When Uranus Lets Out A Toot

To all my readers, I apologize in advance for the goofy title. 
While it may take some brains to write a space blog, maturity is purely optional! :)

As a die-hard space fan, I get a peculiar satisfaction dropping random tidbits of space trivia on people without them even asking for it. Weird flex, I know, but I like seeing the look of intrigue on their faces when people learn something about the cosmos they'd never pondered before. One fact I tell people that often surprises them: even after decades of planetary exploration and all the famous probes we've sent out, only one spacecraft has ever visited the distant worlds of Uranus and Neptune: Voyager 2, in 1986 and 1989!

Why explore space, you ask? For the best memes in the galaxy, of course!!

This means that even in 2020, for any planetary scientist hoping to find a breakthrough about the two ice giants, their best bet is still to delve into data collected over 30 years ago, when Ronald Reagan was president, the Soviet Union still existed, and the first Apple Macintosh and its 128KB of RAM were state of the art! And sometimes, they discover hidden treasure in all that data - in this recent announcement from NASA, scientists discovered that when Voyager 2 flew by Uranus, it actually flew through what's know a plasmoid, a giant bubble of the atmosphere that got ejected into space due to interactions in the planet's magnetic field!

Artist's impression of Voyager 2 zooming by Uranus

According to the researchers, while magnetic fields are crucial in preventing solar wind from stripping away a planet's atmosphere (just look what happened to Mars!), sometimes the complex magnetic field lines get tangled and pinch off a little bubble. It turns out, Uranus is particularly susceptible to this phenomenon as it's the only planet that spins on its side, causing its magnetosphere to "wobble like a poorly thrown football." We're fortunate that Voyager 2 happened to fly by just as Uranus was letting off a little steam! 

Sharing this visualization from the NASA announcement showing Uranus' magnetic field:
 The yellow arrow denotes Uranus' direction of orbit, while the light blue arrow marks the planet's magnetic axis 

So this all begs the question, why not pay Uranus and Neptune another visit? And on top of that, given Voyager 2 already nailed a flyby of both planets, why not send an orbiter to at least one of them so we can maintain constant observation? In fact, there are a number of ambitious proposals being submitted, both for NASA's Planetary Science Decadal Survey and through the European Space Agency, that would answer many longstanding mysteries about the nature of the gas giants. Unfortunately, missions to Uranus and Neptune are far less likely to be chosen than inner planets like Mars and Venus, meaning astronomers will probably be left guessing and combing through Voyager's old data for decades to come

Some proposed missions to Uranus and Neptune. Think any of them will ever see the light of day?

So I'm faced with this harsh reality, fine then. But nothing can stop me from dreaming of a world where NASA has an infinite budget, and in my imagination I'm going to propose an even more ambitious planetary mission: a Uranus/Neptune double orbiter!! By this I mean a single probe that reaches Uranus first, orbits it for a period of time, then departs for Neptune and establishes a new orbit there as well (unlike ODINUS, which would send two separate probes)! Obviously this would require an enormous amount of energy, but how impossible would it be?

I take comfort knowing that my hypothetical mission wouldn't be totally without precedent. The Dawn spacecraft launched in 2007 was the first ever to orbit two different bodies, orbiting the asteroid Vesta for 14 months before departing for Ceres, where it orbited for several years! But Uranus and Neptune would be significantly more challenging, being so far away...

Animation of Dawn's mission trajectory, from Wikipedia
Pink = Dawn   |   Dark Blue = Earth   |   Yellow = Mars   |   Light Blue = Vesta   |   Green = Ceres

Bust out the calculators!

In investment banking jargon, this is what we call a "back-of-the-envelope calculation" (remember I don't have a PhD in astrophysics, just my simple bachelors in finance!). But the exercise I'll demonstrate here should roughly give the change in velocity (i.e. delta-v) needed to travel from an orbit around Uranus to an orbit around Neptune, using a Hohmann transfer orbit. It's essentially the same calculation I demonstrate for my trajectory to Mars, so check that out here for additional detail!

Here's the general diagram of what we're trying to achieve

Some constants we're going to need:
Gravitational constant:  $G = 6.674 \cdot 10^{-11} \thinspace m^3\cdot kg^{-1}\cdot s^{-2}$
Mass of the Sun:  $M = 1.989 \cdot 10^{30} \thinspace kg$

Radius of Uranus' orbit:  $r_{Uranus} = 2.963 \cdot 10^{9} \thinspace km$
Radius of Neptune' orbit:  $r_{Neptune} = 4.477 \cdot 10^{9} \thinspace km$

Some useful formulas:
Vis-viva equation (used for calculating orbital velocities):  $V = \sqrt{GM(\frac{2}{r} - \frac{1}{a})}$
Special case (circular orbits):  $V = \sqrt{\frac{GM}{r}}$

Great, now we're ready to begin. Since our probe is starting off in a stable orbit around Uranus, we need to calculate its initial velocity relative to the Sun. In other words, what's Uranus' orbital velocity as it orbits the Sun? Assuming the orbit of Uranus is circular (a reasonable approximation; we'll do the same for Neptune later), we can calculate it as follows:

$V_{Uranus} = \sqrt{\frac{GM}{2.963 \cdot 10^{9}}} = 6.693 \thinspace km/s$

Now in order to start our probe on its path to Neptune, we need to accelerate our probe with an engine burn right at the point where Uranus' orbit intersects the Hohmann transfer orbit. Since this point is the periapsis of the elliptical orbit, we can calculate it as follows:

$a = \frac{2.963 \cdot 10^{9} + 4.477 \cdot 10^{9}}{2} = 3.720 \cdot 10^{9} \thinspace km$

$V_{periapsis} = \sqrt{GM(\frac{2}{2.963\cdot 10^{9}} - \frac{1}{3.720 \cdot 10^{9}})} = 7.343 \thinspace km/s$

Using this, we can determine our first delta-v:

$\Delta V_1 = 7.343 - 6.693 = 0.650 \thinspace km/s$

Awesome, so we've started our probe on its path to Neptune. But when it finally reaches Neptune's orbit, how fast will it be going? With the same vis-viva formula we used to calculate velocity at periapsis, we can do the same at apoapisis:

$V_{apoapsis} = \sqrt{GM(\frac{2}{4.477\cdot 10^{9}} - \frac{1}{3.720 \cdot 10^{9}})} = 4.860 \thinspace km/s$

But remember, our goal is to get into orbit around Neptune, so we have to match Neptune's orbital velocity around the Sun:

$V_{Neptune} = \sqrt{\frac{GM}{4.477 \cdot 10^{9}}} = 5.445 \thinspace km/s$

Now we can determine the second delta-v:

$\Delta V_2 = 5.445 - 4.860 = 0.585 \thinspace km/s$

So upon reaching the orbit of Neptune, the probe needs to fire its engine again and accelerate to match Neptune's orbital speed

There you have it: two engine burns generating a total delta-v of 1.235 km/s to get us from an orbit around Uranus to an orbit around Neptune. I've done the math for you NASA, let's get this mission approved!!

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