Energia orbital específica

No problema gravitacional dos dois corpos, a energia orbital específica de dois corpos orbitantes é a soma constante das suas mútuas energias potenciais ( $\epsilon _{p}\,\!$ ) e das suas energias cinéticas ( $\epsilon _{k}\,\!$ ), dividida pela sua massa reduzida. De acordo com a equação de conservação de energia orbital, também conhecida como equação vis-viva, não varia com o tempo:

\epsilon =\epsilon _{k}+\epsilon _{p}\!

\epsilon ={v^{2} \over {2}}-{\mu  \over {r}}=-{1 \over {2}}{\mu ^{2} \over {h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}

onde

$v\,\!$ é a velocidade orbital relativa;
$r\,\!$ é a distância orbital entre os corpos;
$\mu ={G}(m_{1}+m_{2})\,\!$ é o parâmatro gravitacional padrão dos corpos;
$h\,\!$ é o vetor momento angular orbital, no sentido do momento angular relativo dividido pela massa reduzida;
$e\,\!$ é a excentricidade orbital;
$a\,\!$ é o semi-eixo maior.

Análise energética para o modelo de dois corpos

Expressando a magnitude da velocidade em função da forma do angulo vetor do momento orbital , e, em seguida, dependendo da reta assimilada, é possível chegar a uma energia orbital específica dado como uma função exclusiva do semi-eixo maior da órbita:

\epsilon =-{\mu  \over {2a}}

em que $a$ é o semi-eixo maior da órbita.

Então:

Para uma órbita elíptica a energia total específica é negativa ( $\varepsilon <0$ );
Para uma órbita parabólica a energia total específica é nula ( $\varepsilon =0$ );
Para uma órbita hiperbólica a energia total específica é positiva ( $\varepsilon >0$ ).

Bibliografia

Wie, Bong (1998). «Orbital Dynamics». Space Vehicle Dynamics and Control. Col: AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. p. 220. ISBN 1-56347-261-9

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