# 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