I can't quite visualize how a force that acts tangentially to the path can produce radial deviations.

Consider a satellite orbiting the earth on a circular orbit. If there would be no air drag, then the attractive gravitational force ($F_g=-\frac{GMm}{r}$) and the repulsive centrifugal force ($F_c=mr\dot{\phi}^2$) would perfectly balance. And hence the radius $r$ stays constant, giving a circular orbit.

Now add some air drag. The drag force is acting horizontally. That means that the angular speed ($\dot{\phi}$), and hence the centrifugal force ($F_c=mr\dot{\phi}^2$) will decrease. This destroys the balance between centrifugal force and gravitational force, so that there is now a small net force pointing down towards the earth.

I can't quite visualize how a force that acts tangentially to the path can produce radial deviations.

Consider a satellite orbiting the earth on a circular orbit. If there were no air drag, the attractive gravitational force ($F_g=-GMm/r$) and the repulsive centrifugal force ($F_c=mr\dot{\phi}^2$) perfectly balance. Hence, the radius $r$ stays constant, giving a circular orbit.

Now, add some air drag. The drag force is acting horizontally. That means that the angular speed ($\dot{\phi}$), and hence the centrifugal force ($F_c=mr\dot{\phi}^2$) will decrease. This destroys the balance between centrifugal force and gravitational force, so that there is now a small net force pointing down towards the earth.

Consider a satellite orbiting the earth on a circular orbit. If there would be no air drag, then the attractive gravitational force ($F_g=-\frac{GMm}{r}$) and the repulsive centrifugal force ($F_c=mr\dot{\phi}^2$) would perfectly balance. And hence the radius $r$ stays constant, giving a circular orbit.

Now add some air drag. The drag force is acting horizontally. That means that the angular speed ($\dot{\phi}$), and hence the centrifugal force ($F_c=mr\dot{\phi}^2$) will decrease. This destroys the balance between centrifugal force and gravitational force, so that there is now a small net force pointing down towards the earth.

I can't quite visualize how a force that acts tangentially to the path can produce radial deviations.

Consider a satellite orbiting the earth on a circular orbit. If there would be no air drag, then the attractive gravitational force ($F_g=-\frac{GMm}{r}$) and the repulsive centrifugal force ($F_c=mr\dot{\phi}^2$) would perfectly balance. And hence the radius $r$ stays constant, giving a circular orbit.

Now add some air drag. The drag force is acting horizontally. That means that the angular speed ($\dot{\phi}$), and hence the centrifugal force ($F_c=mr\dot{\phi}^2$) will decrease. This destroys the balance between centrifugal force and gravitational force, so that there is now a small net force pointing down towards the earth.