**2.3. The Baryon Rest Mass Budget**

The entries in this category refer to the baryon rest mass: one must add the negative binding energies to get the present mass density in baryons. The binding energies are small and the distinction purely formal at the accuracy we can hope for in cosmology, of course, with the conceivable exception of the baryons sequestered in black holes.

We begin with the best-characterized components, the stars, star remnants, and planets. We then consider the diffuse components, and conclude this subsection with discussions of the baryons in groups and the intergalactic medium and the lost baryons in black holes.

This is an update of the analysis in FHP.
Following the same methods, we estimate the baryon mass in stars from
the galaxy luminosity density and the stellar mass-to-light ratio,
*M*_{stars} / *L*, along with a stellar initial mass
function (the IMF) that allows us to
estimate the mass fractions in various forms of stars and star remnants.

The integrated luminosity densities from the SDSS broad-band galaxy luminosity function are (Yasuda et al. in preparation)

(21) |

where _{B} is inferred
from the densities in other color bands.

Kauffmann et al. (2003)
present an extensive analysis of the stellar
mass-to-light ratio based on *ugriz* photometry for
10^{5} SDSS galaxies and a population synthesis model
(Bruzual & Charlot
2003)
that is meant to take account of the stellar
metallicities and the star formation histories.
Their estimate of the stellar mass-to-light ratio
is *M*_{stars} / *L*_{z}
1.85 for luminous
galaxies, with magnitudes
*M*_{z} < *M*_{z}^{*} -
0.8, and it decreases gradually to
*M*_{stars} / *L*_{z}
0.65 for
fainter galaxies with
*M*_{z}
*M*_{z}^{*} + 3. Our estimate of the
resulting luminosity-function-weighted mean is

(22) |

for the IMF Kauffmann et al. used. Equation (22) represents the present mass in stars and stellar remnants, and it excludes the mass shed by evolving stars and returned to diffuse components.

The estimate of
*M*_{stars} / *L* assumes a universal IMF,
which is not thought to seriously violate
the observational constraints. The IMF is not tightly constrained, and
it is particularly uncertain at the subsollar masses that make little
contribution to the light but can make a considerable contribution to
the mass. We consider two continuous broken power law models, of the form

(23) |

where, in the first model,

(24) |

and, in the second model,

(25) |

The first line in the first model is from
Burgasser et al. (2003),
and the second line is from
Reid, Gizis, & Hawley
(2002).
The third line, for *m* > 1
*m*_{}, is
the standard
Salpeter (1955)
IMF. The second model is from
Kroupa (2001).
For our model IMF we take the Salpeter index for
*m*
1 *m*_{}. It
is known that the Salpeter-like slope gives the correct UBV
colour and H
equivalent widths for normal galaxies
(Kennicutt 1983).
For the subsolar IMF, we take the geometric
mean of the above two models, and we take the
difference as an indication of the error (± 18%).

The stellar mass-to-light ratio in equation (22)
assumes the IMF in equation (25). With our adopted IMF
the stellar mass-to-light ratio is 1.18 times
the number in equation (22).
^{4} Thus we get our fiducial
estimate,

(26) |

This translates to *M* / *L*_{B}
2.4, or 0.7 times that
used in FHP, which employed the subsolar mass IMF of
Gould, Bahcall &
Flynn (1996).
In the Salpeter IMF, with *x* = 1.35 cut off at 0.1
*m*_{},
the mass-to-light ratio is 1.48 times our adopted value. The
Kennicutt (1983)
IMF results in 0.81 times equation (26).

The IMF at substellar masses, *m* < 0.08
*m*_{}, is more
uncertain, but recent observations of T dwarfs indicate *x* < 0
(e.g.,
Burgasser et al. 2003).
In the two IMF models quoted above the substellar mass is
6 to 9% of the mass integral for 0.08 < *m* < 1
*m*_{}. We
adopt 8% and assign an error of 50%.

We estimate the mass density locked in stars, including those
in dead stars, to be
_{stars} =
0.0025 ± 0.0007.
A comparable estimate is derived from *b*_{J}, *J*,
*K*_{s} multicolour
photometry of 2MASS combined with 2dF data by
Cole et al. (2001),
_{stars} =
0.0029 ± 0.0004 with the
IMF we have adopted. For the energy inventory we
take the mean of our present number and that of Cole et al.:

(27) |

This means that the stars contain 6.0 ± 1.3% of the total baryons.
The FHP estimate is
_{stars} =
0.0019 - 0.0057. Equation (27) also is consistent within the
errors with the more recent estimates by
Salucci & Persic
(1999),
Kochanek et
al. (2001), and
Glazebrook et
al. (2003),
and with
Shull's (2003)
baryon inventory.

