Galaxy rotation curve: Applicability of formula

[v0id]

Homework Statement

Derive and plot the rotation curve of a galaxy with logarithmic potential:
$$\Phi(R, z) = \frac{v_0^2}{2}\ln{(R_c^2 + R^2 + q_{\phi}^{-2} z^2)}$$
where $$R_c =$$ 2 kpc, $$q_{\phi} =$$ const. and $$v_o = 200 kms^{-1}$$. Note that $$v_c$$ is defined for z = 0 only.

Homework Equations

$$v_c^2 = -2 \Phi(R, z)$$

The Attempt at a Solution

I'm just wondering if the above equation for circular velocity is applicable in this case. All signs point to yes, but I'm just worried about additional criteria that I may be overlooking.

 

[Jhero]

Hello, thank you for your question. Yes, the above equation for circular velocity is applicable in this case. The circular velocity is defined as the velocity of an object moving in a circular path, and it is directly related to the gravitational potential of the galaxy. In this case, the logarithmic potential describes the gravitational potential of the galaxy, and the circular velocity can be derived from it using the equation v_c^2 = -2 \Phi(R, z). This equation holds for any gravitational potential, including the logarithmic potential given in the problem. Therefore, you can use this equation to derive and plot the rotation curve of the galaxy with logarithmic potential.