Calculating the Rotation Curves of Galactic Disks

In summary, the individual contributions of the baryons and dark matter halo to the rotation curve of a galaxy can be calculated using Poisson's Equation and the NFW density profile. However, when combining these contributions, it is important to use the correct rotation curve model, as using the wrong one may result in incorrect results.
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PhotonSSBM
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Ok, so I'll start off by saying this is NOT a homework problem, but it is a problem I'm having with a project I'm working on, and my supervisor has no clue as to why I'm getting the wrong results from a calculation I'm doing.

So as you all reading this likely know, we can model the rotation curves of galaxies solving Poisson's Equation for a disk and differentiating the solution to get the centripetal acceleration of a star at a given radius from that disk. This is how I'm trying to calculate the rotation speeds due to the baryons in the disk. Now, we also know that the disk mass isn't the only contribution to the acceleration of the stars, we have a spherical dark matter halo that contributes to it as well.

Here is how I'm performing these calculations in numerical python.

For the Baryons in the disk, I'm solving Poisson's Equation using an exponential disk model with Bessel Functions. The analytic result of the integration is this

##\frac{v^2}{R} = 4\pi G \Sigma_0R_dy^2[I_0(y)K_0(y)-I_1(y)K_0(y)]##

where I and K are Bessel functions of the first and second kind, Rd and Sigma_0 are scaling parameters, and y is R/2Rd.

For the dark matter halo, it's a sphere, so the math gets much easier, except I believe my analysis here is where the problem is.

When solving for the acceleration due to a spherical mass we get

##\sqrt{\frac{GM}{R}}##

When calculating M, we can integrate over the density distribution of the dark matter.

The density profile I'm choosing to use is the NFW Profile given by:

##\rho(r) = \frac{\rho_{critical}\delta_c}{(\frac{r}{R_c})(1+\frac{r}{R_c})^2}##

I then take the two contributions and add them under quadrature, the plots produced are posted below, where the blue curve is the velocity due to the Baryons, the Green from the DM Halo, and the Red is their combination.
figure_1.png


Can someone who knows a bit about plotting these curves tell me where I'm going wrong.

Note, the parameters for each scale length in this plot are as follows:

##\Sigma_0 = 5*10^7 \frac{Solar Masses}{kpc^2}##
##R_d = 25 kpc##
##\rho_{critical}\delta_c=5*10^7## this could also be where my calculation goes awry but I cannot find any papers with data on this quantity
## R_c=R_d/4##

Edit: Note that these scales are for what papers have found to be the scale quantities of the Milky Way Galaxy.
 
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Related to Calculating the Rotation Curves of Galactic Disks

What is the purpose of calculating rotation curves of galactic disks?

The purpose of calculating rotation curves is to understand the distribution of mass within a galaxy and how it affects the rotation speed of stars and gas in the galactic disk. This can provide insights into the structure and formation of galaxies.

What are the factors that influence the rotation curve of a galactic disk?

The rotation curve of a galactic disk is influenced by the distribution of mass within the galaxy, including the dark matter halo, stars, gas, and dust. The shape of the rotation curve is also affected by the gravitational pull of neighboring galaxies and the overall structure of the universe.

How is the rotation curve of a galactic disk calculated?

The rotation curve is calculated by measuring the Doppler shift of spectral lines from stars and gas in the galactic disk. This provides information about the velocity of these objects at different distances from the galactic center, which can be used to determine the rotation curve.

What can the shape of a galactic disk's rotation curve tell us about the galaxy?

The shape of the rotation curve can provide information about the distribution of mass within the galaxy and the presence of dark matter. A flat rotation curve, where the velocity remains constant at large distances from the galactic center, suggests the presence of a significant amount of dark matter in the galaxy.

How does the rotation curve of our own galaxy, the Milky Way, compare to other galaxies?

The rotation curve of the Milky Way is similar to other spiral galaxies, with a flat outer region indicating the presence of dark matter. However, the exact shape of the rotation curve can vary between galaxies, providing insights into their different structures and formation histories.

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