Deprojecting Sersic Profiles for Arbitrary Triaxial Shapes: Robust Measures of Intrinsic and Projected Galaxy Sizes

The spatial distribution of stars in a galaxy encodes key information about its formation history, whether dissipative or dissipationless processes dominated and whether angular momentum has been retained or lost. The half-light radius is the simplest observational measure of stellar mass distribution. From an empirical perspective, this quantity has played a central role in defining the nature of galaxies through examining and interpreting their scaling relations (e.g., Kormendy, 1977; Djorgovski & Davis, 1987; Dressler et al., 1987) and tracking the build up of galaxies through cosmic time (e.g., Trujillo et al., 2004; van der Wel et al., 2014a). From a theoretical perspective, galaxy sizes have repeatedly revealed that essential physical elements are missing in galaxy formation models (e.g., Navarro & Steinmetz, 2000), leading to the implementation of (astro)physical processes such as feedback.

Measurements of galaxy sizes as traced by stellar light have now reached a point where 0.1 dex accurate statements about two-dimensional (2D) projected galaxy sizes across cosmic time can be made with great precision (e.g., Mowla et al., 2019; van der Wel et al., 2014a). Currently, we are limited by the interpretation of the data, not its scarcity – almost a unicum in the field of galaxy evolution. One limiting factor is the missing link between the projected size (the half-light radius that we measure) and a physically more directly meaningful quantity such as the average or median distance of a star to the center of its galaxy. This conversion factor, from projected to three-dimensinonal (3D) intrinsic size, can vary by large factors given the range of possible galaxy shapes and projections. As a result, comparisons with simulations have remained indirect and prone to systematic effects (e.g., Parsotan et al., 2021; Ludlow et al., 2019; Genel et al., 2018).

We note that we take the word size to mean the 2D projected or 3D intrinsic half-light radius (or, ideally, half-mass radius) that serves to quantify what we define as the mean radius of the galaxy. Other definitions of size adopt a fixed surface brightness or density threshold, aiming to quantify the full extent of the galaxy, which is a useful but qualitatively different quantity from the mean radius.

To our knowledge attempts at the deprojection of measured 2D light profiles so far have assumed spherical symmetry (e,g, Bezanson et al., 2009), symplifying the calculation, but failing to account for the fact that in reality most galaxies are far from spherical: even most early-type galaxies have an intrinsic short-to-long axis ratio of no more than $\sim 0.3$ (e.g., Vincent & Ryden, 2005), which more recently has been shown to be the case across cosmic time (e.g., Chang et al., 2013). In this paper we present the analytical framework and numerical implementation for the conversion of 2D light profiles to 3D light distributions for galaxies of arbitrary triaxial shape. Only with this machinery can we take full advantage of the available high-quality data and make accurate comparisons with theoretical predictions.

This papers is organised as follows: Section 2 describes how triaxial shapes project in 2D; Section 3 depicts the deprojection of Sérsic profiles; Section 4 introduces the definition and derivation of the median radius, $r_{\mathrm{med}}$; Section 5 outlines how to infer, in practice, $r_{\mathrm{med}}$ from projected size and shape measuremts; Section 6 concludes with a brief description of implications of our findings for galaxy size estimates.

We consider intrinsic density distributions $\rho(x,y,z)=\rho(m)$ that are constant on ellipsoids

with $a\geq b\geq c$. The major semi-axis length $a$ is a scale parameter, whereas the intermediate-over-major ($p\equiv b/a$) and minor-over-major ($q\equiv c/a$) axis ratios determine the intrinsic shape. In the oblate or prolate axisymmetric limit we have $a=b>c$ (pancake-shaped) or $a>b=c$ (cigar-shaped), respectively, while in the spherical limit $a=b=c$ (so then $m=r/a$). Note that $m$ is a dimensionless ellipsoidal radius.

