这是一份UIUC宇宙学课程的作业，学生同时选了好几门物理课，课业压力大，这份作业交给了我们写手完成加答疑。
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:
PRIMORDIAL GRAVITATIONAL WAVES
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.
Links to research groups and facilities: Jeffrey Filippini, SPIDER
Primordial Nucleosynthesis and Particle Dark Matter
The lightest and most abundant elements in the universe were forged from a primordial soup of subatomic particles during the first three minutes of cosmic time. Our group performs state-of-the-art calculations of the primordial abundances of the elements. By combining these theoretical predictions with astronomical observations of light elements and of the cosmic microwave background radiation, we wield the earliest reliable probe of the cosmos. At even earlier times, even higher-energy interactions likely gave rise to exotic particles that gave rise to dark matter today. We use primordial nucleosynthesis and other astrophysical observations to probe dark matter particle physics.
Links to research groups and facilities: Brian Fields
Faculty working in Cosmology
Peter J. AdsheadAssistant Professor of Physics[email protected]Matias Carrasco KindResearch Assistant Professor[email protected]Patrick I DraperAssistant Professor of Physics[email protected]Brian D. FieldsProfessor[email protected]Jeffrey P FilippiniAssistant Professor of Physics[email protected]Gilbert HolderProfessor of Physics[email protected]Xin LiuAssociate Professor[email protected]Felipe MenanteauResearch Associate Professor[email protected]Gautham NarayanAssistant Professor[email protected]Paul M. RickerProfessor[email protected]Stuart L. ShapiroProfessor of Physics[email protected]Jessie F SheltonAssistant Professor of Physics[email protected]Yue ShenAssociate Professor[email protected]Joaquin VieiraAssociate Professor[email protected]Helvi WitekAssistant Professor[email protected]Dr. Nicolas YunesProfessor of Physics[email protected]
