**2.8. Products of Stellar Evolution**

It is in principle straightforward to compute the integrated outputs of stellar evolution - energy, neutrinos, helium, and heavy elements - given models for the IMF, the star formation history, and stellar evolution. Since the details of the results of stellar evolution computations are not easily assembled, we use approximate estimates by procedures similar to those developed in Section 2.3.1 and 2.3.2 for the stellar population and its evolution. The results add to the checks of consistency of our estimates of the stellar production of helium and heavy elements and the resulting total energy release, and are used to estimate the inventory entries for the neutrino cosmic energy density.

All stars with masses *m* < 1
*m*_{} are
still on the main sequence.
We assume that on average 5% of the hydrogen in these subsolar stars
has been consumed, with energy production efficiency 0.0071. Most of
the stars with masses *m* > 1
*m*_{} have
already undergone full evolution and left compact remnants, while the
fraction ~ (*m* / *m*_{})^{-2.5} is still on the main sequence. We
assume that in the latter stars on average 5% of the hydrogen has been
consumed, as for subsolar stars. For the evolved stars we do not
attempt to follow the details of nuclear burning and mass
loss. Instead, we adopt estimates of the nuclear fuel consumed or mass
lost in a few discrete stages of evolution, in a similar fashion to the
approach used in Section 2.3.1 to
tally stellar remnants.

When the amount of hydrogen consumed in stellar burning is 10% of the
mass of a star it leaves the main sequence. In the model in
Section 2.3.1, stars with masses in
the range 1 < *m* <
8*m*_{}
eventually produce white dwarfs that mainly consist
of a carbon-oxygen core. In standard stellar evolution models, hydrogen
burning extends outward to a shell after core hydrogen exhaustion, and
helium burning similarly continues in a shell after core helium
exhaustion. That leaves a CO core with the mass given
by equation (69). The helium layer outside the CO core is
thin for *m* < 2.2
*m*_{}, for
which helium ignition takes place as a flash, but in stars with masses
*m* > 2.2
*m*_{} a
significant amount of helium is produced outside the
core, transported by convection to the envelope, and liberated. A
5 *m*_{} star
liberates
0.4 *m*_{} of
helium and produces a core with mass
0.85 *m*_{}
(e.g.,
Kippenhahn, Thomas &
Weigert 1965).
We model the helium production as a
function of the initial star mass *m* as

(114) |

in solar mass units. The energy production in hydrogen shell burning is the product of this mass with the post-stellar hydrogen mass fraction, 0.71 (eq. [85]), and the efficiency factor, 0.0071. Helium burning in the core produces energy with efficiency factor 0.0010.

For stars with masses *m* >
8*m*_{} we
adopt the helium yield from Table 14.6 of
Arnett (1996),
^{9} which we parametrise as

(115) |

This connects to equation (114) at
8.7 *m*_{}.
We take the CO core mass as a function of the initial stellar mass from
Arnett (1996):

(116) |

We use an interpolation of eqs. [69] and
[116] for stellar masses between 8 and
13*m*_{}.
The energy release is 0.0071 × 0.71
per unit mass for He production, and 0.0014 for CO core formation
and the further heavy element production.

The energy output obtained by integration over the IMF and PDMF is

(117) |

The partition into each phase of stellar evolution and stellar mass range is given in Table 5, where the numbers are normalised to equation (117). About 60% of the energy is produced in the evolved stages.

stage of stellar evolution | 0.08 - 1
m_{} |
1 - 8
m_{} |
8 - 100
m_{} |
sum |

main sequence | 0.11 | 0.20 | 0.12 | 0.43 |

H shell burning | 0.18 | 0.29 | 0.48 | |

core evolution | 0.05 | 0.04 | 0.09 | |

sum | 0.11 | 0.43 | 0.46 | 1.00 |

^{a} Normalised to
= 5.3 ×
10^{-6}. |

The estimate of the total energy generation in
equation (117) is in satisfactory agreement with our
estimate of the nuclear binding energy,
_{BE} = 5.7
× 10^{-6} (eq. [97]),
and the energy production required to produce our estimate of the
present radiation energy density,
_{ + } = (5.1± 1.5) ×
10^{-6} (eq. [106] corrected for neutrino
emission, as discussed in
Section 2.8.2).
That is, our models for stellar formation and evolution are
consistent with our estimates of the accumulation of heavy elements and
the stellar background radiation.

