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\begin{document}

\title{Galaxies HW 4}


\author[ ]{Ryan Farber}

\affil[ ]{}
\vspace{-1em}


\date{}

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\section{Definitions}\label{definitions}

\subsection{Mass (within a sphere)}\label{mass-within-a-sphere}

\[ M(< r) = 4 \pi \int_{0}^{r} r^{2} \rho \text{ dr} \]

\subsection{Gravitational potential}\label{gravitational-potential}

\[ \Phi(r) \equiv \frac{1}{m} \int_{\infty}^{r} F dr \] \[ \Phi(r) = \frac{1}{m} \int_{\infty}^{r} \frac{G m M(< r)}{r^{2}} dr \] \[ \Phi(r) = \int_{\infty}^{r} \frac{G M(< r)}{r^{2}} dr \]

\subsection{Circular Velocity}\label{circular-velocity}

\[ \frac{m v_{circ}^{2}}{r} = \frac{G m M(< r)}{r^{2}} \] \[ v_{circ} = \sqrt{\frac{G M(< r)}{r}} \]

\section{Derivations}\label{derivations}

\subsection{Point mass with mass M}\label{point-mass-with-mass-m}

Density profile (by definition): \[ \rho = M \delta(r=0) \] where
\( \delta \) is the Dirac delta function.

Gravitational potential: \[ \Phi(r) = \int_{\infty}^{r} \frac{G M}{r^{2}}dr = -GM \Big(\frac{1}{r} - \frac{1}{\infty} \Big) \] \[ \Phi(r) = -\frac{GM}{r} \]

Circular velocity: \[ v_{circ} = \sqrt{\frac{G M(< r)}{r}} \] \[ v_{circ} = \sqrt{\frac{G M}{r}} \]

\subsection{Spherically symmetric Singular isothermal sphere with mass
M, truncated at radius
a}\label{spherically-symmetric-singular-isothermal-sphere-with-mass-m-truncated-at-radius-a}

Circular velocity (by definition): \[ v_{circ} = v_{H} \]

Cumulative mass (with r \(\leq\) a): \[ v_{circ} = \sqrt{\frac{G M(< r)}{r}} \]
\[ M(< r) = \frac{v_{H}^{2} r}{G} \] \( \text{if } r > a, M(< r) = \frac{v_{H}^{2} a}{G} \)

Density profile (with r \(\leq\) a): \[ M(< r) = 4 \pi \int_{0}^{r} r^{2} \rho \text{ dr} \]
\[ \frac{dM(< r)}{dr} = 4 \pi r^{2} \rho \] \[ \frac{v_{H}^{2}}{G} = 4 \pi r^{2} \rho \] \[ \rho = \frac{v_{H}^{2}}{4 \pi G r^{2}} \]
\( \text{if } r > a, \rho = 0 \)

Gravitational potential (for r \(\leq\) a):
\[ \Phi(r) = \int_{\infty}^{r} \frac{G M(< r)}{r^{2}} dr \] \[ \Phi(r) = \int_{\infty}^{a} \frac{G M(< r)}{r^{2}} dr + \int_{a}^{r} \frac{G M(< r)}{r^{2}} dr \] \[ \Phi(r) = \int_{\infty}^{a} \frac{v_{H}^{2} a}{r^{2}} dr + \int_{a}^{r} \frac{v_{H}^{2} r}{r^{2}} dr \]
\[ \Phi(r) = v_{H}^{2} \left[ -a \left( \frac{1}{a} - \frac{1}{\infty} \right) + \Big( \ln(r) - \ln(a) \Big) \right] \] \[ \Phi(r) = v_{H}^{2} \left[ \ln \left( \frac{r}{a} \right) - 1 \right] \] \( \text{if } r > a, \Phi(r) = -\frac{G M(< r)}{r} = -\frac{v_{H}^{2} a}{r} \)

If instead one assumes the density profile (r \(\leq\) a)
(Sparke \& Gallagher p.~114) \[ \rho = \frac{\rho_{0} r_{0}^{2}}{r^{2}} \] \( \text{if } r > a, \rho = 0 \)

Then the cumulative mass (r \(\leq\) a) is:
\[ M(< r) = 4 \pi \int_{0}^{r} r^{2} \rho \text{ dr} \] \[ M(< r) = 4 \pi \int_{0}^{r} r^{2} \frac{\rho_{0} r_{0}^{2}}{r^{2}} \text{ dr} \] \[ M(< r) = 4 \pi \rho_{0} r_{0}^{2} r \]
\( \text{if } r > a, M(< r) = M \)

And hence the circular velocity is: \[ v_{circ} = \sqrt{\frac{G M(< r)}{r}} \]
\[ v_{circ} = \sqrt{\frac{4 \pi G \rho_{0} r_{0}^{2} r}{r}} \] \[ v_{circ} = 2 r_{0} \sqrt{\pi G \rho_{0}} \]

Thus \( v_{H} = 2 r_{0} \sqrt{\pi G \rho_{0}} \)

Solving for the constants \(\rho_{0} r_{0}^{2}\) (at r=a):
\[ M = 4 \pi \int_{0}^{a} r^{2} \frac{\rho_{0} r_{0}^{2}}{r^{2}} \text{ dr} \] \[ M = 4 \pi a \rho_{0} r_{0}^{2} \] \[ \rho_{0} r_{0}^{2} = \frac{M}{4 \pi a} \]

