ESTIMAR LOS RECURSOS DE LAS ACTIVIDADES

In this section, I describe the measures of galaxy structure and morphology used in this work, including several traditional measures found in literature that had been widely used in studies of galaxy structure as well as a new morphological indicator designed to measure asymmetric structures in the outskirts of galaxies, developed in this work.

2.4.1

Sérsic index

The distribution of light within galaxies can be well described by fitting the Sérsic function (Sérsic, 1963) to the galaxy intensity (or surface-brightness) profile:

I(R) =Ieexp § −bn • R R_e ‹1/n −1 ˜ª (2.5)

where I_e stands for the intensity at the galaxy’s effective radius,R_e, enclosing half of the total galaxy light, and the constant b_n is determined by the Sérsic index,n(see Graham & Driver (2005) for more details). The Sérsic index itself is a parameter that describes the shape of the intensity profile and has been widely used in discriminating between early- (n_∼4) and late-type (n_∼1) galaxies.

In this work, I used a single-component Sérsic fit to a one-dimensional intensity profile measured within circular annuli (Section 2.3.1). The fit was performed by utilising an IDL function,mpfitfun.pro1and it was weighted using the uncertainty in the photon counts within each annulus:

∆I=Æ((I+I_{sk y})/g+N(DV +∆I_{sk y})), (2.6)

where I and I_{sk y} are the intensity of the source and sky background, respectively, summed over all pixels within an annulus, and ∆I_{sk y} is the uncertainty in the latter. The number of pixels in an annulus is given byN, the number of photo-electrons is related to the pixel counts through the gain, g, and the contributions to noise from the dark current and read noise are

2.4. Standard measures of galaxy structure

contained within the dark variance,DV.

Prior to the fitting, the model had to be adjusted to account for the imaging limitations caused by the atmospheric turbulences, aberration and diffraction of light by the telescope’s aperture. Therefore, at each step of the fitting procedure, the Sérsic function was convolved with the appropriate point-spread function (PSF). To represent the PSF I used the sum of two Gaussian functions with widths and relative amplitudes specified by the corresponding SDSS image header parameters.

2.4.2

Central concentration index

An alternative way of quantifying the light distribution in a galaxy is to measure its concen- tration index,C (Kent, 1985; Abraham et al., 1994; Bershady et al., 2000). This parameter is computed by considering the growth curve radii of a galaxy. As shown by Bershady et al. (2000), the most robust definition ofC, suitable for studying galaxies in nearby universe as well as those at higher redshifts, and with a large dynamical range, is:

C =5log₁₀ _r 80 r₂₀ ‹ (2.7)

withr20andr80being the radii containing 20% and 80% of the total galaxy light. C typically ranges between 2 and 5, with highest values characteristic of highly concentrated early-type galaxies (C >4) and decreasing toC ∼3 for later types (eg. Bershady et al. 2000; Conselice 2003; Hernández-Toledo et al. 2008).

In this work, the growth curve radii were calculated within rma x, as described in Section

2.3.3, rather than the commonly used factor of 1.5 of the Petrosian radius,r_p.

2.4.3

Asymmetry parameter

Galaxies with bursty star formation and those which have undergone a recent interaction or a merger tend to have irregular morphologies and show peculiar features in their structure. Such features are likely to appear asymmetric to the observer. Quantifying the degree of asymmetry in galaxies has been a subject of many studies. In particular, Schade et al. (1995), Abraham et al. (1996), Conselice et al. (2000), Conselice (2003), Hernández-Toledo et al. (2008) in-

vestigated a parameter that measures the rotational asymmetry of galaxy images, defined as:

A= Σ|I0−Iθ |

2Σ|I0| −

A_{b g r}, (2.8)

where I_θ is the flux from an individual pixel in the galaxy image rotated by a given angle, most often 180o, about a chosen centroid, and I_θ is flux from a pixel at the same position in the original, un-rotated image. The sum is computed over all pixels within a symmetric extraction region, usually defined in therms of the galaxy radius. The second term is a noise correction: the asymmetry computed on a blank region of sky of the same size as the extraction region (see Conselice et al. 2000 for details). The choice of the centre of rotation can have a significant impact on the value of Aand should be chosen such as to minimiseAfor a given object (Conselice et al., 2000).

