COLLISION PROBABILITIES FOR KEPLERIAN ORBITS
DOYLE T. HALL
and MARK J. MATNEY
The Boeing Co., 1250 Academy Park Loop, Suite 120, Colorado Springs, CO 80910, USA
(Tel.: +1-719-638-5038; Fax: +1-719-638-5301; E-mail: firstname.lastname@example.org)
NASA Orbital Debris Program Ofﬁce, 2400 NASA Road 1 – Mail Code C-104,
Johnson Space Center, Houston, TX 77058, USA (Tel.: +1-281-244-2258; Fax: +1-281-244-5031;
(Received 2 January 2002; Accepted 14 December 2002)
Abstract. We present a new derivation of the probability of collisions between spherical satellites occupying
Keplerian orbits. The equations follow from the central concept of the instantaneous collision rate,an
expression that describes the occurrence of collisions by using a Dirac δ-function. The derivation proceeds
by showing how this instantaneous collision rate can be averaged over orbital mean anomaly angles and,
additionally, over orbital precession angles to generate expressions appropriate for intermediate and long time
scales. Collision rates averaged over mean anomalies tend to be non-zero during relatively brief collision
seasons, when the peak collision probability may exceed the long-term average by several orders of magnitude.
Derived precession-angle averages have a functional form similar but not identical to the collision probability
expression derived using the spatial density approach of Kessler (Icarus, 48: 39–48, 1981), and the two
methods have been found to yield numerical results to within 1% for all cases examined.
Keywords: impact risk analysis, orbital debris environment, probability of collision
Kessler (1981) derived equations for collision probabilities of orbiting objects that provide
the basis for many computer models of Earth’s evolving orbital debris population (Kessler
and Cour-Palais, 1978; Kessler, 1990; Reynolds and Eichler, 1997). The Kessler (1981)
method involves calculating the time-averaged spatial densities of each orbiting object, and
integrating over the volume where collisions can potentially occur. A fundamental assump-
tion in the spatial density approach is that satellite arguments of pericenter (ω) and longitudes
of ascending node () vary uniformly in time, so that there is a uniform probability of each
having a value between 0 and 2π . This assumption is employed because, for most terrestrial
artiﬁcial satellites, secular perturbations caused primarily by the J
component of Earth’s
gravity-ﬁeld induce orbital precession in the form of near-uniform variations in both ω and
(see Taff, 1985; Danby, 1988).
The assumption is also employed when evaluating col-
lision probabilities in other satellite systems, such as asteroids in heliocentric orbit, which
experience gravitational perturbations from Jupiter and other planets that induce similar
variations in ω and (Wetherill, 1967; Greenberg, 1982). When considering potential col-
lisions between two precessing satellites, either the spatial density method of Kessler (1981)
For convenience in this analysis the angles ω and are collectively referred to as precession angles.
Space Debris 2; 161–198, 2000.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.