Bate, Mueller, White:(adsbygoogle = window.adsbygoogle || []).push({}); Fundamentals of Astrodynamics(Dover 1971), p. 29, equation 1.6-4 for the eccentricity of a satellite's orbit:

[tex]e=\sqrt{1+\frac{2\mathcal{E}h^2}{\mu^2}}[/tex]

[tex]e \text{, eccentricity}[/tex]

[tex]\mathcal{E} \text{, specific mechanical energy}[/tex]

[tex]h \text{, magnitude of specific angular momentum}[/tex]

[tex]\mu \text{, gravitational parameter } = GM[/tex]

(Gthe gravitational constant,Mthe mass of the planet which the satelite is orbiting.) The book describes the shape of orbits for specific mechanical energy negative (circle, ellipse), zero (parabola) and positive (hyperbola).

My question: how do the definitions of these variables guarrantee that

[tex]\frac{2 \mathcal{E} h^2}{\mu^2} \geq -1[/tex]

for all physically possible orbits (given the simplifying assumptions: two bodies, the satellite much less massive than the planet, mass of planet spherically distributed, no other forces significant)?

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# Eccentricity and specific mechanical energy

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