We attempt to partition the stars into their species.
Our estimates of the mass fraction in stars on the main sequence
(MS), and the mass fractions in stellar remnants, including white
dwarfs (WD), neutron stars (NS) and stellar mass black holes (BH), are
shown in Table 2. Stars on the MS are represented
by the present-day mass function (PDMF), which
for 1 < *m* < 100
*m*_{}
is constructed by multiplying the IMF by *m*^{-2.5} to
account for the lifetime of
massive main sequence stars; for *m* < 1
*m*_{} the
PDMF is the same as the IMF.

initial mass range | fate | remnant ^{a} |
mass fraction | mass consumed ^{b} |

0.01 < m < 0.08 |
SS | ... | 0.052 | 0.052 |

0.08 < m < 100 ^{c} |
MS | ... | 0.769 | 0.769 |

1 < m < 8 |
WD | 0.62 | 0.135 | 0.463 |

8 < m < 25 |
NS | 1.35 | 0.019 | 0.186 |

25 < m < 100 |
BH | 7.5 | 0.025 | 0.146 |

sum | 1.0 | 1.616 | ||

^{a} Mass in units of Solar
masses |
||||

^{b} The gas consumed to make
unit mass of stars now present. |
||||

^{c} For the mass range
1 < m < 100
m_{} those
stars burning today are counted. |

As indicated in the third line of Table 2, we
take the average mass of a white dwarf to be <*m*> = 0.62
*m*_{}, for
consistency with the model we describe in
section 2.5.2 (eq. [70]). This is
close to the value from recent observations, 0.604
*m*_{}
(Bergeron & Holberg, in preparation; Bergeron private
communication). We assume all stars with initial masses in the range
1 < *m* < 8
*m*_{} end up
as white dwarfs with the adopted average mass,
and the rest of the mass returns to the interstellar medium. With our
adopted IMF this model predicts that the ratio of masses
in white dwarfs to main sequence stars is
(*WD*) /
(*MS*) =
0.176. This can be compared to the observations. The 2dF survey
(Vennes et al. 2002),
with the use of our mean mass
0.62 *m*_{},
yields a measure of the mass density
of DA white dwarfs, (*DA*-*WD*, *local*) = (4.2 ± 2.3)
× 10^{-3}
*m*_{}
pc^{-3}.
This is multiplied by 1.3 to account for DB, DQ and DZ white dwarfs
(Harris et al. 2003),
to give the total white dwarf mass density
(*WD*,
*local*) = (5.5 ± 3.0) × 10^{-3}
*m*_{}
pc^{-3}. This divided by the local density of main
sequence stars,
(*MS*,
*local*) = 0.031 ± 0.002
*m*_{}
pc^{-3}, from
Reid, Gizis, & Hawley
(2002),
gives (*WD*) /
(*MS*) =
0.18 ± 0.10, which agrees with our model prediction, albeit with a
large uncertainty.

We assume all stars in the initial mass range

(28) |

end up as neutron stars with mass
1.35 *m*_{}
(Nice, Splaver, &
Stairs 2003;
Thorsett &
Chakrabarty 1999),
and the rest of the star mass is recycled. Estimates of
the lower critical mass for stellar core collapse vary from 6 to
10 *m*_{}
(Reimers & Koester
1982;
Nomoto 1984;
Chiosi, Bertelli, &
Bressan 1992).
The upper critical
mass for the formation of a neutron star is more uncertain; our choice
in equation (28) is taken from
Heger et al. (2003).
We assume all stars with main sequence mass above the limit in
equation (28) end up as stellar black hole remnants with mass
*m*_{} = 7.5
*m*_{}, with
the rest of the mass recycled.
This again follows Heger et al. The remnant mass is quite uncertain,
and Heger et al. also indicate that some stars with masses above the
range in equation (28) may produce neutron stars.

The last column in Table 2 is the total amount of mass that has gone into stars, including what is later shed, normalised to the mass in column (4). According to these estimates, 1.6 times the mass in observed stars was used (and reused) in star formation.

