Since Navarro et al. (1996, 1997) showed from numerical cosmological dark-matter-only simulations that the density profiles of dark matter halos are similar to each other with only little dependence of mass (see also Navarro et al. 2004), the broken power law that they introduced to describe the shape of those dark matter halos (see Sec. 1.4.2 for more details on the profile) has been used in a multitude of studies on the dark matter content of galaxies, from cluster environments down to isolated galaxies.
The so called NFW-profile,
ρNFW(r)= ρ0
(r/rs)(1+r/rs)2, (5.1)
wherersis the characteristic scale radius andρ0 is the characteristic density (see also Eq. 1.30), in-creases monotonically towards the center as d lnρ/d lnr ≈ −1 and is thus shallower than isothermal, while in the outskirts it follows d lnρ/d lnr ≈ −3. In simulated halos, however, the two “free” pa-rameters of this profile, rs andρ0, are not uncorrelated, in fact they reflect the accretion history of the individual halos, with more concentrated halos living in denser environments (e.g., Bullock et al., 2001). In other words, the earlier a halo starts to assemble, the higher its central density.
For stellar systems, there exists a number of density profiles in the literature that are used to ana-lytically describe collisionless systems: the Hernquist profile (see Eq. 1.29) and the Jaffe profile (see Eq. 1.27), for example, are often used to describe the bulges of spiral galaxies and ellipticals, as they reproduce successfully the de Vaucouleur profile (see Eq. 1.17) when projected; the Plummer profile (see Eq. 1.26) is often used to describe the stellar distributions within globular clusters. Less accurate, however still used for its simplicity, is the density profile of an isothermal sphere (see Eq. 1.22).
As a first approach to find a universal description for the density profile of the stellar halos of galaxies, we fit these profiles to our stacked density profiles in the entire radius range, as shown in Fig. 5.8. For the dark matter density (upper left panel), we find that the NFW-profile (dash-dotted line) is a good approximation to our density profile, as expected, and the power-law (short-dashed black line) and the isothermal sphere (dotted black line) profiles fit parts of the profile but show strong deviations in the inner and outer parts. While the Jaffe profile (dash-dot-dot-dotted line) provides a rather good fit to the dark matter density, the Hernquist (long-dashed line) and the Plummer (solid thin black line) profiles both fail to fit the dark matter density profiles properly.
For the stellar halo density profiles with (upper right panel) and without (lower left panel) sub-structures, none of these standard profiles works. The closest approximation to the stellar halo density with substructures is a Jaffe profile, and the stellar halo without substructures is best represented by a Plummer profile, but both fits are crude and do not mirror the curved behaviour of the stellar halo.
The crudeness of those fits originates from the fact that all models have two or even only one free parameter, and none of them has a continuous change of slope with radius from a flat inner profile to a very steep outer profile, as needed for a successful description of the stellar halo density profile.
5.3. THE EINASTO PROFILE 123
Figure 5.8: Fits of different density profiles to the stacked dark matter (black solid lines) and stellar density profiles including satellites (blue solid lines) and without satellites (red solid lines) for halos with Mtot ≈ 1×1012Mfrom Box4 uhr at z = 0: power-law fits at40 kpc to100 kpcradius (short-dashed lines), NFW profile fits (dash-dotted lines), Jaffe profile fits (dash-dot-dot-dotted lines), Hernquist profile fits (long-dashed lines), Plummer profile fits (thin solid lines), isothermal sphere density fits (dotted lines) and Einasto profile fits (green dotted lines). Upper left panel: Fits to the stacked dark matter density profile. Upper right panel: Fits to the stacked stellar density profile including satellites.Lower panel: Fits to the stacked stellar density profile without satellites.
Figure 5.9: Parameters resulting from fits of the Einasto profile to the individual density profiles for the 24 halos with Mtot≈1×1012Mfrom Box4 uhr for dark matter (black dashed line/open circles) and stars without substructures (red solid line/open diamonds)Left panel: Histogram of the scale radii rcEin. Middle panel:
Histogram of the slopeαEinof the fits of the Einasto profile.Right panel:αEinversus rcEin.
