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).

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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