Entries 3.3 to 3.8 in Table 1 are
based on the partition of
_{stars}
among main sequence stars and star remnants in the fourth
column of Table 2, with the FHP partition between
the two galaxy population types in the proportion

(29) |

of mass in ellipticals plus bulges of spirals to the mass in disks plus
irregulars. The objects in the mass range 0.01 < *m* < 0.08
*m*_{} are
classified as `brown dwarfs', or substellar objects (SS), in
Tables 1 and 2.

**2.3.2. Consistency with the Star Formation
Rate**

We can compare our estimate of the mass in stars to the mass
integrated over the star formation rate density (the SFR) from
high redshift to the present. The
H luminosity density at zero
redshift measured from an SDSS nearby galaxy sample is
(*H*
) =
10^{39.310.11-0.08} *h* erg
s^{-1} Mpc^{-3}
(Nakamura et al. 2004).
This agrees with the earlier value obtained by
Gallego et al. (1995).
Within the star formation models
Glazebrook et al. (1999)
explored, 1
*m*_{}
yr^{-1} of star formation produces
H luminosity
*L*_{} =
(2.00^{+0.92}_{-0.24}) × 10^{41} erg
s^{-1} for our adopted IMF. The ratio of
(*H*
) to
*L*_{}
is an estimate of the present-day SFR,

(30) |

The evolution of the SFR at *z*
1 is now
observationally well determined
(Lilly et al. 1996;
Heavens et al. 2004;
see Glazebrook et
al. 1999
for a summary of the measurements up to that time), and may
be approximated as

(31) |

where

(32) |

and *t*_{o} - *t* is the time measured back from the
present in the
CDM cosmology
(Fukugita & Kawasaki
2003;
see also
Glazebrook et al. 1999).
The situation at higher redshift is still controversial
(Madau et al. 1996;
Steidel et al. 1999).
We shall suppose that the SFR is
constant from *z* = 0.85 back to the start of star formation at
redshift *z*_{f}, at the value

(33) |

In this model the integrated star formation is

(34) |

There is a significant downward uncertainty,
_{stars} ~ -
0.0025, arising from the uncertainty in the star formation rate (eq. [30]).
Equation (27) and Table 2
indicate that the mass processed into stars, including what was later
shed, is
= 0.0027 ×
1.62 = 0.0044. This is consistent
with the cumulative star formation in equation (34).
There would be a problem with the numbers if there were reason to
believe that the SFR continued to increase with increasing redshift well
beyond *z* = 1. A constant or declining SFR at *z* > 1 is
well accommodated with our estimates for the baryon budget and current
ideas about the star formation history at high redshift.

The star formation history determines the average reciprocal redshift factor (as in eq. [100]) for the effect of redshift on the integrated comoving energy density of electromagnetic radiation (and other forms of relativistic mass) generated by stars. In our model for the SFR the correction factor is

(35) |

where (*t*) is the
bolometric luminosity density per comoving
volume, which we are assuming is proportional to the star formation rate
density. The numerical value assumes the redshift cutoff is in the
range *z*_{f} = 2 to 5. For the purpose of comparison of
the accumulation of stellar products to the present rate of production,
another useful quantity represents the integrated comoving density of
energy radiated in terms of the effective time span
normalised by the present-day luminosity density,

(36) |

The value assume the range of models for the star formation history in equation (34).

We can compare the rate of stellar core collapse in our model to
observations of the supernova rate. (See
Fukugita & Kawasaki
2003 and
Madau, della Valle &
Panagia 1998
for similar analyses). The present SFR in
equation (30) and the critical minimum mass
8 *m*_{} in
equation (28) imply that the present rate of formation of neutron stars
and stellar black holes is expected to be

(37) |

Our estimate of the observed supernova rate is

(38) |

This is the geometrical mean of the rates from three surveys for
Type II and Type Ib/c supernovae, 0.037, 0.018 and 0.017, in units of
*h*^{3}(100 yr)^{-1} Mpc^{-3}
(Tammann, Löffler
& Schröder 1994;
Cappellaro et al. 1997;
van den Bergh &
McClure 1994;
see also
Cappellaro, Evans &
Turatto 1999).
We conclude that, if most stars with
initial masses greater than about
8 *m*_{}
produced Type II and Ib/c supernovae,
then our model for the star formation history would pass this
consistency check.

