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Rigorous Science What is the highest theoretical escape velocity a 2 billion metric ton vehicle can achieve using a >20$^{\circ}$ gravitational slingshot?

I want to send a flying saucer (a $\frac{1}{10}$ scale Independence Day City Destroyer - calculated as a sphere cap 112m high with 1.2km segment radius, $\rho$ about 43.3kg/m3) away from the sun as...

1 answer · posted 5y ago by Vogon Poet‭ · edited 4y ago by HDE 226868‭

Question orbital-mechanics

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HDE 226868‭ · 2020-06-02T18:33:32Z (over 4 years ago)

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HDE 226868‭ · 2020-06-02T18:33:18Z (over 4 years ago)
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I want to send a flying saucer (a $\frac{1}{10}$ scale <a href="https://medium.com/@brandonweigel/the-largest-ships-in-science-fiction-d3dc9c2da3f" rel="nofollow noreferrer">Independence Day City Destroyer</a> - calculated as a sphere cap 112m high with 1.2km segment radius, $ ho$ about 43.3kg/m3) away from the sun as fast as possible, accelerating it from some initial velocity $V_i$ to escape velocity $V_f$ by <a href="https://en.m.wikipedia.org/wiki/Gravity_assist" rel="nofollow noreferrer">gravity assist</a>, and learn the $\Delta V$ gained. It's mass is about $2\times 10^9$ metric tons. Parameters are:
~~<ul>~~
<li>The assist will deflect the initial trajectory by no less than 20Â° ($\Delta$V is $\delta$ or the angle between the hyperbolic asymptotes). This allows a useable steering benefit.</li>
~~<li>The entry vector $V_i$ is an indeterminate variable with less velocity than the exit vector.</li>~~
~~<li>Any celestial body in our solar system can be used for the assist except the sun (the sun's gravity can't assist something escaping itself?)</li>~~
~~<li>The mass cannot enter any atmosphere (no friction calculations or concerns) so $\text r_0 = $ the assist planet's upper atmosphere.</li>~~
~~<li>No thrust burn, no Oberth effect.</li>~~
~~</ul>~~
The entire problem is centered around the sun's frame of reference, so the plot needs an answer with the largest rate of separation from the sun as it's trajectory approaches that asymptote, not the assist planet. So there's a little vector translation I need to figure out.
I suppose the most useful solution has the $V_f = f(m)$ and $V_i = f(m)$ allowing for more diverse applications. This way we know the required initial velocity along with the exit velocity achieved.
I believe the problem can be stated, find the maximum translated V$_f$ with $ ho > 20Â°$ as $r_0$ approaches the upper atmosphere from above.
~~<hr>~~
~~<h1>Premises:</h1>~~
~~<ul>~~
~~<li>The saucer is an interstellar bulk carrier that wants to save fuel and travel time.</li>~~
~~<li>This is for an opportunistic use not routine benefit. It's an energy cost benefit analysis.</li>~~
~~<li>I realize the diminishing returns from steering the vehicle along a vector inclined to the planet's orbital plane.</li>~~
~~<li>Jupiter seems to be the obvious choice but I don't know if getting closer to a fast-moving planet with less atmosphere, like Mercury, gives more oomph?</li>~~
<li>I also realize a smaller angle will allow a greater $V_f$, but the $V_i$ I feel will be prohibitive ($\Delta$V will get too small).</li>
~~</ul>~~

I want to send a flying saucer (a $\frac{1}{10}$ scale <a href="https://medium.com/@brandonweigel/the-largest-ships-in-science-fiction-d3dc9c2da3f" rel="nofollow noreferrer">Independence Day City Destroyer</a> - calculated as a sphere cap 112m high with 1.2km segment radius, $ ho$ about 43.3kg/m3) away from the sun as fast as possible, accelerating it from some initial velocity $V_i$ to escape velocity $V_f$ by <a href="https://en.m.wikipedia.org/wiki/Gravity_assist" rel="nofollow noreferrer">gravity assist</a>, and learn the $\Delta V$ gained. It's mass is about $2\times 10^9$ metric tons. Parameters are:
<ul>
<li>The assist will deflect the initial trajectory by no less than 20$^{\circ}$ ($\Delta$V is $\delta$ or the angle between the hyperbolic asymptotes). This allows a useable steering benefit.</li>
<li>The entry vector $V_i$ is an indeterminate variable with less velocity than the exit vector.</li>
<li>Any celestial body in our solar system can be used for the assist except the sun (the sun's gravity can't assist something escaping itself?)</li>
<li>The mass cannot enter any atmosphere (no friction calculations or concerns) so $\text r_0 = $ the assist planet's upper atmosphere.</li>
<li>No thrust burn, no Oberth effect.</li>
</ul>
The entire problem is centered around the sun's frame of reference, so the plot needs an answer with the largest rate of separation from the sun as it's trajectory approaches that asymptote, not the assist planet. So there's a little vector translation I need to figure out.
I suppose the most useful solution has the $V_f = f(m)$ and $V_i = f(m)$ allowing for more diverse applications. This way we know the required initial velocity along with the exit velocity achieved.
I believe the problem can be stated, find the maximum translated V$_f$ with $ ho > 20^{\circ}$ as $r_0$ approaches the upper atmosphere from above.
<hr>
<h1>Premises:</h1>
<ul>
<li>The saucer is an interstellar bulk carrier that wants to save fuel and travel time.</li>
<li>This is for an opportunistic use not routine benefit. It's an energy cost benefit analysis.</li>
<li>I realize the diminishing returns from steering the vehicle along a vector inclined to the planet's orbital plane.</li>
<li>Jupiter seems to be the obvious choice but I don't know if getting closer to a fast-moving planet with less atmosphere, like Mercury, gives more oomph?</li>
<li>I also realize a smaller angle will allow a greater $V_f$, but the $V_i$ I feel will be prohibitive ($\Delta$V will get too small).</li>
</ul>

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