Equation of the center

In two-body, Keplerian orbital mechanics, the equation of the center is the angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the same period. It is defined as the difference true anomaly, ν, minus mean anomaly, M, and is typically expressed a function of mean anomaly, M, and orbital eccentricity, e.

Discussion

Since antiquity, the problem of predicting the motions of the heavenly bodies has been simplified by reducing it to one of a single body in orbit about another. In calculating the position of the body around its orbit, it is often convenient to begin by assuming circular motion. This first approximation is then simply a constant angular rate multiplied by an amount of time. However, the actual solution, assuming Newtonian physics, is an elliptical orbit (a Keplerian orbit). For these, it is easy to find the mean anomaly (and hence the time) for a given true anomaly (the angular position of the planet around the sun), by converting true anomaly

ν

{\displaystyle \nu }

to "eccentric anomaly":

E

=

atan2

⁡

(

1

−

e

2

sin

⁡

ν

,

e

+

cos

⁡

ν

)

{\displaystyle E=\operatorname {atan2} \left(\ {\sqrt {1-e^{2}}}\sin \nu ,\ e+\cos \nu \right)}

where atan2(y, x) is the angle from the x-axis of the ray from (0, 0) to (x, y), having the same sign as y (note that the arguments are often reversed in spreadsheets), and then using Kepler's equation to find the mean anomaly:

M

=

E

−

e

sin

⁡

E

{\displaystyle M=E-e\sin E}

If

M

{\displaystyle M}

is known and we wish to find

E

{\displaystyle E}

and

f

{\displaystyle f}

then Kepler's equation can be solved by numerical methods, but there are also series solutions involving sine of

M

{\displaystyle M}

.

In cases of small eccentricity, the position given by a truncated series solution may be quite accurate. Many orbits of interest, such as those of bodies in the Solar System or of artificial Earth satellites, have these nearly-circular orbits. As eccentricity becomes greater, and orbits more elliptical, the accuracy of a given truncation of the series declines.