We remark that in our model the comoving number density of supernovae of types II and Ib/c integrated back to the start of star formation is

(39) |

where the effective time span is given in equation (36).

**2.3.3. Planets and Condensed Matter**

The mass in planets that are gravitationally bound to stars must be
small, but
it is of particular interest to us as residents of a planet. Marcy
(private communication; see also
Marcy & Butler
2000)
finds that about 6.5% of nearby FGKM stars have
detected Jovian-like planets, and that an
extrapolation to planets at larger orbital radii might be expected to
roughly double this number. In our model for the PDMF the ratio of the number density of stars in the mass range 0.08 to
1.6 *m*_{} to
the mass density in stars is
*n* / =
2.1 *m*_{}^{-1}. The product of this quantity with the
mass density in stars (eq. [27]), the fraction 0.13, and the ratio of
the mass of Jupiter to the Solar mass is

(40) |

Marcy indicates that stars with lower metallicity have fewer planets, but that may not introduce a serious error because there are fewer low metallicity stars. There is a population of interstellar planets that have escaped from stars that have shed considerable mass. The mass fraction in escaped planets might be expected to be comparable to the mass fraction of stars that have exploded, and hence subdominant to equation (40). The large uncertainty is whether the stellar neighborhood a fair sample. This leads us to enter the order of magnitude in Table 1.

We consider separately the mass in objects small enough to be held
together by molecular binding energy rather than gravity. The dominant
amount of the former is interstellar dust. It is known
(Draine 2003)
that 90%
of silicon atoms in interstellar matter are condensed into grains,
probably dominantly in the forms of enstatite (MgSiO_{3}) and
forsterite (Mg_{2}SiO_{4}) with the former likely three
to four times more abundant
(Molster et al. 2002).
This means the mass in this form
is *Z*(*silicates*) / *Z* = 0.17 times the mass in heavy
elements.
Draine (2003)
suspects that the dominant contribution to the
carbonaceous material is in the form of polycyclic aromatic
hydrocarbon with the inferred abundance C/H = 6 × 10^{-5}.
Adding this to the silicates, we find that the mass fraction becomes
*Z*(*dust*) / *Z* = 0.20. The formation of dust with iron or
other forms of carbonates could increase this number. Entry 3.12 is the
product of the density parameter in cool gas (entries 3.9 plus 3.10)
with the mean metallicity discussed below (and displayed as
eq. [86]) and the 20 percent heavy element mass
fraction in dust. Our estimate, which assumes that what we know about
the Milky Way applies to other galaxies, is crude, but it seems likely
that the mass in dust exceeds the mass in planets.

There are larger objects in which molecular binding is important. The
gravitational binding energy of a roughly spherical object with mass
*m* and radius *r* is
~ *Gm*^{2} / *r*, and the molecular binding energy
is roughly 1 eV per atom. The molecular binding energy is larger than the
gravitational energy when

(41) |

where the mass density is
and the mean
atomic weight is . For
silicates, this bound is *m* = 8 × 10^{26} g. That is,
gravitational binding energy dominates in the Earth and molecular
binding energy dominates in the Moon and asteroids.
Trujillo, Jewitt, &
Luu (2001)
estimate that the mass in the Kuiper Belt objects is
about a tenth of an Earth mass, or 10^{-6.5}
*m*_{}. A
comparable mass is in the moons in the Solar system. If this mass
fraction were common to all stars the density parameter in these objects
would be

(42) |

a small fraction of the mass in dust.

The recent blind HI surveys are a significant advance over the data used by FHP to estimate the mass density in neutral atomic gas. The largest survey, HIPASS, with 1000 galaxies (Zwaan et al. 2003), gives

(43) |

The increase over the value quoted in FHP (1.5 times the upper end value of FHP) illustrates the advantage of blind surveys over observations of programmed galaxies. The molecular hydrogen abundance from the CO survey of Keres, Yun & Young (2003) is

(44) |

The sum of these two values is multiplied by 1.38 to accommodate helium.

We place in entry 3.9 the atomic hydrogen and the helium abundance belonging to atomic and molecular hydrogen. Entry 3.10 is the molecular hydrogen component. The mass in this neutral gas is 1.7± 0.4% of the total baryon mass.

