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Suppose a planet is moving in an elliptical orbit with the Sun at one of its focii.

I know that the forces in existence will be gravity which provides the necessary centripetal force. Now my book says that in such a case only angular momentum of planet is conserved and not linear momentum. I cannot understand why. Can someone please explain?

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That's because there's the force of gravity acting in the planet. Since there's a net force acting on the planet, its velocity changes which means its linear momentum changes. In fact, the absolute value of linear momentum changes too since the planet's speed is variable as it goes around in its elliptical orbit. But the angular momentum about the sun is conserved since the torque of gravitational force is zero as $\mathbf{F}_g \times \mathbf{r}=\mathbf 0\;.$ From any other point, angular momentum will not be conserved.

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