In our models for stellar formation and evolution the mass of helium that has been liberated to interstellar space is

(118) |

This may be compared to our estimate from the production of heavy
elements and the associated production of helium,
_{Y} =
1.7± 1 × 10^{-4} (eq. [94]).
The heavy element production is calculated in a
similar way. The current wisdom is that a neutron star or
black hole is left at the end of the evolution of a
star with mass *m* > 8
*m*_{}, and
the rest of the mass is returned to the interstellar medium.
On subtracting the remnant mass (indicated in Table 2) from the total
heavy element mass produced, we find that in our model the heavy
element production is

(119) |

The more direct estimate in equation (92) is
_{Z}'
8 × 10^{-5}.

We suspect the checks on helium
and heavy element production in equations (118)
and (119) are as successful as could be expected. In
particular, equation (116) for the CO core mass is not
tightly constrained. One must consider also the possibility that some
supernovae do not produce compact remnants, as is suggested by the
cases of Cas A and SN1987A. The maximum heavy element production
when there are no remnant neutron stars or black holes is three times
the value in equation (119), and larger than
_{Z}'
(eq. [92]).

**2.8.2. Neutrinos from Stellar Evolution**

We need the fraction
*f*_{} of the
energy released as neutrinos by the
various processes of nuclear burning. In stars with masses *m* < 1.4
*m*_{},
energy generation in hydrogen burning is dominated by
the slow reaction
*p* + *p*
*d* + *e*^{+} +
_{e}. This produces
neutrinos with mean energy 0.265 MeV, which amounts to
*f*_{} = 0.020
times the energy generated
in helium synthesis. In a Solar mass star electron capture of
^{7}Be adds 0.005 to the fraction
*f*_{}. The
neutrino energy emission fraction is larger in higher
mass (*m* > 1.4
*m*_{})
main-sequence stars in which the CNO cycle
dominates. In this process, the neutrinos produced in the
^{+}
decays of ^{13}N and ^{15}O carry away the energy
fraction *f*_{} =
0.064. The fraction increases to 0.075 for
*m*
2*m*_{}, when
the subchain
^{15}N-^{16}O-^{17}F-^{17}O-^{14}N
starts dominating and neutrinos are produced by the
beta decay of ^{17}F. This sidechain also dominates during shell
burning.

The integral of these factors over the stellar IMF
normalised to the present-day mass density
_{star} =
0.0027 (eq. [27]), together with our
prescription for energy generation in Table 5,
yields the neutrino energy production,

(120) |

The temperatures and densities that are reached up to carbon burning are low enough that there is negligible neutrino energy loss from neutrino pair production processes.

The temperatures after carbon burning are high enough that the
neutrino energy loss dominates, that is,
*f*_{}
1
(Weaver et al. 1978).
Thus we may take it that the extra binding energy
(BE)
of the elements heavier than
^{20}Ne with respect to the binding energy of carbon represents the
neutrino energy emitted in the late stages
of stellar evolution. On multiplying the heavy element mass
(the sum of entries 1, 2 and 4 to 9 in
Table 3) by
(*Z*_{i} /
*Z*) (*BE*)_{i} (where
<(*BE*)>
= 0.0004 for the solar mix of element abundances, and
*Z*_{i} /
*Z* = 0.35 is from
Grevesse & Sauval
(2000),
we obtain

(121) |

The sum of equations (120) and (121) is 7% of the total energy production (eq. [117]). The present energy density of the stellar neutrinos, in entry 8.1 in Table 1, is the product of this sum with the redshift loss factor ~ 0.5 (eq. [35]).

**2.8.3. White Dwarfs and Neutron Stars**

Most of the gravitational energy liberated in white dwarf formation goes to neutrinos, and in the latest stage to X-rays. Since the latter is small the contribution to the neutrino energy density (entry 8.2 in the inventory) is the product of the gravitational binding energy in entry 5.3 with the redshift factor.

More than 99% of the energy released in core collapse also is carried away by neutrinos. Thus we similarly obtain entry 8.3 by multiplying entry 5.4 by the redshift factor.

^{9} This
helium yield is significantly larger than Arnett's helium
core mass, *m*_{He} = 0.43*m* - 2.46.
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