The Observed Growth of Massive Galaxy Clusters II: X-ray Scaling Relations
This is the second in a series of papers in which we derive simultaneous constraints on cosmology and X-ray scaling relations using observations of massive, X-ray flux-selected galaxy clusters. The data set consists of 238 clusters with 0.1–2.4 luminosities , and incorporates follow-up observations of 94 of those clusters using the Chandra X-ray Observatory or ROSAT (11 were observed with both). The clusters are drawn from three samples based on the ROSAT All-Sky Survey: the ROSAT Brightest Cluster Sample (78/37 clusters detected/followed-up), the ROSAT-ESO Flux-Limited X-ray sample (126/25), and the bright sub-sample of the Massive Cluster Survey (34/32). Our analysis accounts self-consistently for all selection effects, covariances and systematic uncertainties. Here we describe the reduction of the follow-up X-ray observations, present results on the cluster scaling relations, and discuss their implications. Our constraints on the luminosity–mass and temperature–mass relations, measured within , lead to three important results. First, the data support the conclusion that excess heating of the intracluster medium (or a combination of heating and condensation of the coldest gas) has altered its thermodynamic state from that expected in a simple, gravitationally dominated system; however, this excess heat is primarily limited to the central regions of clusters (). Second, the intrinsic scatter in the center-excised luminosity–mass relation is remarkably small, being bounded at the per cent level in current data; for the hot, massive clusters under investigation, this scatter is smaller than in either the temperature–mass or –mass relations (10–15 per cent). Third, the evolution with redshift of the scaling relations is consistent with the predictions of simple, self-similar models of gravitational collapse, indicating that the mechanism responsible for heating the central regions of clusters was in operation before redshift 0.5 (the limit of our data) and that its effects on global cluster properties have not evolved strongly since then. Our results provide a new benchmark for comparison with numerical simulations of cluster formation and evolution.
keywords:large-scale structure of Universe – X-rays: galaxies: clusters.
Establishing the relationship between total mass and observable quantities is a crucial step in deriving cosmological constraints from the growth of cosmic structure using galaxy clusters. Not only can these scaling relations provide useful proxies for mass, but they are also fundamentally important in accounting for selection effects such as Eddington bias and Malmquist bias.
The construction of X-ray flux-selected cluster samples out to redshift and beyond has now enabled investigations of dark energy using these data (Mantz et al. 2008, hereafter M08; Vikhlinin et al. 2009b). However, the task of calibrating X-ray scaling relations has become correspondingly more complex; the evolution with cosmic time of the scaling relations and their scatter must be well understood, since such evolution can be degenerate with the effects of dark energy. The effect on cosmological constraints of systematic uncertainties in the scaling relations has been discussed in the context of future surveys by, e.g., Sahlén et al. (2009).
Fortunately, as we describe below, it is possible to simultaneously constrain both the evolution of the scaling relations and cosmological parameters, using a flux-limited sample of which some clusters have been targeted by detailed, follow-up X-ray observations. To distinguish X-ray methods from other measures of the growth of cosmic structure, including those using optically selected clusters, we refer to the resulting data set as the cluster X-ray luminosity function (XLF), although, strictly speaking, it contains a great deal more information than the luminosity function alone.
This is the second of a series of papers in which we address these issues. In a companion paper (Mantz et al., 2009, hereafter Paper I) we describe the statistical methods required to simultaneously constrain the scaling relations and cosmology, self-consistently accounting for all selection effects111Throughout this paper, we refer to “selection effects” or “selection biases” relative to a mass-limited sample, since we are primarily interested in deriving scaling relations as a function of cluster mass. and systematic uncertainties, and present the cosmological results from our analysis. This paper focuses on the reduction of the follow-up X-ray observations, and the constraints on the scaling relations from the simultaneous analysis. In Papers III (Rapetti et al., 2009) and IV (Mantz, Allen, & Rapetti, 2009), we respectively apply our analysis to investigations of modified gravity and neutrino properties.
In addition to their utility for cosmological investigations, cluster scaling relations are of significant astrophysical importance. Of primary interest is the heating mechanism that prevents cooling gas in dense cluster cores from condensing into stars and molecular gas at much higher rates than are observed (for reviews, see Peterson & Fabian, 2006; McNamara & Nulsen, 2007). The shape and evolution of the scaling relations, and specifically any departures from the simplest predictions for gravitationally dominated systems, can provide information on the physical mechanisms responsible for averting strong cooling and star formation.
