Colonialidad del poder y del ser

While for the Milky Way a direct measurement of the motions of individual stars is possible (e.g. Gillessen et al. 2009), no single stars can be resolved in the cen- tres of other galaxies. Determining the stellar kinematics of these galaxies there- fore is restricted to the velocity distribution of a large number of unresolved stars along the line of sight. A galaxy spectrum can be described as the convolution of the spectrum of the stars S(λ) (assuming all stars have the same spectrum) with

a broadening function, the so-called line-of-sight velocity distribution (LOSVD)

3.1. EXTRACTION TECHNIQUES

G(lnλ) =S(lnλ)⊗ L(v). (3.1)

Many techniques to reconstruct the LOSVDs from galaxy spectra have been de- veloped (see review below), all of them require the use of one or several stellar templates – spectra of nearby stars of similar type as the stars in the galaxy, which contribute most to the luminosity. Early methods extract the LOSVDs in Fourier space. This is a reasonable approach as a convolution in real space is simply a multiplication in Fourier space, thus the LOSVD can be recovered easily without large demands regarding computing power. More recent methods fit the spectra directly in pixel space, which has the advantage that parts of the spectrum (e.g. emission lines and bad pixels) can easily be excluded. All methods can be fur- ther classified with respect to whether a non-parametric LOSVD is derived or an a priori assumption on the LOSVD shape is made. A Gaussian-shaped LOSVD is certainly a reasonably good first approximation in most cases, however, there is no physical reason why stars in a galaxy should have a purely Gaussian veloc- ity distribution. Real LOSVDs indeed always show deviations from a Gaussian, which provide important information on the orbital structure and therefore are essential for the derivation of the mass of a central black hole. Most parametric techniques therefore use the Gauss-Hermite parametrization byvan der Marel & Franx(1993) and Gerhard(1993):

L(v) =γ α(y) σ N X i=0 h_iH_i(y), (3.2) with y= v−v0 σ and α(y) = 1 p 2πe −y₂2_,

whereγ is the line strength,v₀the recession velocity, v andσ the measured veloc-

ity and velocity dispersion, andH_i(y)are the Hermite polynomials. The Hermite

coefficients h_i describe the deviations from a Gaussian. If the LOSVD does not deviate too strongly from a Gaussian shape it is usually sufficient to truncateL(v)

ati =4:

L(v) =γ α(y)

CHAPTER 3. STELLAR KINEMATICS with H₃(y) = _p1 6 € 2p2y3−3p2yŠ (3.4) H₄(y) = _p1 24 4y 4_−12y2_{+3. (3.5)}

This results in four free parameters that describe the LOSVD: the velocity v, the velocity dispersion σ, and the higher-order moments h₃ and h₄. h₃ describes asymmetric deviations from a Gaussian profile (skewness). Symmetric deviations (kurtosis) are described by the parameter h₄. Some example LOSVDs are shown in Fig. 3.1.

In the following a selection of LOSVD extraction techniques are presented in chronological order. Methods used later in this work are explained in greater detail.

- The Fourier Quotient method (FQ, Sargent et al. 1977) is a parametric

method which fits a Gaussian LOSVD to the quotient of the galaxy spec- trum and the template spectrum in Fourier space. This method has sev- eral disadvantages, e.g. it is very sensitive to template mismatch (i.e. when the spectral type of the template does not match that of the galaxy spec- trum), and the errors are strongly correlated and thus complicated to esti- mate. Later methods based on FQ but with a different error analysis are, e.g., the Fourier Difference (Dressler,1979) and the Fourier Fitting (Franx et al.,1989) method.

- In theCross-Correlationmethod (CC,Simkin 1974;Tonry & Davis 1979),

the galaxy spectrum is cross-correlated against a stellar template spectrum. The peak of the galaxy-template correlation function is fitted by a Gaussian. The position of the peak gives the velocity and the velocity dispersion can be derived from the peak width. CC is less affected by template mismatch than FQ and the error estimation is easier.

- The Fourier Correlation Quotientmethod (FCQ,Bender 1990) is an im-

provement of the previous methods. It has the advantages that it is less sensitive to template mismatch than FQ and it is non-parametric, i.e. it recovers the full broadening function without a priori assumptions on the shape. This hybrid approach is based on the deconvolution of the peak of

3.1. EXTRACTION TECHNIQUES −300 0 300 −300 0 300 −300 0 300 v (km/s) h3 h4 -0.1 0 0.1 -0.1 0 0.1

Figure 3.1: Illustration how the LOSVD shape changes with varying h3 and h4. The central LOSVD is purely Gaussian (h₃=h₄=0). Forh₃6=0 the LOSVDs are skewed

and forh₄6=0 the LOSVDs are more peaked or more flattened.

the template-galaxy correlation function with the peak of the autocorrela- tion function of the template. In contrast to the previous methods, where the complete correlation functions are deconvolved, deconvolving only the peaks is what makes this method so insusceptible to template mismatch. High-frequency components of the template-galaxy correlation function, caused by noise, would be strongly amplified during deconvolution. There- fore a Wiener filter is implemented which suppresses these components. This filter is obtained by fitting the template-galaxy correlation function with a Gaussian, using the fitting parameters to construct a model of its Fourier transform, and add a model for the noise. A smoothing parameter

W determines the width of the Gaussian. W =1 corresponds to “optimal”

CHAPTER 3. STELLAR KINEMATICS

- Another non-parametric method is Direct Fitting(Rix & White,1992) in

pixel space, where the galaxy and template spectra are assumed to consist of several components (continuum, noise, absorption lines) and the best-fitting combination of components is determined by a least-squares minimisation. - Winsall & Freeman (1993) developed a Modified Fourier Quotient tech-

nique which is able to deal with non-Gaussian LOSVDs. This method is parametric and works in Fourier space.

- The Unresolved Gaussian Decomposition (Kuijken & Merrifield, 1993)

parametrizes the broadening function as a sum of pure Gaussians uniformly spaced in velocity.

- In the Gauss-Hermite fitting method a Gauss-Hermite expansion is used

to parametrize the LOSVD and a least-squares minimisation is performed either in Fourier space (van der Marel & Franx,1993) or in pixel space (van der Marel et al.,1994).

- Saha & Williams(1994) developed aBayesian Methodto extract the broad-

ening function. It is able to cope with low-SN data and template mismatch- ing, can be used either in a parametric or a non-parametric mode and the fitting is done in pixel space.

- The Maximum Penalized Likelihood method of Merritt (1997) is non-

parametric and works in pixel-space. Here the broadening function is found by minimizing the penalized log likelihood

−logL_p=X

i

[G(λ_i)₋(_{L ⊗}S)_i]2+αP , (3.6)

where α is the smoothing parameter and P is a penalty function, which

is large for noisy velocity distributions and zero for Gaussian ones. This method can also be applied to low-S/N data as even a large degree of smooth- ing does not bias the solutions too much.

- The likewise non-parametric Maximum Penalized Likelihood method

(MPL,Gebhardt et al. 2000a) is based onSaha & Williams(1994) andMerritt

(1997), but the LOSVDs and the weights of the template stars are simulta- neously fitted. Both the LOSVDs and the template weights are constrained to be non-negative at all points. An initial binned velocity profile is con- volved with a linear combination of the template spectra and the residuals