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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.

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.