# Introduction

Ender's Game, by Orson Scott Card, is set in the near future when space travel and colonization are both possible. One of the more prominent advances in technology that is described throughout the book is artificial gravity. As soon as Ender arrives at the battle school he notices the artificial gravity being created. Dap describes the ship soon after Ender arrives, "The ship is spinning, and that's what makes it feel like [the outer ring] is down. The floor actually curves around in that direction. Keep going long enough that way, and you come back to where you started." The spinning of the ship is what creates the gravity. How does it create gravity? A look into centripetal and centrifugal force will provide the answer.

# Centripetal and Centrifugal Force

Centripetal Force is the force acting on a mass that keeps it moving in a circular motion. Newton's third law states that for every action, there is an equal and opposite reaction. This explains centrifugal force. Centrifugal force is the tangential force felt by a mass that is in circular motion. In the diagram below, the red line represents the centripetal force, which accelerates the mass always towards the center of the circle of rotation. The orange line represents the centrifugal force, which is the force that is felt by the mass as a reactive force to the centripetal force. The green arrow represents tangential velocity, the velocity at which the mass moves on a straight-line path that is tangential to the circular motion.

So on the battle-school ship, the force that is required to create artificial gravity is the centrifugal force that is created by the rotation of the ship around a central axis. The book does not describe weather or not the gravity felt on the ship is equal to Earths's gravity but for the sake of demonstration, Earth's gravity (9.81m/s2) will be used.

In the article Adapting to Artificial Gravity (AG) at High Rotational Speeds, it is described that the human tolerance for centrifugal artificial gravity is between four and six rotations per minute. For the sake of this article, four rotations per minute will be used. The question that arises is: How big of a circle does it take to create Earth's gravity and rotate at four revolutions per minute?

The formula for centripetal force is same as the formula for centrifugal force except centrifugal force is equal to and opposite centripetal force. The equation is as follows:

Fc = mv2/r

Fc is the centripetal force, v is the tangential velocity, and r is the radius of the circle. Since force is equal to mass multiplied by acceleration,Fc can be substituted by mg (mass and the acceleration of gravity).

mg = mv2/r → g = v2/r

Since tangential velocity is also unknown, the formula for tangential velocity must be used within the equation.

vt = rω → g = (rω)2/r → g = r2ω2/r → g = rω2

ω is the angular velocity in radians per second. Four rotations per minute are equal to 8π radians per minute, which is equal to 2π/15 radians per second. Since we want to solve for radius, the formula must be re-arranged in terms of r.

r = g/ω2

Now we can input the known values:

r = (9.81m/s2)/(2π/15 radians per second)2

r = 55.91m!

If we convert meters to feet, that means in order to rotate at four rotations per minute and create a centrifugal acceleration equal to Earth's gravity, the battle ship would have to be 367 feet in diameter!

Is that a possibility to create with today's current technology? Maybe. NASA lists the international space station (ISS) at a length of 167 feet.

Keep in mind that the diameter calculated above was only for four rotations per minute. What would happen if we increased or decreased the radius? If gravity is held constant, the radius is a function of rpm. Therefore if we increase the rpm, the radius decreases. The table below shows the relationship between rpm and radius. Keep in mind that the higher the RPM, the greater the effects of the Coriolis effect, which is discussed here.

RPMDiameter (ft)
21467.53
3652.24
4366.88
5234.80
6163.06
7119.80
891.72
972.47
1058.70
1148.51
1240.76

Let's take a look at how fast the ISS would have to spin using the size of the crew living quarters. NASA lists the ISS module at 41.2 feet.If we look at the table above, we can see that the module would have to rotate at approximately 12 rotations per minute to sustain a centrifugal force equal to Earth's gravity. We can assume from the text of Ender's Game, that the battle school ship is fairly large because it has several rooms and barracks and Ender describes how the barracks curved upward ever so slightly.Therefore the battle school must be a very large structure. Is a structure this size even feasible?

# NASA's Artificial Gravity

## Conceptual Piloted Torus

In 2001, NASA proposed an idea of a spacecraft that mimicked the spacecraft in 2001: A Space Odessey.The ship was designed to feature living quarters as seen below in a video clip from 2001: A Space Odessey:

The vehicle was a conceptual piloted torus that used Nuclear Fusion Propulsion. The vehicle was designed to carry a 172-megaton payload from Earth to Jupiter in 118 days and would carry a nuclear fusion reactor that would serve as a method of propulsion. The crew payload was designed to be at the farthest end of the 240m vehicle. The article goes into detail describing the crew payload as it was "comprised of three rotating Laboratory/Habitation modules" that would rotate around the longitudinal axis of the vehicle and would create 0.2 g's of artificial gravity. So the artificial gravity created by this conceptual vehicle does not even compare to the artificial gravity created on the battle school ship in Ender's Game. When the article was published, NASA had determined that more research was required to determine if Nuclear Fusion Propulsion was feasible.

# Coriolis Effect

Another problem that arises with the creation of artificial gravity through the use of centrifugal acceleration is the Coriolis effect.The Coriolis effect is technically an "imaginary" force because it is based off of the relative frame of the observer. The Coriolis effect is described perfectly in the article The Need for Artificial Gravity, "The Coriolis effect gives the illusion that mysterious forces are at work. The truth is that as one moves towards the centre of rotation the linear speed required to maintain the same rotational period." For example, if you were Ender standing on one of the floors of the battle school ship and you dropped a pencil onto the floor, your observation of the pencil falling to the floor would show that it does not fall directly to the floor. The pencil will be deflected in the opposite direction that battle school ship is rotating. This could prove to be problematic because any force that is applied perpendicular to the tangential direction of rotation will undergo the effects of the Coriolis effect. This includes movement of the head and the vestibular system.

In the article Adaptation in a rotating artificial gravity environment, the authors describe how a subject can become used to the rotational effects of centrifugal force, however, "if the subject makes an arm or head movement, he or she will be exposed to a transient Coriolis force, the same as in a rotating space vehicle. This force is absent prior to the movement and at the end of the movement because it is a velocity-dependent force." This would cause a subject whom is experiencing the effects of centrifugal force to become dizzy and disoriented if they make any head movement.

Ender's Game does not account for any technology that counters the Coriolis effect therefore it would be impossible to live in the battle school ship without always becoming dizzy and disoriented. It would also be difficult to perform everyday tasks that require vertical movement. For example, any time Ender took a shower throughout the book, the water wouldn’t have flowed directly down to the floor.

The video below demonstrates how the Coriolis effect works:

# Conclusion

Artificial Gravity has already been proven to work using centrifugal force, however it is not feasible with today's technology. Any ship that would be built to simulate artificial gravity would need to be large enough to counter any adverse conditions imposed by centrifugal acceleration. The ship that is described in Ender's Game is plausible but not feasible…yet. It would require several years of construction to create a ship massive enough to imitate the ship described in Ender's Game.

Created by Cameron Koch

Embry-Riddle Aeronautical University

Aeronautical Science