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Simplifying total energy equation orbital mechanics

  1. Apr 2, 2013 #1
    I am trying to show that
    $$
    \frac{\Delta v}{v_{c_1}} = \frac{1}{\sqrt{R}} - \frac{\sqrt{2}(1 - R)}{\sqrt{R(1+R)}} - 1
    $$
    where ##R=r_2/r_1##.

    ##r_1## is the radius of the circular orbit 1 and ##r_2## is the radius of the circular orbit 2. Similarly, ##v_{c_1}## is the velcotiy of the circular orbit 1 and so on for ##v_{c_2}##.
    The velocity of a circular orbit is
    $$
    v_{c_k} = \sqrt{\frac{\mu}{r_k}}
    $$
    where ##k = 1,2##.
    By the Law of Cosine,
    $$
    (\Delta v)^2 = v_{c_2}^2 + v^2 - 2v_{c_2}vv_{\theta}
    $$
    where ##v## is the velocity from the Hohmann ellipse transfer and ##v_{\theta} = \frac{h}{r} = \frac{\sqrt{\mu p}}{r}##.
    The velocity of the ellipse is ##v = \sqrt{\mu\left(\frac{2}{r} - \frac{1}{a}\right)}##.
    $$
    a = \frac{1 + R}{2},\quad r = r_2 - r_1,\quad p = a(1 - e^2),\quad e = \frac{r_2 - r_1}{r_2 + r_1} = \frac{R - 1}{R + 1}
    $$

    So I took ##\frac{(\Delta v)^2}{v_{c_1}^2}##.
    $$
    \frac{(\Delta v)^2}{v_{c_1}^2} = \frac{1}{R} + \frac{2}{R - 1} - \frac{4r_1}{R + 1} - 2r_1\sqrt{\frac{2}{r_2r} - \frac{1}{r_2a}}\frac{\sqrt{a\mu(1 - e^2)}}{r}
    $$
    At this point, I can't seem to make any useful head way towards the result.
     
  2. jcsd
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