Motion blur

Motion blur arises when a detector averages changing image positions during a finite exposure rather than recording an instantaneous position. For a stationary harmonic optical trap with position variance $k_{B}T/k$ and relaxation time $\tau$, rectangular exposure of duration $W$ reduces that variance by the factor $2[\alpha -1+e^{-\alpha }]/{\alpha }^2$, where $\alpha =W/\tau$. Integrating the exponential position covariance over both exposure times proves this formula. Ignoring the reduction makes a naive equipartition stiffness estimate too large.^[ref]