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Fall 2013
Phys 260: University Physics III
Prof. Anca Constantin
Assignment #3
Due date: 9/18/12 at the start of class.
Please use a different piece of paper to show your calculations and your answers.
Problem 1:
The distribution of stars in the Galaxy’s disk (fyi: Galaxy = Milky Way; galaxy = any other galaxy ) can be approximated by an exponential function. If stars dominate the mass distribution of the disk, we can then say that the mass distribution of the disk follows an exponential function, like this:
Σ(r) = Σ0 e−r/h , where h is just a scaling factor, a constant.
Using this function, derive an analytic expression for how the orbital (circular) velocity of the stars dependence on the distance to the center of the galaxy, i.e., Vc = Vc (r) (which is called the rotation curve of the galaxy) should look like (ignore the bulge and halo of the Galaxy for this calculation). Remember that for a disk of material, the mass interior to a radius r is given by
(integral over rings of thickness dr):
M (r) =
r
0
2πrΣ(r) dr
a) After you calculated the mass of the disk of the Galaxy enclosed by a circle of radius r, your expression for the rotation curve at that r, should look something like:
Vc (r) = f (Σ0 , h, r). I did not mention universal constants (e.g., G) in this function, they should be there somewhere.
b. Now, with the following notation 1 pc = 3 × 1013 km (a very useful step-ladder in astronomy), and understanding that 1kpc = 103 pc, and if Σ0 = 780 Msun /pc2 and h = 3.5 kpc, plot what the rotation curve of the galaxy should look like from r = 0 to r = 30 kpc. (Hint: if it looks flat, you screwed up.)
c. You probably know already that the observed rotation curve is flat (see figure on the upper right corner for some real data for two galaxies, not The Galaxy this time); at r = 30 kpc, the circular velocity is still ∼ 220 km/s. What is the mass needed to give this circular velocity? How much disk mass is there inside r =
Phys 260: University Physics III
Prof. Anca Constantin
Assignment #3
Due date: 9/18/12 at the start of class.
Please use a different piece of paper to show your calculations and your answers.
Problem 1:
The distribution of stars in the Galaxy’s disk (fyi: Galaxy = Milky Way; galaxy = any other galaxy ) can be approximated by an exponential function. If stars dominate the mass distribution of the disk, we can then say that the mass distribution of the disk follows an exponential function, like this:
Σ(r) = Σ0 e−r/h , where h is just a scaling factor, a constant.
Using this function, derive an analytic expression for how the orbital (circular) velocity of the stars dependence on the distance to the center of the galaxy, i.e., Vc = Vc (r) (which is called the rotation curve of the galaxy) should look like (ignore the bulge and halo of the Galaxy for this calculation). Remember that for a disk of material, the mass interior to a radius r is given by
(integral over rings of thickness dr):
M (r) =
r
0
2πrΣ(r) dr
a) After you calculated the mass of the disk of the Galaxy enclosed by a circle of radius r, your expression for the rotation curve at that r, should look something like:
Vc (r) = f (Σ0 , h, r). I did not mention universal constants (e.g., G) in this function, they should be there somewhere.
b. Now, with the following notation 1 pc = 3 × 1013 km (a very useful step-ladder in astronomy), and understanding that 1kpc = 103 pc, and if Σ0 = 780 Msun /pc2 and h = 3.5 kpc, plot what the rotation curve of the galaxy should look like from r = 0 to r = 30 kpc. (Hint: if it looks flat, you screwed up.)
c. You probably know already that the observed rotation curve is flat (see figure on the upper right corner for some real data for two galaxies, not The Galaxy this time); at r = 30 kpc, the circular velocity is still ∼ 220 km/s. What is the mass needed to give this circular velocity? How much disk mass is there inside r =