We introduce a new Cartesian coordinate system $(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime})$, with $x^{\prime\prime}$ and $y^{\prime\prime}$ in the plane of the sky and the $z^{\prime\prime}$-axis along the line-of-sight. Choosing the $x^{\prime\prime}$-axis in the $(x,y)$-plane of the intrinsic coordinate system (cf. de Zeeuw & Franx, 1989) and their Fig. 2), the transformation between both coordinate systems is known once two viewing angles, the polar angle $\vartheta$ and azimuthal angle $\varphi$, are specified. The intrinsic $z$-axis projects onto the $y^{\prime\prime}$-axis; for an axisymmetric galaxy model the $y^{\prime\prime}$-axis aligns with the short axis of the projected density,¹¹1For an oblate galaxy, the alignment ($\psi=0$) follows directly from equation (2) since $T=0$. For a prolate galaxy it is most easily seen by exchanging $a$ and $c$ ($c>b=a$), so that the $z$-axis is again the symmetry axis (instead of the $x$-axis). but for a triaxial galaxy model the $y^{\prime\prime}$-axis is misaligned by an angle $\psi\in[-\pi/2,\pi/2]$ such that (cf. equation B9 of Franx, 1988)

where $T$ is the triaxiality parameter defined as $T=(a^{2}-b^{2})/(a^{2}-c^{2})$. A rotation through $\psi$ transforms the coordinate system $(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime})$ to $(x^{\prime},y^{\prime},z^{\prime})$ such that the $x^{\prime}$ and $y^{\prime}$ axes are aligned with the major and minor axes of the projected density (respectively), while $z^{\prime}=z^{\prime\prime}$ is along the line-of-sight.

Projecting $\rho(m)$ along the line-of-sight yields a surface density $\Sigma(x^{\prime},y^{\prime})=\Sigma(m^{\prime})$ that is constant on ellipses in the sky-plane,

where we have used $z^{\prime}=abc\,\sinh(u)/(a^{\prime}b^{\prime})+\mathrm{constant}$, and $m=m^{\prime}\cosh(u)$. The sky-plane ellipse is given by

The projected major and minor semi-axis lengths, $a^{\prime}$ and $b^{\prime}$, depend on the intrinsic semi-axis lengths $a$, $b$, and $c$ and the viewing angles $\vartheta$ and $\varphi$ as follows:

where $A$ and $B$ are defined as

It follows that $A=a^{\prime}b^{\prime}$ is proportional to the area of the ellipse.

The flattening $q^{\prime}\equiv b^{\prime}/a^{\prime}$ of the projected ellipses actually depends on the viewing angles $(\vartheta,\varphi)$ and the intrinsic axis ratios $(b/a,c/a)$, and is independent of the scale length. Since the intrinsic and projected semi-major axis lengths are directly related via the left equation of (5), the scale length can be set by choosing either $a$ or $a^{\prime}$.

In what follows, we will describe the density and related quantities in terms of mass, including intrinsic mass density, surface mass density and enclosed mass. Without loss of generality, these quantities may also be expressed in terms of light, i.e., intrinsic luminosity density, surface brightness, and enclosed luminosity. However, mixing of mass and light quantities requires an additional conversion with the mass-to-light ratio which in general varies with galactic radius.

It has long been known that the stellar surface density profiles of early-type galaxies and of spiral galaxy bulges are well described by a Sersic (1968) profile $\Sigma(R)\propto\exp[-(R/R_{e})^{1/n}]$, with the effective radius $R_{e}$ enclosing half of the total stellar mass. A key conceptual difference from cusped models is that the profile does not converge to a particular inner slope on small scales. Unfortunately, the deprojection of the Sersic surface density profile cannot be expressed in common functions (for special functions see Baes & van Hese, 2011), but is well approximated by the analytic density profile of Prugniel & Simien (1997)

where the inner negative slope is given by

The enclosed mass for the Prugniel-Simien model is

where $\gamma[p;x]$ is the incomplete gamma function, which in the case of the total luminosity reduces to the complete gamma function $\Gamma[p]=\gamma[p;\infty]$.

