(a) Calculate the orbital speed of each planet assuming that the orbits are perfectly circular. Report these speeds in AU/year.
(b) Recall that Kepler’s Third Law has the form:
\[P^2=\frac{4\pi^2a^3}{GM}\]
where P is the orbital period, a is the semimajor axis, M is the sum of the two masses in the system, and \(G=6.67\times 10^{-8}cm^3 f^{-1} s^{-2}\) . Calculate the orbital speeds of the planets predicted by Kepler’s Third Law for each planet.
(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).
\[P^2=\frac{4\pi^2a^3}{GM}\]
where P is the orbital period, a is the semimajor axis, M is the sum of the two masses in the system, and \(G=6.67\times 10^{-8}cm^3 f^{-1} s^{-2}\) . Calculate the orbital speeds of the planets predicted by Kepler’s Third Law for each planet.
(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).
(a) Calculate the orbital speed of each planet assuming that the orbits are perfectly circular. Report these speeds in AU/year.
If the orbits are completely circular, then each planet passes through a distance of \(2\pi a)\ AU per orbit since the is simply the circumference of a circle.
\[Velocity=\frac{Distance}{Time}=\frac{2\pi a}{P}\]
Plugging in numbers for Mercury, we get:
\[V=\frac{2\pi a}{P}=\frac{2\pi (0.39 AU)}{0.24 years}=10.2AU/Year\]
The rest of the planets follow the same system, yielding:
(b) Recall that Kepler’s Third Law has the form:
\[P^2=\frac{4\pi^2a^3}{GM}\]
where P is the orbital period, a is the semimajor axis, M is the sum of the two masses in the system, and \(G=6.67\times 10^{-8}cm^3 f^{-1} s^{-2}\) . Calculate the orbital speeds of the planets predicted by Kepler’s Third Law for each planet.
\[P^2=\frac{4\pi^2a^3}{GM}\]
where P is the orbital period, a is the semimajor axis, M is the sum of the two masses in the system, and \(G=6.67\times 10^{-8}cm^3 f^{-1} s^{-2}\) . Calculate the orbital speeds of the planets predicted by Kepler’s Third Law for each planet.
There is a far easier way to do this. Since our units are AU and years, we can ignore all of the constants and get:
\[P^2=a^3\]
From earlier we established that \(V=\frac{2\pi a}{P}\). Solving for P we get:
\[P=\frac{2\pi a}{V}\]
Plugging this into Kepler's 3rd Law, we get:
\[\frac{4 \pi^2 a^2}{V^2}=a^3\]
Solving for V yields:
\[V=2\pi a^{-1/2}\]
Solving for Mercury yields:
\[V=2\pi a^{-1/2}=2\pi (0.39AU)^{-1/2}=10.06AU/Year\]
The rest of the planets follow the same equation yielding:
(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).
The curve corresponds to an exponential decay defined by \(a^{-1/2}\).
I worked with B. Brzycki, G. Grell, and N. James on this problem.
I worked with B. Brzycki, G. Grell, and N. James on this problem.
nice job
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