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Consider a star orbiting around the center of its galaxy at a radius R with a velocity vc, so that the orbital angular frequency is w = vc/R.

Consider a star orbiting around the center of its galaxy at a radius R with a velocity vc, so that the orbital angular frequency is w = vc/R.

Assume the galaxy is well described by an isothermal sphere model, which yields a at rotation curve: v(R) = vc. In this case, the gravitational potential is (R) = v^2c ln(R) + a constant. What is the epicycle frequency of small radial oscillations? How does it compare to w in this case? Hint: The equation for K uses the eff ective potential, which includes a term for specifi c angular momentum; that quantity is straightforward to calculate for the circular orbit assumed here
SOLUTION:-
230 ± 15 km s−1

b)The Sun orbits in the Milky Way with a speed of 220 km/s at a radius of R = 8.2 kpc. What are the orbital frequency K and epicycle frequency ? Whatare the respective periods? You can assume an isothermal sphere model for the Milky Way and use your results from part (a).
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