The Oxygen Abundances of Luminous and Ultraluminous Infrared Galaxies
ABSTRACT Luminous and ultraluminous infrared galaxies (LIRGs and ULIRGs) dominate the star formation rate budget of the universe at z > 1, yet no local measurements of their heavy element abundances exist. We measure nuclear or near-nuclear oxygen abundances in a sample of 100 star-forming LIRGs and ULIRGs using new, previously published, and archival spectroscopy of strong emission lines (including [O II] 3727, 3729 A) in galaxies with redshifts ~ 0.1. When compared to local emission-line galaxies of similar luminosity and mass (using the near-infrared luminosity-metallicity and mass-metallicity relations), we find that LIRGs and ULIRGs are under-abundant by a factor of two on average. As a corollary, LIRGs and ULIRGs also have smaller effective yields. We conclude that the observed under-abundance results from the combination of a decrease of abundance with increasing radius in the progenitor galaxies and strong, interaction- or merger-induced gas inflow into the galaxy nucleus. This conclusion demonstrates that local abundance scaling relations are not universal, a fact that must be accounted for when interpreting abundances earlier in the universe's history when merger-induced star formation was the dominant mode. We use our local sample to compare to high-redshift samples and assess abundance evolution in LIRGs and ULIRGs. We find that abundances in these systems increased by ~0.2 dex from z ~ 0.6 to z ~ 0.1. Evolution from z ~ 2 submillimeter galaxies to z ~ 0.1 ULIRGs also appears to be present, though uncertainty due to spectroscopic limitations is large. Comment: To appear in 1 Dec 2007 issue of ApJ; 23 pages, 13 figures
- Hα emission line equivalent width, in˚ A, uncorrected for underlying absorption Hβ stellar absorption line equivalent width..
arXiv:0708.1766v1 [astro-ph] 13 Aug 2007
ApJ, 1 Dec 2007, v670n2, in press
Preprint typeset using LATEX style emulateapj v. 03/07/07
THE OXYGEN ABUNDANCES OF LUMINOUS AND ULTRALUMINOUS INFRARED GALAXIES
David S. N. Rupke, Sylvain Veilleux
Department of Astronomy, University of Maryland, College Park, MD 20742-2421
and Andrew J. Baker
Department of Physics and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019
Draft version February 1, 2008
Luminous and ultraluminous infrared galaxies (LIRGs and ULIRGs) dominate the star formation
rate budget of the universe at z ? 1, yet no local measurements of their heavy element abundances
exist. We measure nuclear or near-nuclear oxygen abundances in a sample of 100 star-forming LIRGs
and ULIRGs using new, previously published, and archival spectroscopy of strong emission lines
(including [O II]λλ3727, 3729) in galaxies with redshifts ?z? ∼ 0.1. When compared to local emission-
line galaxies of similar luminosity and mass (using the near-infrared luminosity-metallicity and mass-
metallicity relations), we find that LIRGs and ULIRGs are under-abundant by a factor of two on
average. As a corollary, LIRGs and ULIRGs also have smaller effective yields. We conclude that the
observed under-abundance results from the combination of a decrease of abundance with increasing
radius in the progenitor galaxies and strong, interaction- or merger-induced gas inflow into the galaxy
nucleus. This conclusion demonstrates that local abundance scaling relations are not universal, a
fact that must be accounted for when interpreting abundances earlier in the universe’s history when
merger-induced star formation was the dominant mode. We use our local sample to compare to high-
redshift samples and assess abundance evolution in LIRGs and ULIRGs. We find that abundances in
these systems increased by ∼0.2 dex from z ∼ 0.6 to z ∼ 0.1. Evolution from z ∼ 2 submillimeter
galaxies to z ∼ 0.1 ULIRGs also appears to be present, though uncertainty due to spectroscopic
limitations is large.
Subject headings: galaxies: abundances — galaxies: evolution — galaxies: interactions — galaxies:
ISM — galaxies: kinematics and dynamics — infrared: galaxies
Mid-infrared and submillimeter observations show that
luminous and ultraluminous infrared galaxies (LIRGs
and ULIRGs)1host most of the star formation in the
universe at z ? 1 (Le Floc’h et al. 2005; Chapman et al.
2005; Daddi et al. 2005; Wang et al. 2006; Caputi et al.