Thus, we need a profile with an additional free parameter to represent the curvature of the stellar halo density profile. One such profile is the Einasto profile, first introduced by Einasto (1965) and used to fit stellar profiles of nearby galaxies like the Milky Way and Andromeda (Einasto, 1974), which is characterised by its power-law logarithmic slope:
ρ(r)=ρ−2 exp (
− 2 αEin
"
r r−2
!αEin
−1
# )
, (5.2)
where αEin controls the curvature, ρ−2 is the density and r−2 the radius at which ρ(r) ∝ r−2, see Retana-Montenegro et al. (2012) and Sec. 1.4.2 for more details. This profile can equivalently be written as
ρEin(r)=ρ0 exp (
− r rcEin
!αEin)
, (5.3)
whereρ0=ρ−2(r)e2/αEin is the central density and rcEin= r−2Ein
2 αEin
!αEin (5.4)
is the scale length (i.e., the radius at which the density has decreased to 1/eof its central value), as shown by Retana-Montenegro et al. (2012).
As demonstrated in Fig. 5.8, the Einasto profile is a much more accurate fit to all three stacked density profiles (see green dotted lines), and even for the dark matter density it provides a better fit than the commonly used NFW profile. For the dark matter density it has already been shown that there are systematic deviations from the NFW profile in several different simulations, and that the Einasto profile is a much better fit (Navarro et al. (2004), Merritt et al. (2006), Gao et al. (2008), Stadel et al.
5.3. THE EINASTO PROFILE 125
Figure 5.10:Mean value of the slopeαEinof the Einasto profile fits within the three halo mass ranges studied in this work versus the total mass Mtot, for the dark matter fits (black diamonds) and the stellar halo fits (red diamonds). Error bars show the standard deviations within our sample. For comparison, the mean values for the six dark-matter-only Aquarius halos from Navarro et al. (2010) are shown as blue diamond, and the values presented in Gao et al. (2008) for the Millenium simulation dark matter halos are shown as green diamonds (for z=0).
(2009), Navarro et al. (2010) & Klypin et al. (2014)). Navarro et al. 2004, 2010 showed that the NFW profile underestimates the dark matter density in the inner regions, and especially for massive halos it is far offthe true profile (Klypin et al., 2014), causing a systematic bias in the concentration measurements (Gao et al., 2008). In contrast, the Einasto profile is more successful in describing the density profiles of the dark matter halos, even if the slopeαEinis kept at a constant average value, i.e., the profile is effectively reduced to two free parameters, see Navarro et al. (2010).
As mentioned by Merritt et al. (2006) and Retana-Montenegro et al. (2012), the Einasto pro-file mimics the Sersi´c propro-file. Thus, one might expect it to be a good fit to the stellar halo of galaxies as well. We fitted the dark matter and stellar halos for our galaxies of Milky Way mass (Mtot ≈ 1×1012M) with Einasto profiles, using the same logarithmic binning as before. The re-sulting distributions of the fitted parametersrcEinandαEinare shown in the left respective the middle panel of Fig. 5.9, while the right panel shows the correlation between both parameters.
We find that the values for both parameters are similar for the dark matter and the stellar halo, with slight trends to larger values for both parameters for the stellar component. The slope-parameterα∗Ein
Figure 5.11:Left panel: SlopeαEinversus scale radius rcEinof the Einasto profile fits to the DM (filled circles) and stellar (filled diamonds) density profiles of all 449 halos with total masses between Mtot≈1×1012Mand Mtot≈1×1014Mfrom Box4 uhr (DM dark grey, stars light grey). Red and blue symbols are the parameters for the fits to the halos with total mass of Mtot≈1×1012Mand Mtot≈1×1013M, respectively.Right panel:
Slope αEinversus the radius r−2Ein where the fitted Einasto profile has the value ofρ−2(r) ≈ r−2. Colors and symbols as in the left panel. Green circles show the values for the six dark matter Aquarius halos presented in Navarro et al. (2010).
peaks aroundα∗Ein≈ 0.20, with a mean value ofα∗Ein=0.17±0.08, and for the dark matter we find a peak at αDMEin ≈ 0.15 with a mean value ofαDMEin = 0.15±0.07. Our values for the dark matter are in excellent agreement with the results presented by Navarro et al. (2010) for the Aquarius simulations (Springel et al., 2008), where they analysed the density profiles for six high-resolution dark matter re-simulations of Milky-Way like galaxies, and found values for the Einasto slope parameters of 0.130± 0.0200< αDMEin <0.173±0.0123.
