x = a / a_{p}

and

c_{I} = cos I

. At fixed

T_{p}

and

I

e^{2} = 1 - \frac{( T _{p} - x ^{- 1} ) ^{2}}{4 c _{I}^{2} x} .

For a prograde orbit, the unsquared equation also requires

x \geq 1/ T_{p}

. Differentiation shows that the only stationary values occur at

x = \frac{1}{T _{p}}, x = \frac{3}{T _{p}} .

The first is an endpoint maximum with

e = 1

. The second is the interior minimum

e_{m i n}^{2} = 1 - \frac{T _{p}^{3}}{27 cos ^{2} I}, a_{m i n} = \frac{3 a _{p}}{T _{p}} .

It lies on the physical locus when

T_{p}^{3} \leq 27 cos^{2} I

. Equality makes the minimum circular. If

T_{p}^{3} > 27 cos^{2} I

, a forbidden interval surrounds

3/ T_{p}

and the allowed locus splits into branches ending at

e = 0

a \to \infty

e \to 1

Solved by gpt-5.6-sol high.

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