Ellipticity

Although there are many mathematical definitions of ellipticity used in astrophysics, it can generally be thought of as how ellipticial a shape is. A circle is not elliptical and has ellipticity 0. A line, on the other hand, is perfectly elliptical (width in one direction and none in the direction perpendicular to it); it has an ellipticity of 1.

A nice geometrical ellipse has a semi-major axis and a semi-minor axis. One can define an ellipticity vector that lies along the semi-major axis, which is in the direction of elongation, at an angle of $\theta$ with respect to the x-axis.

A simple way to find the angle and magnitude of this vector is to first calculate the quadruopole moments of the objects as prescribed by Bacon [1]

$$I_{ij}=\int d^{2}x w(\mathbf{x}) x_{i}x_{j}I(\mathbf{x})$$ (1)

ellipticity components, $e_{1}$ and $e_{2}$ defined by

$$e_{1}=\frac{I_{11}-I_{22}}{I_{11}+I_{22}} \quad\quad e_{2}=\frac{I_{12}}{I_{11}+I_{22}}$$ (2)

From these components, the ellipticity magnitude and angle are given by

$$e=(e_{1}^{2}+e_{2}^{2})^{1/2} \quad\quad \tan 2\theta = e_{2}/e_{1}$$ (3)

so that $e_{1}=e\cos 2\theta$ and $e_{2}=e\sin 2\theta$. The $2\theta$ is quite tricky; it has to do with the fact that the ellipticity is actually a bi-directional vector. But this definition allows the proper addition of ellipticities and averaging of the angles over a field.

Another way to find this angle is to rotate the coordinate system in which the ellipticity is defined in order to maximize $e_{1}$. With this procedure, angle of rotation is given by

$$\theta=sign({e}_{2})\arcsin \sqrt{\frac{1}{2}(1-sign({e}_1)(1-\frac{{e}_{2}^{2}}{{e}^{2}})^{1/2})}$$ (4)

In this frame, $e$ lies along the same direction as $e_{1}$, namely, the x-axis, and $e_{2}$ lies along the y-axis. By applying the right trigonometric identites and considering the 4 cases dealt by the signs of $e_{1}$ and $e_{2}$ , one finds that the same angle is obtained with both procedures.