2007). Understanding local examples of these sources is
thus a window to star formation and accretion onto su-
permassive black holes at the epoch of highest star for-
mation rate and active galactic nucleus (AGN) density
(e.g., Madau et al. 1998; Schmidt et al. 1995).
A great deal is known about nearby ULIRGs; for a
recent review see Lonsdale et al. (2006).
possess strong starbursts, and many also host optically-
visible AGN (Veilleux et al. 1999). The starbursts and
AGN are on average heavily obscured (Genzel et al.
1998; Hao et al. 2007), and optically-invisible AGN may
be present (Lutz et al. 1999; Armus et al. 2007). The
prevalence of optically-visible AGN increases with in-
creasing infrared luminosity (Veilleux et al. 1999) and
as the merger progresses (Veilleux et al. 2002).
thus hypothesized that ULIRGs play a role in the
evolution of quasars.When obscuring dust is re-
moved from a buried AGN (by starburst- or AGN-
driven outflows; see, e.g., Rupke et al. 2005a,c), a bright
Electronic address: firstname.lastname@example.org
1LIRGs are defined by 1011< LIR/L⊙< 1012and ULIRGs by
1012< LIR/L⊙< 1013, where LIRis the ‘total’ infrared luminosity
from 8 − 1000 µm.
quasar is left (Sanders et al. 1988).
sis is under scrutiny; comparison to the fundamental
plane of ellipticals shows that many ULIRGs are evolv-
ing into moderate-mass ellipticals (Genzel et al. 2001;
Veilleux et al. 2002; Tacconi et al. 2002; Dasyra et al.
2006a,b), similar to but slightly less massive than the
hosts of optically bright quasars (Dasyra et al. 2007).
LIRGs host less intense starbursts than ULIRGs, and
the frequency of occurrence of optically-visible AGN in
LIRGs is much smaller (Veilleux et al. 1995). Photomet-
ric studies divide LIRGs into two groups (Ishida 2004).
The most luminous are early in the merger sequence of
two roughly equal-mass galaxies, and thus may be the
progenitors of ULIRGs (Arribas et al. 2004; Ishida 2004).
Other LIRGs are unequal-mass mergers or isolated disk
galaxies which may or may not be experiencing an inter-
action (Ishida 2004). Kinematic studies confirm that the
pair mass ratios in at least some LIRG interactions are
of order 1 − 3 (Rothberg & Joseph 2006; Dasyra et al.
The interstellar medium in LIRGs and ULIRGs is
in a kinematically extreme state, dominated by inflows
(e.g.,Barnes & Hernquist
1996), outflows (Heckman et al. 2000; Rupke et al.
2002, 2005a,b,c; Martin 2005, 2006; see Veilleux et al.
2005 for a recent review), and turbulent motions
(Downes & Solomon 1998).
the power to significantly alter the chemical states
of theprogenitor galaxies
K¨ oppen & Edmunds 1999; Dalcanton 2007). Ongoing,
1996;Mihos & Hernquist
These gas motions have
2 Rupke, Veilleux, & Baker
intense star formation in LIRGs and ULIRGs is also
producing and redistributing heavy metals at a prodi-
gious rate. In this paper we describe the first compre-
hensive study of the oxygen abundances of local LIRGs
and ULIRGs (§§2 − 3).This will allow us to assess
the effects of these processes on the gas-phase abun-
dances of infrared-selected, interacting galaxies. Studies
of optically-selected mergers suggest that gas motions do
alter nuclear abundances (Kewley et al. 2006a). Here we
present evidence that this is true also of strong mergers
with high star formation rates.
In order to understand the chemical evolution of LIRGs
and ULIRGs, it is crucial to compare these sources to
weakly- or non-interacting galaxies with modest star for-
mation, which represent the progenitors of LIRGs and
ULIRGs. To this end, we compare LIRGs and ULIRGs to
published luminosity-metallicity, mass-metallicity, and
mass-effective yield relations (§§4−6). We discuss these
results in §7.
There exist a few measurements of oxygen abundances
in high-redshift infrared-luminous galaxies. Liang et al.
(2004) measure the abundances of ∼20 z = 0.4 − 0.9
LIRGs selected at 15 µm by the Infrared Space Obser-
vatory (ISO). Abundance measurements also exist for a
handful of z ∼ 2 submillimeter-selected galaxies that
have total infrared luminosities greater than or equal
to those of ULIRGs (Tecza et al. 2004; Swinbank et al.
