# Futurama Physics

The rotting corpses of sunbeams cause global warming.

Good news, everyone! While rummaging through all my old stuff at home, I
found my long-lost copy of *Toto IV*. Huzzah for me! This is entirely
unrelated to what I wanted to talk about, but I have it on good
authority that Toto’s
*Africa* syncs up *really*
well with this post [1]. I’ll tell you when to press play. Anyway, what
I really wanted to talk about was a fairly well-posed problem in
*Futurama.* In the episode “Crimes of the Hot,” all of the Earth’s
robots vent their various “exhausts” into the sky at the same time,
using the thrust to push the Earth into an orbit slightly further away
from the sun. As a result of this new orbit, the year is made longer by
“exactly one week.” Anything that quantitative is pretty much asking to
be analyzed. Let’s explore this problem a bit more then, why not? [
*Those wishing to get the full aural experience of this post should
press*play *on their cassette players … now ]* * First, a little
background. In this episode, it is learned that all the robots
(especially Hedonism Bot) emit the greenhouse gases responsible for
Global Warming. The previous solution (detailed
here) is no longer viable,
so it is decided that all robots must be destroyed (especially Hedonism
Bot). The disembodied head of Richard Nixon rounds up all the world’s
robots on the Galapagos [2] to have a “party” so that they may be
destroyed by a giant space-based electromagnetic pulse cannon. In a
last-ditch effort to save the robots, Professor Farnsworth has all the
robots blast their exhausts into the sky, using the thrust to push the
Earth into an orbit further away from the sun, thus solving the problem
of global warming once and for all. As a result of changing the Earth’s
orbit, the year is “exactly one week longer.” First Pass Through Ok, so
what can we say about the new orbit if all we know is that its orbital
period is exactly one week longer? Well, we know from our good buddy
Kepler that the square of the period of a bound orbit is proportional to
the cube of its semi-major axis [3], so * is the semi-major axis of the
orbit and

*k*=

*GMm*. So the difference in Earth’s energy before and after the robo-boost is

Robots blasting from the Galapagos (which now appear to be in China…)

Alright, so how much effort would it take to give that kind of boost to
the Earth? We can quantify this effort in terms of a force or in terms
of the energy difference. Let’s do both. For the force, we have *Futurama*episode.
In the red orbit, the boost was made parallel to the orbital velocity of
the Earth. In each case, the boost was applied at the point labeled with
an “X.” One thing that jumps out from this figure is that the Earth is
always further away from the sun on the red orbit than it was on the
initial (dashed black) orbit. But on the blue orbit, the Earth is
further away from the Sun than it was initially for only half the orbit.
On the other half, the blue orbit would actually make the Earth’s
temperature *higher* than it was on the old orbit! The temperature
calculation we made earlier should hold pretty well for the red orbit,
since it is essentially a circle. It would be a little more tricky for
the blue orbit, as one would need to get a time-averaged value of the
flux over the course of the whole orbit. A hundred Quatloos to anyone
that does the calculation. Wrap-Up So what have we found out here? Well,
it seems that there are certain scenarios in which boosting the Earth
out to a new orbit with period of 1 year + 1 week could cool the Earth
by a few degrees. Granted, we have made some simplifications (the Earth
is not a blackbody), but the general idea of the thing should still
hold. I had some fun playing around with this problem and I thought it
was neat that there was a good deal of information to get started with
from the episode. The *Futurama* people gave an exact period and at
least a visual representation of the direction the robots apply their
push. So 600 Quatloos for the writers! Not Quite As Useless as Usual
Footnotes [1] At least I think it says so if you play this
song backwards. [2]
According to the Wikipedia page for the
episode,
the location of the Galapagos for the party was chosen because the
writers felt that it would be most convenient to push the Earth near the
equator. [3] The semi-major axis of an ellipse is half of the longest
line cutting through the center of the ellipse. Likewise, the semi-minor
axis is half of the shortest line drawn through the center of the
ellipse. Check this out for
some more fun stuff on ellipses. [4] This is technically incorrect,
since the semi-major axis is measured from the center of the ellipse,
but the sun is located on one of the foci. However, this requires
information on the eccentricity of the orbit, which we are currently
glossing over right now. Our method is then approximate, but becomes
exact in the case where both orbits are circles. The effect, however, is
minor. At worst, it is semi-minor. Zing! [5] My source for this billion
robots number comes from Professor Farnsworth himself. When it looks
like the robots will all be destroyed the Professor says “A billion
robot lives are about to be extinguished. Oh, the Jedis are going to
feel this one!” [6] Well, sort of. The height of Everest is \~10^4 m,
so this gives a volume of \~10^12 m^3. The density of most metals is
around \~10 g/cm^3, which is \~10^4 kg/m^3. This gives a mass of
\~10^16 kg. The weight is then \~10^17 N. Each robot exerts a force of
\~10^16 N. So not quite, but hey it was the first thing I thought of
and it almost worked out so I’m sticking with it!

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