Gravitation: launching a craft out of the solar system.

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Homework Statement

Let's suppose that we wanted to launch a spacecraft of mass m out of the Solar System.
a) If we want to launch the spacecraft directly from Earth, what boost Δv would be required
and what direction relative to the Earth's velocity about the Sun would this boost be pointed
in?
b) Now let us assume that we wanted to save energy (reduce the magnitude of Δv) by using a
gravitational slingshot off of a single planet. Assume that all the planets are on circular
orbits with the same orbital plane as the Earth, and for now assume that the planets are point
masses. Which planet would you choose and why? The necessary information about
planets is available on Wikipedia. What would be the magnitude and direction of Δv?
c) Real planets have surfaces, and bad things happen to spacecraft (and astronauts) that crash
into the surfaces of planets. Does this change your answer to part b of this problem? Why
might it?

Homework Equations

$E=\frac{1}{2}μv^2-\frac{GMμ}{r}=\frac{GMμ}{2a}; a=r_1+r_2$

The Attempt at a Solution

Part A)

Let $E_p = \frac{1}{2}μ(v_p)^2-\frac{GMμ}{r_1}=\frac{GMμ}{r_1+r_2}$ be the energy at the periapsis.

Then the corresponding magnitude of the velocity of the spacecraft at periapsis is: $v_p = \sqrt{2GM(\frac{1}{r_1}-\frac{1}{r_1+r_2})}$

Also the magnitude of the velocity around the 1st circular orbit is : $v_1=\sqrt{\frac{GM}{r_1}}$

So in order to boost from the circular orbit around Earth to the transitional elliptical transfer orbit requires $v_1+Δv=v_p$ or $Δv=v_p-v_1$

Thus $Δv = \sqrt{\frac{GM}{r_1}}(\sqrt{\frac{2r_2}{r_1+r_2}}-1)$

As $r_2 →∞ , Δv→\sqrt{\frac{GM}{r_1}}(\sqrt{2}-1)$

Now for the direction: $(v_p)^2 = \vec{v_p}\cdot\vec{v_p} = \vec{v_1}\cdot\vec{v_1}+\vec{Δv}\cdot\vec{Δv}+2\vec{v_1}\cdot\vec{Δv}$ which is clearly maximized for $Δv$ in the same direction as $v_1$

(everything in part A makes sense to me, but I want to be sure I didn't make any careless errors)

Part B) I know I need to minimize the first expression for $Δv$ with respect to $r_2$ i.e $\frac{d}{dr_2}Δv=0$ and then verify that that second derivative of this expression evaluated at the extrema is positive. Then I need to check which planet fits the radial orbit I find from minimizing $Δv$ wrt $r_2$. However, before I do this Is my expression for $Δv$ correct?

Part C) I personally don't see any reason why the planets having surface areas affects anything in part B, but just because I don't see a reason it doens't mean there isn't one. Any clues on this one?

Thank you in advance!

 

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Actually I just realized the answer to part B) lies in the gravitational slingshot: in the reference frame where the planet is still the initial velocity of the craft is equal to the final velocity (well the magnitudes are equal, not the directions) $\|\vec{v_c}\|=\|\vec{v'_c}\|$

If the craft moves along with the planet (which is moving at velocity $v_p$ then $\|\vec{v_p}+\vec{v_c}\|<\|\vec{v_p}+\vec{v'_c}\|$ so in order to minimize $Δv$ I need to choose a planet with a high orbital velocity about the sun. So in the approximation that the planets are points masses Mars is the best choice.

It also just occurred to me that if I were to use the result from part b the craft would crash directly into Mars since Mars is not a point mass and has surface area so I would need to use a more ecentric elliptical orbit in order to properly use the sling shot i.e increase the semi-major axis by increasing $r_2$