2004). Finally, we describe in this paper a handful of
new moderate-redshift measurements.
While these high-z measurements are valuable on their
own, they are also sparse and, in many cases, uncer-
tain. The more robust measurements of local LIRGs and
ULIRGs that we report here facilitate two comparisons:
we can now compare the enrichment histories of infrared-
luminous and infrared-faint galaxies in the local universe
(§§ 4.2, 4.3, and 5); and we can look for evidence of chem-
ical evolution among infrared-luminous populations as a
function of redshift (§4.4).
We summarize our work and discuss its consequences
Throughout the paper, we use the notation (O/H) to
refer to the ratio of the number densities of O and H
atoms in the ISM. The variable Z refers instead to the
mass fraction of O relative to the total mass of gas. These
two variables are related by a constant: Z = 16/C(O/H),
where C ∼ 1.4 is the ratio of total to H gas masses.
Where appropriate, we use (O/H) and Z interchange-
ably. For the solar oxygen abundance, we use the recent
value from Asplund et al. (2004): 12+log(O/H)⊙= 8.66.
For all calculations, we assume the standard cosmology
of H0= 75 km s−1Mpc−1, Ωm= 0.3, and ΩΛ= 0.7.
2. SAMPLE SELECTION
Abundance diagnostics of star-forming galaxies rely
mostly on emission lines. The emission lines of many
LIRGs and ULIRGs, however, may include contributions
from an AGN and/or strong shocks (Kim et al. 1998;
Veilleux et al. 1999). Abundance diagnostics are gener-
ally calibrated using galaxies with modest star formation,
whose emission line fluxes do not contain strong contribu-
tions from either of these ionization mechanisms. Thus,
choosing galaxies whose lines are starburst-dominated is
important for computing accurate abundances.
This decision is complicated by the multiple options
for defining a star-forming galaxy in the phase space
of optical emission line flux ratios. For instance, using
the classic diagnostics of Veilleux & Osterbrock (1987),
60% of LIRGs and one-third of all ULIRGs are ‘H II-
region-like’ galaxies (Veilleux et al. 1995, 1999). More
recent work updates this empirical classification scheme
using ∼105galaxies from the Sloan Digital Sky Sur-
vey (SDSS; Kauffmann et al. 2003; Kewley et al. 2006b).
Many LIRGs and ULIRGs classified as H II galaxiesusing
the Veilleux & Osterbrock (1987) scheme lie away from
the bulk of local star-forming galaxies; instead, they lie
in the region of the [O III]λ5007/Hβ vs. [N II]λ6583/Hα
flux ratio diagram that is between the outer boundary of
the locus of SDSS galaxies (Kauffmann et al. 2003) and
the line delineating the maximum line ratios achievable
by starbursts, according to theory (Kewley et al. 2001).
In §3.3, we discuss in detail the abundance uncertain-
ties that arise from such physical effects. For now, we
adopt loose initial selection criteria. We include in our
sample all galaxies classified as H II galaxies or low ion-
ization nuclear emission-line regions (LINERs) under any
scheme. We also require that line flux uncertainties be
smaller than 50%; almost all fluxes are far more certain
than this, but a few fall near the limit. In Figure 1,
we place the galaxies in the current sample on several
line-ratio diagrams, with various classification schemes
Our ULIRGs are primarily from the 1 Jy sample,
which is a complete, flux-limited, northern-hemisphere
sample drawn from the Infrared Astronomical Satellite
(IRAS) database (Kim & Sanders 1998).
that our galaxies have measured [O II]λλ3727, 3729
fluxes.The moderately-high-resolution spectra of
Rupke et al. (2002, 2005b), from Keck or the MMT,
have broad enough wavelength coverage for this pur-
pose.We supplement this data set with a handful
of spectra of 1 Jy objects from the Fifth Data Re-
lease of the SDSS (York et al. 2000; Strauss et al. 2002;
Adelman-McCarthy et al. 2007). To improve our statis-
tics, we also include a few ULIRGs with published [O II]
fluxes from the Revised Bright Galaxy Sample (RBGS),
the Warm Galaxy Survey (WGS), and the 2 Jy survey
(Kim et al. 1995; Wu et al. 1998). The final local sam-
ple of 31 galaxy nuclei has ?z? = 0.14, with a maximum
redshift of 0.27. However, in addition we include mea-
surements for a few galaxies with z = 0.4− 0.5 from the
FIRST-FSC catalog (Stanford et al. 2000; Rupke et al.
