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Perigee and Apogee

The distance r of the particle from the center of forces becomes minimum (or 1/r becomes maximum) when θ = θ₀, cos (θ – θ₀) = 1. That is, we define the distance of closest approach as,

The point P of closest approach is called the perigee of the orbit. If we choose the axis θ = 0 as the line joining O to P, then we have θ₀ = 0). Hence, the equation of orbit becomes

The farthest separation for the particle is infinite for unbounded orbit. If e = 1 (parabola), we find as θ ± π. Similarly, if e = eh > 1 (hyperbola), we get , when

(1 – eh cos θ) = 0 , or cos θ = 1/eh

However, for bounded orbits, r can never exceed a maximum value. In fact, if e = 0 (circle), r = p, constant.

For e < 1 (ellipse), the point opposite to perigee represents the farthest distance; this point is called apogee.

That is, apogee is the point Q for which θ = π, cos θ = – 1 and hence r is maximum (or 1/r is minimum)

Apogee represents the farthest point of the elliptic orbit from origin O. We get

At the points of closest and farthest separations of the particle from O, the radial velocity of the particle must vanish, i.e. v_r = dr/dt = 0, velocity vector v is now perpendicular to r.