The estimate of the plasma mass in rich clusters of galaxies depends on
a convention for the cluster radii and masses. We use the mass
*M*_{200}
contained by the nominal virial radius, *r*_{200}.
This definition is more faithful to a physical definition of the part
of a cluster that is close to dynamical equilibrium, and it also traces
the X-ray radius which is the definition of our hot gas. In the
limiting isothermal sphere model the relation to the mass within the
Abell radius *r*_{A} = 1.5*h*^{-1} Mpc is

(45) |

This agrees with the estimates of the two masses given by Reiprich & Böhringer (2002), Table 4. We adopt the Reiprich & Böhringer cluster mass density parameter,

(46) |

for the cluster mass limit

(47) |

The Bahcall & Cen
(1993)
Abell mass function is consistent with the
Reiprich & Böhringer estimate. We caution that the
Bahcall et al. (2003)
SDSS mass function is about half what we adopt,
perhaps because SDSS samples a subset of the clusters.
Note also that the integral over the cluster mass within the Abell
radius gives a substantially larger value for
_{cl}
because *M*_{A} > *M*_{o}
at the masses
*M*
10^{14}
*m*_{} that
dominate the integral for a given mass function.

The abundance of hot baryons in clusters is obtained from the expression

(48) |

The change from the value in FHP is mainly due to the definition of the cluster mass, as also noted by Reiprich & Böhringer.

We follow the standard idea that the massive objects in the centers of galaxies are black holes that formed by the accretion of baryons. Baryons entering black holes are said to lose their identity, but for our accounting it is appropriate to consider them to be sequestered baryons.

We define the characteristic efficiency factor
_{n} for black
hole formation (where the subscript is meant to distinguish the massive
black holes in the centers of galaxies from stellar mass black holes)
by the black hole mass produced out of an initial diffuse baryon mass
*m*_{b},

(49) |

where the energy released in electromagnetic radiation, neutrinos, and kinetic energy is

(50) |

The baryon mass sequestered in massive black holes,
*m*_{b} = *m*_{bh} / (1 -
_{n}), could
be substantial if
_{n} were
close to unity. However, if an appreciable part of the
binding energy were released as electromagnetic radiation then the
bounds on the radiation background (in category 7) and
those of the CMBR distortion (eq. [14]) would require that
the energy was released at very high redshift, *kT* >
10^{7} K. It seems much more likely that
_{n} is
small, as is assumed in entry 3.13. The efficiency factor
_{s} typical
of stellar mass black holes does not appear in entry 3.7, because this
estimate is based on an
analysis of the progenitor star masses. The estimate of the mean mass
density in massive black holes that is used for entry 3.13 is discussed
in Section 2.5.3.

Entry 3.1, for the baryon mass outside galaxies and clusters of galaxies, is the difference between our adopted value of the baryon density parameter (eq. [16]) and the sum of all the other entries in category 3. Within standard pictures of structure formation this component could not be in a compact form such as planets, but rather must be a plasma, diffuse enough to be ionized by the intergalactic radiation or else shocked to a temperature high enough for collisional ionization, but not dense and hot enough to be a detectable X-ray source outside clusters and hot groups of galaxies.

In our baryon budget 90% of the baryons are in this intergalactic
plasma. It is observed in several states. Quasar
absorption lines show matter in low and high atomic ionization states
in the halos of
*L* ~ *L*_{*} galaxies, extending to radii ~
200 kpc
(Chen et al. 2001
and references therein). Absorption lines of HI and
MgII reveal low surface density photoionized plasma at kinetic
temperature *T* ~ 10^{4} K, which can be well away from
*L* ~ *L*_{*}
galaxies, as discussed by
Churchill, Vogt, &
Charlton (2003),
and Penton, Stocke &
Shull (2004).
The latter authors estimate that 30% of
the baryons are in this state. Absorption lines of O VI around galaxies
and groups of galaxies
(Tripp, Savage &
Jenkins 2000;
Sembach et al. 2003;
Shull, Tumlinson, &
Giroux 2003;
Richter et al. 2003),
reveal matter that may
be excited by photoionization by the X ray background radiation and by
collisions in plasma at the kinetic temperature
*T* ~ 10^{6} K
characteristic of the motion of matter around galaxies
(Cen et al. 2001).
Improvements in the constraints on the amount of
matter in these various states of intergalactic baryons will be
followed with interest.

