Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 2

Use a steady, isotropic, self-gravitating dark matter halo, with $V$ its constant one-dimensional velocity dispersion. The Spherical Jeans equation and spherical mass conservation readFor the scale-free outer region, put $\rho =A{r}^{-p}$. Its enclosed mass scales as $r^{3-p}$, so $GM(r)/r$ is constant only if $p=2$; the Spherical Jeans equation then gives $2V^2=GM(r)/r=4\pi GA$. Thus the singular isothermal sphere hasIts circular speed is $v_c=\sqrt{2}V$. If a speed denoted $V$ were instead the circular speed, the coefficient would be $V^2/(4\pi G)$, but the printed variable is a velocity dispersion.

The assumptions matter. A constant velocity dispersion and spherical shape alone do not specify a unique central mass density or orbital distribution. Even with isotropy, the Spherical Jeans equation integrates to $\rho ={\rho }_0\exp [-(\Phi -{\Phi }_0)/V^2]$, and the self-gravitating Poisson equation for Newtonian gravity givesThere are finite-central-density isothermal solutions as well as the singular one; the $r^{-2}$ expression is the intended outer scale-free halo approximation. A finite halo must also end or steepen at large radius, since the untruncated singular isothermal sphere has infinite total mass.

Let $m_{\chi }$ be the dark matter particle mass, so its number density is $n_{\chi }=\rho /m_{\chi }$. Interpret $ϵ$ as total emitted power per particle, and assume isotropic optically thin dark-matter decay radiation; volume emissivity is $j_d=ϵ\rho /m_{\chi }$. At projected radius $R$, a line-of-sight coordinate $z$ has $r^2=R^2+z^2$. Writing $A=V^2/(2\pi G)$, the surface brightness isHere $z=R\tan \theta$ turns the integral into $\pi /R$. For the isothermal outer halo,If emissivity was defined per unit solid angle, the factor $1/(4\pi )$ is already included in $ϵ$; this only changes the normalization. The radial law holds between any core and outer truncation. For example, a phenomenological cored profile $\rho =A/(r^2+r_c^2)$ gives $I_d=ϵA/[4m_{\chi }\sqrt{R^2+r_c^2}]$: finite central surface brightness, but the same $R^{-1}$ outer law. This cored interpolation is not being asserted to be the exact isotropic constant-dispersion equilibrium.

A two-body annihilation requires two nearby particles, so the dark-matter annihilation radiation has emissivity proportional to the square of the mass density. With a spatially constant rate coefficient, writefor identical self-conjugate particles, where $E_a$ is the radiated energy per annihilation; the factor one-half counts unordered pairs. Particle and antiparticle populations instead require their respective number densities, without that identical-pair factor. For the singular isothermal sphere, the surface brightness becomesThe substitution $z=R\tan \theta$ gives $R^{-3}{\int }_{-\pi /2}^{\pi /2}{\cos }^2\theta d\theta =\pi /(2R^3)$. Hence annihilation gives the steeper projected profileFor the illustrative cored profile used above, $I_a=C_a{A}^2/[8(R^2+r_c^2{)}^{3/2}]$. These expressions assume a smooth dark matter halo, negligible absorption and constant $C_a$; unresolved density clumps or a velocity-dependent annihilation coefficient change the projected shape. The projection of a spherical power-law emissivity summarizes why three-dimensional powers $r^{-2}$ and $r^{-4}$ produce $R^{-1}$ and $R^{-3}$.

Measure the logarithmic surface brightness slope in resolved radial annuli after accounting for instrumental blurring and foreground emission. Between a halo core and its outer boundary, dark-matter decay radiation predicts $d\ln I/d\ln R=-1$, whereas dark-matter annihilation radiation predicts $-3$; doubling radius therefore reduces the respective surface brightness by factors $2$ and $8$. These are quantitative templates to compare with independently mapped stellar and gaseous components.

An exponential galactic disk has $I_{\ast }(R)\propto e^{-R/R_d}$, so $d\ln I_{\ast }/d\ln R=-R/R_d$ and $I_{\ast }(2R)/I_{\ast }(R)=e^{-R/R_d}$. At $R=5R_d$ the latter is about $0.0067$, much faster decline than either halo template, and the projected emission should also be flattened along the galactic disk. A galactic bulge described by a Sérsic profile has $d\ln I_{\ast }/d\ln R=-(b_n/n)(R/R_e{)}^{1/n}$; for $n=4$ and $b_4\simeq 7.67$, this is about $-3.4$ at $10R_e$ and continues to steepen, whereas a pure halo power law keeps a fixed slope.

For a stellar halo with outer luminosity density proportional to $r^{-p_{\ast }}$, the projection of a spherical power-law emissivity gives $I_{\ast }\propto R^{1-p_{\ast }}$. Illustratively, $p_{\ast }=3$--$4$ gives projected slopes $-2$--$-3$: then $I_d/I_{\ast }\propto R^{p_{\ast }-2}$ rises strongly outward, but annihilation can resemble the steeper stellar template. Gas-related gamma emission must be compared with observed gas columns and the cosmic ray distribution; it is not always proportional to optical light.

A shallow, extended, approximately spherical $R^{-1}$ component is particularly distinctive; an $R^{-3}$ component needs additional evidence. Fit the stellar, gas and halo templates together over a broad radial range, including morphology and emission spectra. A measured slope alone cannot uniquely identify dark matter, since stellar or gas profiles can coincide with a halo slope over a limited interval.