page_title: Orbit modeling
section_title: Cowell's method
hierarchical_section_title: Orbit modeling / Orbit simulation methods / Cowell's method
caption_attribution_description: English: Vector diagram of Cowell's method of perturbation analysis, depicting forces on a perturbed object by two other objects.
context_page_description: Orbit modeling is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.
context_section_description: Cowell's method is perhaps the simplest of the special perturbation methods;
  mathematically, for  mutually interacting bodies, Newtonian forces on body  from the other bodies  are simply summed thus,
  where
   is the acceleration vector of body 
   is the gravitational constant
   is the mass of body 
   and  are the position vectors of objects  and 
   is the distance from object  to object 
  with all vectors being referred to the barycenter of the system. This equation is resolved into components in , ,  and these are integrated numerically to form the new velocity and position vectors as the simulation moves forward in time. 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.
  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.
caption: Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body i (red), and this is numerically integrated starting from the initial position (the epoch of osculation).
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