Blog #8, Worksheet #3.1, Problem 1, Orbital Speeds

1. Below are the orbital distances and periods of solar system planets.

a) Calculate the orbital speed of each planet assuming that the orbits are perfectly circular. Report these speeds in AU/Year.

We calculate the orbital speed by first finding the circumference of each planet's orbit using \(d = 2 \pi a\) then dividing the circumference by the orbital period to obtain the orbital speed. We then add it to the given chart.

 
b) Recall that Kepler’s Third Law has the form:

Where P is the orbital period, a is the semimajor axis, \(M_{tot}\) is the sum of the two masses in the system. Calculate the orbital speeds of the planets predicted by Kepler’s Third Law for each planet.

We begin by solving for an equation for v, the orbital speed. We know that the orbit speed takes the form \[v = \frac{d}{p}\]We know that \[d = 2 \pi a \] and p is found from taking the square root of kepler's equation above. Thus to solve for our speed, v, we get \[v = \frac{2 \pi a}{(\frac {4 \pi^2 a^3}{G M_{tot}})^{\frac{1}{2}}}\]When simplified, this comes out to \[v = (\frac {G M_{tot}}{a})^{\frac{1}{2}}\]Solving for the the semimajor axis of each planet, we get the chart below. The orbital speeds from Kepler's equation are incredibly close to the observed speeds. 

c) Plot the observed orbital speeds against the semimajor axis of each planet. In the same graph, plot the curve predicted by Kepler’s Third Law. Describe the shape of the resultant graph. What you have plotted is a rotation curve for the solar system, and the shape you observe is characteristic of Keplerian systems where one central mass dominates (e.g. the Sun).