured with respect to the mutual
line of nodes |

In general, the heliocentric distances *r* of the ascending or descending
nodes are given by

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

We emphasize that, due to the finite size of real objects, collisions
may occur not only on this curve but also in its immediate vicinity,
over a range measured in the
-plane perpendicular to the curve of
intersection. For spherical particles is proportional
to the combined radius of interaction *R*=*R _{1}*+

The total collision probability is then calculated by integrating
along the curves of intersection, choosing small line elements of each
curve, determining their dimensions and summing their resulting
contributions to the total probability. The direction along each curve
of intersection is chosen using from
Eq. (9). The stepping can be mirrored to allow for the
symmetry, but differing geometry (associated with the particles'
relative velocities at collision) means that *P* and may be different for different mirrored points.

Determination of and requires a model of the encounter geometry. We adopt a simple vector approach in which each body is assumed to move at a constant velocity, denoted and respectively, during the close encounter. The respective vectors (see Fig. 3) are determined using the classical formulae for heliocentric elliptical motion (Roy 1988):

(10) |

(11) |

(12) |

(13) |

(14) |

Moving the origin of the coordinate system to the point at which the
two paths cross, the motion of the two particles near the mutual node
or collision point can be described by

(15) |

(16) |

We now determine the magnitude of the range perpendicular to the curve of intersection. We define to be
the vector that describes the relative motion of the two encounter
vectors as and are varied normal to the curve of
intersection, and introduce a separation parameter *S*, analogous to
the impact parameter *b*, whose value equals *R* at the extremum
. We thus have

(17) |

(18) |

Therefore is simply calculated from

(19) |

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