The details of the cluster sample selection, the follow-up observations, and their reduction are discussed in Section 2. Sections 3 describes the scaling relation model and Section 4 our statistical method, which is more comprehensively detailed in Paper I. In Section 5, we present constraints on the scaling relations, and investigate various extensions to the simplest model, including possible evolution with redshift and asymmetric scatter. Section 6 contains a discussion of the influence of cool, X-ray bright gas in cluster centers on the scaling relations. Implications of our results for the astrophysics of the intracluster medium (ICM) are discussed in Section 7.
Unless otherwise noted, masses, luminosities and distances in tables and figures are calculated with respect to a reference cosmology, defined to be spatially flat, with Hubble constant , present mean matter density with respect to the critical density , and dark energy in the form of a cosmological constant. We adopt the conventional definition of cluster radius in terms of the critical density of the Universe; thus, is the radius within which the mean density of the cluster is times the critical density at the redshift of the cluster.
2.1 Cluster samples
Our data are drawn from three wide-area cluster samples derived from the ROSAT All-Sky Survey (RASS; Trümper, 1993): the ROSAT Brightest Cluster Sample (BCS; Ebeling et al., 1998), the ROSAT-ESO Flux-Limited X-ray sample (REFLEX; Böhringer et al., 2004), and the bright sub-sample of the Massive Cluster Survey (Bright MACS; Ebeling, Edge, & Henry 2001; Ebeling et al. 2010). Each sample covers a distinct volume of the Universe: the BCS covers the northern sky at , REFLEX covers the southern sky222The coverage of REFLEX actually extends slightly into the northern hemisphere. In this study, we restrict REFLEX to declinations and adjust the sky coverage fraction accordingly. at , and Bright MACS covers higher redshifts, , at declinations . The distributions of cluster detections from each sample in redshift and luminosity are shown in Figure 1 (here luminosities in the rest frame 0.1–2.4 band are calculated straightforwardly from the survey fluxes, assuming our reference cosmology).
The flux limits, , defining the samples are respectively , and in the 0.1–2.4 keV ROSAT energy band. In order to restrict the data set to massive clusters, for which the scaling relations can be described simply, we select only clusters whose flux in this band satisfies
approximately corresponds to a fixed, intrinsic luminosity of in the 0.1–2.4 band for our reference cosmology. Using this selection criterion, the BCS, REFLEX and MACS samples respectively contribute 78, 126 and 34 clusters to our data set.333Our previous analysis in M08 included 130 REFLEX sources using similar selection criteria. Published catalog entries RX J0117.8-5455 and RX J2251.7-3206 were eliminated from the present study, since high-resolution Chandra imaging reveals their emission to be point-like rather than extended. Source RX J0507.6-0238 satisfies our criteria after correcting its redshift (Dale Kocevski, private communication) from that published in Böhringer et al. (2004), and was therefore added. Three other sources (all Abell clusters), RX J0552.8-2103, RX J1959.1-3454 and RX J2331.2-3630, were removed because they do not satisfy the flux limit defined in Equations 1 and 2, reflecting a small difference between our K-correction and that used by Böhringer et al. (2004), and the fact that Equation 2 is not precisely identical to a fixed luminosity. The BCS and MACS sources selected are the same as in M08. Within the redshift and flux range defined above, all three samples are thought to be approximately 100 per cent complete and pure.444In the sense that the published selection function for each sample, the likelihood for cluster detection as a function of redshift and flux, is accurate. See additional comments in M08.
2.2 Follow-up X-ray observations
Additional information about a subset of the flux-selected clusters is available from follow-up X-ray observations. These observations enable more precise measurements of cluster luminosities, as well as measurements of morphology, temperature, and gas mass, which is an excellent proxy for total mass.
