Physics:Effective radius

Half light radius R_e encloses half of the total light emitted by an object

The effective radius or half-light radius ([math]\displaystyle{ R_e }[/math]) of a galaxy is the radius at which half of the total light of the system is emitted. This assumes the galaxy has either intrinsic spherical symmetry or is at least circularly symmetric as viewed in the plane of the sky. Alternatively, a half-light contour, or isophote, may be used for spherically and circularly asymmetric objects.

[math]\displaystyle{ R_e }[/math] is an important length scale in [math]\displaystyle{ \sqrt[4] R }[/math] term in de Vaucouleurs law, which characterizes a specific rate at which surface brightness decreases as a function of radius:

[math]\displaystyle{ I(R) = I_e \cdot e^{-7.67 \left( \sqrt[4]{\frac R {R_e}} - 1 \right)} }[/math]

where [math]\displaystyle{ I_e }[/math] is the surface brightness at [math]\displaystyle{ R = R_e }[/math]. At [math]\displaystyle{ R = 0 }[/math],

[math]\displaystyle{ I(R=0) = I_e \cdot e^{7.67} \approx 2000 \cdot I_e }[/math]

Thus, the central surface brightness is approximately [math]\displaystyle{ 2000 \cdot I_e }[/math].