Problem 1.

The rotation curve of a thin exponential disk, with mass density (given in terms of cylindrical coordinates)
$$\rho_{\text {disk }}(\boldsymbol{r})=\rho(R, z, \phi)=\delta(z) \frac{M}{2 \pi R_{0}^{2}} \exp \left(-\frac{R}{R_{0}}\right)$$
can be approximated by calculating the rotation curve of a spherically symmetric mass distribution $\rho_{\mathrm{sph}}(r)$ with the same mass enclosed in any (spherical) radius, i.e., $M_{\text {sph }}(r<R)=M_{\text {disk }}(<R)$. Find an expression for the rotation curve $v_{\mathrm{circ}}(r)$ that would be produced by this “matched” spherical density distribution.
Plot the rotation curve for the case that $M=2 \times 10^{11} M_{\odot}$ and $R_{0}=5 \mathrm{kpc}$ and hence obtain (numerical) values for the radius where the rotation curve peaks and the circular speed at this radius.

The mass enclosed within cylindrical radius $R$ for the disk-like distribution is
\begin{aligned} M_{\text {disk }}(<R) &=\int_{0}^{R} \mathrm{~d} R^{\prime} 2 \pi R^{\prime} \rho_{\text {disk }}\left(R^{\prime}, 0,0\right) \ &=\frac{M}{R_{0}^{2}} \int_{0}^{R} \mathrm{~d} R^{\prime} R^{\prime} \exp \left(-\frac{R^{\prime}}{R_{0}}\right) \ &=M \int_{0}^{R / R_{0}} \mathrm{~d} x x \exp (-x) \ &=0-\frac{R}{R_{0}} \exp \left(-\frac{R}{R_{0}}\right)+1-\exp \left(-\frac{R}{R_{0}}\right) \ &=M\left[1-\left(1+\frac{R}{R_{0}}\right) \exp \left(-\frac{R}{R_{0}}\right)\right] \end{aligned}
where the second-last step uses integration by parts. The spherical distribution hence satisfies
$$M_{\mathrm{sph}}(<r)=M\left[1-\left(1+\frac{r}{R_{0}}\right) \exp \left(-\frac{r}{R_{0}}\right)\right]$$
The resultant rotation curve is
$$v_{\text {circ }}(r)=\left{\frac{G M}{r}\left[1-\left(1+\frac{r}{R_{0}}\right) \exp \left(-\frac{r}{R_{0}}\right)\right]\right}^{1 / 2}$$
The rotation curve for For $M=2 \times 10^{11} M_{\odot}$ and $R_{0}=5 \mathrm{kpc}$ is:

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An epoch of rapid expansion in the early universe, known as “inflation”, is thought to have set the stage for the evolution of cosmic structure. This process should also have seeded the universe with a faint hum of primordial gravitational waves, undetectable today but visible as a faint pattern in the polarization of the cosmic microwave background (CMB).  Illinois collaborates on instrumentation and data analysis for leading teams seeking to constrain this elusive signature from the South Pole (BICEP) and stratospheric balloons (SPIDER), as well as on future efforts to probe fundamental physics with novel low-temperature detector technology.

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