Radial Density Profiles: Deprojecting Observations

To describe the intrinsic three-dimensional structure of a galaxy, it is useful to construct models of galaxies for which it is possible to analytically calculate the depth and shape of the potential well and the velocity distributions from the intrinsic mass density distribution. The two-dimensional projec-tions of those models should furthermore fit the properties of observed galaxies. There are several different models discussed in the literature, of which we will in the following present those that will

be important in the course of this work. All of the commonly used models assume the systems to be spherically symmetric, and in an equilibrium state. While it is known from observations that most systems are more likely (slightly) elliptical or even triaxial, those models are still a good first approxi-mation. Since their properties can be solved analytically in most cases, which is not the case for most triaxial and elliptical models, they are nevertheless a useful tool to understand the properties of the potentials and the mass (and light) distributions of galaxies.

Single Power Law Profiles: An Isothermal Sphere

A first simple approach to model the density distribution of a spheroidal galaxy is to assume a density which constantly decreases with radius. In this case, the density can be described by a power law of the form

ρ(r)=ρ₀ r r_s

!γ

, (1.21)

with the characteristic densityρ₀and the power law slopeγ <0. The special case ofγ=−2 is called the singular isothermal sphere:

ρ_Iso(r)=ρ₀ r r_s

!−2

. (1.22)

Using the fact that the mass distribution can be calculated from the density profile as M(r)=4π

Z r

0

r⁰²ρ(r⁰) dr⁰, (1.23)

the rotational velocity profile of the isothermal sphere equals v²_circ= G M(r)

r =4πGρ₀r_s². (1.24)

Thus, the rotational velocity of the isothermal sphere is constant at all radii. One benefit of the single power law profiles is that they can be easily transformed from two to three dimensions and vice versa asρ_2D∝r^γ−1.

While it is possible to always find a radius range in the observed surface brightness profiles of observed galaxies where a single power law profile is a good fit, the full radius regime usually cannot be fit by a single power law profile. Nevertheless, the rotational velocity profile of the isothermal sphere is flat, which resembles the observations. Since the observed rotational velocity profiles reflect the contributions of all mass components of a galaxy, i.e., the luminous and dark matter, this could indicate that the total (stellar and dark matter) radial density profile can be described by a single power law profile, even if the individual components alone follow different laws. One of the main goals of this thesis is to address this issue, and to understand the interplay between the dark and luminous components of (spheroidal) galaxies.

Double Power Law Profiles

As the single power law is not sufficient in describing the full radius range of the observed surface brightness profiles, double (broken) power laws have been discussed to solve the issue. These double

1.4. THE RADIAL (SURFACE) DENSITY PROFILES OF GALAXIES 31

Figure 1.10: Examples for the different radial density profiles, normalized toρ₀ =1and r_s =1, andα=1 in case of the Einasto profile. Shown are the profile of the singular isothermal sphere (yellow dotted line), the Plummer profile (green dashed line), the Jaffe profile (cyan dash-dot-dot-dotted line), the Hernquist profile (blue long-dashed line), the NFW profile (red dash-dotted line) and the Einasto profile (grey solid line).

power laws are generally given by

ρ(r)= ρ₀

r r_s

!β1 





1+ r r_s

!β3







(β2−β1)/β3 , (1.25)

withρ₀the characteristic density andr_sthe scale radius. They follow one power law withρ∝r^−β1 in the inner part and a second power law with ρ ∝ r^−β2 in the outskirts with a smooth transition in the mid-radius regime. The third exponentβ₃measures the sharpness of the transition.

As for the isothermal sphere, the broken power laws also only apply to spherically symmetric systems in equilibrium. The most commonly used profiles, for which the corresponding mass and gravitational potentials have analytical expressions, are:

• The Plummer Profile: Already before the first measurements of the surface brightness pro-files of extragalactic nebula, Plummer (1911) presented a profile to describe the radial density

distribution of stars in globular clusters:

ρ_Plum(r)= ρ₀







1+ r r_s

!2







5 2

. (1.26)

This is a broken power law profile with exponentsβ₁ = 0, β₂ = 5, and β₃ = 2. It has a flat inner core and decreases steeply outside of the scale radiusr_s, as shown as green dashed line in Fig. 1.10.

• The Jaffe Profile: One of the first intrinsic profiles to describe the density distributions of spheroidal galaxies and bulges was introduced by Jaffe (1983):

ρ_Jaffe(r)= ρ₀ r r_s

!2

1+ r r_s

!2 . (1.27)

This profile, with β1 = 2, β2 = 4, and β3 = 1, has an inner slope which is identical to the isothermal sphere, but steepens at the outskirts to fall offwith a power law slope of−4 (cyan dash triple-dotted line in Fig. 1.10). In projection, this profile is similar to the de Vaucouleurs R^1/4profile in the outskirts but deviates strongly in the inner parts. With its steep inner slope it resembles the power-law-excess-galaxies and is therefore sometimes still used today to model less massive spheroidals, however, it is not sufficient to describe the more massive, cored galax-ies.

