Galaxies HW 4

Definitions

Mass (within a sphere)

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

Gravitational potential

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

Circular Velocity

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

Derivations

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

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):

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

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

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

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

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:

Figure 1. Density profiles