4. Detection of the FIR emission from SDSS galaxies by stacking analysis 25
4.2 Stacking analysis
4.2.2 Radial profiles of galaxies
32CHAPTER 4. DETECTION OF THE FIR EMISSION FROM SDSS GALAXIES BY STACKING ANALYSIS
0 5 10 15 20 25 30
50 100 150
Ng,pix
Ar,SFD [mmag]
Figure 4.8: Mean values of Ar,SFD of pixels containing Ng,pix plotted against Ng,pix. The quoted error bars indicate the corresponding rms.
4.2. STACKING ANALYSIS 33 expected to be independent of mr and computed from the PDF of the extinction P(A) (see Figure 4.3) as
C =⟨A⟩ ± σA
√Ng, (4.6)
where Ng is the number of stacked galaxy images, and the mean and rms are given by
⟨A⟩ =
∫ ∞
0
AP(A)dA, (4.7)
σ2A =
∫ ∞
0
A2P(A)dA− ⟨A⟩2. (4.8)
As we see below, however, this is not the case. Therefore we treat C as a free parameter for each magnitude bin in the fitting analysis described below. Figure 4.8 plots ⟨A⟩with quoted error-bars of σA as a function of Ng,pix.
The clustering term is written as Σcg(θ;mr) =
∫∫
dm′dφ Σsg(θ−φ;m′)wg(φ;m′, mr)dNg(m′)
dm′ , (4.9)
where wg(φ;m′, mr) is the angular galaxy cross-correlation function between magnitudes m′ and mr, and dNg(m′)/dm′ is the differential galaxy number density.
Given the large smoothing length of the SFD map (6′.1 FWHM), a single galaxy profile is expected to be approximated by the circular Point Spread Function (PSF), independently of its intrinsic profile. Thus we adopt the Gaussian PSF profile:
Σsg(θ;mr) = Σs0g (mr) exp (
− θ2 2σ2
)
, (4.10)
where σ is the Gaussian width of the PSF. The Gaussian approximation of the PSF is justified in Appendix A. Also we assume that the angular cross-correlation function is given as
wg(φ;m′, mr) =K(m′, mr)(φ/φ0)−γ, (4.11) where the constants φ0 and γ are assumed to be independent of m′ and mr. We adopt γ = 0.75 (Connolly et al., 2002; Scranton et al., 2002), which is valid for φ < 1◦. With equations (4.10) and (4.11), equation (4.9) reduces to
Σcg(θ;mr) = Σc0g (mr) exp (
− θ2 2σ2
)
1F1 (
1−γ 2; 1; θ2
2σ2 )
, (4.12)
where 1F1(α;β;x) is the confluent hypergeometric function, and Σc0g (mr) = 2πσ2
( φ0
√2σ )γ
Γ (
1−γ 2
) ∫
dm′Σs0g (m′)K(m′, mr)dNg(m′)
dm′ . (4.13) Equation (4.12) results in the extended tail due to the clustering term in addition to the Gaussian tail of the single central galaxy. The latter is negligible at θ ≫ σ, and the observed tail of the profile around galaxies is basically dominated by the clustering term.
34CHAPTER 4. DETECTION OF THE FIR EMISSION FROM SDSS GALAXIES BY STACKING ANALYSIS
16 18 20
2.5 3.0 3.5
mr [mag]
σ [arcmin]
Figure 4.9: Best-fit values of the Gaussian PSF width, σ, in the case that σ is treated as a free parameter separately for different magnitudes. The error-bars computed from 400 jackknife resamplings. The dashed line indicates the error-weighted average of the all magnitude bins.
The average radial profiles of the stacked images centered at photometric galaxies are plotted in Figure 4.10. Filled circles and triangles correspond to galaxies in the different r-band magnitude ranges in Figure 4.7, and the quoted error-bars represent rms in each circular bin of ∆θ = 0′.66. The signal-to-noise ratio evaluated by equation (4.4) is S/N = 23.6 for the brightest sample (15.5< mr<16.0), and S/N = 22.8 for the faintest sample (20.0< mr <20.5).
We fit the observed radial profiles by equations (4.5), (4.10), and (4.12), treating Σs0g , Σc0g , andCas free fitting parameters for each magnitude bin. The Gaussian width of PSF, σ, is also an uncertain, since the SFD map is constructed after smoothing the IRAS data in a complicated fashion. Therefore we determine the value ofσby the radial profile fitting as follows. We first perform the model-fit to the observed profile treating σ as another free parameter separately for different magnitudes, σ(mr), in addition to Σs0g , Σc0g , and C. The resulting best-fit values of Σs0g and Σc0g are shown in Figure 4.11. The results return small negative values for Σs0g , in the cases of mr > 19.0. This would be simply due to the fact the total signal is dominated by the clustering term; the unambiguous extraction of the single galaxy contribution in those cases is difficult if we add another degree of freedom in σ for each magnitude bin. Figure 4.9 shows the best-fit values and the statistical errors of σ. In the case of mr < 19, the best-fit values of σ are indeed almost independent of mr and σ = 3′.1 as expected from our model assumption. This value of σ = 3′.1 is reasonable, given the resolution of the SFD map (2′.59 in Gaussian width) and the additional smoothing due to the 2′.37 pixelization and our cloud-in-cell interpolation. This is why we constrainσ to be independent of mr and fix as σ= 3′.1 in the actual fitting procedure below. We will discuss to what extent the result of the profile fit is affected by the fixed value ofσ later.