The expression for the surface density is, to high accuracy, the Sérsic profile

Given the enclosed projected mass

the requirement that the total intrinsic and projected mass have to be equal yields a normalisation

The value of $b_{n}$ depends on the index $n$ and the choice for the scale length. The latter is commonly chosen to be the effective radius $R_{e}$ in the stellar surface density profile, which contains half of the total stellar mass. We adopt a similar convention requiring that the ellipse $m^{\prime}=1$ contains half of the projected mass. This choice results in the relation $\Gamma[2n]=2\,\gamma[2n,b_{n}]$, which to high precision can be approximated by (cf. Ciotti & Bertin, 1999)

We suggest as a robust measure for the size of a galaxy the median spherical radius. To infer this value numerically we have to find the spherical radius which encloses half of the stellar mass. The mass enclosed within a spherical radius $r$ follows from

with ellipsoidal radius $m(r,\theta,\phi)=(r/a)\,C_{pq}(\theta,\phi)$, where we have defined

and as before $p=b/a$ and $q=c/a$. Substituting the deprojected Sérsic density of equation (8), we see that the integral over radius is an incomplete gamma function, so that

With the total stellar mass given by equation (10) for $m\to\infty$, the median radius normalized by the scale length $r_{\mathrm{med}}/a$, follows upon numerically solving

When $p=1$ in oblate axisymmetry the integral over $\phi$ can be discarded. In case of prolate axisymmetry, it is easiest to exchange $a$ and $c$, so that again $p=1$ and the integral over $\phi$ can be discarded, whereas $q>1$ and the resulting $r_{\mathrm{med}}/c$ has to be divided by $q$ to obtain $r_{\mathrm{med}}/a$.

As can be seen from Figures 1 and 2, the resulting $r_{\mathrm{med}}/a$ depends on the intrinsic axis ratios $p$ and $q$, but very little on Sérsic index $n$. This implies that the median radius is robust against measurement errors in $n$.

In practice we cannot measure the intrinsic scale length $a$, but instead we measure the projected semi-major axis length $a^{\prime}$ and semi-minor axis length $b^{\prime}$. Using the relations (5), we infer the ratio of the median radius to these projected length scales. Figure 3 shows the resulting $r_{\mathrm{med}}/a^{\prime}$ (solid curves), $r_{\mathrm{med}}/b^{\prime}$ (dotted curves), and $r_{\mathrm{med}}/R_{e}$ (dashed curves) with the adopted definition $R_{e}^{2}=a^{\prime}b^{\prime}$, as function of the polar viewing angle $\vartheta$.

In the axisymmetric case (top-left panel), the latter polar angle is the inclination $i$, ranging from $\cos i=0$ for edge-on (side-on) to $\cos i=1$ for face-on (end-on) in the oblate (prolate) axisymmetric case, with intrinsic flattening $q<1$ ($q>1$) indicated by the different colors.

In the triaxial case, the polar angle $\vartheta$ ranges from viewing the short axis from the side ($\cos\vartheta=0$) to from the top ($\cos\vartheta=1$). In addition, the projection depends on the azimuthal viewing angle $\varphi$, which ranges from viewing from along the long axis ($\varphi=0$^∘, top-right panel), intermediate between long and short axis ($\varphi=45$^∘, bottom-left panel) to along the intermediate axis ($\varphi=90$^∘, bottom-right panel). The intrinsic short-to-major axis ratio is fixed to $q=0.5$, whereas the colors indicate different intrinsic intermediate-to-major axis ratios $p$.

In all four panels the blue curves correspond to the same oblate axisymmetric case $(p=1,q=0.5)$, which is independent of the azimuthal angle $\varphi$. Even though the red curves also correspond to the prolate case with the same intrinsic flattening, the orientation in the top-left panel is such that its symmetry long-axis is along the $z$-axis whereas in the remaining panels it is along the $x$-axis. Finally, all curves are for a Sérsic profile with index $n=2$, but we have seen before that there is little variation with $n$.