- How good is the microgravity onboard the ISS? If a 1 kg test mass is placed at the centre on gravity of the ISS then the centripetal force required to keep it in orbit is provided exactly by the gravitational force of the Earth. Would there be any difference for test mass fixed to the fabric of the ISS one metre above, one metre below the centre of gravity? Try to make your answer quantitative if you can. Assume that the ISS orbits at an altitude of 400 km.

First of all qualitatively: Each particle that makes up the ISS is attracted, totally independently, to the centre of the Earth by its gravitational pull. This force is determined by the distance (or more accurately the displacement) from the centre of mass of the Earth, for most practical purposes this is the same as the centre of the Earth. The centripetal force required to keep it in orbit however is determined by the distance and the velocity and that velocity is determined by the motion of the space station and so if a particle is not at the centre of gravity then the two may not be exactly equal. As a result there will be a residual acceleration That will reduce the quality of microgravity on the ISS.

Since the particle will be orbiting as part of the ISS then its angular velocity must be the same (think of the various parts of a spinning roundabout). The radius will however be different across the ISS.

The formula for the gravitational force is:

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And the formula for the centripetal force is:

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Only r, the distance between the centre of the Earth and the particle will change. If the particle is ‘above’ the centre of mass then r will increase and so the gravitational force will decrease whilst the centripetal force will increase. That means that a larger centrally pointing force will be required than can be provided by gravity alone. The only place this force can come from is the contact force between the particle and the rest of the ISS structure; the quality of microgravity will no longer be perfect. If the particle is ‘below’ the centre of gravity then the situation will be reversed and the ISS will provide a support force, directed away from the centre of the Earth.

To handle the problem quantitatively is a little more complex. First we need an equation a formula for the angular velocity of the test mass.

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Where *F0* is the force on the test mass when it is at the centre of mass of the ISS.

The residual force is the difference between the gravitational force provided by the Earth and the centripetal force needed to keep the particle in motion:

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But ∆r << r0, so:

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Really we would prefer our answer to be in terms of the weight of the mass *on the Earth’s surface*, so:

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Instead of talking about forces we can refer to accelerations. So the residual acceleration is:

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Importantly, we set out to answer what the microgravity conditions are inside the ISS and produced a general equation for the quality of microgravity on any Earth satellite.

If the test mass were placed to the side of the centre of mass there would be a slightly different effect since the centripetal force and the gravitational force would be at a slight angle to one another:

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For the plane that is parallel to the Earth’s surface and runs through the ISS’s centre of gravity the difference between the distance to the centre of the orbit of the particle and the distance to the centre of the Earth will be so small as to be insignificant. The important thing is the angle θ between the two forces.

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Where d is the off centre distance parallel to the Earth’s surface and r is the orbital radius of the ISS. So for a test mass placed 1 m to the side of the centre of mass the residual acceleration would be:

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Again, we would prefer the measurement to be compared to the forces we experience at ground level and we would also prefer to work in accelerations rather than forces, so:

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So for a test mass positioned 1 m to the side of the centre on gravity of the station:

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This is exactly a third of the value calculated for a radial shift of 1 m. Looking at the equations it should be clear that the consistency of microgravity across a satellite improves as the radius of the orbit increases. The quality of microgravity across the ISS is summed up by the following diagrams. The laboratory facilities on the ISS lie as close as possible to the central area, where the microgravity quality is highest.

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