Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 72 / 3

For a singular isothermal sphere, spherical integration of the mass density givesThe spherical shell theorem makes the radial attraction $GM(<R)/R^2$. Balancing it with circular centripetal acceleration yieldsThus the model has a flat galaxy rotation curve, resembling the outer galaxy rotation curves of many spiral galaxies. Central rotation curves usually rise, and not all real curves are flat throughout. A pressure-supported elliptical galaxy does not itself rotate at this test-particle circular speed.

The model has divergent central mass density, although finite enclosed mass at each finite radius, and an infinite total mass because $M(<R)\propto R$. Its potential is logarithmic, requiring an outer cutoff for a finite escape energy. A globally constant isotropic velocity dispersion, spherical symmetry, and one common profile for all gravitating components are further idealizations. Stellar light commonly has a quite different radial profile from the dark matter halo; real disks are flattened, and halos are neither exact isothermal spheres nor infinite. Tidal truncation also changes the distribution near the satellite boundary.

An orbital prescription is needed to fix the coefficient in a tidal radius. Adopt a low-mass satellite on a circular orbit of separation $D$, with $r_t\ll D$. The host singular isothermal sphere hasIn the corotating frame, subtract the host force on the satellite centre and add the differential centrifugal force. Along the line joining the centres, the outward effective acceleration to first order in displacement $r$ is $({\Omega }^2-{\Phi }_h^{\prime \prime })r=4{\sigma }_h^{2}r/D^2$. Equate this to the dwarf's inward acceleration at its edge:This gives the Jacobi truncation of two singular isothermal spheresIt is the distance to the radial escape saddle, not an exact spherical surface: the actual tidal lobe is distorted. For an eccentric orbit, separation alone does not specify an instantaneous effective angular speed or the stripping history; a pericentric estimate is commonly used with further assumptions.

A simpler order-of-magnitude convention equates mean internal and host densities, $m_d/r_t^3=M_h(D)/D^3$, giving $r_t\sim D{\sigma }_d/{\sigma }_h$. Equivalently it retains the host gradient without the circular-orbit centrifugal contribution. Both reproduce the scaling $r_t\propto D{\sigma }_d/{\sigma }_h$, but their coefficients differ by $\sqrt{2}$. The familiar point-mass factor three is not appropriate for this extended host, whose enclosed mass grows linearly with radius. The circular-orbit Jacobi tidal radius will be used consistently below, with the rough convention's numerical values also given.

Integrate the dwarf singular isothermal sphere only to its truncation radius:ConsequentlyThe dimensions are $(\text{length})(\text{speed}{)}^2/G$, as required for mass. At fixed host dispersion and separation the mass scales as ${\sigma }_d^3$, rather than the luminosity's fourth-power scaling used later. Under the rough equal-density tidal radius convention, the corresponding mass is $2D{\sigma }_d^3/(G{\sigma }_h)$, larger by $\sqrt{2}$.

Using the supplied approximate conversion factors, $D=50\times 10^3\times 3\times 10^{16}=1.5\times 10^{21}\mathrm{m}$, ${\sigma }_d=10^4\mathrm{m}{\mathrm{s}}^{-1}$, and ${\sigma }_h=3\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$. The circular-orbit Jacobi tidal radius isThe dwarf mass isTo one significant figure,The rough equal-density convention instead gives $r_t=1.67\mathrm{kpc}$ and $m_d=7.14\times 10^7{M}_{\odot }$, or $2\mathrm{kpc}$ and $7\times 10^7{M}_{\odot }$ to one significant figure. The unrounded results are retained for subsequent ratios. For the circular prescription $r_t/D\simeq 0.024$, so the local expansion is well controlled.

Use the classical Faber-Jackson relation $L\propto {\sigma }^4$ as the explicitly requested extrapolation. The supplied host fixes its normalization:With the mass derived using circular-orbit Jacobi truncation of two singular isothermal spheres, the dynamical mass-to-light ratio isHence to one significant figureThe rough equal-density truncation convention gives ${{\rm Y}}_d\simeq 6\times 10^2{M}_{\odot }/L_{\odot }$ instead. Both conclusions indicate a very large dynamical mass-to-light ratio compared with a normal old stellar population.

Within their bright stellar bodies, typical giant elliptical galaxies have optical dynamical mass-to-light ratios of order a few to ten in solar units, with variations in band, initial mass function, and dark matter. Large outer apertures can give substantially larger ratios. Many dwarf spheroidal galaxies have much larger inferred ratios, from tens to hundreds and, for some very faint systems, still larger values. Thus the computed ratio is plausible as the scale of a very dark-matter-dominated faint dwarf, not as a purely stellar ratio. Dynamical inference depends on member identification, binary motions, equilibrium, and the assumed mass profile.

The stipulated Faber-Jackson relation is not a reliable universal extrapolation down to every dwarf spheroidal galaxy. Also, the fundamental plane of elliptical galaxies contains radius and surface brightness as well as velocity dispersion; the two-variable Faber-Jackson relation is a projection of that information, not an interchangeable definition of the full plane.

The host mass inside $D$ follows from the same singular isothermal sphere:Using the supplied host luminosity givesThis treats essentially all the stated optical light as contained in that aperture; a measured luminosity profile would refine that convention. Keeping consistent unrounded quantities,Thus the dwarf ratio is about twenty times the parent ratio; the rough equal-density tidal convention would give a factor thirty.

There is no logical contradiction in two objects sharing $L\propto {\sigma }^4$ but having different mass-to-light ratios. A virial theorem scaling gives $M\propto {\sigma }^{2}R$, so that $M/L\propto R/{\sigma }^2$. Equal ratios additionally require homologous relevant radii scaling as $R\propto {\sigma }^2$. Here the satellite's tidal radius scales as ${\sigma }_d$ in the fixed host field, not as ${\sigma }_d^2$. Equating the two ratios would require $r_t=D({\sigma }_d/{\sigma }_h{)}^2\simeq 56\mathrm{pc}$, far smaller than the derived tidal radius.

The heuristic derivation of the Faber-Jackson relation with common mass-to-light ratio and common surface brightness made auxiliary structural assumptions; the empirical relation alone cannot enforce them. Furthermore, the aperture dependence of mass-to-light ratio matters: a satellite's whole tidal mass and a host's $50\mathrm{kpc}$ halo mass are not homologous half-light-aperture masses. Dark matter need not follow stellar light, and a dwarf spheroidal galaxy is not automatically a scaled giant elliptical galaxy. The model calculates total gravitating mass: reading its density literally as stars alone would imply an implausibly large purely stellar ratio, exposing the limits of that interpretation.