I don't know too much about the upcoming moving Elysium, but it looks pretty cool. Here is the trailer. The first interesting thing I saw was the rotating space station.
How Do You Make Fake Gravity?
Why are you weightless in space? I have already answered this question in great detail. The short answer is that you don't really feel gravity anyway since it acts on all parts of your body equally and on the space station. Both the space station and all parts of your body have the same acceleration which means there is no interaction forces between them. What you actually feel is not the gravitational force, but rather the force that other objects push on you. On the surface of the Earth, you are mostly at rest. This means that the ground has to push up on you to balance the gravitational force and you feel "weight".
Ok, I'm not going to spend more time talking about "weightlessness". If you aren't happy with my short answer, read my other post. For now, let's say that you have a person in a space station. The space station could be in orbit around the Earth or in deep space where there are no strong gravitational forces. Either way, the person will be weightless.
How do you make a person "feel weight"? One way is to accelerate the person so that there is a force from the "ground" of the space station that is similar to Earth. Here is a force diagram of a person on the surface of the Earth and a person in the space station with "fake gravity".
The person on the ground has the two forces that are equal in magnitude (thus no acceleration). I am calling FN the normal force. This is the force the person feels and calls "weight". Now, we want the astronaut person to have this same FN force. Since there is no gravitational force, this one force would make the person accelerate in that same direction.
That is our fake gravity force - make the person accelerate. There are two ways this person could accelerate in that direction. The person could increase in speed in that direction, this would be an acceleration. However, this would also change the velocity. For a station in orbit, this wouldn't work but it would be fine for a spaceship traveling to another star and accelerating. No, for the space station we need to do something else. The answer is to make the acceleration perpendicular to the velocity of the person so that the person moves in a circle.
When an object moves in a circle, the velocity vector is always changing and the direction of this change in velocity is towards the center of the circle. Acceleration is the rate of change of the velocity vector. It's really that simple. Here is how this would work.
And this is your basic spinning space station. You have seen this in countless space movies - because it is an idea that would actually work. How do you change the acceleration of the object moving in a circle? There are two things to change, the radius of the circle and the speed that you move in a circle. The magnitude of this acceleration is:
Why are there two versions? The first expression is in terms of the linear velocity of the person and the second is in terms of the angular velocity of the space station. Really, they are the same thing. To increase the fake gravity force you could either spin with a faster angular velocity or make a bigger space station. Bigger is better (because you don't have to spin so fast).
How Big Is the Space Station?
Now for the fun part. I was initially going to calculate the required angular speed of the Elysium space station. However, doing that would require me to know the size. Sure, I could make some estimates from the trailer, but I found a better way. There is one shot in the trailer that shows the space station rotating.
There are only very few frames shown, but it is enough for a rough (super rough) estimate using Tracker Video analysis. I will approximate part of the outer space station structure as having an angular size of 0.45 radians. Maybe I should report this as 0.45 +/- 0.8 radians. That should cover the actual size. With this, I get the following plot for a point on the spinning station.
The y-axis should really be the angle, not y. This gives an angular velocity of 0.017 rad/sec (the sign doesn't matter). If I take into account the uncertainty in the angular size of the scene, I get an angular velocity (using the crank three times uncertainty method) of 0.017 +/- 0.003 rad/s.
Now I can solve for the circular radius of the space station assuming the people would have an acceleration of 9.8 m/s2 (which would give a normal force the same as on Earth).
Adding in the uncertainty, I get a space station radius of (3.39 +/- 1.28) x 104 m. That's a pretty big range of sizes - but they are all big space stations. Of course, not big enough to be confused for a small moon.
Here is what that size space station would look like. I created this in VPython since it was super easy to do.
That looks about the same size as in the movie trailer. Yes, VPython is indeed awesome.
Here is a bonus plot. What if you want to build your own space station? It could happen. Here is a chart that shows the required angular speeds for different sized disks and for different levels of fake gravity.
Here you can see what really matters when designing a space station - size. Even if you make the sacrifice of 60% fake gravity, you would still have to spin this thing fairly fast for a radius less than 1 km. Of course, as you get over a radius of 50 km you can also see that you don't gain too much by making your space station much bigger. Well, it might look cooler but the change in angular speed isn't really that great.
Homework
There are too many unanswered questions. Instead of trying to answer them all, I will assign them as homework.
- For a space station radius of 3.4 x 104 m and an angular speed of 0.017 meters rad/s, how much energy would it take to make this rotate? Oh, you need some more info? Save this question for after you answer the following.
- If the space station is 2 km wide, how much material would it take to create? You are going to have to pick the material and some other parameters. However, after this you can answer the first question.
- Make a plot of rotational kinetic energy as a function of space station radius.
- If multiple ATV (Automated Transfer Vehicles) were used to get the space station spinning, how long and how many would it take? Yes, there is more than one answer. This other homework assignment might be useful.
I can probably think of some more questions, but that should be enough for now.
Throwing a Ball on the Space Station
A mere mortal physics blogger would stop with all the stuff above. I'm not stopping.
Let's say you want to play basketball on your nice fancy Elysium space station. How would that work? First, I am going to with the assumption of a station with a radius of 3.39 x 104 m and spinning with an angular velocity of 0.017 rad/s.
If I throw a basketball in the space station, there will be zero forces on after it leaves the shooter's hand. However, the space station will move while the ball travels in a straight line. This will give it the appearance of not traveling in a straight such as what would happen when a force is on it. The easiest way to deal with this problem is to look at the ball from the view point of the basketball players. This means that we will use an accelerating reference frame which will introduce fake forces.
For accelerating reference frames, we can add in fake forces to make everything work. The fake force would be:
There is a problem though. If you are in the Elysium space station and you throw a ball, it will go "up" where "up" means closer to the center of rotation. When the ball gets closer to center, it will be in a different reference frame than it was before. The best way to account for this is to include the coriolis force as well as the centrifugal force. I can write this coriolis force as:
It's pretty complicated - yes, that "x" is the cross product. Instead of going into much detail, let me just model the motion of a basketball both with and without these fake forces. I am going to make an assumption that might be devastating - but I'm going to anyway. I am going to assume that this part of the space station is "locally flat" and that the up direction is constant. Of course, this isn't really true. If you move over a little bit, the "up" direction would also change. I think this will still work though.
In this VPython output, I have two basketballs. Both are thrown with a speed of 10 m/s at 45°. The green trajectory shows what the path would be like on Earth and the orange is in the Elysium space station.
The difference between the two ending locations is just 8.5 cm - so not a big deal. Of course, I neglected air resistance but that might be ok.
What about golf? If Elysium is really the fancy place it tries to be, they have to have a golf course. Let's get serious with this one. That means doing the same thing but also including air resistance (but not lift). Here is the same comparison for a golf ball on Earth and in the space station.
You can't tell, but the difference in range is just 1.2 meters. Not a big deal. I guess those Elysium snobs can continue to play golf.
But wait. There's more homework.
- For a golf ball that goes 76 meters (as in the example above), how much does the ground deviate from a flat plane (due to the curved nature of the space station).
- How small would the space station have to be such that you could no longer play basketball? You might want to start by making the space station quite a bit smaller and calculating the deviation of a basketball.
That's just two homework questions - so stop complaining.
