Cowell's formulation Cowell's formulation (so named for Philip H. Cowell, who, with A.C.D. Cromellin, used a similar method to predict the return of Halley's comet) is perhaps the simplest of the special perturbation methods. In a system of mutually interacting bodies, this method mathematically solves for the Newtonian forces on body by summing the individual interactions from the other bodies: where is the acceleration vector of body , is the gravitational constant, is the mass of body , and are the position vectors of objects and respectively, and is the distance from object to object , all vectors being referred to the barycenter of the system. This equation is resolved into components in , , and and these are integrated numerically to form the new velocity and position vectors. This process is repeated as many times as necessary. The advantage of Cowell's method is ease of application and programming. A disadvantage is that when perturbations become large in magnitude (as when an object makes a close approach to another) the errors of the method also become large. However, for many problems in celestial mechanics, this is never the case. Another disadvantage is that in systems with a dominant central body, such as the Sun, it is necessary to carry many significant digits in the arithmetic because of the large difference in the forces of the central body and the perturbing bodies, although with modern computers this is not nearly the limitation it once was. Encke's method Encke's method begins with the osculating orbit as a reference and integrates numerically to solve for the variation from the reference as a function of time. Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification. Encke's method is similar to the general perturbation method of variation of the elements, except the rectification is performed at discrete intervals rather than continuously. Letting be the radius vector of the osculating orbit, the radius vector of the perturbed orbit, and the variation from the osculating orbit, and are just the equations of motion of and where is the gravitational parameter with and the masses of the central body and the