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I know that torque is the rate of change of angular momentum.

Suppose a planet is revolving around the sun. Then its angular momentum about the common center of mass of the planet and the sun will remain conserved, as the external torque acting on the planet is zero.

However, if we consider the angular momentum of the planet about a point present on its orbit, the angular momentum of the planet will not remain conserved. For example when the planet is on the point itself, at that instant position vector is zero so angular momentum of planet is zero. But as planet progresses on its orbit angular momentum will now be non zero. This implies that there must be an external torque acting on the planet.

Therefore, the conservation of angular momentum in this case depends on the choice of the point about which we are calculating the angular momentum.

Am I interpreting the things correctly or not? Please help me understand.

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The total angular momentum about any point of bodies in a closed system is conserved, provided that the forces between the particles of the system are central forces – that is acting along the line joining the particles. [Show on a diagram the forces (central and obeying Newton's third law) between 2 particles and it will be obvious that their total torque about any point is zero.]

So, in your example, the total angular momentum of Sun and planets is constant, if the Solar System can be taken as a closed system.

The angular momentum of a planet by itself is conserved about just one point: the centre of mass of the planet-Sun system (which is nearly at the centre of the Sun), assuming that we can neglect the (much smaller) forces on the planet from other planets.

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A central force will apply a torque to the orbiting body when calculated about any point that is not the barycenter. Of course there is an equal and opposite torque applied to the central mass, so total angular momentum is conserved.

Sometimes angular momentum, torque, and the like, are called pseudo-vectors, because their value is origin dependent. This term can also be used to mean axial-vector: vectors that don't change sign under parity inversion..because they're really the 3-independent components of an antisymmetric rank-2 tensor.

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