(a) Show that Kepler’s 3rd can be expressed in terms of the orbital frequency Ω =2π/P (i.e. orbits/time) and the distance from the center r3Ω2=GMtot
(b) Now, assume that the Milky Way has a spherical mass distribution – this is a good approximation when talking about the total mass distribution. Using what you learned from Problem 2, rewrite the above for an object orbiting a radius r from the center of the galaxy.
(c) Next, let’s call the velocity of this object at a distance r away from the center, v(r). Use Kepler’s Third Law as expressed above to derive v(r) for a mass m if the central mass is concentrated in a single point at the center (with mass Menc), in terms of Menc, G, and r. This is known as the Keplerian rotation curve. As you saw earlier, it describes the motion of the planets in the solar system, since the Sun has nearly all of the mass.
(a) Show that Kepler’s 3rd can be expressed in terms of the orbital frequency Ω =2π/P (i.e. orbits/time) and the distance from the center r3Ω2=GMtot
P2=4π2r3GM
Next Let's solve our new equation for P.
P=2πΩ
Now we substitute the second equation into the first.
4π2Ω2=4π2r3GM
The 4π2's cancel, leaving us with our answer:
r3Ω2=GM
(b) Now, assume that the Milky Way has a spherical mass distribution – this is a good approximation when talking about the total mass distribution. Using what you learned from Problem 2, rewrite the above for an object orbiting a radius r from the center of the galaxy.
In Problem 2 we determined that the mass inside a certain radius of a spherical mass distribution is defined as:
Menc=43πρr3
Where ρ is a mass density constant.
Substituting in for M, we get:
r3Ω2=G43πρr3
Ω=(43Gπρ)1/2
(c) Next, let’s call the velocity of this object at a distance r away from the center, v(r). Use Kepler’s Third Law as expressed above to derive v(r) for a mass m if the central mass is concentrated in a single point at the center (with mass Menc), in terms of Menc, G, and r. This is known as the Keplerian rotation curve. As you saw earlier, it describes the motion of the planets in the solar system, since the Sun has nearly all of the mass.
Let's start with Kepler's Third as usual.
r3Ω2=GMenc
Earlier we defined Ω=2πP. With a simple modification of a factor of r, we can relate it to V. Ωr=2πrP=V
Substituting in, we get:
V2(r)r=GMenc
V(r)=(GMencr)1/2
I worked with B. Brzycki, G. Grell, and N. James on this problem.
nice
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