We can then also write (all for r \(\leq\) a):

\( \text{Density} = \frac{M}{4 \pi a r^{2}} \)

\( \text{Cumulative Mass (< r)} = M \frac{r}{a} \)

\( \text{Circular velocity} = \sqrt{\frac{G M}{a}} \)

\subsection{Constant density sphere with mass M, truncated at radius
a}\label{constant-density-sphere-with-mass-m-truncated-at-radius-a}

Density profile (by definition): \[ \rho(r \leq a) = \rho_{0} = \frac{M}{\frac{4}{3} \pi a^{3}} \] \[ \rho(r > a) = 0 \]

Cumulative mass (r \(\leq\) a): \[ M(< r) = 4 \pi \int_{0}^{r} r^{2} \rho_{0} \text{ dr} \]
\[ M(< r) = \frac{4}{3} \pi r^{3} \rho_{0} \] \[ M(< r) = \frac{r^{3}}{a^{3}} \] \( \text{if } r > a, M(< r) = \frac{4}{3} \pi a^{3} \rho_{0} = M\)

Gravitational potential (for any r \(\leq\) a):
\[ \Phi(r) = \int_{\infty}^{r} \frac{G M(< r)}{r^{2}} dr = \int_{\infty}^{a} \frac{G M(< r)}{r^{2}} dr + \int_{a}^{r} \frac{G M(< r)}{r^{2}} dr \] \[ \Phi(r) = \int_{\infty}^{a} \frac{\frac{4}{3} \pi G a^{3} \rho_{0}}{r^{2}} dr + \int_{a}^{r} \frac{\frac{4}{3} \pi G r^{3} \rho_{0}}{r^{2}} dr \] \[ \Phi(r) = \frac{4}{3} \pi G \rho_{0} \left[ a^{3}\left(-\frac{1}{a} + \frac{1}{\infty} \right) + \left( \frac{r^{2}}{2} - \frac{a^{2}}{2} \right)  \right] \]
\[ \Phi(r) = \frac{4}{3} \pi G \rho_{0} \left[ -\frac{3}{2}a^{2} + \frac{r^{2}}{2}  \right] \] \[ \Phi(r) = -2 \pi G \rho_{0} \left[ a^{2} - \frac{r^{2}}{3}  \right] \] \[ \Phi(r) = -2 \pi G \rho_{0} \left[ a^{2} - \frac{r^{2}}{3}  \right] \]
\[ \Phi(r) = -\frac{3}{2} \frac{G M}{a} + \frac{1}{2} \frac{G M r^{2}}{a^{3}} \] \( \text{if } r > a, \Phi(r) = -\frac{G M(< r)}{r} = -\frac{\frac{4}{3} \pi G \rho_{0} a^{3} }{r} \)

Circular velocity (for r \(\leq\) a): \[ v_{circ} = \sqrt{\frac{G M(< r)}{r}} \]
\[ v_{circ} = \sqrt{\frac{\frac{4}{3} \pi G r^{3} \rho_{0}}{r}} \] \[ v_{circ} = 2r \sqrt{\frac{\pi}{3} G \rho_{0}} \] \[ v_{circ} = r \sqrt{\frac{GM}{a^{3}}} \]
\( \text{if } r > a, v_{circ} = 2a \sqrt{\frac{\frac{\pi}{3} G a \rho_{0}}{r}} \)

\section{Plots}\label{plots}

We next show comparison plots of the three systems. Note that the units
are funky. The total mass of each object was set equal to 1.0. That
choice constrains \(r_{0} = 1/\sqrt{12} \) and \(\rho_{0} = 6/\pi \). G was also
set equal to one. The radius of truncation, a, was set to 0.5. Without
further ado, the plots:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/density-profiles/density-profiles}
\caption{{Figure 1. Density profiles%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mass-profiles/mass-profiles}
\caption{{Figure 2. Cumulative mass profiles%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/circular-velocity-curves/circular-velocity-curves}
\caption{{Figure 3. Circular velocity curves%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/graviational-potentials/graviational-potentials}
\caption{{Figure 4. Gravitational potentials%
}}
\end{center}
\end{figure}

\section{Figure 5.21}\label{figure-5.21}

Since we are looking at the rotation curves, flat lines indicate
singular isothermal sphere profiles, and linearly increasing segments
indicate constant density profiles.

\subsection{Sb}\label{sb}

The Sb galaxy has a mass density profile which is predominantly singular
isothermal sphere profile but has a constant density region from about
4.5 kpc - 6.5 kpc scale lengths.

\subsection{Sab}\label{sab}

The Sab galaxy has a spatially small, very high constant density inner
region but acts like a point mass thereafter.

\subsection{LSB}\label{lsb}

The LSB galaxy also has a spatially small, very high constant density
inner region which afterwards acts as a combination of constant density
plus singular isothermal sphere density profile.

\subsection{Irr}\label{irr}

The irregular galaxy acts as a mix of a constant density profile and a
singular isothermal sphere for its entire rotation curve, except the
last kpc or so it trends down slightly, indicating the point mass
profile taking over.

The total mass profile is not the same for all the
rotationally-supported systems; the mass profile depends on the density
profile and will differ significantly from Irr to Sb (since their
density profiles are quite different).

None of the profiles look very similar to a NFW profile; none of them
exhibit the gentle decline after maximum; however, all but the Irr are
similar to the NFW profile in that the initial part is a steep increase
until maximum and then a flat region.

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