# Massimo Capaccioli » 5.Apparent and true flattening of galaxies

### Elliptical isophotes

Isophotes of ETGs are reasonably fit by ellipses. Assume for now that, within the same galaxy, these ellipses are similar (same axis ratio, $b/a$), homocentric (same position angle, P.A., between the major axis and the direction of the North, counted Eastwards), and coaxial. In absence of internal extinction, this isophotal pattern is the projection of a 3D light distribution where the surfaces of equal light density are similar, homocentric, and coaxial spheroids (cfr. G. Galletta, Ap.Sp.Sc., 92, 335, 1983; B.S. Ryden, MNRAS, 253, 742, 1991).

### Elliptical isophotes

Spheroids are figures obtained by rotating an ellipse about one main axis: $\displaystyle{ \frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{c^2} = 1}.$

If $c < a$, the spheroid is oblate (sort of flattened sphere), if $c > a$ it is prolate (cigar like). These two shapes [not distinguishable in the projected image. Why?] are remarkably different! In the oblate figure the symmetry axis coincides with that of the highest moment of inertia [if light density is proportional to mass density, though!].

For a prolate figure it is just the opposite. In general, for a homogeneous ( $\rho =$ const.) ellipsoid: $\displaystyle{ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 },$

with mass: $m=\rho\times\left(4/3\right)\pi abc$, the moments of inertia are: $I_{xx}=m(b^2+c^2)/5 ,I_{yy}=m(c^2+a^2)/5, I_{zz}=m(a^2+b^2)/5 .$

### Elliptical isophotes

Let us remind that: an ellipse projects into another ellipse but, if the line of nodes (cross section of the planes containing the true and the projected ellipses) is not aligned with the major axis of the true ellipse, then the major axis of the projected ellipse is not the projection of the major axis of the true ellipse; when a spheroid projects into an ellipse, for reasons of symmetry either the major or the minor axis of the latter figure is parallel to the line of nodes, and if it is an oblate spheroid, its projections are a series of coaxial ellipses; an ellipsoid projects also into an ellipse but, if the line of nodes does not coincide with one of the principal axes, the axes of the projected figure are not in simple relation with those of the ellipsoid.

With this in mind, let’s calculate the relation between the intrinsic flattening $\zeta =c/a$  of a spheroid: $(\zeta x)^2 + (\zeta y)^2 + z^2=c^2,$

and that of the projected ellipse as a function of the inclination angle $\theta$ (between the line of sight and the symmetry axis $z$).

Assuming the $y$ axis (normal to the plane of the figure) to coincide with the line of nodes, the apparent ellipse has an axis of length $a =c/\zeta$, and another of length $A$.

### Projection of a spheroid

Since the projected figure is an ellipse, whatever it is the original spheroid, we will always measure an apparent axial ratio $q$ which is $\le 1$: $\displaystyle{ q= \frac{A}{ (c/\zeta)}=\frac{\zeta A}{c}\le 1}\hspace{10pt}\mbox{\it oblate case}}$ $\displaystyle{ q=\frac{(c/\zeta)}{A}=\frac{c}{\zeta A}\le1}\hspace{12pt}\mbox{\it prolate case}} .$

The slope $s$ of the line of sight: $(\zeta x)^2+z^2=c^2 ,$  tangent to the ellipse in $\left(x_0, y_0\right),$ is: $\displaystyle{ s=\frac{dz}{dx} = \cot\theta=-\frac{\zeta^2 x_0}{z_0}.$

By squaring and substituting: $(\zeta x_0)^2=c^2-z_0^2$,  we obtain: $\displaystyle{ \cot^2\theta=-\frac{\zeta^4 x_0^2}{z_0^2}=\zeta^2\left(\frac{c^2}{z_0^2}-1\right) }$,

from where: $\displaystyle{ \frac{c^2}{z_0^2}=\frac{\cot^2\theta}{\zeta^2} +1 }.$

### Projection of a spheroid

Since the intercept of the line of sight with the $z$ axis is $\left(0, C\right)$, from the figure it is: $\displaystyle{ C=\frac{A}{\sin\theta}=z_0-x_0s=z_0+\frac{\zeta^2x_0^2}{z_0}=\frac{c^2}{z_0} },$

or, using the ratio $c^2/z_0^2$  given above: $\displaystyle{ A^2=C^2\sin^2\theta=c^2\left(\frac{c^2}{z_0^2}\right)\sin^2\theta= c^2\left(\frac{\cos^2\theta}{\zeta^2}+\sin^2\theta\right)}.$