2005b) for the purpose of assessing redshift evolution
We have selected our LIRGs primarily from the
Revised Bright Galaxy Sample,
complete,flux-limited IRAS sample (Sanders et al.
2003).Our RBGS data come from a variety of
spectroscopic surveys:(1) Kim et al. (1995);
Liu & Kennicutt (1995); (3) Wu et al. (1998);
Rupke et al. (2002, 2005c); (5) the Fourth Data Re-
lease of the SDSS (Adelman-McCarthy et al. 2006); and
(6) Moustakas & Kennicutt (2006, nuclear spectra only).
Again, to improve our statistics, we include published
fluxes of galaxies from the Warm Galaxy Survey (WGS)
and 2 Jy sample (Kim et al. 1995; Wu et al. 1998). The
sample of 65 galaxy nuclei has ?z? = 0.04. (We also have
measurements for one LIRG with z = 0.48, which we use
which is also a
Abundances of Luminous Infrared Galaxies3
Fig. 1.— Emission-line ratio diagrams, with fluxes corrected for extinction. Blue stars are luminous infrared galaxies and red circles
are ultraluminous infrared galaxies. The solid lines separate star-forming galaxies from LINERs (Veilleux & Osterbrock 1987); the dotted
lines denote the phase space limits of the Kewley et al. (2001) starburst models in all plots except the bottom right, where they separate
starbursts, LINERs, and AGN (Kewley et al. 2006b); and the dashed line denotes the outer boundary of the bulk of star-forming galaxies
in the SDSS (Kauffmann et al. 2003).
The selected galaxies are representative of the local
(z ? 0.2), star-forming, infrared-luminous galaxy pop-
ulation. The 1 Jy sample, 2 Jy sample, RBGS, WGS
are complete samples. The particular galaxies which
appear in this paper are effectively a random selection
from these samples, since they are culled from their par-
ent spectroscopic studies only on the basis of spectral
type and signal-to-noise ratio. The parent studies from
which our spectroscopic data were collated impose vari-
ous constraints on sky location, emission-line sensitivity,
and (to a minor degree) redshift. However, these have
no effect on the physical properties of the galaxies cho-
sen, as shown by comparisons among Figure 1 and simi-
lar figures from the parent studies (e.g., Kim et al. 1995;
Veilleux et al. 1999).
Some of the galaxies in our sample are multiple-nucleus
systems2. For the 1 Jy ULIRGs, the nuclei are charac-
terized using sensitive optical and near-infrared imag-
ing (Kim et al. 2002; Veilleux et al. 2002).
sources, we specify nuclei using unique designations from
the NASA/IPAC Extragalactic Database (NED; see Ta-
ble 1). We treat each nucleus separately in our analysis,
since the apertures for our spectra are nuclear or near-
nuclear (§3.3).Accordingly, for each multiple-nucleus
system we divide the total system infrared luminosity be-
tween component nuclei based on resolved IRAS imaging
(Surace et al. 2004) or the near-infrared luminosities of
the component galaxies (§4.1). The fraction of the to-
tal infrared luminosity of the system belonging to each
nucleus (or equivalently fractional near-infrared luminos-
2For some multiple-nucleus systems, only one nucleus enters our
sample. This is due to a lack of data or because the other nucleus
has a Seyfert optical spectral type.
4Rupke, Veilleux, & Baker
ity) for those systems where the NIR luminosity is used
to divide LIR is listed in Table 1. Seven ULIRG nu-
clei descend into the LIRG category due to their nuclear
luminosity. This reclassification does not impact our re-
In total, there are 100 galaxies or nuclei in our sam-
ple. Table 1 lists the basic properties of each galaxy or
The strong emission lines in the Rupke et al. (2002,
2005b) and SDSS flux-calibrated spectra were measured
using the IRAF task SPLOT. The average measurement
uncertainty in the brightest lines (e.g., [O II] or [N II]) is
5% or less. For weak or noisy lines ([O I], [O III], or [S II]
in a few cases) or those affected by a continuum that
has strong stellar absorption, the uncertainty rises to ∼
20 − 30%. Table 2 lists the line fluxes newly measured
for this study.