We incorporate Chandra data for clusters at where it is available. Below this redshift, the Chandra Advanced CCD Imaging Spectrometer (ACIS) field of view is typically too small to allow measurements at , a canonical radius at which simulations indicate that the scaling relations are well behaved (Evrard et al., 2008). For , we do, however, incorporate follow-up ROSAT Position Sensitive Proportional Counter (PSPC) observations. The ROSAT and Chandra observations were cross-calibrated using clusters in the redshift range that were observed with both instruments (Section 2.2.4). Details of the follow-up observations can be found in Tables 1–4; the total good exposure time of the follow-up data, including both ROSAT and Chandra observations, is 3.3 Ms, distributed over 94 clusters. The reduction and analysis of these data are described in the following sections.
|Name||RA (J2000)||Dec (J2000)||Date||Detector||Mode||Exposure (ks)|
|Abell 2256||17 04 02.3||+78 38 14||1990||Jun||17||PSPC||16.||4|
|Abell 1795||13 48 53.1||+26 35 38||1991||Jul||01||PSPC||15.||1|
|Abell 401||02 58 57.2||+13 34 46||1992||Jan||23||PSPC||5.||3|
|Abell 2029||15 10 55.9||+05 44 44||1992||Aug||10||PSPC||8.||7|
|Abell 2255||17 12 50.0||+64 03 43||1993||Aug||24||PSPC||11.||8|
|Abell 478||04 13 25.2||+10 28 00||1991||Aug||31||PSPC||21.||9|
|Abell 2142||15 58 21.1||+27 13 45||1992||Aug||25||PSPC||5.||9|
|Abell 2244||17 02 42.0||+34 03 24||1992||Sep||21||PSPC||2.||9|
|Abell 2034||15 10 12.6||+33 30 45||1993||Jan||31||PSPC||4.||9|
|Abell 1068||10 40 44.4||+39 57 13||1992||Nov||30||PSPC||9.||8|
|Abell 2204||16 32 47.2||+05 34 33||1992||Sep||04||PSPC||5.||2|
|Abell 2218||16 35 53.0||+66 12 36||1991||May||25||PSPC||31.||9|
|Abell 1914||14 26 01.0||+37 49 37||1992||Jul||20||PSPC||6.||3|
|Abell 665||08 30 58.1||+65 51 02||1991||Apr||10||PSPC||32.||0|
|Abell 520||04 54 09.0||+02 55 18||1993||Sep||06||PSPC||4.||8|
|Abell 963||10 17 03.6||+39 02 52||1993||Oct||28||PSPC||9.||1|
|RX J0439.0+0520||04 39 02.2||+05 20 42||2000||Aug||29||ACISI||VFAINT||9.||6|
|Abell 1423||11 57 17.4||+33 36 40||2000||Jul||07||ACISI||VFAINT||9.||9|
|Zwicky 2701||09 52 49.2||+51 53 06||2001||Nov||04||ACISS||VFAINT||25.||9|
|Abell 773||09 17 52.7||+51 43 36||1991||Oct||28||PSPC||11.||0|
|Abell 2261||17 22 27.0||+32 07 58||1999||Dec||11||ACISI||VFAINT||6.||7|
|Abell 1682||13 06 50.7||+46 33 30||2002||Oct||19||ACISI||VFAINT||1.||8|
|Abell 1763||13 35 19.0||+40 59 59||1992||Jun||23||PSPC||12.||7|
|Abell 2219||16 40 20.3||+46 42 30||1993||Aug||02||PSPC||8.||5|
|Abell 2111||15 39 41.1||+34 25 07||1993||Jul||23||PSPC||6.||7|
|Zwicky 5247||12 34 22.1||+09 47 05||2000||Mar||23||ACISI||VFAINT||9.||3|
|Abell 267||01 52 42.2||+01 00 30||1999||Oct||16||ACISI||FAINT||7.||1|
|Abell 2390||21 53 37.1||+17 41 45||1993||Nov||13||PSPC||8.||4|
|Zwicky 2089||09 00 36.8||+20 53 40||2006||Dec||23||ACISI||VFAINT||9.||1|
|RX J2129.6+0005||21 29 39.7||+00 05 18||2000||Oct||21||ACISI||VFAINT||8.||7|
|RX J0439.0+0715||04 39 00.6||+07 16 00||1999||Oct||16||ACISI||FAINT||6.||3|