### Apparent to true flattening in projected spheroids

From: $A^2=\displaystyle{c^2\left(\frac{\cos^2\theta}{\zeta^2}+\sin^2\theta\right)}$,

remembering that: $q=\fracA}{\displaystyle(c/\zeta)}=\frac\zeta A}c}\le 1\hspace{10pt}\mbox{oblate case}$ $\displaystyle{q=\frac{(\displaystyle c/\zeta)}A}=\fracc}\zeta A}\le1\hspace{10pt}\mbox{prolate case}}$

we obtain the relation between the apparent flattening $q\le 1$  of the ellipse into which a spheroid of the intrinsic flattening $\zeta$ is projected: $\zeta^2\sin^2\theta+cos^2\theta= q^2 \hspace{12pt}\mbox{\it oblate case},$ $\displaystyle{\zeta^2\sin^2\theta+cos^2\theta=\frac1}q^2} \hspace{10pt}\mbox{\it prolate case}}$

The above formulae, derived as early as 1926 by Hubble, apply to spheroids with negligible internal extinction.

In flat (degenerate oblate) disks: $\zeta=0 \hspace{5pt} \rightarrow \hspace{5pt}q=\cos\theta\hspace{5pt}( \theta=0 \ \ \mbox{is face-on}, \theta= \pi/2}\ \ \mbox{is egde-on}).$

### Distribution of the apparent flattening

Let us now ask the following question: what will it be the normalized distribution (probability function) $P(q\vert\zeta)$ of the apparent flattening $q$ of $N$ spheroidal (axisymmetric) galaxies with intrinsic fattening distribution $n(\zeta)$, if their symmetry axes are randomly oriented in space?

The last assumption implies that there is no correlation between the flattening $\zeta$ and the angle $\theta$[principal value] that the symmetry axis forms with a fixed direction $z$.

From the figure it is apparent that the number of the $dN=n(\zeta)d\zeta$ galaxies with intrinsic flattening between $\zeta$ and $\zeta +d\zeta$ which assume an inclination between $\theta$ and $\theta +d\theta$ is proportional to $\sin(\theta)d\theta$
[note the normalization to half of the sphere due to the fact that the symmetry axes are not oriented].

### Apparent to true flattening in projected spheroids

We have seen that, if $\zeta$ is assigned, $q$ depends on $\theta$: $q=q(\theta)$. Thus, the probability that a galaxy with an intrinsic flattening $\zeta$ appears with a flattening $q$ is: $P(q\vert\zeta) =\sin(\theta)d\theta = \sin(\theta)\left\vert \displaystyle{\frac{d\theta}{dq}\right\vert dq} = \displaystyle{ \frac{\sin(\theta)dq}{\vert dq/d\theta\vert}}.$

The total number of galaxies with apparent flattening between $q$ and $q +dq$ is: $f(q)dq=\displaystyle{ \int d\zeta\ n(\zeta)P(q\vert\zeta)} = \displaystyle{ \int d\zeta\ n(\zeta) \frac{\sin \theta dq}{|dq/d\theta|}}$

or: $f(q) = \displaystyle{ \int d\zeta\ n(\zeta)\frac{\sin(\theta)}{\vert dq/d\theta\vert}} .$

Solving this integral equation yields to the distribution of intrinsic flattening.

### Statistics of flattening for spheroids

Remembering that: $\zeta^2\sin^2\theta+cos^2\theta$ $= q^2,$

for the oblate case, $\zeta^2\sin^2\theta+cos^2\theta$ $= 1/q^2},$

for the prolate case,it is:

a) oblate case $\left. \begin{array}{l} \sin(\theta) = \displaystyle{\left[\frac{ (1-q^2)}{ (1-\zeta^2)}\right]^{1/2}}\\ [10pt]\displaystyle{\left\vert dq/d\theta\right\vert} = \displaystyle{q^{-1}\left[(1-q^2)(q^2-\zeta^2) \right]^{1/2}}\end{array} \right\}\hspace{10pt}f(q)=\displaystyle{ q\int_0^q \frac{N(\beta)d\beta }{\left[(1-\beta^2)(q^2-\beta^2)\right]^{1/2}}}\hspace{10pt}\beta=\zeta\le1,$

b) prolate case: $\left. \begin{array}{l} \sin(\theta) = \displaystyle{q^{-1}\left[\frac{ (1-q^2)}{ (\zeta^2-1)}\right]^{1/2}}\\ [10pt]\displaystyle{\left\vert dq/d\theta\right\vert} =q\left[(1-q^2)(q^2\zeta^2-1)\right]^{1/2}\end{array}\right\}\hspace{10pt}f(q)=\frac{1}{q^2}\displaystyle{ \int_0^q\frac{N(\beta) \beta^2 d\beta }{\left[(1-\beta^2)(q^2-\beta^2)\right]^{1/2}}}\hspace{10pt}\beta=\frac{1}{\zeta}\le1.$

### Statistics of flattening for spheroids

You must pay attention, in the prolate case, that the measured flattening $a$, which is always $\le 1$  irrespective of the intrinsic shape of the spheroid [the projected ellipse is always "flat"?], enters in the formula as $1/q$ in view of the a priori knowledge of the shape of the spheroid.