Similar results for halos in the Milky-Way mass range were also found by Gao et al. (2008) who fitted Einasto profiles to dark matter halos selected from the Millenium Simulation (Springel et al., 2005c). They also found an interesting trend of the slope αDMEin with the total mass of a halo: for halos with larger masses, the slope αDMEin tends to be generally larger than for less massive halos.
These systematic variations of the density profiles with the total halo mass had already been found by Merritt et al. (2006) from a study of six cluster mass halos and four galaxy mass halos, and have been recently confirmed by Klypin et al. (2014) for halos selected from the MultiDark simulations (Prada et al., 2012).
As shown in Fig. 5.10, we find a similar trend with mass if we include the halos from the other two mass ranges studied in the previous section, namelyMtot≈ 3×1012MandMtot ≈1×1013M, see the black diamonds. Error bars show the standard deviations within our sample. In addition, we included the mean value for the six Aquarius halos (blue diamond) and the values from the Millenium
5.3. THE EINASTO PROFILE 127
Figure 5.12:Left panel: Slopesα∗Einfrom Einasto profile fits to the stellar halos versus those from fits to the dark matter halos,αDMEin. Halos in the Milky Way mass range are shown as large symbols, with colors according to their morphology as in Fig. 5.6. The full sample of Magneticum halos with more than Mtot ≈1×1012M
is shown as filled small grey circles. The dashed line marks the 1:1 correlation.Right panel: Same as the left panel but for the radii r−2Ein.
halos atz=0 as presented in the left panel of Fig. 2 of Gao et al. (2008) (green diamonds). The dark matter halos from Magneticum agree very well with the previous results. In addition, we show the mean values ofα∗Einfor our three halo mass ranges as red diamonds. We clearly see that for all three mass rangesα∗Einis larger thanαDMEin, indicating that the stellar halo profiles are curved more strongly than their dark matter counterparts.
While the slope parameters of the Einasto profile have been discussed in the literature in some detail (at least for dark matter halos due to the lack of sufficiently large simulations including baryons with high enough resolution to study the stellar density profiles), less attention has been given to the scale radii. As already shown in the right panel of Fig. 5.9 for our halos in the Milky Way mass range, there is a correlation between the scale radiusrcEinand the slopeαEinfor both dark matter and stellar halos, i.e., for larger values ofαEin the scale radius is larger. The correlation is the same for dark matter and stars, but the values forαEinare larger for the stars, as discussed before, while the values for the scale radii cover nearly the same range for both components.
This behaviour can also be seen in the left panel of Fig. 5.11, where we showαEinandrcEinfor all halos selected from Magneticum with a total massMtot >1×1012Mas light grey diamonds (stellar halo fits) and dark grey circles (dark matter halo fits). The halos with a total mass ofMtot≈1×1012M
are shown in red, the halos with a total mass ofMtot≈1×1013Mare shown in blue. We find a clear trend with the total halo mass in the correlation of the two parameters as well, with more massive stellar and dark matter halos having smaller values ofαEin at a givenrcEin than their less massive counterparts.
As the scale radii of the Einasto profiles are far inside the centers of the galaxies and thus not clearly part of the visible halos, it is more useful for the understanding of its meaning to use Eq. 5.4 and calculater−2Ein, the radius at which the fitted Einasto profile has a slope ofρ(r)∝ r−2. The result
is shown in the right panel of Fig. 5.11, using the same colors as before. While the correlation was rather narrow for slope versus scale radius, the scatter inr−Ein2 for both dark matter and stellar halo is much larger. But despite the larger scatter, we still see that there is an obvious trend for stellar halos with largerr−2Einto have larger slopes, and for less massive halos to have smallerr−2Ein. This implies that the less massive galaxies have less extended stellar halos, and more compact stellar halos have generally smaller slopes.