• The Hernquist Profile: In 1990, L. Hernquist introduced a density profile that resembled the de Vaucouleurs R^1/4 profile better than the Jaffe profile (Hernquist, 1990). The equation he presented was

ρ(r)= M 2π

r_s r

1

(r+r_s)³, (1.28)

which can be rewritten as ρ_Hern(r)= ρ₀

r

r_s 1+ r r_s

!3 , (1.29)

with β₁ = 1, β₂ = 4, and β₃ = 1 (blue long dashed line in Fig. 1.10). For the Hernquist profile, the stellar half-mass radius (i.e., the radius which contains half of the stellar mass of a galaxy) can be calculated from the scale radius as r_1/2 = (1+ √

2)r_s. With its excellent resemblance of the de Vaucouleurs R^1/4 profile in projection it is the density profile which is most commonly used to especially model the mass distributions of the bulges in late-type galaxies, but also massive ellipticals. The effective radiusr_eff, which can be calculated from the de Vaucouleurs R^1/4 profile, can thus be calculated directly from the Hernquist profile as r_eff ≈ 1.8153 r_s ≈ 3/4 r_1/2 (Hernquist, 1990). Both, the Jaffe and the Hernquist profiles, are part of the family of Dehnen profiles, which include all density profiles withβ₂ = 4 and β₃ = 1, and an arbitraryβ₁ (Dehnen, 1993). Generally, properties like mass distribution and intrinsic velocity dispersion can be solved analytically for many Dehnen models (especially with 0 6 β₁ 6 3), in some cases even for the projected properties (e.g., β₁ = 0,1,2), which makes those models particularly useful for modeling galaxies.

1.4. THE RADIAL (SURFACE) DENSITY PROFILES OF GALAXIES 33

• The NFW Profile: The previously discussed profiles were all motivated by the wish to an-alytically model the observed surface brightness profiles of galaxies (or stellar clusters), and intended to understand the distribution and behaviour of the luminous matter in galaxies. Using cosmologicalN-Body dark-matter-only simulations, Navarro et al. (1996, 1997) found that the radial density distributions of dark matter halos in their simulations could always be fitted by profiles of a similar shape. The profile they presented is

ρ_NFW(r)= ρ₀ r

r_s 1+ r r_s

!2, (1.30)

which resembles a double power law profile with β₁ = 1, β₂ = 3, andβ₃ = 1. In contrast to the Jaffe and Hernquist profiles, the NFW profile has a total mass that diverges logarithmically for r → ∞, while it is similar to the Hernquist profile at small radii (see red dash-dotted line in Fig. 1.10). Interestingly, as shown by Navarro et al. (1996), the two free parameters of the NFW profile, ρ₀ andr_s, are correlated. Thus, calculating the radius within which the density of the dark halo is larger than 200 times the critical density of the Universe (ρ_crit) and thus the mass of the halo within this radiusr₂₀₀, enables to define a concentration parameterc_NFW

c_NFW≡ r₂₀₀

r_s . (1.31)

For a given mass of a halo, this concentration parameter is nearly the same for all halos, and it decreases with increasing mass, indicating that more massive dark matter halos are less concen-trated than less massive ones. This has been confirmed by observations of the hot gas content in galaxy clusters through its X-ray emissions by Pointecouteau et al. (2005), who also found a decreasing concentration parameter of their best-fitting NFW profile with cluster mass. Those observations of X-ray properties of galaxy clusters have furthermore confirmed that the NFW profile is generally a proper description of the density profiles of galaxy clusters (e.g., Pratt &

Arnaud 2005; Pointecouteau et al. 2005).

In summary, the double power law profiles can, in their projected form, successfully explain some of the observed surface brightness profiles: the Hernquist profile is, in projection, a good approximation to the de VaucouleursR^1/4 profile, and the Plummer profile can describe (in some cases) the density distribution of globular clusters. The NFW profile seems to be a good description of the dark mat-ter density distributions, as indicated for example by strong lensing observations of galaxy clusmat-ters.

However, none of the profiles can actually mimic the S´ersic profile which has been most successful in describing the surface brightness profiles of observed spheroidals.

A More Realistic Approach: Exponential Profiles

None of the double power law profiles can mimic all configurations of the S´ersic profile because the S´ersic profile has three free parameters, and with the S´ersic indexnit has a parameter which modifies the curvature of the profile and thus the profile is not limited to one clear shape as it is the case for the single and double power-law profiles. A similar, however three dimensional, profile was introduced by Einasto (1965) in an approach to fit stellar profiles of galaxies like Andromeda and the Milky Way

Figure 1.11:The Einasto profile for four different combinations of the free parameters r_candα, with the third free parameter,ρ₀, is fixed to the value ofρ₀ = 1 for all test cases since it only shifts the profile in vertical direction. The green solid line shows the Einasto profile for rc=1andα=1, as in Fig. 1.10. The blue dashed line shows the profile for a smaller core-radius of rc =0.5andα =1, the profile with rc =1andα=0.5is shown ad red dotted line, and rc = 0.5andα = 0.5is shown as cyan dash-dotted line. As can be seen, the scale radius shifts the profile in horizontal direction, while the power-law slope parameterα=1determines the profiles curvature.

(Einasto, 1974). TheEinasto profileis characterised by its power-law logarithmic slopeα_Ein: ρ_Ein(r)=ρ₀ exp

(

− r r_c^Ein

!αEin)

, (1.32)

where ρ₀ is the central density andr_c^Ein is the radius at which the density has decreased to 1/e of its central value (see also Retana-Montenegro et al., 2012). Forρ₀ = 1,r_c^Ein = 1 andα_Ein = 1, the Einasto profile has a flat inner part similar to the Plummer profile, while it bends towards very steep slopes at large radii, steeper than all other three-dimensional profiles discussed before, as shown as grey solid line in Fig. 1.10.

Fig. 1.11 shows how the profile changes if the parameters are varied: A smaller core radius r_c^Ein = 0.5 causes only a shift in horizontal direction towards smaller radii (blue dashed line). A smaller curvature parameterα_Ein =0.5, on the other hand, significantly changes the curvature of the

Im Dokument The outer halos of elliptical galaxies (Seite 43-49)