u3 approaches and is much larger than or . Hence, is approximately For the same reasons, the denominator of Δφ is approximately Since the modulus is close to zero, the period K can be expanded in powers of ; to lowest order, this expansion yields Substituting these approximations into the formula for Δφ yields a formula for angular advance per radial oscillation For an elliptical orbit, and represent the inverses of the longest and shortest distances, respectively. These can be expressed in terms of the ellipse's semi-major axis and its orbital eccentricity , giving Substituting the definition of gives the final equation Bending of light by gravity In the limit as the particle mass m goes to zero (or, equivalently if the light is heading directly toward the central mass, as the length-scale a goes to infinity), the equation for the orbit becomes Expanding in powers of , the leading order term in this formula gives the approximate angular deflection δφ for a massless particle coming in from infinity and going back out to infinity: Here, is the impact parameter, somewhat greater than the distance of closest approach, : Although this formula is approximate, it is accurate for most measurements of gravitational lensing, due to the smallness of the ratio . For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 arcseconds, roughly one millionth part of a circle. Relation to Newtonian physics Effective radial potential energy The equation of motion for the particle derived above can be rewritten using the definition of the Schwarzschild radius rs as which is equivalent to a particle moving in a one-dimensional effective potential The first two terms are well-known classical energies, the first being the attractive Newtonian gravitational potential energy and the second corresponding to the repulsive "centrifugal" potential energy; however, the third term is an attractive energy unique to general relativity. As shown below and elsewhere, this inverse-cubic energy causes elliptical orbits to precess gradually by an angle δφ per revolution where is the semi-major axis and is the eccentricity. The third term is attractive and dominates at small values, giving a critical inner radius rinner at which a particle is drawn inexorably inwards to ; this inner radius is a function of the particle's angular momentum per unit mass or, equivalently, the length-scale defined above. Circular orbits and their stability The effective potential can be re-written