Space Station Math

We can lick gravity, but sometimes the paperwork is overwhelming.

— Wernher von Braun:

Introduction

Figure 1: United Space Structures Proposal for a Rotating Space Station.

Figure 1: United Space Structures Proposal for a Rotating Space Station.

I read an article this morning on a space station proposal concept from United Space Structures (Figure 1). I find it interesting that so many proposals for space station structures are appearing now. These proposal appear to be driven by recent discussions of asteroid capture , sending a married couple on a Mars flyby, and a Mars surface exploration mission.

When I was a child (late 1950s and early 1960s), there were many discussions of about a rotating space "wheel"Wernher von Braun was a key proponent for developing this type of space station (Figure 2). There was even a horrible feature film made about the subject, Conquest of Space – a truly awful plot with pretty good special effects.

Figure 2: Rotating Space Wheel Proposal.

Figure 2: Rotating Space Wheel Proposal.

I found the new discussions interesting because they are discussing rotation rates and the fraction of Earth's gravity they wish to achieve. NASA even has a project called Nautilus-X that they want to use to determine the requirements for an artificial gravity system.

In this post, I will derive a pair of expressions for determining:

  • rotation rate as a function of desired artificial gravity level (i.e. percentage of the Earth's gravity).
  • artificial gravity level as a function of space station radius and rotation rate.

Background

All of these space station proposals generate artificial gravity using centrifugal force. Equation 1 gives the centrifugal acceleration as a function of velocity and radius. This formula forms the basis of my mathematical argument.

Eq. 1 \displaystyle a=\frac{{{{v}^{2}}}}{R}=\frac{{{{{\left( {\omega \cdot R} \right)}}^{2}}}}{R}={{\omega }^{2}}\cdot R={{\left( {2\cdot \pi \cdot f} \right)}^{2}}\cdot R

where

  • ω is the angular velocity of the space station [radians/sec].
  • f is the revolutions per minute of the space station.
  • v is the tangential velocity of a person on the space station rim.
  • R is the radius of the space station.

Analysis

Scope

I will look at the space station characteristics from three proposals:

      • Original von Braun proposal.
      • United Space Structures proposal
      • Nautilus-X table of diameters and rates

Derivations

Figure 3 shows how I derived the formulas I needed using Mathcad.

Figure M: Derivation of the Rotation Rate and Artificial Gravity Formulas.

Figure 3: Derivation of the Rotation Rate and Artificial Gravity Formulas.

Formula Derivation

Example Calculations

Figure 4 shows my calculations for the three examples mentioned above.

Figure M: Calculation for 3 Examples.

Figure 4: Calculation for 3 Examples.

Von Braun Proposal Von Braun Proposal Nautilus-X Proposal

I was able to duplicate all of their calculated values to a reasonable level of accuracy.

Conclusion

I have seen this discussion occur before – I hope this time someone actually builds a rotating space station. It is hard to believe, but after all these years we really do not have human data on the effects of generating artificial gravity using centrifugal force. There was one case where micro-gravity was generated during Gemini 11 in 1966 (Gemini spacecraft tethered to Agena target vehicle) and there is a proposal to use a tethered booster-Soyuz combination to run an artificial gravity experiment as part of an International Space Station resupply mission.

 
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