Past exam of the short options paper of the physics Prelims of the University of Oxford / 2021 / S02 / 3

For the estimate implied by the supplied Hubble constant, use the low-redshift Hubble law $D\simeq cz/H_0$. ThenWith ${\theta }_E=\pi /10800$ radians, inversion of the Einstein angular radius relation givesThe corresponding projected Einstein radius is $R_E=D_L{\theta }_E\simeq 233\mathrm{kpc}$.

This is the projected enclosed mass of the galaxy cluster, including all matter, not just its stars. Strictly, the gravitational lensing formula uses angular diameter distances, and at $z=0.5$ the low-redshift approximation is only an estimate. An exact value requires a specified cosmological expansion history and a consistent lens-to-source angular diameter distance; it cannot be determined from $H_0$ alone. The later estimates use the same distance approximation consistently.

Assume virial equilibrium, structurally homologous early-type galaxies, a fixed relation between the measured central velocity dispersion ${\sigma }_c$ and the global random stellar motions, and a constant total mass-to-light ratio ${\rm Y}$. The virial theorem gives $M=k_v{\sigma }_c^2{R}_e/G$, where the dimensionless structure factor $k_v$ is common to the sample. With ${\mathcal{I}}_e$ denoting mean linear surface brightness inside the effective radius, $L=2\pi R_e^2{\mathcal{I}}_e$. ThusThe usual observed fundamental plane of elliptical galaxies has a tilt relative to these exponents because mass-to-light ratio varies with stellar population, chemical composition, and dark matter fraction; structural and dynamical nonhomology can also change $k_v$.

Printed convention. The source places the logarithm of a value expressed in $\mathrm{mag}\mathrm{arcse}{\mathrm{c}}^{-2}$ in the empirical fit. This differs from the usual logarithm of linear surface brightness. If ${\mu }_e$ is the quoted magnitude surface brightness,A coefficient $-0.8$ on $\log {\mathcal{I}}_e$ therefore becomes $+0.32$ on ${\mu }_e$, with an adjusted intercept; it does not become $-0.8\log {\mu }_e$. For a concrete distinction, changing ${\mu }_e$ from 20 to 25 decreases linear surface brightness by a factor 100, so the linear-intensity fit changes $R_e$ by $100^{0.8}\simeq 39.8$, whereas the literal printed logarithm changes it by $(25/20{)}^{-0.8}\simeq 0.837$.

The following part evaluates the printed numerical fit literally and labels that convention. If the fit was intended for linear surface brightness, its calibration must be specified; a conditional result for using the same coefficients with linear units is also given. No equality between these two conventions is assumed.

Evaluate the source's printed logarithm using the numerical magnitude value 21.0, with base-ten logarithms:This is the literal printed-fit convention identified above. Convert the mean visual surface brightness to a linear luminosity per projected area. At 10 parsecs, one square arcsecond subtends $100/(206264.806{)}^2\mathrm{p}{\mathrm{c}}^2$. The definition of absolute magnitude consequently givesThe constant $21.572=5{\log }_{10}(206264.806)-5$ is the conversion between the angular and physical projected area. Since the effective radius encloses half the total luminosity,For the stipulated stellar population of Solar-type stars, approximate the stellar mass-to-light ratio by $M_{\odot }/L_{V,\odot }$. Summing 100 such galaxies then givesThis assumes negligible interstellar extinction and that the supplied surface brightness is an intrinsic or already corrected value; an observed cosmological surface brightness requires additional dimming and passband corrections before this conversion.

For comparison only, if the empirical coefficients and intercept were instead meant for ${\log }_{10}({\mathcal{I}}_e/[L_{V,\odot }\mathrm{p}{\mathrm{c}}^{-2}])$, substituting $140.9$ in that fit would give $R_e=1.09\mathrm{kpc}$, $L_V=1.06\times 10^9{L}_{V,\odot }$, and $M_{\ast ,\mathrm{cluster}}=1.06\times 10^{11}{M}_{\odot }$. Those are conditional alternative results, not an unannounced correction of the printed units.

The gravitational lensing mass includes dark matter, hot intracluster gas, stars, and any other material in the projected aperture. Most is not stellar: in the literal convention above the stellar fraction is only $2.22\times 10^{12}/(5.89\times 10^{14})\simeq 0.00376$.

Assume the cluster shares the cosmic cosmic baryon fraction, $f_b={\Omega }_b/{\Omega }_m=1/6$, and compare the same aperture for all components. ThenNeglecting stars entirely would give $9.82\times 10^{13}{M}_{\odot }$. Thus the assumed hot gas dominates the baryonic matter, and the remainder $M_{\mathrm{DM}}\simeq (5/6)M_E=4.91\times 10^{14}{M}_{\odot }$ is dark matter. The budget is consistent: stars are about $2.26\%$ of the inferred baryonic matter, and all components sum to $M_E$.

For a thermal scale, take gas extent $R_g\simeq 2R_E=465\mathrm{kpc}$ and mean particle mass $\mu m_p$ with $\mu \simeq 0.61$ for ionized gas. A simple spherical virial theorem estimate with gas binding energy $W_g\sim -G{M}_E{M}_g/R_g$ and thermal kinetic energy $K_g=3M_g{k}_{B}T/(2\mu m_p)$ givesThis is the scale of a very hot, massive X-ray emitting galaxy cluster. The numerical prefactor depends on the radial mass density profile, gas distribution and boundary pressure; for example an isothermal-sphere estimate $k_{B}T\sim \mu m_{p}GM/(2R_g)$ gives about $2.0\times 10^8\mathrm{K}$. Moreover $M_E$ is a projected aperture mass at $R_E$, whereas the estimate needs a spherical enclosed mass at $2R_E$; the given data do not determine that conversion. The quoted temperature is therefore an order-of-magnitude estimate, not a uniquely measured value. Nor do the data determine the total gas mass outside the Einstein ring without a profile. The gas budget above pertains to the matched aperture under the cosmic-fraction assumption.