The Cost of Future Collisions in LEO - Star Technology and Research

26
The conditional probability that the two objects will be within “touching
distance” in the direction of the intersection line is obtained by integration
P r ≈
∫ H2
H 1
P A (h) P B (h) (D 1 + D 2 ) dh, (A3)
where D 1 and D 2 are the typical dimensions of the objects, and (H 1 , H 2 ) is the
range of overlapping altitudes.
Fig. A2. Relative motion near the intersection.
The conditional probability that the two objects will be within “touching
distance” in projection onto a reference sphere near the intersection line with given
positions of the ascending nodes of the orbits is approximated as
P ϑ ≈ D 1 + D 2
2πR sin ϑ ,
(A4)
where R is the average orbit radius, and ϑ is the angle between the tangent to the
trajectory of the object A and the velocity of the object B relative to the object A
near the intersection, as shown in Fig. A2. The reasoning is very simple. When the
center of the object B passes the intersection line, the distance between the center
of the object A and the intersection line measured along the orbit of the object A
can be anywhere between 0 and approximately πR, but a collision can occur only
when the distance is less than (D 1 + D 2 )/2 sin ϑ.
If the objects maintain certain attitude, their dimensions in formulas (A1)
and (A4) could have more specific meaning. For example, if we consider two gravitationally
stabilized cylinders, the dimensions in formula (A1) will represents their
heights, while the dimensions in formula (A4) will represents their diameters.
Averaging over all possible nodal positions yields
P ϑ ≈ D 1 + D 2
2πR
β, (A5)
where β is the “inclination pairing” coefficient, as defined by Carroll [6],
β = 1 ∫ 2π
2π 0
dφ
sin ϑ ,
(A6)