Consider a circle $C_4$ with radius $R$ that rolls inside $C_2$ in such a way that the two circles always touch in the point opposite to the touching point of $C_2$ and $C_3$. Then the circles $C_3$ and $C_4$ follow each other and make the same number of revolutions, and so we will assume that the astronaut is inside the circle $C_4$ instead. But the touching point of $C_2$ and $C_4$ coincides with the touching point of $C_1$ and $C_2$. Hence the circles $C_4$ and $C_1$ always touch each other, and we can disregard the circle $C_2$ completely. Suppose the circle $C_4$ rolls inside $C_1$ in counterclockwise direction. Then the astronaut revolves in clockwise direction. If the circle $C_4$ had rolled along a straight line of length $2\pi nR$ (instead of the inside of $C_1$), the circle $C_4$ would have made $n$ revolutions during its movement. As the path of the circle $C_4$ makes a $360^\circ$ counterclockwise turn itself, the total number of revolutions of the astronaut relative to the ground is $n-1$.