The Einstein–Infeld–Hoffmann equations of motion, jointly derived by
Albert Einstein,
Leopold Infeld and
Banesh Hoffmann, are the
differential equations describing the approximate
dynamics of a system of point-like masses due to their mutual gravitational interactions, including
general relativistic effects. It uses a first-order
post-Newtonian expansion and thus is valid in the limit where the velocities of the bodies are small compared to the speed of light and where the gravitational fields affecting them are correspondingly weak.
Given a system of N bodies, labelled by indices A = 1, ..., N, the
barycentric acceleration vector of body A is given by:
and the
big O notation is used to indicate that terms of order c−4 or beyond have been omitted.
The coordinates used here are
harmonic. The first term on the right hand side is the Newtonian gravitational acceleration at A; in the limit as c → ∞, one recovers Newton's law of motion.
The acceleration of a particular body depends on the accelerations of all the other bodies. Since the quantity on the left hand side also appears in the right hand side, this system of equations must be solved iteratively. In practice, using the Newtonian acceleration instead of the true acceleration provides sufficient accuracy.[1]
References
^Standish, Williams. Orbital Ephemerides of the Sun, Moon, and Planets, Pg 4.
"Archived copy"(PDF). Archived from
the original(PDF) on 2011-02-05. Retrieved 2010-04-03.{{
cite web}}: CS1 maint: archived copy as title (
link)