# Gravitation: launching a craft out of the solar system.

• Wavefunction
In summary, launching a spacecraft out of the Solar System requires a boost of Δv and a specific direction relative to the Earth's velocity about the Sun. Using a gravitational slingshot off of a single planet can save energy, with Mars being the best choice due to its high orbital velocity. However, the presence of planet surfaces may affect the choice of the planet and require the use of a more eccentric elliptical orbit.
Wavefunction

## 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?

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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 $r_2$

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## 1. How does gravitation affect the launch of a craft out of the solar system?

Gravitation plays a crucial role in the launch of a craft out of the solar system. The gravitational pull of the sun and other celestial bodies can either assist or hinder the craft's trajectory. For example, the gravitational pull of the sun can be used to slingshot the craft and increase its speed, while the gravitational pull of other planets may cause the craft to deviate from its intended path.

## 2. What factors influence the trajectory of a craft launched out of the solar system?

The trajectory of a craft launched out of the solar system is influenced by several factors, including the initial velocity of the craft, the gravitational pull of the sun and other celestial bodies, and the presence of any intervening objects or obstacles. The angle at which the craft is launched also plays a significant role in determining its trajectory.

## 3. How does the mass of the craft affect its ability to escape the solar system's gravitational pull?

The mass of the craft does not significantly impact its ability to escape the solar system's gravitational pull. The gravitational force between two objects is determined by their masses and the distance between them. However, the force required to overcome the gravitational pull of the sun and other celestial bodies is primarily dependent on the craft's initial speed and direction.

## 4. Can a craft be launched out of the solar system without using the sun's gravitational pull?

Yes, a craft can be launched out of the solar system without using the sun's gravitational pull. This can be achieved by using powerful propulsion systems, such as rocket engines, to provide the necessary force to overcome the gravitational pull of the sun and other celestial bodies.

## 5. What are some of the challenges scientists face when launching a craft out of the solar system?

One of the main challenges scientists face when launching a craft out of the solar system is calculating and predicting the trajectory of the craft accurately. This requires a thorough understanding of the gravitational forces at play and precise measurements of the craft's initial conditions. Other challenges include designing and building a craft that can withstand the extreme conditions of outer space, such as high levels of radiation and temperature fluctuations.

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