Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 41 / 4

For approximately circular motion in a spherically dominated Newtonian gravitational field, $v_c^2(r)=GM(<r)/r$. If most gravitating mass followed the concentrated luminous galaxies, then beyond their main light distribution the enclosed mass would nearly saturate, giving a declining Keplerian galaxy rotation curve, $v_c\propto r^{-1/2}$.

Instead, extended galaxy rotation curves commonly remain roughly flat. The spherical mass profile for a flat rotation curve then requiresThe inferred mass continues growing where the luminous contribution is small, so the mass-to-light ratio increases outward. Under the assumed gravitational dynamics this is evidence for an extended dark matter halo. Measurements of neutral-gas motion can probe radii beyond the bright stellar disk. A realistic disk contribution must be computed using its flattened geometry; the spherical formula describes a halo-dominated region and does not turn every disk rotation curve into a spherical mass profile.

The three probes constrain the same deep gravitational potential by different observations. A large member-galaxy velocity dispersion requires a large binding mass. Applying the virial theorem, with a gravitational scale radius $R$ and approximately isotropic orbits, gives a cluster velocity-dispersion mass estimate of order $M\sim R{\sigma }_v^2/G$, with a profile-dependent numerical coefficient. The inferred mass-to-light ratio is much larger than the stellar value. Membership, orbital anisotropy and equilibrium assumptions affect the precision.

The hot intracluster medium radiates through thermal bremsstrahlung and line emission. Its X-ray spectrum measures temperature, while its surface brightness constrains gas density. Confining this gas requires a potential with characteristic $GM/R\sim k_{B}T/(\mu m_p)$. Combining resolved profiles through the cluster hydrostatic mass estimator gives the total mass, not just the gas mass. The detected gas contributes substantial baryonic mass but does not normally account for the full gravitating mass. Nonthermal pressure or a merging cluster can invalidate a purely thermal hydrostatic equilibrium estimate.

Finally, gravitational lensing measures mass through light deflection. Strong gravitational lensing supplies multiple images and arcs; weak gravitational lensing measures coherent background-image distortion over larger radii. The lensing convergence is projected surface mass density divided by the critical surface density, irrespective of whether the material emits light. Cluster gravitational-lensing mass estimates agree broadly with the high masses implied by galaxy motions and hot gas, while having different equilibrium assumptions. Together these observations support a dominant nonluminous component in galaxy clusters, after accounting for their stars and observed gas.

Use conservation of energy for an isolated spherical perturbation of fixed mass $M$, with negligible initial random motion and no significant surface-pressure term. For a homologous density profile its Newtonian gravitational potential energy is $U=-\alpha G{M}^2/R$, with the same positive structure coefficient $\alpha$ at both stages; a uniform sphere has $\alpha =3/5$.

At spherical-collapse turnaround, coherent radial motion momentarily vanishes, so $E=U_{\max }=-\alpha G{M}^2/R_{\max }$. In the final virialized state, the virial theorem gives $2T_{\mathrm{vir}}+U_{\mathrm{vir}}=0$, hence $E=T_{\mathrm{vir}}+U_{\mathrm{vir}}=U_{\mathrm{vir}}/2$. ThereforeThis is the virial radius from turnaround energy in the spherical-collapse model. It is an idealized virial-radius estimate rather than the radius of the formal pressureless point collapse. If the profile coefficient changes, the same argument instead gives $R_{\mathrm{vir}}/R_{\max }={\alpha }_{\mathrm{vir}}/(2{\alpha }_{\max })$. Significant energy loss, mass loss or boundary work also changes the result.

Let ${\rho }_g(r)$ be the gas density and $M(<r)$ the total gravitating mass. For a spherically symmetric gas in hydrostatic equilibrium, radial force balance isUsing the ideal gas law $P={\rho }_g{k}_{B}T/(\mu m_p)$, with constant mean molecular weight $\mu$, differentiate the product rather than setting $T$ constant:Substitution and cancellation of ${\rho }_g$ give the cluster hydrostatic mass estimatorHere $k_B$ is the Boltzmann constant, denoted $K$ in the question. The prefactor has units of mass; a pressure decreasing outward makes the bracket negative and the inferred mass positive. For an isothermal gas with ${\rho }_g\propto r^{-\beta }$, this reduces to $M(<r)=\beta k_{B}Tr/(\mu m_{p}G)$. The density in the logarithmic derivative is the gas density, not the total density. A varying $\mu$ or nonthermal pressure would require the corresponding extra terms.

Measure the cluster's baryonic mass, dominated by its intracluster medium with a smaller stellar contribution, and divide it by an independently inferred total mass. If this cluster baryon fraction $f_b$ is representative of the cosmic baryon fraction, then $f_b={\Omega }_b/{\Omega }_m$. Thus the cluster baryon-fraction estimate of matter density isBig Bang nucleosynthesis constrains the baryon-to-photon ratio and hence ${\Omega }_b{h}^2$ through primordial light-element abundances. A specified $h$ converts this to ${\Omega }_b$; the cluster mass and gas-mass estimates also have distance-dependent $h$ factors, which must be used consistently. This method estimates the total nonrelativistic matter density, including baryonic and dark matter. It does not add radiation or vacuum energy to that inferred ${\Omega }_m$; in a matter-dominated interpretation the question's $\Omega$ is this matter parameter.

Primordial deuterium as a baryon-density indicator works because increasing baryon density makes the destruction of deuterium into heavier nuclei more efficient during Big Bang nucleosynthesis. Smaller inferred primordial D/H therefore implies larger ${\Omega }_b$, with the other nucleosynthesis assumptions fixed. For a fixed measured cluster baryon fraction, a smaller primordial deuterium abundance raises the inferred matter density. A low abundance caused instead by stellar processing is not a lower primordial abundance; depletion, gas ejection and the representativeness of clusters must also be assessed before identifying the cluster fraction with the cosmic one.