The perpendicular distance between a body's initial path and the center of a gravitational source.The impact parameter is the perpendicular distance between a body's initial velocity vector and the center of a gravitational source, such as a star or planet. It is a key quantity in scattering problems, describing how 'off-center' a collision or flyby is. In gravitational physics, the impact parameter determines the trajectory's shape: a small impact parameter leads to a strong deflection or even a bound orbit, while a large one results in a nearly straight path. It is also used in nuclear and particle physics to describe the geometry of collisions between particles.
The concept was introduced in the context of classical scattering theory in the 18th and 19th centuries, notably by Pierre-Simon Laplace in celestial mechanics. It was later applied to atomic and nuclear scattering by Ernest Rutherford in 1911, who used it to explain the gold foil experiment.
In a two-body gravitational encounter, the impact parameter b, along with the relative speed and masses, sets the angular momentum of the system. For a given initial speed, a smaller b means the body passes closer to the source, experiencing a stronger gravitational force and a larger deflection angle. The relationship is quantified by the Rutherford scattering formula (adapted to gravity), which gives the scattering angle as a function of b. In general relativity, the impact parameter also determines the closest approach and whether a photon or particle is captured by a black hole; the critical value for photon capture is related to the photon sphere.
It is used to design gravitational assist maneuvers for spacecraft, to predict the trajectories of near-Earth objects, and to analyze particle collisions in accelerators and cosmic ray interactions.
For a black hole, the critical impact parameter for photon capture is about 2.6 times the Schwarzschild radius, meaning light passing closer than that is inevitably swallowed.
| b | variable | impact parameter — impact parameter metre |
| q | variable | distance of closest approach in absence of gravity — distance metre |
| M | variable | mass of the gravitating body — mass kilogram |
| v_infinity | variable | speed of the test particle at infinity — velocity metre per second |
| G | constant | Universal constant of gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| sqrt | operator | takes the square root of the argument — square root |
| δθ | variable | centroid shift — angular centroid shift radian |
| u | variable | impact parameter in units of Einstein radius — impact parameter |
| θ_E | variable | Einstein radius — einstein radius radian |
| r | variable | radial coordinate — radial coordinate metre |
| a | variable | specific angular momentum (spin parameter) — angular momentum per unit mass metre |
| Δ | variable | horizon function — horizon function metre squared |
| Δ' | variable | derivative of horizon function with respect to r — derivative of horizon function metre |
| ξ | variable | critical impact parameter (azimuthal) — impact parameter metre |
| η | variable | critical impact parameter (polar) — impact parameter metre squared |
| d/dr | operator | computes the rate of change of Δ with respect to r — derivative operator |