Our Keck, MMT, and SDSS spectra are typically of
high enough resolution for us to fit simultaneously a
Voigt absorption and Gaussian emission component to
Hβ. In the few cases where this was not possible, we used
lower order Balmer lines to estimate the expected absorp-
tion in Hβ. We corrected the Hα emission line in these
galaxies for stellar absorption by extrapolating from the
absorption equivalent widths of lower order Balmer lines,
using the relative values for different lines predicted by
the oscillator strengths (Menzel 1969):
In this equation, HNis the Balmer transition from which
the equivalent width of Hα is to be calculated; Hirepre-
sents the Balmer transition with upper principal quan-
tum number i (e.g., H4 = Hβ); and f(Hi) is its oscil-
lator strength. Patris et al. (2003) tabulate the coeffi-
cient values that we used. Table 2 lists the (absorption-
uncorrected) Hα emission line and Hβ absorption line
equivalent widths for newly measured data.
The emission lines were corrected for extinction using
the Balmer decrement. We assume an intrinsic Hα/Hβ
flux ratio of 2.86 (Hummer & Storey 1987), an effective
foreground screen, and the starburst attenuation curve of
Calzetti et al. (2000). Table 2 lists the E(B−V ) values
for newly measured data.
The data quality was checked through cross-correlation
of objects common to different data sets. The results
showed remarkable consistency, given the number of ref-
erences from which the data were drawn. The typical dis-
crepancy between two measurements of the same galaxy
nucleus is of the order ∼0.1 dex. Where multiple data
were available for a given source, we chose the spectrum
with highest spectral resolution and signal-to-noise. Ex-
ceptions are sources not drawn from the 1 Jy sample or
RBGS that have both published and SDSS spectra; for
consistency, we chose the published data even though the
SDSS data may be of higher spectral resolution or sen-
sitivity. We decided to use the SDSS data only to sup-
plement the number of spectra available from the 1 Jy
sample and RBGS (or for data quality checking) in order
to keep the sample selection fairly clean and the size of
the sample manageable. (I.e., there are a large number
of infrared-luminous galaxies in the SDSS that we did
not include; e.g., Pasquali et al. 2005.)
3.1. Comparison of Diagnostics
Numerous strong-line abundance diagnostics exist,
each relying on different combinations of the strongest
emission lines measurable in optical spectra. Each diag-
nostic in turn has different absolute calibrations, based
on photoionization models, electron temperature (Te)
measurements (i.e., weak-line diagnostics), or a combi-
nation of the two. Different diagnostics, or different cali-
brations of the same diagnostic, can give vastly different
abundances for the same galaxy or group of galaxies. For
instance, models calibrated on measurements of electron
temperatures of H II regions in nearby galaxies differ
from theoretical calibrations by factors of a few (see, e.g.,
Kennicutt et al. 2003, and references therein).
To help the reader understand the uncertainty that at-
taches to the choice of a particular diagnostic/calibration
combination, we have investigated several different op-
tions. One of these (the Tremonti et al. 2004 calibration)
we employ in §§ 4−6 to compare our data with published
luminosity-metallicity (L−Z), mass-metallicity (M−Z),
and mass-effective yield relations. The others provide
a useful baseline comparison and illustrate some of the
uncertainties due to choice of calibration and physical
1. Pilyugin & Thuan
tion of O32 ≡ f([O III]λλ4959, 5007)/f([O II]).
The latter is a proxy for ionization parameter,
which is the ratio of ionizing photons to hydrogen
nuclei present in gas.
5007)}/f(Hβ) and a func-
2. The photoionization models of McGaugh (1991)
also use both R23 and O32 as parameters in the
diagnostic. The calibration is updated and printed
in analytic form by Kuzio de Naray et al. (2004);
we use their semi-empirical version.
3. Tremonti et al. (2004, hereafter T04) use the mod-
els of Charlot & Longhetti (2001, hereafter CL01)
to compute the abundances of ∼105galaxies from
the SDSS. The original models are a suite of ana-
lytic functions involving different combinations of
strong emission lines; the choice of diagnostic then
depends on the spectral data available. However,
T04 directly cross-correlate their data with model
spectra to find the best-fit abundance. They then
fit a simple analytic function to the upper branch
of the abundance vs. R23relation.
