Simplifying total energy equation orbital mechanics

[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.

 

[KataKoniK]

Hello,

Thank you for sharing your progress so far! It looks like you have made some good initial steps towards proving the given equation.

One suggestion I have is to try using the expression for $v_{c_k}$ in terms of $r_k$, which is given by $v_{c_k} = \sqrt{\frac{\mu}{r_k}}$. This may help simplify some of the expressions involving r1 and r2.

Additionally, you may want to consider using the relationship between the semi-major axis $a$ and the radius of the circular orbit $r_c$, which is given by $a = \frac{r_c}{2}$. This may also help simplify some of the expressions.

I hope these suggestions are helpful and good luck with your further analysis!