where is the effective radius, or the radius within which the galaxy emits half its brightness, is the surface brightness at and is a positive parameter that, for a given , can be determined from the definition of and . The value of determines the degree of concentration of the profile, quantified e.g. by the fraction of energy emitted within a given number of effective radii, the profile being steeper or less concentrated for higher and conversely flatter or less concentrated for lower . Particularly interesting special cases are the bulge-like profile for and the disk-like exponential profile for , which will be discussed in greater detail in Sections A.2 and A.3.

According to Equation 17, the brightness integrated within a given radius is given by

where is the incomplete gamma function. The total brightness predicted by the profile is

where is the gamma function. This relation, remembering that, by definition of effective radius, it is , can be used to obtain an equation linking and . After cancellation of common terms, one obtains

(20) |

a non-linear equation which can only be solved numerically, e.g. via the Newton method (see Section 9.7 in [Press