= 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> with
$$
\ddot{\mathbf r}=-\mu_\odot\frac{\mathbf r}{r^3},\qquad\mu_\odot=GM_\odot,
$$
where $\mathbf r$ is the body's <position> relative to the <Sun>, $r=\|\mathbf r\|$, $G$ 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.<Introduction to Celestial Mechanics by Richard Fitzpatrick>[Online Orbital elements, Sun-centred coordinates and explicit neglect of interplanetary interactions]{ref}
For a bound <elliptic orbit> with <semi-major axis> $a$ and <orbital eccentricity> $e$, the <specific orbital energy> and <orbital period> obey
$$
\varepsilon=-\frac{\mu_\odot}{2a},\qquad T=2\pi\sqrt{\frac{a^3}{\mu_\odot}},\qquad r_{\min}=a(1-e),\quad r_{\max}=a(1+e).
$$
For 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 $2^{3/2}=2\sqrt2$ times that of a circular solar <orbit> of radius one <astronomical unit>, independent of its <orbital eccentricity> in this approximation.<Introduction to Celestial Mechanics by Richard Fitzpatrick>[Online Elliptic orbits, equations 4.51–4.63, radius, energy and orbital-period relations]{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.
Back to article page
