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A satellite moving in an elliptical orbit will increase in angular velocity as it nears a planet. I understand that this is consistent with angular momentum. But what causes the increase in angular velocity if there is no torque acting on the satellite?

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There is no torque (with respect to the focus) which implies the angular momentum is constant, but since the angular momentum is $L=mr^2\omega$ and the radius $r$ changes, the angular velocity is not constant. The mechanism resulting this change can be understood as follows.

In an elliptical orbit (with non zero eccentricity) the gravitational force has in general a component along the tangential direction which accelerate the body.

The red arrows in the figure bellow show the gravitational attraction where the blue ones show the components along the tangent of the orbit. Notice that as you go from A to B (counter clockwise orbit) the tangential acceleration increases the tangential velocity. On the other hand, from B to A the tangential acceleration opposes the velocity. That is what changes the angular velocity of the body.

enter image description here

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The gravitational force is a central force so there is no change in the angular momentum of the planet about the Sun but that does not mean that the angular velocity cannot change.
You can think of it as the moment of inertia of the planet about the Sun getting smaller as the planet gets closer to the Sun thus increasing its angular velocity.

As the planet is moving faster the planet has gained kinetic energy at the expense of a loss of gravitational potential energy due to the planet being closer to the Sun.

The force of attraction on the planet due to the Sun is doing work to increase the kinetic energy of the planet because as the planet is getting closer to the Sun there must be a component of the planet's displacement in the direction of the gravitational force.

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The gravitational force of the host planet causes the change in angular velocity.

If you have no issues with circular orbit, then elliptical orbit is no different except that the satellite comes closer to the planet and higher gravitational attraction causes the sharper turn. GR (and Newtonian) math, both describe these orbits.

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Consider an object moving straight past a point that doesn't gravitate at all.

In uniform linear motion, the angular velocity with respect to this point starts out very small, then grows to a finite value, and then decreases towards 0 again -- all without any force at all acting on the moving object.

Change in angular velocity doesn't need a cause other than the relative positions of the involved objects changing.

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