Mass-to-light ratio ( $Υ = M / L$ )

The mass-to-light ratio compares the mass assigned to an astronomical system with its luminosity in a specified band, often in solar units. A stellar value uses only stellar mass and depends on the stellar population, age, composition and band. A dynamical value can include gas and dark matter, so it need not equal the stellar value. In a halo-to-galaxy mapping,

L = M_{*} / Υ_{*}

requires the stellar ratio and not the total dark matter halo ratio.

Table of contents
- Aperture dependence of mass-to-light ratio Mass-to-light ratio
- Mass-to-light ratio of a power-law stellar population Mass-to-light ratio

Aperture dependence of mass-to-light ratio ( $Υ (< R) = M (< R) / L (< R)$ )

 0  0

A dynamical mass-to-light ratio must specify its aperture and whether the mass includes dark matter. A growing dark halo can increase

Υ (< R)

outside the bright stellar body even when the stellar ratio is constant. Comparing a tidal-radius satellite mass with a large-aperture host mass is different from comparing both within their half-light radii. From

M \propto σ^{2} R

and

L \propto σ^{4}

, the inferred ratio scales as

R / σ^{2}

, so equal ratios additionally require homologous radius scaling.

Mass-to-light ratio of a power-law stellar population

 0  0

Write

x = m / M_{⊙}

, the initial mass function as

d N = A x^{- α} d x

, and a main-sequence mass-luminosity relation as

L / L_{⊙} = x^{β}

. On

a < x < b

, the mass-to-light ratio in solar units is

\frac{Υ}{M _{⊙} / L _{⊙}} = \frac{\int _{a}^{b} x ^{1 - α} d x}{\int _{a}^{b} x ^{β - α} d x} .

(1)

For

α = β = 3

this is exactly

1/ (ab)

. The result concerns the specified unevolved stellar population; stellar remnants, evolved luminous stars and cooling substellar objects require different accounting.