|Abell 1835||14 01 01.9||+02 52 40||1993||Jul||03||PSPC||6.||1|
|Abell 68||00 37 05.9||+09 09 36||2002||Sep||07||ACISI||VFAINT||10.||0|
|Name||RA (J2000)||Dec (J2000)||Date||Detector||Mode||Exposure (ks)|
|MS J1455.0+2232||14 57 15.1||+22 20 33||2000||May||19||ACISI||FAINT||9.||9|
|Abell 697||08 42 57.6||+36 21 57||1991||Apr||10||PSPC||9.||6|
|Zwicky 3146||10 23 39.6||+04 11 12||1993||Nov||13||PSPC||8.||1|
|Abell 781||09 20 25.3||+30 30 11||2000||Oct||03||ACISI||VFAINT||8.||7|
|Name||RA (J2000)||Dec (J2000)||Date||Detector||Mode||Exposure (ks)|
|Abell 3558||13 27 58.4||31 30 04||1991||Jul||17||PSPC||24.||0|
|Abell 85||00 41 50.1||09 18 22||1992||Jun||11||PSPC||3.||2|
|Abell 3667||20 12 31.2||56 49 47||1992||Oct||09||PSPC||6.||3|
|Abell 3266||04 31 21.5||61 26 21||1992||Apr||30||PSPC||6.||6|
|Abell 3112||03 17 58.2||44 14 07||1992||Dec||17||PSPC||7.||0|
|Abell 2597||23 25 19.6||12 07 27||1991||Nov||27||PSPC||6.||7|
|Abell 3921||22 49 57.0||64 25 53||1992||Nov||15||PSPC||11.||5|
|MS J1111.8||11 14 12.7||38 11 25||1993||Jan||11||PSPC||18.||5|
|Abell 1689||13 11 29.6||01 20 28||1992||Jul||18||PSPC||13.||5|
|Abell 2163||16 15 46.0||06 08 54||1992||Feb||28||PSPC||3.||1|
|Abell 209||01 31 53.1||13 36 48||2000||Sep||09||ACISI||VFAINT||8.||7|
|RX J1504.10248||15 04 07.6||02 48 16||2004||Jan||07||ACISI||FAINT||12.||0|
|RX J0304.13656||03 04 03.3||36 56 30||2008||Mar||16||ACISI||VFAINT||19.||9|
|RX J0237.42630||02 37 27.4||26 30 28||2008||Mar||03||ACISI||VFAINT||7.||4|
|Abell 2667||23 51 39.7||26 04 60||1992||Dec||05||PSPC||2.||0|
|RX J0638.75358||06 38 47.3||53 58 28||2008||Apr||11||ACISI||VFAINT||19.||9|
|RX J0220.93829||02 20 56.5||38 28 52||2008||Feb||29||ACISI||VFAINT||19.||9|
|Abell 521||04 54 07.4||10 13 24||1999||Dec||23||ACISI||VFAINT||36.||8|
|RX J0307.02840||03 07 02.0||28 39 56||2008||Mar||13||ACISI||VFAINT||18.||1|
|RX J2011.35725||20 11 27.2||57 25 10||2004||Jun||08||ACISI||VFAINT||22.||3|
|RX J0232.24420||02 32 17.7||44 20 55||2004||Jun||08||ACISI||VFAINT||7.||9|
|RX J0528.93927||05 28 53.3||39 28 19||2004||Mar||10||ACISI||VFAINT||15.||9|
|RX J0043.42037||00 43 24.8||20 37 24||2008||Feb||02||ACISI||VFAINT||19.||7|
|1ES 0657558||06 58 27.5||55 56 32||1997||Feb||28||PSPC||4.||5|
|Abell 2537||23 08 22.0||02 11 30||2004||Sep||09||ACISS||VFAINT||36.||2|
|Name||RA (J2000)||Dec (J2000)||Date||Detector||Mode||Exposure (ks)|
|MACS J2245.0+2637||22 45 04.6||+26 38 04||2002||Nov||24||ACISI||VFAINT||13.||9|
|MACS J1131.81955||11 31 55.6||19 55 45||1993||Jun||27||PSPC||7.||2|
|MACS J0014.33022||00 14 18.8||30 23 18||1992||Jun||16||PSPC||13.||6|
|MACS J2140.22339||21 40 15.2||23 39 40||1993||Nov||07||PSPC||9.||2|
|MACS J0242.52132||02 42 35.9||21 32 26||2002||Feb||07||ACISI||VFAINT||9.||6|
|MACS J1427.62521||14 27 39.5||25 21 03||2002||Jun||29||ACISI||VFAINT||14.||6|
|MACS J0547.03904||05 47 01.5||39 04 26||2002||Oct||20||ACISI||VFAINT||20.||9|
|MACS J0257.62209||02 57 41.3||22 09 13||2001||Nov||12||ACISI||VFAINT||20.||5|
|MACS J2049.93217||20 49 55.3||32 16 49||2002||Dec||08||ACISI||VFAINT||22.||9|
