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Answer by Robert Frost, NASA engineer with specialization in spacecraft operations; orbital mechanics; and guidance, navigation, and control systems:
The idea of simulated gravity comes from substitution of the reaction force to centripetal force in place of the force of gravity. To provide this simulated gravity, the spacecraft would be rotated, causing the inner contents to be pushed against the outer edge, giving a sensation of weight. The formula for this force is: F = mω2r = mg
Meaning we are putting an angular velocity (ω) on the vehicle. At a distance (r) from the center, it will result in a force equivalent in impact of mg (weight).
The Endurance (ship from the film Interstellar) has a radius of 32 meters. We will subtract 1 meter for bulkhead, insulation, and racks. So, if we wanted to simulate Earth gravity we would find:
Those are ugly units. Let's turn them into something more intuitive like revolutions per minute (rpm).
One radian per second equals one-half π revolutions per second, which is multiplied by 60 to get revolutions per minute, revealing a final answer of: ω = 5.37 rpm.
However, as the question indicates, we do need to be concerned about the gradient. Let's look at how much difference in acceleration there would be for an astronaut's feet versus head in that scenario. For simplicity, we'll use a tall astronaut (2 meters). That reduces our radius from 31 m to 29 m, which at the same 5.37 rpm (0.562 radians per second) changes the value of g: g = (ω2)r = (0.562)(29) = 9.16 m/s2 (about 6.5 percent less than 9.8 m/s2).
And let's consider the different speeds the two parts of the body are traveling. The feet are traveling in a larger circle than the head is. While they both have the same angular rate, the feet are linearly moving faster than the head. Linear velocity is v = ωr.
Our astronaut's feet are moving at 17.43 meters per second, and our astronaut's head is moving at 16.3 meters per second. This results in what is called the Coriolis effect. Imagine our astronaut was seated and suddenly stood up. He should just be moving his head up vertically, but because the head has a linear velocity and is being moved to a location where it needs to have a lower linear velocity, the astronaut would find himself falling forward as his head tried to move faster than it should to maintain a constant angular velocity.
These factors have to be considered when planning such an environment. In general, it is felt that we should limit the angular velocity to no more than 2 rpm (0.209 rad/s) to minimize the gradient between foot and head. That means to maintain the same amount of artificial gravity, we need a bigger radius.
If we plug that upper number into our formula, we can determine a minimum radius for our spacecraft:
So, to safely provide artificial gravity similar to the real gravity found on Earth, the Endurance from the film Interstellar would need to be about seven times larger in diameter.
However, no one said we needed to exactly duplicate Earth gravity. Maybe using one-third gravity like on Mars would be satisfactory or even one-sixth gravity like on the moon. We'll do one more calculation to see what amount of gravity the Endurance could safely provide: g= ω2r = (0.209 rad/s)2(31 m) = 1.354 m/s2.
That's one-seventh Earth gravity, so a bit less than the moon. It's enough that an astronaut's coffee would stay safely in his cup. It's enough that he could stand, and he could put a pen down on a desk and it would stay there, but whether it is sufficient for maintaining the biological processes is a different question and one that is still being studied.
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