A revolving astronaut

The concept of orbital velocity tells us that there must be a minimum velocity for a satellite to revolve around earth and the velocity should be such that the gravitational force of earth provides centripetal acceleration to that velocity. So how is it that even a stationary body like an astronaut outside his spaceship revolves around the earth?? i mean shouldn't he get pulled into its surface? (the concept of escape velocity is quite different.....that makes a body escape the gravitational field of earth).

• How does mass come into the equations? – Bernhard Dec 15 '13 at 13:37
• The astronaut is stationary wrt to the spaceship, but how fast is the ship going? – David H Dec 15 '13 at 13:38
• Why gravity would rotate the space-ship only not the astronaut. i.e "if astronaut should get pulled into the surface" then by the same logic "spaceship should also get pulled into the surface". Don't you think so? – user31782 Dec 15 '13 at 14:05

If I understand your question correctly you assume that the spaceship is driven by some kind of engine giving it the necessary speed to revolve around the earth. As the astronaut does not have such an engine you believe he should fall back on the earth. If this interpretation is not correct, maybe you could make your point a little more clear.

There are two things you maybe got wrong:

1. The spaceship does not need an engine as the atmosphere in most used orbits is very thin. Thus the spaceship is nearly not slowed down. So if it has the necessary speed once, it will keep it for a while. The same holds for the astronaut: The time he leaves his spaceship he has the same speed and will revolve on the same trajectory.

2. There is no such "minimum speed" to revolve around the earth (at least not in the sense I think you believe). If a body in an orbit slows down it will come closer to earth but only until it reaches a closer equilibrium radius. So even if the astronaut slows down more than the spaceship he will only "fall down" in an stable orbit closer to earth, not immediately fall down to the surface.

• what you thought i was thinking is true......but i somehow had this idea that astronaut does not fall due to inertia.......thanxx – vas Dec 16 '13 at 15:06
• just a point, if a body slows down it will not go into a lower circular orbit, it will go into an elliptical orbit which will pass closer to the orbiting body at one tip of the ellipse, and return to the same distance as it started at after every revolution (the case where it's losing speed constantly is more complex but still not circular, there are ways to get into a lower stable orbit, but they require specific impulses applied at specific times) – Zephyr Dec 16 '13 at 15:10

Remember that for the astronaut's spaceship to be in a steady orbit, it must be moving around the earth at the appropriate velocity $v$, where $\frac{GMm}{R^2} = \frac{mv^2}{R}$ (i.e. the gravitational pull of earth is matched by the force needed to accelerate the astronaut in a circular orbit), where $M$ is the mass of the earth, $m$ of the ship, and $R$ is the distance from the center of the earth.

We can rearrange this to find $v = \sqrt{\frac{GM}{R}}$, notice that this is independent of the spaceship's mass.

When the astronaut leaves his ship, he may have zero velocity relative to the ship, but this means that to any other observer he is going at the same speed as the ship.

Thus, because the necessary velocity is independent of mass, he also is going at the required speed for a circular orbit, and will continue to orbit indefinitely (assuming both astronaut and ship are well above the atmosphere).

• i thought so that this happens due to inertia thanx a ton....... – vas Dec 16 '13 at 15:00