|MACS J2229.72755||22 29 45.2||27 55 36||2002||Nov||13||ACISI||VFAINT||12.||9|
|MACS J1319.9+7003||13 20 07.5||+70 04 37||2002||Sep||15||ACISI||VFAINT||8.||7|
|MACS J0520.71328||05 20 42.2||13 28 47||2002||Feb||10||ACISI||VFAINT||19.||0|
|MACS J1931.82634||19 31 49.6||26 34 34||2002||Oct||20||ACISI||VFAINT||13.||1|
|MACS J0035.42015||00 35 26.2||20 15 46||2003||Jan||22||ACISI||VFAINT||20.||8|
|MACS J0947.2+7623||09 47 13.0||+76 23 14||2000||Oct||20||ACISI||VFAINT||11.||7|
|MACS J1115.8+0129||11 15 51.9||+01 29 55||2003||Jan||23||ACISI||VFAINT||8.||7|
|MACS J0308.9+2645||03 08 56.0||+26 45 35||2002||Mar||10||ACISI||VFAINT||23.||4|
|MACS J0404.6+1109||04 04 32.7||+11 08 11||2002||Feb||20||ACISI||VFAINT||17.||6|
|MACS J1532.8+3021||15 32 53.8||+30 20 59||2001||Aug||26||ACISS||VFAINT||9.||4|
|MACS J0011.71523||00 11 42.8||15 23 22||2002||Nov||20||ACISI||VFAINT||20.||8|
|MACS J0949.8+1708||09 49 51.8||+17 07 08||2002||Nov||06||ACISI||VFAINT||14.||3|
|MACS J1720.2+3536||17 20 16.7||+35 36 23||2002||Nov||03||ACISI||VFAINT||19.||8|
|MACS J1731.6+2252||17 31 39.2||+22 51 50||2002||Nov||03||ACISI||VFAINT||18.||5|
|MACS J2211.70349||22 11 45.9||03 49 42||2002||Oct||08||ACISI||VFAINT||15.||2|
|MACS J0429.60253||04 29 36.0||02 53 06||2002||Feb||07||ACISI||VFAINT||22.||1|
|MACS J0159.80849||01 59 49.4||08 49 60||2002||Oct||02||ACISI||VFAINT||17.||4|
|MACS J2228.5+2036||22 28 32.8||+20 37 15||2003||Jan||22||ACISI||VFAINT||19.||9|
|MACS J0152.52852||01 52 33.9||28 53 33||2002||Sep||17||ACISI||VFAINT||17.||3|
|MACS J1206.20847||12 06 12.3||08 48 06||2002||Dec||15||ACISI||VFAINT||23.||5|
|MACS J0417.51154||04 17 34.3||11 54 27||2002||Mar||10||ACISI||VFAINT||11.||2|
|MACS J2243.30935||22 43 21.4||09 35 43||2002||Dec||23||ACISI||VFAINT||20.||2|
|MACS J1347.51144||13 47 30.8||11 45 09||2000||Mar||05||ACISS||VFAINT||8.||9|
|MACS J0358.82955||03 58 51.2||29 55 22||2009||Oct||18||ACISI||VFAINT||8.||7|
|MACS J2311.5+0338||23 11 35.3||+03 38 25||2002||Sep||07||ACISI||VFAINT||6.||5|
2.2.1 Chandra data analysis
The standard level-1 event files distributed by the Chandra X-ray Center (CXC) were reprocessed in accordance with CXC recommendations, using the ciao software package555http://cxc.harvard.edu/ciao/ (version 4.1.1, CALDB 4.1.2). This processing includes removal of bad pixels, corrections for cosmic ray afterglows and charge transfer inefficiency, and application of standard grade and status filters, using appropriate time-dependent gain and calibration products. The extra information available for observations taken in VFAINT mode (the majority) was used to improve cosmic ray rejection. The data were cleaned to remove times of high or unstable background using the energy ranges and time bins recommended by the CXC. Blank-field data sets made available by the CXC were tailored to each observation and cleaned in an identical manner to the real data. The normalizations of these blank-sky files were scaled to match the count rates in the target observations measured in the 9.5–12 keV band.
