4.3 Tests
4.3.1 Analytic models
extent of the galaxy and determination of the projected half-light radius, i.e. depends on the global galaxy properties. It should be noted that although R1/2 is one of the most basic galaxy characteristics, it is not always uniquely determined from observations (see, e.g., Cappellari et al. 2013 for discussion). R1/2 is sensitive to (i) quality and depth of photometric data, (ii) used radial range, (iii) methodology applied. Even for nearby galaxies various groups report different values of the effective (half-light) radius. A notable example is a well-studied massive nearby elliptical galaxy M87 (NGC 4486). On the basis of high-quality photometric data Kormendy et al. (2009) deriveR1/2 ≃194′′for M87, while Chen et al. (2010) report R1/2 ≃107′′ (see also Figure 11 in Chen et al. 2010).
Notice that both estimators give exactly the same characteristic radius and circular speed estimate for the pure power-law surface brightness profile I(R) ∝ R−α (the 3D stellar density scales as j(r) ∝ r−α−1) in the isothermal gravitational potential. For this case the characteristic radius isRchar =R2 =r3and the circular speed is well approximated byVc =√
3σp(R2)≡q 3
σ2p .
Main properties of Churazov et al. and Wolf et al. estimators are summarized in Table 4.1.
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Table 4.1: Main properties of Churazov et al. and Wolf et al. estimators.
Estimator Assumptions Data Characteristic radius σp k
local dyn. equilibrium log-slope of I(R) Rsweet σp(Rsweet) p
1 +α(Rsweet) +γ(Rsweet)
sph. symmetry log-slope of σp(R) or R2 or or
no rotation σp(R2) p
1 +α(R2) +γ(R2)
isothermal Φ(r) or p
1 +α(R2) global dyn. equilibrium σp(R) over whole galaxy r3
σp2 √ 3 sph. symmetry deprojection of I(R) orr1/2
no rotation or determination ofR1/2 or 4/3R1/2
flat σp(R)
Figure 4.1: Typical profiles considered for a sample of analytical models. Upper left: as-sumed stellar densityj(r), anisotropyβ(r) and circular speedVc(r) as functions ofR/R1/2, where R1/2 is a projected half-light radius. Upper right: surface brightness I(R) cor-responding to the projection of j(r) (under the assumption of spherical symmetry) and velocity dispersion profile σp(R) from the spherical Jeans equation. Low panel: profiles, used to get the simple Vc-estimates. Logarithmic slopes of j(r) and I(R) are shown in the top panel as dark green and purple curves respectively. r3 and R2 are marked as dark green dashed and purple dotted lines. The simple circular speed estimates are shown as opened and filled symbols of different colors: the filled dark green square is for the global Vc-estimate at r3, the filled cyan square - at r1/2 ≈ 4/3R1/2, the open blue square shows the local estimate atRsweet and the open purple square - atR2, the open purple star shows the simplified version of the local estimator (eq. 4.8).
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Figure 4.2: Circular speed estimates for ‘ideal’ model galaxy with surface brightness de-scribed by S´ersic profile (S´ersic index n = 4), constant anisotropy β = −0.25, flat Vc(r) (left side) or flatσp(R) (right side). The upper panels show the log-slopes of the 3D stellar density α3D (in dark green) and of the surface brightness α (in purple), the projected ve-locity dispersion is plotted in the middle panels, and the true circular speed profile as well as simple Vc-estimates are shown in the low panels. The symbols in the low panels are the same as in Figure 4.1.
(rising, roughly flat and decreasingVc). rs varies from 0 to 90 half-light radius, rc changes from 0 to 12 half-light radius.
The luminosity-weighted average of the projected velocity dispersion profile is calcu-lated as
σp2
=
R I(R)Rσ2p(R)dR
R I(R)RdR over [0.1R1/2; 10R1/2]. Figure 4.1 illustrates all steps of the analysis and shows typical profiles considered for analytical models.
Performance in an ideal case.
First we apply the simple estimators to the ideal ‘model’ galaxy, which meets all the as-sumptions used to derive the formulae: dynamical equilibrium, spherical symmetry, no streaming motions, constant anisotropy, flat Vc(r) for Churazov et al. formula and con-stant σp(R) for Wolf et al. estimator. Figure 4.2 demonstrates the typical profiles for the
‘ideal’ galaxy and derived estimates. If the circular speed is assumed to be flat, then the de-rived projected velocity dispersion should monotonically decrease with radius. For the flat velocity dispersion profile and constant anisotropy it is possible to infer the circular speed from a deprojection ofσp(R) andI(R) and solving the Jeans equation (Mamon and Bou´e 2010). As expected both estimators work well when applied to ‘ideal’ galaxies which meet
all the assumptions. For the flat Vc(r) and decreasing σp(R) Wolf et. al formula slightly underestimates the true circular speed with typical deviation of≈ −3%, for constantσp(R) and growing Vc(r) Churazov et al. approach tends to overestimate the true Vc by≈ −3%.