4.2. STACKING ANALYSIS 35
mr = 15.5 − 16.0
16.0 − 16.5 16.5 − 17.0
17.0 − 17.5
17.5 − 18.0 18.0 − 18.5 18.5 − 19.0 19.0 − 19.5
19.5 − 20.0
20.0 − 20.5
0 5 10 15 20
79.0 79.5 80.0 80.5 81.0 81.5 82.0 82.5 83.0 83.5
θ [arcmin]
Σgtot (θ ; mr) [mmag]
Figure 4.10: Radial profiles of stacked galaxy images corresponding to Figure 4.7. Solid curves indicate the best-fit model of equation (4.5), (4.10), and (4.12).
36CHAPTER 4. DETECTION OF THE FIR EMISSION FROM SDSS GALAXIES BY STACKING ANALYSIS
Σgs0
Σgc0
(σ: free) Σgs0 + Σgc0
Σgs0
Σgc0
Σgs0 + Σgc0 (σ = 3’.1)
16 18 20
0.01 0.1 1.0
mr [mag]
Σg (θ = 0 ; mr ) [mmag]
Figure 4.11: Best-fit parameters characterizing the FIR emission of galaxies against their r-band magnitude. Blue crosses, red triangles, and black circles indicate the best-fit value of Σs0g , Σc0g , and Σs0g +Σc0g , respectively, assumingσ = 3′.1 andγ = 0.75. The same symbols in cyan, magenta, and gray indicate the best-fit values in the case that σ is treated as a free parameter separately for each magnitude bin.
16 18 20
1.0 10.0 100.0
mr [mag]
Σgc0(mr) / Σgs0(mr)
Figure 4.12: Ratio of the clustering term and the central galaxy contribution as a function of the r-band magnitude of the central galaxy.
4.2. STACKING ANALYSIS 37
all
restricted (+25.0 mmag)
16 18 20
79 80 81 82
mr [mag]
C [mmag]
(all)
Σgs0
Σgc0
Σgs0 + Σgc0 (restricted)
16 18 20
0.01 0.1 1.0
mr [mag]
Σg (θ = 0 ; mr ) [mmag]
Figure 4.13: Left panel; The background noise level C against the r-band magnitude of the central galaxy. Black crosses indicate the results for all our sample and red ones are restricted in the inner regions shown as yellow lines in Figure 4.1. Right panel; The best-fit values of Σs0g (cyan), Σc0g (magenta), and Σs0g + Σc0g (gray) for the restricted region.
The shaded regions indicate the best-fit values for all our sample as the same as shown in Figure 4.11.
The solid curves in Figure 4.10 indicate the best-fit model of equations (4.5), (4.10), and (4.12), where we treat Σs0g , Σc0g , and C as the free parameters for each magnitude bin. The best-fit parameters for Σs0g (mr) and Σc0g (mr) are plotted in Figure 4.11. The statistical uncertainties of the best-fit parameters are evaluated from the 400 subsamples of the random jackknife resampling. The quoted error bars for each parameter are computed by marginalizing the other parameters. Since we fix the value of σ = 3′.1, the error bars are smaller than those for the case that we treatσas a free parameter. Even at the central position of the stacked images, the FIR signals are indeed dominated by the clustering term Σcg rather than the single galaxy term (see Figure 4.12).
The fitted values of the background offset term C are plotted against mr in the left panel of Figure 4.13. Although our model assumes that C is independent of mr, it is not the case at all; a systematic decrease of C against mr is clearly seen. We repeated the same analysis by selecting those galaxies located in the inner contiguous regions (160◦ < α < 220◦, 5◦ < δ < 80◦; see Figure 4.1). The results are plotted in red crosses after shifting 25 mmag, just for the ease of visual comparison. While the values of C is sensitive to the region of the map and their dependence on mr is weaker in this case, the best-fit values for other quantities are hardly changed due to the particular choice of subregions in the SFD map as shown in the right panel of Figure 4.13. We discuss the possible origins of the systematic dependence of C onmr in Appendix B.
Incidentally the small value ofC with respect to the general trend at 16.0< mr <16.5 is the reason why the corresponding profile in Figure 4.10 does not follow the systematic trend of the other profiles.
Figure 4.14 shows to what extent the results of the profile fit are affected by the choice ofγandσ. Since the value ofγis somewhat uncertain and also depends on magnitudes,mr
38CHAPTER 4. DETECTION OF THE FIR EMISSION FROM SDSS GALAXIES BY STACKING ANALYSIS
γ=0.75 Σgs0
Σgc0
Σgs0 + Σgc0
γ = 0.65 (filled) γ = 0.85 (open)
16 18 20
0.01 0.1 1.0
mr [mag]
Σg (θ = 0 ; mr ) [mmag]
γ = 0.75 γ = 0.65 (filled) γ = 0.85 (open)
16 18 20
79 80 81 82 83
mr [mag]
C [mmag]
σ=3’.1 Σgs0
Σgc0
Σgs0 + Σgc0
σ = 2’.9 (filled) σ = 3’.3 (open)
16 18 20
0.01 0.1 1.0
mr [mag]
Σg (θ = 0 ; mr ) [mmag]
σ = 3’.1 σ = 2’.9 (filled) σ = 3’.3 (open)
16 18 20
79 80 81 82 83
mr [mag]
C [mmag]
Figure 4.14: Best-fit parameters of the radial profile fit by varying the values ofγ (upper panels) and σ (bottom panels). Left panels indicate the best-fit values of Σs0g (squares), Σc0g (triangles), and Σs0g + Σc0g (circles). Right panels indicate the best-fit values of C.
The best-fit values shown as filled (open) symbols for γ = 0.65 (0.85) andσ = 2′.9 (3′.3).
The shaded regions indicate the best-fit value for all our sample as the same as shown in Figure 4.11.