Orbital Energies

According to Equations (1.320) and (1.321), when $\theta=\pi$ an object moving in the Sun's gravitational attains its closest distance to the Sun,

$$r_p = \frac{h^2}{G\,M\,(1+e)}.$$ (1.324)

This distance is known as the perihelion distance. At the point of closest approach to the Sun, the object's instantaneous radial velocity, $\dot{r}$, is zero (because $r$ attains a minimum value at this point). Hence, making use of Equations (1.288) and (1.312), the object's speed at the perihelion distance is

$$v^2 = r_p^{\,2}\,\skew{5}\dot{\theta}^{\,2} = \frac{h^2}{r_p^{\,2}}.$$ (1.325)

Thus, according to Equation (1.280) and (1.324), the object's energy per unit mass at the perihelion distance is

$${\cal E} = \frac{h^2}{2\,r_p^{\,2}} - \frac{G\,}{r_p} = \frac{G\,M\,(1+e)}{2\,r_p} - \frac{G\,M}{r_p},$$ (1.326)

which reduces to

$${\cal E} = \frac{G\,M}{2\,r_p}\,(e-1).$$ (1.327)

Of course, because ${\cal E}$ is a conserved quantity, the previous expression specifies the energy per unit mass of the object at all distances from the Sun. We conclude that an object in an elliptic orbit ($e<1$) has a negative total energy, whereas an object in a parabolic orbit ($e=1$) has zero total energy, and an object in a hyperbolic orbit ($e>1$) has a positive total energy. This makes sense, because in a conservative system in which the potential energy at infinity is set to zero [see Equation (1.279)], we expect bounded orbits to have negative total energies, and unbounded orbits to have positive total energies. (See Section 1.3.6.) Thus, elliptical orbits, which are clearly bounded, should indeed have negative total energies, whereas hyperbolic orbits, which are clearly unbounded, should indeed have positive total energies. Parabolic orbits are marginally bounded (i.e., an object executing a parabolic orbit only just escapes from the Sun's gravitational field), and thus have zero total energy.