Mass versus rotation Curves Is there an equation that describes the relationship between the mass of the Galaxy and  rotation curve?
I found V versus R graphs and equations that describe their relationship (kind of). But I wonder how mass would affect the rotation curves. For instance, if the Milky Way had more mass what would be the rotation curve look like or if it had less mass etc.
 A: If the galaxy is axisymmetric and $\Phi = \Phi(R, z)$ is the potential then
$$
v_c^2 = \left.R\frac{\partial \Phi(R, z)}{\partial R}\right|_{z = 0}
$$
The trick is now in getting the potential $\Phi$. For the Milky Way you have a bunch of components
$$
\Phi = \Phi_{\rm halo, ~DM} + \Phi_{\rm disk} + \Phi_{\rm halo, \star} + \Phi_{\rm bulge} + \Phi_{\rm BH}
$$
you can even include the gas disk, or the hot gas halo. Now, to answer your question: there's a relationship between mass and potential. For example, if the component is spherical (e.g. the dark halo), then
$$
\Phi(r) = -\int_r^{+\infty}{\rm d}r'\frac{G M(r')}{r'^2}
$$
where $M(r)$ is the enclosed mass at a given radius
$$
M(r) = 4\pi \int_0^r{\rm d}r' r'^2 \rho(r')
$$
For the disk the expression is a bit more complicated, but the idea is the same: the circular velocity depends on the gradient of the potential, which in turn depends on the enclosed mass at a given radius.

EDIT The above clearly depends on the election of the model for the components. To give you an example, consider a Hernquist Dark Matter halo with density
$$
\rho_{\rm halo,~DM} = \frac{\rho_0}{r / r_{\rm halo} (1  + r/r_{\rm halo})^3}
$$
and an exponential razor-thin disk with density
$$
\Sigma_{\rm disk}(R) = \Sigma_0 e^{-R / R_{\rm disk}}
$$
It is not very complicated to calculate the circular velocity for these two components
$$
v_{c, \rm halo}^2 = \frac{GM}{2R_{\rm halo}} \frac{(R / R_{\rm halo})}{(1 + R/R_{\rm halo})^2}
$$
and
$$
v_{c, \rm disk}^2 = \frac{2 G M_{\rm disk}}{R_{\rm disk}} y^2[I_0(y) K_0(y) - I_1(y)K_1(y)]
$$
with $y = R / (2 R_{\rm disk})$. The curve below shows a model with $R_{\rm disk} = 3$ kpc, $R_{\rm halo} = 30$ kpc, $M_{\rm disk} = 10^{10}~M_{\odot}$ and $M_{\rm halo} = 3\times 10^{11}~M_{\odot}$

These are just to give you an example of the numbers
A: import math
from scipy import special
import matplotlib.pyplot as plt

#Constants
G = 4.302*(10**(-3)) # in Pc MS-1 (km/s)
R_halo = 30000 #in pc
M_disk = 10**10 # in solar mass
M_halo = 3*10**11 # in solar mass
R_disk = 3000 # in pc
Radius = []
Velocity = []
V_H = []
V_D = []

for R in range(1,30000,100):
    y = R/(2*R_disk)
    F = (special.iv(0, y)*special.kv(0, y))-(special.iv(1, y)*special.kv(1, y))
    v_halo = (G*M_halo*(R/R_halo)) / (2*R_halo*((1+(R/R_halo))**2))
    v_disk = ((2*G*M_disk*(y**2)*F)/R_disk)
    t = v_halo+v_disk
    Velocity.append(t**(1/2))
    Radius.append(R)
    V_H.append(v_halo**(1/2))
    V_D.append(v_disk**(1/2))


plt.plot(Radius,Velocity,"r")
plt.plot(Radius,V_H,"g")
plt.plot(Radius,V_D,"p")

plt.xlabel("Radius (pc)")
plt.ylabel("Velocity (km/s)")
plt.minorticks_on()
plt.grid(b=True, which='major', color='k', linestyle='-')
plt.grid(b=True, which='minor', color='r', linestyle='-', alpha=0.2)
plt.show()

