How do planets orbiting the sun and skaters conserve angular momentum?

Planets orbiting the sun and skaters conserve angular momentum by increasing their speed and decreasing their moment of inertia or vice versa.

##L## is a quantity conserved in something as small as a spinning top, the motion of the planets in the solar system, and even the swirling of all the stars in our galaxy!

##L=I## X ##omega##

where ##I## is the of an object and ##omega## is the angular velocity or how quickly the object is rotating. To talk about the skater and the planets, we must first discuss moment of inertia...

The figure skater is a continuos body, so the moment of inertia will be an integral for every part that is spinning about the axis, the axis itself is running straight down the denter of the body starting at the top of the head. You can see the trend of mass distribution and moment of inertia from the basic shapes below. As you can see, the more mass that is concentrated further from the axis of rotation, the larger the moment of inertia becomes! So as the skater brings their arms in, they are effectively lowering their moment of inertia because they are reducing the distance that arm/leg mass is from the axis of rotation...but why do they increase in speed? Angular momentum just like linear momentum is a conserved quantity.

##L_1=L_2## ##I_1*omega_1=I_2*omega_2##

As the moment of inertia decreases, the angular velocity must increase, so they spin faster!

Now onto the planets... So as the planet gets closer, the distance from the axis of rotation decreases (just like the figure skater bringing the arms inwards), when this happens, in order for angular momentum to be conserved, the angular velocity must increase!