Let us now see how to use this result.

In 1970 Sandage et a. (Ap.J., 160, 831) solved the oblate case integral equation above for samples of Shapley-Ames galaxies grouped in the three families of ellipticals, S0s, and spirals. The inputs were the normalized histograms of the average flattening, relative frequencies of the mean $q$ within discrete bins mimicking $f(q)$. The equation was either inverted [see the original paper for the technique] or solved forcing $n(\zeta)$  to be Gaussian. The outcomes are in the next figure.

### Binney and de Vaucouleurs analysis

In 1981, Binney and de Vaucouleurs (MNRAS, 194, 679) repeated the analysis using 1750 galaxies from the Second Reference Catalogue of Bright Galaxies (all with a diameter $> 2\ arcmin$), grouped in 9 morphological bins (E to Im). For ellipticals they considered also the prolate case and presented the formalism for the triaxial case. It is found that the modal true axial ratio for Es is 0.62 (Hubble type E4) irrespective of whether ellipticals are oblate or prolate.

There are problems in fitting S0 with axisymmetric flat bulges.

Flattening of spirals increases with the Hubble type from Sa to Sc.

### Binney and de Vaucouleurs analysis True flattening distributions for early spirals (a), intermediate and late (b), and very late (c) according to Binney and de Vaucouleurs (MNRAS, 194, 679, 1981). Note how flattening increases along the Hubble sequence.

### Isophotes are more complex

Actually, even for elliptical galaxies, the assumption that they are simple homocentric coaxial spheroids does not work. Observations prove that:

1. isophotes have properties which may depend on color;

2. their flattening $q$ may vary with the distance $r$ from the galaxy center – flattening profile $q(R)$;

3. their orientation may vary with $r$, i.e. the position angle of their major axis is not constant – orientation profile $PA(R)$;

4. their centers may not coincide (off-centering of the isophotes).

How do you prove this? By fitting isophotes with ellipses.

### Isophotal fitting

Isophotal fitting can be done in many ways (see, for instance, R. Jedrzejewski, MNRAS, 226, 747, 1987).

One way is to fit the general expression for a conic, the quadratic curve: $c_1x^2+c_2xy+c_3y^2+c_4x+c_5y+c_6=0,$

to the set of N points $(x_i,y_i)$ sampling the isophote by minimizing the sum of the algebraic distances: $D(c_1,\dots, c_5) = \sum_1^N\left(c_1x_i^2+c_2x_iy_i+c_3y_i^2+c_4x_i+c_5y_i\right)^2 .$
[This is a good time to recall the Least Squares Method and the orthogonal polynomials.]

The N sample $(x_i,y_i)$ is elementarily derived from a digital image by searching for the pairs of adjacent pixels whose brightness clips the brightness value chosen for the isophote.

### Isophotal fitting formulae

The 5 best-fitting parameters $c_i$ [why 5?] are related to the ellipse parameters: $a$, $b$, $PA = \theta$, and the coordinates of the center $(x_c,y_c)$, by the following relations [derive them by translation and rotation of the canonical equation of an ellipse]: $\displaystyle{ x_c=\frac{c_3c_4-c_2c_5}{c_2^2-c_1c_3}},$ $\displaystyle{ y_c=\frac{c_1c_5-c_2c_4}{c_2^2-c_1c_3}},$ $\displaystyle{ a=\sqrt{\frac{2\left( c_1c_5^2+c_3c_4^2+c_5c_2^2-2c_2c_4c_5-c_1c_3c_6\right)}{\left( c_2^2-c_1c_3\right)}\left[\sqrt{(c_1-c_3)^2+4c_2^2}-(c_1+c_3) \right]},$ $\displaystyle{ b=\sqrt{\frac{2\left(c_1c_5^2+c_3c_4^2+c_5c_2^2-2c_2c_4c_5-c_1c_3c_6\right)} {\left(c_2^2-c_1c_3\right)\left[-\sqrt{(c_1-c_3)^2+4c_2^2}-(c_1+c_3\right]}} },$ $P.A=\left\{\begin{array}{ll}0\; \text{ for } b=0\;\text{ and } a>c), \\ \frac 1 2 cot^{-1}\left(\frac{c_1-c_3}{2c_2}\right)\; \text{ for } b\ne 0\;\text{ and } a
It is a good exercise to compute the uncertainties in these various quantities from the errors on $c_i$ provided by the Least Square fitting algorithm.
It is a safe practice to run the algorithm twice, using the first set of parameters to translate and rotate the isophotal coordinates.

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