After identifying and masking point sources, flat-fielded images and background-subtracted surface brightness profiles were prepared in the 0.7–2.0 energy band. The emissivity in this band is largely insensitive to the gas temperature, , provided (as is the case for all the clusters studied here), making it the preferred energy range for determining the gas mass. The center of each cluster was identified with the centroid of 0.7–2.0 emission after masking point sources.
Our spectral analysis is a two-stage process. In the first stage, spectra in the 0.8–7.0 band were extracted in an annulus about the cluster centers. The inner radii of all annuli were set to 100 in order to prevent relatively cool gas in the cores from strongly influencing the spectral fits. The outer radii were initially chosen to be the radii at which the signal-to-noise ratio apparent in the 0.8–7.0 surface brightness profiles falls to 2. Background spectra were extracted from the blank-sky fields for targets at , and from source-free regions of the detector for targets. When required, e.g. due to the presence of strong excess soft emission in the field, a model for additional, soft thermal emission was included in the spectral modeling of the background. Photon-weighted response matrices and effective area files were generated for each observation using calibration files for the appropriate period.
The spectral analysis was performed using xspec.666http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/ The spectra were fit to a single-temperature, optically thin thermal emission model evaluated with the mekal code (Kaastra & Mewe 1993; Mewe, Kaastra, & Liedahl 1995, incorporating the Fe-L calculations of Liedahl, Osterheld, & Goldstein 1995), including a model for photoelectric absorption due to Galactic hydrogen (Balucinska-Church & McCammon, 1992). Column densities were fixed to the Galactic values determined from the H survey of Kalberla et al. (2005), unless the published value exceeded , in which case they were fit as free parameters.777We note that the choice of column density does not have a significant impact on our derived luminosities, temperatures and masses. In particular, the energy bands used (), and the use of locally determined backgrounds for the surface brightness analysis (Section 2.2.3) and, when possible, for the spectral analysis, make our results relatively insensitive to uncertainties in the soft X-ray background. Residual uncertainties are within the systematic tolerances defined in Section 2.2.3. The abundances of all metals were assumed to vary with a common ratio, , relative to solar values (Anders & Grevesse, 1989), which was a free parameter in the fit for each cluster. We minimized the modified C-statistic (Cash, 1979; Arnaud, 1996) rather than , as is appropriate for spectra with few counts per bin. Once a best fitting model was identified, the parameter space was explored using Markov Chain Monte Carlo; these Monte Carlo samples were used to propagate the uncertainty due to the fit in the conversion of 0.7–2.0 count rate to 0.1–2.4 flux for each observation. We note that, since both the count rate and flux in this conversion are at soft energies, the conversion is only weakly dependent on temperature and has negligible dependence on metallicity, provided that the temperature is sufficiently high ().
The second stage of the spectral analysis uses an estimate for , obtained as described in Section 2.2.3 below. New spectra in the annuli (0.15–0.5) and (0.15–1) were extracted and fit as above. For observations of nearby clusters where a large fraction of the (0.15–1) region falls outside the detector, only the (0.15–0.5) region was analyzed; we then estimated the (0.15–1) temperature using the relation , fit from the other clusters. (This result is consistent with that of Vikhlinin et al. 2009a, who followed a similar procedure; see Section 4.1.1 and Figure 6 of that work.) This final step is motivated by the desire to measure a temperature within , for consistency with the mass and luminosity measurements (Section 2.2.3), while still reliably excising the cool core, if one is present. We note, however, that this level of detail is not entirely necessary; the per cent reduction in temperature within compared with the temperature measurement from the first stage of the analysis is within statistical errors for most systems and has negligible impact on the determination of masses and luminosities. We therefore do not further iterate the determination of using this new temperature estimate. The resulting temperature measurements are referred to as in the following, the subscript indicating “center-excised”, and are listed in Tables 5–7.
We note that the uncertainties on temperatures measured within are necessarily somewhat larger than those on our initial estimates from higher signal-to-noise regions. Ultimately, however, this has negligible effect on our results, since this additional uncertainty is smaller than the systematic allowances associated with instrument calibration and mass estimations (Section 2.2.3).