Statistics – Computation
Scientific paper
Feb 2008
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008apj...674..613l&link_type=abstract
The Astrophysical Journal, Volume 674, Issue 1, pp. 613-614.
Statistics
Computation
1
Scientific paper
In our original paper, an error was introduced in the computation of the differential column density distribution function, f(NHI)=m/(ΔNHIΣΔX)=CHIN-βHI, where the total redshift path (ΣΔX=2.404) was mistakenly omitted. The constant CHI (in Table 7) should therefore be corrected by -0.38 dex. We attach a revised version of Figure 14, where f(NHI) is plotted against logNHI (note that in the original figure, for the histogram representation, systems with σN/N<=0.4 were considered; we correct this so that only systems with σb/b<=0.4 and σN/N<=0.4 are considered).
This error directly affects the estimate of the baryon density of the cool photoionized Lyα forest, since Ω(NLA)~CHI: the values of Ω(NLA) reported in § 6.1 should therefore be corrected by a factor of 1/2.404. As a reminder, we explicitly write the equation to estimate Ω(NLA) using the Schaye (2001) method,Ω1(NLA)~3.1×10-9h-170Γ1/312T0.594((fg)/0.16)1/3(CHI)/4/3-β[N4/3-βmax-N4/3-βmin, (1)where h70≡H0/(70 km s-1 Mpc-1), Γ12=0.05 is the H I photoionization rate in units of 10-12 s-1, T4 is the IGM temperature in units of 104 K, fg=0.16 is the fraction of mass in gas, and Nmin and Nmax are the minimum and maximum column densities used in the calculation.
S. V. Penton et al. (ApJ, 658, 680 [2007]) used a different methodology and made different assumptions to estimate Ω(NLA). In particular, they assume that the clouds have a spherical geometry and an isothermal density profile. Equation (40) in their paper can be written (where their notation dN(NHI)/dz is f(NHI) J. M. Shull 2007, private communication):Ω2(NLA)~5.5×10-11h-170Γ0.512r0.5100(CHI)/3/2-β[N3/2-βmax-N3/2-βmin, (2)where r100=1 is the characteristic absorber size in units of 100 kpc and all the other symbols have the same meaning as in equation (1). In equation (2), the assumption is made that T4=2. This equation can be modified to include the temperature dependency that appears in the hydrogen recombination rate (~4×10-13T-0.764 cm3 s-1):Ω3(NLA)~4.4×10-11h-170Γ0.512r0.5100T0.384(CHI)/3/2-β[N3/2-βmax-N3/2-βmin. (3)
For the Schaye method, Ω(NLA)~(nH/nHI)NHIf(NHI)dNHI. Using the scaling relation nHI~NHI/L, the ionization correction factor is such that (nH/nHI)-1~(NHI/L)1/2. If one introduces this factor in the integrand, one would find the same dependency for the various parameters that are found in equation (3) (but the constant would be a factor ~3 times smaller). J. Schaye (ApJ, 658, 680 [2007]) went a step further by using the Jeans criterion to define a Jeans length that is ~N-1/3HI and therefore (nH/nHI)-1~N2/3HI.
The differences in the methodology to derive Ω(NLA) yield different values for Ω(NLA)/Ωb, which are summarized here in Table A1 for two ranges of Nmin and Nmax and for different values of the assumed temperature in the absorbing gas. The Penton et al. (2000) method (eq. [3]) yields values of Ω(NLA)/Ωb 1.3 to 2.2 times larger than the Schaye (2001) method (eq. [1]), depending on the adopted temperature and the column density interval (but note that the differences can be slightly larger or smaller depending on the adopted parameters in eqs. [1] and [3]). Decreasing the temperature from T4=2 to 0.5 reduces Ω(NLA)/Ωb by a factor of 1.7 and 2.3 for the Penton et al. and Schaye methods (for the adopted parameters), respectively. The smaller temperature (T4=0.5) derived by R. Davé & T. M. Tripp (ApJ, 658, 680 [2007]) is from a comparison of Lyα forest observations and cosmological hydrodynamical simulations and is typically valid for an overdensity ρ/ρ¯<~1 or logNHI<~12.5 at z<~0.4 [remember that the low column density systems dominate Ω(NLA) if they follow the same distribution shown in Fig. 1]. According to the Davé & Tripp (2001) simulations, absorbers with 13.2<~logNHI<~13.5 have a temperature somewhat intermediate between 104 and 2×104 K. The larger temperature also corresponds to low-metallicity gas in photoionization equilibrium.
Note that there are other uncertainties associated with these two methods for estimating Ω(NLA)/Ωb. For example, changing the assumed absorber geometry will have a significant effect on the result. In the case of the Penton et al. method, changing from a spherical geometry to a disk geometry with an aspect ratio of 10 reduces Ω(NLA)/Ωb by a factor of 2.2 (Penton et al. 2000). It will be difficult to eliminate the uncertainties associated with the physical state of the gas and its geometry when estimating Ω(NLA)/Ωb. Therefore, the estimated values of Ω(NLA)/Ωb will likely have large systematic model dependent uncertainties for many years to come. However, the values of Ω(NLA)/Ωb listed in Table A1 clearly reveal that the NLAs (b<=40 km s-1) contain an important share of the baryons at low redshift, especially if there is a population of Lyα forest clouds with logNHI<~12.4 that follows the same differential column density distribution function as the stronger absorbers.
We thank Gerard Williger for calling our attention to a problem with Figure 14, and we apologize for any possible inconvenience.
Lehner Nicholas
Richter Philipp
Savage Blair. D.
Sembach Kenneth Russell
Tripp Todd M.
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