Impact Parameter

The impact parameter b is the perpendicular distance between a scattering center and the straight-line path a projectile would follow if there were no deflection. For a light ray grazing a mass, b is essentially the closest approach the ray would make in the absence of bending; for a planet, comet, or alpha particle it is the offset of the incoming asymptote from the target. It is the geometric input that, together with the strength of the interaction, fixes the deflection.

In gravitational light deflection the impact parameter enters directly: the bend angle is α = 4GM/c²b, falling off as 1/b. A ray that skims the solar limb (b = R_⊙) is deflected by 1.75 arcseconds; a ray passing one solar radius farther out is deflected by half as much. This 1/b dependence is why the 1919 eclipse expeditions targeted stars as close to the Sun's edge as the eclipse would permit — the signal is largest where the light passes closest to the mass.

The concept is general to all scattering problems. In Rutherford's analysis of alpha particles, the impact parameter determines the scattering angle through the Coulomb force; in celestial mechanics it sets the eccentricity of a hyperbolic encounter. In gravitational lensing the impact parameter of each ray, combined with the lens equation, determines the positions and magnifications of the images. It is, in every case, the bridge between the geometry of the encounter and its observable outcome.

The term entered physics through scattering theory in the early twentieth century, notably Rutherford's 1911 analysis of alpha-particle deflection, and was carried over directly into the relativistic treatment of light bending and gravitational lensing.