# Conservation of angular momentum and the behaviour of velocity in orbits

Using $L=mvr$ and conservation of angular momentum, if the radius is halved then the velocity must be doubled and vice versa. But in the case of planetary orbits, this is not the case.

To take the example of the earth orbiting the sun, if the distance is doubled and accounting for constant mass and angular momentum, then the velocity should half which is around 15km/s. But it actually turns out to be closer to 21km/s. This is dividing the initial velocity by the square root of 2 rather than 2.

Why is this?

• Hint: A constant of motion (such as angular momentum) is by definition constant as a function of time. It is not necessarily independent of initial conditions. Feb 5 '17 at 22:37

How do you propose to increase the semi-major axis of the Earth's orbit without applying a torque?

If you apply a torque (for example by slowing down the Earth in its orbit) then angular momentum is not conserved.

A way to conserve angular momentum and increase the semi-major axis is for the Sun to lose mass. Some elementary calculation shows that $J^2 \propto Ma$, where $M$ is the mass of the Sun. Halving the solar mass would double the Earth's orbital radius whilst conserving angular momentum.

• I agree, but am curious. If one did manage to take mass away from Earth, would that not require work and constitute, although indirectly, a type of torque? Feb 6 '17 at 15:19
• @honeste_vivere It is the Sun that would need to lose mass. Isotropic mass loss would not (to first order) exert a torque on the Earth's orbit. Feb 6 '17 at 15:55
• Oh whoops, I completely missed that and thought you meant the Earth. Feb 6 '17 at 15:58

Using $L=mvr$ and conservation of angular momentum, if the radius is halved then the velocity must be doubled and vice versa.

That is not valid. While angular momentum is constant for a given orbit, two quite different orbits will have quite angular momenta. For an object of negligible mass $m$ orbiting circularly about a central body of mass $M$, the angular momentum of the orbiting body is $L=mvr = m\sqrt{GMr}$. (The Earth orbiting the Sun is an example of an object of negligible mass orbiting a central body.) Note that angular momentum increases with the square root of the orbital radius.