Answer
(a) Using the conservation of energy we show that
$v_{per}^2 - v_{ap}^2 = 2GM\left( {r_{per}^{ - 1} - r_{ap}^{ - 1}} \right)$
(b) Using the results from Exercise 13, we show that
$r_{per}^{ - 1} - r_{ap}^{ - 1} = \frac{{2e}}{{a\left( {1 - {e^2}} \right)}}$
(c) We show that $v_{per}^2 - v_{ap}^2 = 4\frac{e}{{{{\left( {1 + e} \right)}^2}}}v_{per}^2$
Using results from part (a) and part (b) we obtain:
${v_{per}} = \sqrt {\left( {\frac{{GM}}{a}} \right)\frac{{1 + e}}{{1 - e}}} $
Work Step by Step
(a) From Exercise 17, we obtain the total energy of a planet orbiting a sun is given by Eq. (8):
$E = \frac{1}{2}m{v^2} - \frac{{GMm}}{{||{\bf{r}}||}}$
Also in Exercise 17, it is shown that the total energy is conserved. Therefore, the total energy at perihelion is equal to the total energy at aphelion:
$\frac{1}{2}mv_{per}^2 - \frac{{GMm}}{{||{{\bf{r}}_{per}}||}} = \frac{1}{2}mv_{ap}^2 - \frac{{GMm}}{{||{{\bf{r}}_{ap}}||}}$
Write $||{{\bf{r}}_{per}}|| = {r_{per}}$ and $||{{\bf{r}}_{ap}}|| = {r_{ap}}$. So,
$\frac{1}{2}mv_{per}^2 - \frac{1}{2}mv_{ap}^2 = \frac{{GMm}}{{{r_{per}}}} - \frac{{GMm}}{{{r_{ap}}}}$
Multiply both sides by $\frac{2}{m}$ we obtain
$v_{per}^2 - v_{ap}^2 = 2GM\left( {r_{per}^{ - 1} - r_{ap}^{ - 1}} \right)$
(b) From Exercise 13, we obtain ${r_{per}} = a\left( {1 - e} \right)$ and ${r_{ap}} = a\left( {1 + e} \right)$.
Evaluate
$r_{per}^{ - 1} - r_{ap}^{ - 1} = \frac{1}{{a\left( {1 - e} \right)}} - \frac{1}{{a\left( {1 + e} \right)}} = \frac{{a\left( {1 + e} \right) - a\left( {1 - e} \right)}}{{{a^2}\left( {1 - {e^2}} \right)}}$
$r_{per}^{ - 1} - r_{ap}^{ - 1} = \frac{{2ae}}{{{a^2}\left( {1 - {e^2}} \right)}}$
$r_{per}^{ - 1} - r_{ap}^{ - 1} = \frac{{2e}}{{a\left( {1 - {e^2}} \right)}}$
(c) From Exercise 15, we obtain
${v_{per}}\left( {1 - e} \right) = {v_{ap}}\left( {1 + e} \right)$
It follows that ${v_{ap}} = {v_{per}}\frac{{1 - e}}{{1 + e}}$.
Evaluate
$v_{per}^2 - v_{ap}^2 = v_{per}^2 - v_{per}^2{\left( {\frac{{1 - e}}{{1 + e}}} \right)^2}$
$ = v_{per}^2\left( {1 - \frac{{{{\left( {1 - e} \right)}^2}}}{{{{\left( {1 + e} \right)}^2}}}} \right)$
$ = v_{per}^2\left( {\frac{{1 + 2e + {e^2} - 1 + 2e - {e^2}}}{{{{\left( {1 + e} \right)}^2}}}} \right)$
Hence, $v_{per}^2 - v_{ap}^2 = 4\frac{e}{{{{\left( {1 + e} \right)}^2}}}v_{per}^2$.
Recall from previous results:
$v_{per}^2 - v_{ap}^2 = 2GM\left( {r_{per}^{ - 1} - r_{ap}^{ - 1}} \right)$
$r_{per}^{ - 1} - r_{ap}^{ - 1} = \frac{{2e}}{{a\left( {1 - {e^2}} \right)}}$
So, $v_{per}^2 - v_{ap}^2 = 2GM\left( {\frac{{2e}}{{a\left( {1 - {e^2}} \right)}}} \right)$
But $v_{per}^2 - v_{ap}^2 = 4\frac{e}{{{{\left( {1 + e} \right)}^2}}}v_{per}^2$. Thus,
$4\frac{e}{{{{\left( {1 + e} \right)}^2}}}v_{per}^2 = 4GM\left( {\frac{e}{{a\left( {1 - {e^2}} \right)}}} \right)$
$v_{per}^2 = GM\frac{{{{\left( {1 + e} \right)}^2}}}{{a\left( {1 - {e^2}} \right)}} = \frac{{GM\left( {1 + e} \right)}}{{a\left( {1 - e} \right)}}$
${v_{per}} = \sqrt {\left( {\frac{{GM}}{a}} \right)\frac{{1 + e}}{{1 - e}}} $
