As distance to the sun decreases, what happens to the orbital velocity?

1 Answer
Dec 10, 2017

As distance to the Sun decreases, orbital velocity increases.

Explanation:

he fact that orbital velocity increases as distance from the Sun decreases can easily be proved using Kepler's laws. We will assume that orbits are circular, which is a good enough approximation to prove the point.

The orbital velocity #v# for a planet orbiting at distance #r# is:

#v=r omega#

Where #omega# is the angular velocity of the planet. Now the angular velocity is related to the orbital period #T#.

#omega = (2pi)/T#

This makes the orbital velocity:

#v=(2 pi r)/T#

Kepler's third law relates orbital period #T# to semi-major axis distance #a=r# as #T^2 prop a^3#. If we add a constant of proportionality #k^2#, which depends on the units of distance and time being used, and take the square root we get:

#1/T=k/(r sqrt(r))#

So the orbital velocity is:

#v=(2 pi k r)/(r sqrt(r))=(2 pi k)/sqrt(r)#

So, the angular velocity is inversely proportional to the square root of the distance.

So, we will choose units such that velocity is in km/s and distance is in AU. Earth at 1AU has an orbital velocity of about 30km/s. This makes the constant #2 pi k=30#.

#v=30/sqrt(r)#km/s

Mercury is at distance 0.39AU which gives a calculated orbital speed of 48km/s.

Neptune at a distance 30AU has a calculated orbital speed of 5.45km/s.

The calculated orbital speeds agree with the measured values. The closer the distance the faster the orbital velocity.