This paper describes the trajectory of an asteroid (or a comet or spacecraft) as it approaches a planet of much greater mass. The solution of this two-body problem is an instructive first approximation to more refined treatments that include the gravitational forces of the Sun and of planets other than the target planet. Detailed properties of encounter trajectories are derived. As an illustration, it is shown that the collision cross section of the planet is greater by a factor F=1+(vescapev0)2 compared to its cross section in the absence of gravitational forces, where vescape is the minimal escape speed from the surface of the planet and v0 is the approach speed of the asteroid at an effectively infinite distance. Sample values of F are given for Earth, Mars, Jupiter, and Saturn.

1.
W. T.
Thomson
,
Introduction to Space Dynamics
(
Wiley
,
New York
,
1961
).
2.
H.
Goldstein
,
C.
Poole
, and
J.
Safko
,
Classical Mechanics
, 3rd ed. (
Addison–Wesley
,
San Francisco
,
2003
), Chap. 3.
3.
J. A.
Van Allen
, “
Gravitational assist in celestial mechanics—a tutorial
,”
Am. J. Phys.
71
(
5
),
448
451
(
2003
).
4.
Allen’s Astrophysical Quantities
, edited by
A. N.
Cox
, 4th ed. (
Springer-Verlag
,
New York
,
1999
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.