Heliocentric orbit

A heliocentric orbit describes a body's trajectory relative to the centre of the Sun. For a low-mass body whose motion is dominated by solar gravity, a useful first approximation is a Kepler orbit withwhere $\mathbf{r}$ is the body's position relative to the Sun, $r=\Vert \mathbf{r}\Vert$, $G$ the gravitational constant and $M_{\odot }$ the solar mass. A heliocentric description specifies an origin; the two-body equation additionally assumes that other gravitational forces can be neglected.^[ref]

For a bound elliptic orbit with semi-major axis $a$ and orbital eccentricity $e$, the specific orbital energy and orbital period obeyFor example, a hypothetical solar orbit with $a=2\mathrm{AU}$ and $e=1/2$ ranges from one to three astronomical units from the Sun. Its orbital period is $2^{3/2}=2\sqrt{2}$ times that of a circular solar orbit of radius one astronomical unit, independent of its orbital eccentricity in this approximation.^[ref]

The Sun itself accelerates as other bodies attract it. Accurate Solar system calculations must therefore include the relevant orbital perturbations and specify the reference frame. A spacecraft orbiting Earth also travels around the Sun, but solar-centred coordinates alone do not make a solar two-body problem an adequate model of that spacecraft's local motion.