Sometimes you just have to move a planet, maybe you want to send the Earth spiraling into the fiery embrace of the Sun, or you want to escape a dying sun, or maybe just maybe you don’t want to put the effort into making your robots more fuel efficient.

In Futurama’s episode “Crimes of the Hot” the Planet Express crew is recruited to battle a massive heat wave. In the episode they learn that to keep the planet habitable for humans it has become necessary to drop giant chunks of ice from Hailey’s comet into the ocean. Alas there is no more ice left, heroic scientists from around the world must gather in Kyoto to help solve the problem of a rapidly heating Earth. Inventor of the Environment and First Emperor of the Moon Al Gore offers a collection of Moon Sapphires to the attendee who can fix the planet. To stop the planet from overheating and to save millions of robots from being deactivated our heroes decide to push the Earth into a higher orbit.

Sounds crazy, but is it as crazy as we might think?

In Short: Yes, yes it is, moving planets is hard, but lets take a quick look at what it might take. (if you just want the answers scroll to the bottom)

To keep this article from getting super long we are going to look at what it would take to change the Earth’s orbit by calculating what is called the Hohman transfer orbit. The Hohman transfer orbit is generally considered the most fuel-efficient way to change orbit*.

Hohman transfer orbits come in 3 parts, the initial change in velocity, followed by a coasting period, and finally a stabilizing burn. If you only do the first 2 parts your orbiting object would be unable to stabilize its new higher orbit as a result you need to burn fuel again to make sure you stay at that new orbit.

Solving an ideal example of a Hohman transfer requires that we know a few critical things, what is the object that we are orbiting, what is our initial orbit, and what is our final orbit?

The dialogue from “Crimes of the Hot” gives us the information we will need to answer all of these questions. At the end of the episode Bender asks “Hey, Professor, now that the Earth's orbit is further from the sun, won't that make the year longer?”. Prof Farnsworth replies “Why, yes! One week longer to be exact.”

This dialogue would appear to indicate that the Earth of the 41st century has gone from having a 365-day year to a 372 day year**.

As there is no indication that the Earth changes which star it orbits in the Futurama universe we can assume that the Earth still orbits our Sun.

The equation for calculating a Hohman Transfer orbit looks like this.

r1= initial orbital distance (in our case the average distance from the Earth to the Sun)

r2= final orbital distance (in our case the distance you would need for the Earth to have a 372 day long year)

μ= the Standard Gravitational parameter (this is the mass of the object being orbited multiplied by the gravitational constant)

Thankfully r1 and μ are both available on Wikipedia. For r1 the Earth is 149.6 million kilometers from the Sun. The Standard Gravitational parameter for the Sun is 1.33*10^20(m^3/s^2) .

The value for r2 needs a little bit more effort, but thankfully there’s an equation that will let us estimate it. Using a version of Kepler’s 3rd law we can determine what orbital radius we would need to achieve a particular orbital period.

Plugging our values for r1, r2, and μ into the Hohmann Transfer orbit equations we can get a sense of how much energy we would need. For our initial burn our math indicates that we would need to increase the Earth’s orbital velocity by 91 m/s or about 203 mph. Increasing the Earth’s velocity over 200 mph sounds like a lot until you remember that the Earth is orbiting the Sun at over 66 thousand mph. The second burn would occur when the Earth had increased its orbit by 1.8 million km. The second burn would further adjust the Earth’s orbital velocity by 86.6 m/s. Overall the Hohman Transfer Orbit would require a net Δv of 178 m/s. Now we can calculate the minimum amount of energy required to change the Earth’s orbit by a week.

The total energy of an orbiting object is the sum of the object’s potential and kinetic energy.

Where M is the sun, m is the Earth, and a is the orbital radius of the Earth (whatever that might be)

To calculate how much energy we would need to change the Earth’s orbit we will want to compare the difference in the total energy of the Earth in its higher orbit vs its lower orbit.

To add 7 additional days to the Earth’s year you would need at least 3.35*10^31 J, to put that number in perspective our Sun produces 384.6 yattawatts of power (3.846*10^26 Joules/S ). Which means if you could magically capture 100% of the Sun’s energy and use it to move the Earth, you would need more than 1 days output of the Sun’s energy. For those of you more destructively minded you would need as much energy as 8.01 Quadrillion Megatons of TNT to move the Earth as much as described in our Futurama scenario.

In Summation

With a mass of 5.97 * 10^24 kg and an average orbit of 1.496 * 10^11 m Farnsworth arnd Robots would need to change the Earth’s orbital energy by at least 33.5*10^30J. to make it possible to add Robot Party Week to the Terrestrial Calendar. The change in energy would raise the Earth’s orbit by 1.8 million km.

The energy change would be greater than what the entire sun puts out each day. You would need the entire sun’s output for 87166 seconds of the suns energy while a day is 86400 seconds.

I hope this was interesting, if you have any questions please feel free to leave a comment

Sorry about the formatting of the equations, it turns out Square’s built in post editor doesn’t really like how Microsoft Word Equation Editor formats things, so I had to manually screen grab things and I’m still trying to get a sense of what resolutions to use.

(1) Frisky Dingo

(2) The Wandering Earth

(3) Futurama Crimes of the Hot

*according to the Wikipedia entries on orbital mechanics there are times where other types of transfer orbits are more efficient, but I mean that sounds like super hard (also I lack the necessary background to really do the topics any kind of justice)

**technically we only know that the Earth year is longer than it was before, but we are not guaranteed that before changing orbits in the episode, that the Terrestrial year was 365 days, but without this assumption we would just be left with variables.

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