<p>You simply cannot obtain <em>any</em> 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:</p>

<ol>
<li><p>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.</p></li>
<li><p>A galaxy consisting of a point source + a uniform disk – that is $I(r) \propto \delta(r) + H(R_2 - r)$.</p></li>
</ol>

<p>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).</p>

<p>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 <em>proportional</em> to the square of the half-light radius, but the proportionality constant will depend on the specific light profile.</p>

CC BY-SA 3.0
2015-09-19T23:08:14.070Z
2015-09-19T23:08:14.070Z
Marco Lombardi
<p>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.</p>

<p>I already know how to obtain $e1$ and $e2$</p>

<p>They are given by the following:</p>

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

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

<p>I am wondering how to obtain the half-light radius, as well. </p>

<p>I have seen:</p>

<p>$2r^2 = Q_{xx} + Q_{yy}$ 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?</p>

<cosmology><gravitational-lensing>
Going from the second moments of an object to its ellipticities and half-light-radius
alvarezcl
# Going from the second moments of an object to its ellipticities and half-light-radius

(Asked by: username_0 on 2014-08-09)

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?

**Tags:** cosmology, gravitational-lensing


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## 1 answers

### Answer by username_1 on 2015-09-19. Score: 1

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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