gaming

physics

maths

Published on 5/28/2021

Last edited on 5/28/2021

Written by Robert Koch

5 min read

My all time favorite video game series is the halo franchise. The lore in the
game spans over 100,000 years from the ancient past to the mid 2500s.

While the games have never been considered nor tried to enter the hard sci-fi
genre the devlopers at Bungie and later 343i have always given consistent
internal explainations for some of the more far fetched concepts seen in the
games. By far the most incredible spectacles included are the namesakes of the
game. In the games lore the rings were created by an ancient race known as the
Forerunners. Possessing technology far beyond humanities comprehension the
Forerunners were able to build megastuctures on a galactic scale.

Installation 04

The games are mainly set on giant ringworlds called Halos, containing a
biosphere capable of supporting life, including oxygen, liquid water, and more
interestingly gravity. However it's not mentioned in the games how the rings are
creating the gravitational force, it's assumed that there's some form of fancy
technology generating the forces on the ring. But the interesting thing is we
know how to generate artificial gravity today, and we've been building
prototypes to test how humans could live and work in space using them.

Using only currently known methods there really is only one way to simulate
gravity, rotational intertia. The idea is fairly simple, when you spin something
really fast the force acting upon the object will be strong enough to push the
object away from the center of rotation.

For humans to live in space for long periods of time and survive we'll need
artificial gravity. This idea has been around for decades and you've probably
seen a giant ringed spacecraft slowly rotating in movies like 2001 or The
Martian. If a large enough spacecraft rotates a habitation ring around a center
axis, it can produce a force similar to gravity.

Space Hotel from 2001: A Space Odyssey

To calculate the required speed the ring needs to be spinning to create the same
gravity felt at sea level on earth (which is $9.81ms^{-1}$) we first need to
understand how large the rings are. Looking through the games the ring in
Halo:CE is reported at $10,000km$ in diameter.

$diameter=10,000km = 1\times10^7m$

So starting with that the radius is half of the diameter.

$radius=5\times10^6m$

Circular motion equations are well known in physics, they're mostly taught in
high school, for a refresher you can find them on
wikipedia. Acceleration is
given as:

$a = \frac{v^2}{r}$

Subsituting in known values can yield an equation for velocity.

$9.81ms^{-2}=\frac{v^2}{5\times10^6m}$

Rearranging and evaluating outputs a result of $7000ms^{-1}$ or 7km a second!
This seems pretty fast, and it is, for context at the equator the earth spins at
$465.10ms^{-1}$.

$v = \sqrt{9.81\cdot5\times10^6} \approx 7000 ms^{-1}$

So that answers the question of how fast the ring needs to spin to simulate
Earth's gravity, but how long does it take for the ring to complete one
rotation? Well the rate of rotation of an object is known as it's angular
frequency and is measured by how many radians (a radian is a unit of angle like
degrees) an object rotates per second.

$\omega = 2\pi f=\frac{2\pi}{T}$

So now we need to solve for omega using values we already know. There is another
definition of angular frequency given by the velocity and radius of the
rotation.

$\omega=\frac{v}{r}=\frac{7000ms^{-1}}{5\times10^6m} = 0.0014 rad s^{-1}$

Now that the angular frequency is known, it can be subsituted back into the
formula, and the rotation time is calculated to be 1.25 hours!

$T=\frac{2\pi}{\omega}=\frac{2\pi}{0.0014} = 4489 s = 1.25 hrs$

So to summarise, a ring with a diameter of 10,000km (which is 70% the size of
the Earth) needs to complete one rotation in less than 90 minutes to simulate
Earth's gravity. That's a pretty short amount of time, but is it so short that
you would feel the negative effects from the spinning and become nauseous?

I'm not too sure about the answer to this to be fairly honest. My suspicion is
that the radius or rotation is large enough such the a human standing on the
ring wouldn't feel nauseous, but they would feel very weord when moving about.
Tom Scott has done a great video on the subject if you want to see.

Another intersting fact about rotational forces, the acceleration is
proportional to the square of the velocity. That means that spinning at half the
speed creates only a quater of the force. You can see this in the interactive
tool I've added below. The diagram on the left shows the relative speed the ring
needs to rotate to generate an acceleration equivalent to gravity (the red
arrow), and the chart on the right shows how the acceleration changes based on
the velocity.

5%

0.02 ms^{-2}

350.18 ms^{-1}

So it seems possible for a ring the size of the ones in Halo to produce enough
of a force to simulate gravity without killing everyone who would be standing on
the ring. The next question is, could you drive a tank on a spinning ring and
accurately hit a target?

Kochie Engineering 2021