I am studying ellipticity distributions of galaxies and am having trouble moving from the second central moment matrix (covariance matrix) to properties for an object such as its half-light radius.

I already know how to obtain $e1$ and $e2$

They are given by the following:

$e = \frac{Q_{xx} - Q_{yy} + 2iQ_{12}}{N}$

Where $N = N(Q)$ and is just the normalization. The real component is $e1$ and the imaginary component is $e2$

I am wondering how to obtain the half-light radius, as well.

I have seen:

$2r^2 = Q_{xx} + Q_{yy}$ for a circular profile, then one can obtain the HLR from $r^2$ but what about for non-circular profiles? Is the above equation even valid?


You simply cannot obtain any information on the half-light radius from the quadrupole moment only of a galaxy, without knowing the galaxy profile. Consider for example two (extreme) light profiles:

  1. A galaxy consisting of a uniform disk of radius $R$ – that is, its surface brightness would be something like $I(r) \propto H(R_1 - r)$, where $H$ is the Heaviside function.

  2. A galaxy consisting of a point source + a uniform disk – that is $I(r) \propto \delta(r) + H(R_2 - r)$.

You can easily find a combination of radii $R_1$ and $R_2$ such that the quadrupole moments of the two galaxies are identical. However, clearly their half-light radii cannot be (for the second galaxy, in particular, the half-light radius vanishes).

From what I can see, the relation you wrote holds for a Gaussian profile, and even then it is not correct: your $r$ is not the half-light radius but the standard deviation of the Gaussian profile. For different profiles there is no guarantee that your relation will work (in general it will not): in general, it will only give you something proportional to the square of the half-light radius, but the proportionality constant will depend on the specific light profile.

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