For the inventory we adopt the measure of the concentration of dark matter around galaxies in equation (13), and the argument discussed in Section 2.3.8 that the baryons are distributed like the dark matter on scales comparable to the virial radii of galaxies. The resulting division into the mass in baryons near the virial radii of normal galaxies outside clusters (entry 3.1a), and the mass well away from galaxies and compact groups and clusters of galaxies (entry 3.1b), is presented as sub-components, because the sum is much better constrained than the individual values.

We comment here on a simple picture for the cooling and settling of
baryons onto galaxies.
The sum of the baryon mass densities belonging to galaxies, in entries
3.3 to 3.13, is
_{b, g} =
0.0035. This is 8% of the total baryon mass. Suppose
_{b, g}
consists of all
baryons gathered from radius *r*_{g} around
*L* ~ *L*_{*} galaxies,
and we can neglect the addition of baryons by settling from further
out and the loss by galactic winds. That is, we are supposing that at
*r* > *r*_{g} the ratio of the baryon density to
the dark matter density is the cosmic mean value, and that the baryons
closer in have collapsed onto the galaxies. In this picture the
characteristic radius of assembly of the baryons satisfies

(51) |

in the limiting isothermal sphere approximation (eq. [9]).

In equation (51) *n*_{g} is a measure of the number density
of luminous galaxies. We record here our choices for this quantity and
related parameters that are used elsewhere. We take

(52) |

where _{r} is the
luminosity density (eq. [21]). The characteristic galaxy luminosity,

(53) |

is the luminosity parameter in the Schechter function, with the power
law index
_{B} = - 1.1,
_{r} = - 1.13,
and _{z} = -
1.14. We refer some estimates of energy densities to what is known about
the Milky Way, for which we need the effective number density of Milky
Way galaxies. In the B band the Milky Way luminosity is
*L*_{MW} = 1.7 *L*_{B}^{*}
(based on the
Sakai et al. 2000
calibration of the B-band Tully-Fisher relation, with
*W*_{20} = 2 × 220 km s^{-1} for the Milky
Way), and the effective number density is

(54) |

Almost the same density follows when referred to the r-band.

With the characteristic velocity dispersion in equation (10) and the characteristic galaxy number density in equation (52) the baryon accretion radius defined by equation (51) is

(55) |

at density contrast 1.3 × 10^{4} and plasma
density *n*_{gas}(*r*_{g}) ~ 0.007
*h*^{3} cm^{-3}.
If plasma at this radius were supported by pressure at the
one-dimensional velocity dispersion
160 km s^{-1}
in equation (10) the temperature would be
*T* 2 ×
10^{6} K. At this
density and temperature the thermal bremsstrahlung cooling time would be
short enough, ~ 4 × 10^{9} yr, that stars would have formed
and disks matured at *z* ~ 1.

We have lower bounds on the cooling radius from the observation that
the neutral atomic hydrogen density around *L*_{*}
galaxies reaches
*N*_{HI} = 1.8 × 10^{20} cm^{-2} at the
effective radius ~ 20 *h*^{-1} kpc
(Bosma 1981),
and Mg II absorption lines are observed at radius
~ 40 *h*^{-1} kpc
(Steidel, Dickinson &
Persson 1994).
For the Milky Way the distribution of RR Lyr stars cuts off sharply at
50 kpc
(Ivezic et al. 2000).
The cooling radius must be larger than these indicators of relatively
cool matter. Equation (55) is not inconsistent with this condition.
Though the history of baryon accretion
by galaxies undoubtedly is complex, we can imagine, as a first
approximation, that a substantial fraction of the baryons now
concentrated in galaxies are there because they were able to cool
and settle from an initial distribution similar to that of the
dark matter.

The relative distributions of baryons and dark matter at distances much
larger than *r*_{g} from galaxies might not be greatly
disturbed from the
primeval condition. If so, then the product of the baryon density
parameter with the virialized dark matter mass fraction in
equation (13) is an estimate of the baryon mass
that resides within the virial radii of normal galaxies, and the remainder,

(56) |

which is presented as entry 3.1b, would be located outside galaxies and remain less than fully documented.

^{4} The
IMF are normalised so that the mass integrals between 0.9 and 2.0
*m*_{}
are equal. The result is virtually identical with those with the
two IMFs normalised at
1*m*_{}.
Back.