While we see a trend with halo mass for the dark matter, in that more massive halos have larger r−2Ein, there is no clear trend with the slope. For comparison, we again included the values of the six Aquarius halos from Navarro et al. (2010), and we again find a good agreement with our results. Thus we conclude that for the dark matter halos the halo mass is the most important quantity in determining the radial extension of the halo, while the whole range of curvatures is covered. For the stars, however, there must be an additional mechanism that causes more compact halos to have smallerαEinand thus be less curved, and we suspect that the gas content during accretion might be the critical ingredient, since gas dissipates energy and thus can reach the galaxy centers, causing star formation there, but thereby “starving” the outer halos compared to the dark matter halos. In this picture, only dry mergers would cause the stellar halo to grow like the dark matter halo, while wet merging would cause the inner core of the galaxy to grow in situ, while growing the stellar halo only slightly.
To understand the correlation between the shape of the density profiles of stellar and dark matter halo, we investigate the correlations betweenαDMEin andα∗Ein, andr−2,DMEin andr−2,∗Ein. As expected, there is no clear correlation betweenr−2,DMEin andr−2,∗Ein (right panel of Fig. 5.12). Within a given total mass range, r−2,DMEin is approximately constant independent of r−2,∗Ein, which can clearly be seen from the colored round symbols in that panel which mark the galaxies withMtot≈1×1012M. We also do not find any correlation inr−2,DMEin andr−2,∗Ein depending on the galaxy morphology. The only slight trend can be found for the dark matter halos, wherer−2,DMEin correlates with the total mass of the halo: the small grey circles show all halos withMtot > 1×1012M, and since most of the grey symbols are at larger values ofr−2,DMEin than the large colored symbols that mark the Milky-Way mass halos, this indicates thatr−2,DMEin increases slightly with total halo mass.
For the slopes of the Einasto profiles, we see a different behaviour, shown in the left panel of Fig. 5.12: stellar halos with a smaller slope seem to reside in dark matter halos with relatively small slopes as well, while large stellar slopes usually coexist with larger dark matter slopes. This indicates that there is, indeed, a correlation between the shapes of both halos. While there is a slight trend for stellar halos to have larger slope values than their dark matter counterparts, there are also several halos which lie on the 1:1 relation for the slope parameters. As indicated by the large filled circles, we do not find any correlation between the mass or morphology of a galaxy and its correlation of slopesαDMEin andα∗Ein.
Since we found no other correlations than with total halo mass for the Einasto profile parameters of the dark matter halos, whereas those of the stellar halos do appear correlated, we checked whether α∗Ein and r−2,∗Ein correlate with other properties of the central galaxies as well. We tested the outer power-law slopeγout, the stellar mass M∗ and the mean age hz∗i for our sample of galaxies with Mtot≈1×1012M. The results are shown in Fig. 5.13. For the outer slopes we find a slight correlation for galaxies with steeper outer slopes to also have larger Einasto slopes and largerr−2,∗Ein, however, there is no obvious trend with morphology (upper panels). This agrees well with our conclusion that the galaxies with the larger Einasto slopes are more compact and thus most likely have had a smaller amount of recent merging.
5.3. THE EINASTO PROFILE 129
Figure 5.13: Different Galaxy properties versus the slopesα∗Ein(left panels) and radii r−2,∗Ein (right panels) of the Einasto profile fits for the 24 individual halos with Mtot≈1×1012Mfrom Box4 uhr:Upper panels: versus the slopesγoutof the power-law fits.Middle panels: Versus the stellar mass M∗of the Galaxies.Lower panels:
Versus the mean age of the stellar populationhz∗i. Colors show the morphological classification according to Sec. 2.4.1 with disks, spheroidals and unclassified galaxies in blue, red and green, respectively.
For the stellar mass, we also find a slight trend for more massive galaxies to have larger Einasto parameters, but this trend is very weak and could be a bias due to the low number of studied galaxies within this mass range. We also find that there is no trend at all of the mean agehz∗iwith either of the Einasto parameters, see lower panels of Fig. 5.13.