4. For comparison with the T04 R23 calibration, we
include the original CL01 suite of analytic func-
tions. We compute abundances using the diagnos-
tics of Cases A through F, where successive diag-
nostic cases rely on less spectral information than
the previous one. For simplicity, we only discuss
Cases A and F to represent the range of possibil-
ities in spectral information. The former relies on
[N II]λ6583/[S II]λλ6717, 6731 and the latter on
[O III]λ5007/Hβ (with O32 as a weak secondary
parameter in each case). We note that the CL01 di-
agnostics use observed fluxes as inputs, unlike most
Abundances of Luminous Infrared Galaxies5
Fig. 2.— Abundances in six different diagnostic/calibration systems. (See §3.1 for a description of each system.) We select a random
sample of LIRGs (left) and ULIRGs (right). Colored lines connect different calibrations for the same galaxy. The heavy open circles and
error bars reflect the median and standard error over the entire sample in each system. We here consider only galaxies that pass the second
emission-line cut (§3.3).
other diagnostics which use fluxes corrected for at-
tenuation by dust.
5. Kewley & Dopita (2002) attempt an optimal cali-
bration by comparing their photoionization models
to previous work and combining several diagnos-
tics. For our galaxies, their ‘combined’ diagnostic
reduces to the [N II]λ6583/[O II]λλ3727, 3729 di-
agnostic in almost all cases.
In each case, we discard galaxies that have log(R23) ≥
1; such values are observed in star-forming regions, but
they are not common (e.g., Pilyugin & Thuan 2005). In
this regime, it is unclear which R23branch is appropriate
and the diagnostics are not well-calibrated. Furthermore,
a very high R23 may be an indication of contributions
to the line emission from processes not related to star
formation. There are nine LIRGs and nine ULIRGs with
We also assume the upper branch for R23-based
diagnostics (Edmunds & Pagel 1984).
[N II]/[O II] flux ratio as a branch indicator, where all
galaxies with f([N II])/f([O II]) > −1 are assumed to lie
on the upper branch (e.g., Kuzio de Naray et al. 2004;
see also the theoretical plots of [N II]/[O II] and R23
vs. abundance in Kewley & Dopita 2002). Eight galax-
ies (five LIRGs and three ULIRGs) fail this test. Seven
of these also surpass the R23 threshold; we exclude the
remaining LIRG from our analysis, to avoid the problem
of stitching together upper and lower branch calibrations
(which exists when two different diagnostics are put to-
gether; e.g., Salzer et al. 2005). The number of galaxies
with low values of [N II]/[O II] is interesting, and may
We use the
suggest an even larger downward spread in abundance
than is found by taking into account only the upper
These two cuts do not strongly impact our analysis.
Once we put aside galaxies with z > 0.27, we have a
working sample of 55 LIRGs and 22 ULIRGs. (We dis-
cuss further the z ∼ 0.4 − 0.5 points in §4.4.)
Different diagnostic/calibration pairs can yield very
different abundances for the same galaxy. Figure 2 shows
the abundance of a random sampling of our galaxies us-
ing each method listed above. The methods are ordered
on the horizontal axis by increasing median abundance.
Over the full sample, the extreme median values (the
CL01 Case A and Pilyugin & Thuan (2005) calibrations)
differ by a factor of 8 for the LIRGs, and correspond to
12 + log(O/H) = 8.2 and 9.1, respectively. The ‘median
of the medians’ is 1.25Z⊙. For the ULIRGs, the extreme
values are 8.1 and 9.0, also a spread of a factor of 8,
with the median being roughly 1.0Z⊙. However, if the
Pilyugin & Thuan (2005) diagnostic (the only one in our
sample based solely on Te abundances) is removed, the
scatter is dramatically reduced. The peak-to-peak varia-
tion among the five remaining diagnostics is only a factor
of 2. The median abundances among these five are 1.6Z⊙
and 1.4Z⊙for the LIRGs and ULIRGs, respectively.
In the next subsections we discuss sources of uncer-
3.2. Uncertainties in Abundance Caused by Choice of
The lowest abundances arise from the use of the
empirically-calibrated diagnostics. This exemplifies the