Grid of analytical models.
We explore ∼30000 analytical models, described by the S´ersic law with S´ersic index 2 <
n < 20, mildly increasing with radius anisotropy profile and circular speed characteristic for (i) dark-matter dominated dwarf spheroidal galaxies (growing Vc) and (ii) massive elliptical galaxies (roughly flat and decreasing with radius Vc). However, we do not aim to explore the whole parameter space, the idea is to understand how sensitive the estimators are to the assumption of a flat Vc(r) or σp(R) and to the mildly varying anisotropy. The resulting histograms for the local estimator (upper row) and the global estimator (lower row) are shown in Figure 4.3. The RMS-scatter for the global estimator is almost twice larger than for the local one, indicating that the latter is less sensitive to the assumption of a constant circular velocity, than the Wolf et al. estimator to the assumption of a flat velocity dispersion.
As the Churazov et al. derivation assumes the isothermal gravitational potential (i.e.
flatVc(r)) it would be interesting to test how accuracy of the estimator depends on the slope of the circular speed profile. Indeed, there is a clear correlation between the deviation ∆ of the estimated circular speed from the true one and the logarithmic slope of the trueVc(r) at Rsweet: ∆≈k×
−dlnVc2 dlnr
(Figure 4.4, the black histogram shows the average deviation).
The details of the mean correlation between ∆ and −dlnVc2
dlnr depends on the sampling of the parameter space. However the main trend for the local estimator seems to be rather universal. For growing/decreasing Vc near Rsweet (which is close to R2) the method tends to overestimate/underestimate the true value by a factor of ≈1 + 0.1
−dlnVc2 dlnr
(order of magnitude estimate). For flat Vc near Rsweet the local estimator is largely unbiased when averaged over the parameter space covered by our grid of analytic models. When deviations are plotted against the logarithmic slope of the projected velocity dispersion γ = −dlnσp2
dlnR at Rsweet, a similar pattern is observed. If σp grows with radius in the vicinity ofRsweet, then the local Vc-estimate overestimates the true circular speed. For flat or moderately falling observed velocity dispersion profiles Churazov et al. method seems to recover almost unbiased Vc-estimate.
The global estimator demonstrates more complex dependence on the slope of the cir-cular speed radial profile, giving a noticeable negative bias for a flat circir-cular speed (for the flat Vc(r) the observed velocity dispersion might vary significantly with radius). As expected, Wolf et al. formula works best for roughly flat line-of-sight velocity dispersion profiles. The global Vc-estimate appears to be significantly overestimated relative to the true one for σp rapidly increasing with radius (γ . 0.3). Large negative deviations are
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Figure 4.3: The histograms of deviations of simple Vc-estimates from the true value for the local (upper row) and the global estimators (lower row) for model spherical galaxies, described by S´ersic surface brightness profile, mildly growing anisotropy, and circular speed profile that is similar in shape to the observed circular velocity curves. Deviations are calculated as ∆ = Vc−Vctrue
Vctrue , where estimated Vc and true Vctrue are taken at the same characteristic radius.
Figure 4.4: Dependence of the error in theVc estimation on properties of the trueVc(r) and observed σp(R) profiles. Left: Deviation ∆ of the estimated circular speed from the true one as a function of a log-slope of the true circular speed−dlnVc2
dlnr taken at a characteristic radius (Rsweet for the local estimator andr3 for the global one). The histogram shows the average deviation in a chosen bin of logarithmic slopes. Right: Deviation as a function of the log-slope of the projected velocity dispersionγ =−dlnσ2p
dlnR at at a characteristic radius.
Only 1000 randomly chosen realizations are shown for clarity.
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present for models with σp(R) showing a bump, i.e. when γ(R) changes a sign.
Figure 4.4 indicates that the local estimator should be applied with cautious to sys-tems with increasing velocity dispersion profiles and/or to syssys-tems that are described by growing circular speed profiles in the vicinity ofR2. As the Wolf et al. estimator relies on global properties of the galaxies, it works well if the velocity dispersion does not change significantly with radius over the whole extent of the system. Roughly speaking, Wolf et al.
formula is appropriate for dwarf spheroidal galaxies (see Kowalczyk et al. 2013 who have tested the global estimator on a sample of simulated dSph) and for a subset of elliptical galaxies with approximately flat velocity dispersion profile, while Churazov et al. estimator works for elliptical galaxies in general.
It should be also noted that for large S´ersic indices (n > 8−10), typical for massive ellipticals sitting at the centers of groups or clusters, log-slope of the surface brightness α is close to 2 over a wide range of radii, and in this radial range the true circular speed is well described by the isotropic oneVciso (eq. 4.6).