# Simplifying total energy equation orbital mechanics

1. Apr 2, 2013

### Dustinsfl

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.