In this work,Awas computed through a 180orotation about such a minimum-asymmetry centroid. As described in Section 2.2.2, the centroid was found by considering many pixels simultaneously, in order to eliminate potential local asymmetry minima (no such local minima were found in the study by Conselice et al. 2000; however, their study did not focus on galaxies with significant morphological disturbance, where this effect could become apparent). The extraction region for the asymmetry measurement was defined by the aperture atrma x, rather

than the commonly used 1.5r_p.

2.4.4

Outer asymmetries

To enhance the signal from the low-surface-brightness regions in the outskirts of the galaxies I used a modified version of the asymmetry measure, which I refer to as the ‘outer’ asymmetry,

A_o. It is computed similarly to the standard asymmetry parameter (Equation 2.8), but with an inner circular aperture containing the brightest 50% of the total flux excluded from the measurement. A comparable definition of ‘outer’ asymmetry was also recently investigated by Wen et al. (2014).

2.4.5

Clumpiness parameter

In star-forming galaxies the distribution of light can become ‘patchy’ as the light intensity of the star-forming regions surpass that coming from other galaxy parts. Conversely, galaxies with little or no star-formation activity tend to have smoother light distributions. This can be quan-

2.4. Standard measures of galaxy structure

tified by considering the ‘clumpiness’ parameter,S (Isserstedt & Schindler, 1986; Takamiya, 1999; Conselice, 2003), designed to measure the amount of light contained within the high- spatial-frequency structures of galaxies, and can be defined as:

S=Σ|Io−Iσ|

Σ|I_o| , (2.9)

following Conselice (2003), whereI_σstands for pixel intensity in the smoothed galaxy image (with the smoothing filter width related to the galaxy radius) and I_o is that in the original image.

In this work, the width of the smoothing filter was set to r20, the growth curve radius enclosing 20% of the total light. Like in the case of A, the extraction region used in this study was defined by an aperture atrma x. As centres of most galaxies are highly concentrated

they could artificially increase the values ofS and therefore, prior to the measurement, the innermost aperture atR20, centred on the highest-intensity centroid,OI ma x, was excluded.

2.4.6

Gini index

The distribution of light in galaxies can be described independently of the definition of their centres by means of the Gini index, a measure of the inequality in the light distribution. This can be particularly useful in the case of morphologically disturbed systems where the definition of the ‘centre’ does not emerge naturally from the galaxy shape and structure.

The Gini index was originally used in economics to study the inequality in the wealth of a population and was adapted in astronomy about a decade ago (Abraham et al., 2003). In image analysis, the Gini index is computed based on the rank-ordered cumulative distribution function of the pixel intensities - the Lorenz curve (Lorenz, 1905). It is given by the ratio of the area between the Lorenz curve and the line of ‘uniform equality’ to the area underneath the curve of ‘uniform equality’. This is illustrated in Figure 2.2. For a discrete distribution, the mathematical expression for computing the Gini index can take the following form (Glasser, 1962): G= 1 2X n(n−1) n X i (2i−n−1)_|Xi | (2.10)

with pixel intensities, Xi, in increasing order, n - the total number of pixels assigned to the

Fraction of pixels F ra ct ion of t ot al fl ux Line of uni form equa lity Lore nz curve 0 1 1

Figure 2.2:A graphical representation of the Gini index. The values on thex-axis represent the fraction of the image pixels, rank-ordered by their intensities from lowest to highest, and the values on they-axis are the corresponding fraction of the total flux contained within the pixels. The dashed line represents the line of uniform equalityand the solid green line represents the light distribution in a galaxy image (Lorenz curve). The Gini index is the ratio of the area shaded in green to the total area under the line of uniform equality. In a case where the light is distributed equally between all pixels the green area is reduced to 0, resulting inG=0; conversely, if all flux is contained within a single pixel, the green area expands to match the total area under the line of uniform equality, leading toG=1.

takes on high values for galaxies with bright nuclei, including highly concentrated early-type galaxies and merging galaxies with multiple nuclei, and its value decreases for systems with more uniform light intensity distributions. UnlikeC, the Gini index can distinguish between galaxies with shallow intensity profiles and those with bright off-centre regions (Lotz et al., 2004).

2.4.7

Moment of light

The degeneracy between early-type and merging galaxies having similarly highG values can be broken by gaining information about the spatial extent of the brightest galaxy regions. This has been done by measuring the M20 parameter - the second order moment of the brightest 20% of the galaxy’s total flux. M₂₀is calculated by summing the individual moments,M_i, over the brightest pixels until the corresponding sum of pixel intensities, f_i, reaches 20% of the total galaxy light, after which stage, the sum is normalised by the total second-order moment