Why do planets' orbital velocities drop off by less than a third when twice as distant from the sun, rather than three-quarters? If gravity is $1/4$?

Question

Looking at the speeds of the planets in our solar system when traveling around the sun and their distances from it (or, using Kepler's third law,), it seems that when a planets distance is doubled, it's velocity drops to about 70.7% of its previous value...

Why is that, if the gravitational strength between the planet and sun is now only a quarter of its previous value? Shouldn't it be moving at 25% the speed it would have if it was half as distant, not 70.7%?...

All I can think of is that the more distant planet doesn't have to expend as much 'gravitational energy' or whatever changing its direction (which is a form of 'acceleration'), because it is moving in a straighter line (less curvature; less angle in its angular velocity) during its orbit....

Answer 1

Consider a particle with mass m, orbiting in a circle a body with mass M. The gravitational force must be the centripetal force causing circular motion, so

$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$

Cancelling the $m$ and $r$ and square rooting gives:

$$ \frac{\sqrt{GM}}{\sqrt{r}} = v$$

So you see the velocity is inversely proportional to the square root of distance $r$. If the distance doubles, the velocity decreases by a factor of $1/\sqrt{2}\approx 0.707$. This is the 70.7% that you observed.