page_title: Orbit modeling
section_title: Encke's method
hierarchical_section_title: Orbit modeling / Orbit simulation methods / Encke's method
caption_attribution_description: English: Vector diagram of Encke's Method of perturbation analysis, showing the osculating orbit, the perturbed orbit, and a perturbing body.
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: 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 than Cowell's method. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification.
  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 perturbed body,  is the perturbing acceleration, and  and  are the magnitudes of  and .
  Substituting from equations (3) and (4) into equation (2),
  which, in theory, could be integrated twice to find . Since the osculating orbit is easily calculated by two-body methods,  and  are accounted for and  can be solved. In practice, the quantity in the brackets, , is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra significant digits.
caption: Encke's method. Greatly exaggerated here, the small difference  δr (blue) between the osculating, unperturbed orbit (black) and the perturbed orbit (red), is numerically integrated starting from the initial position (the epoch of osculation).
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