A&A 389, 702-715 (2002)
DOI: 10.1051/0004-6361:20020602
M. E. Gusakov
Ioffe Physical Technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia
Received 12 March 2002 / Accepted 16 April 2002
Abstract
We calculate and provide analytic fits of
the factors which describe the reduction of
the neutrino emissivity of modified Urca
and nucleon-nucleon bremsstrahlung processes
by superfluidity of neutrons and protons in neutron-star cores.
We consider ^{1}S_{0} pairing of protons and
either ^{1}S_{0}or ^{3}P_{2} pairing of neutrons. We analyze two types of
^{3}P_{2} pairing: the familiar pairing
with zero projection
of the total angular momentum of neutron pairs
onto quantization axis,
;
and the pairing
with
which leads to the gap with
nodes at the neutron Fermi surface.
Combining the new data with those available
in the literature we fully describe
neutrino emission by nucleons from neutron
star cores to be used
in simulations of cooling of superfluid neutron stars.
Key words: stars: neutron - dense matter
It is well-known (e.g., Yakovlev et al. 1999,2001) that cooling of neutron stars depends on the properties of matter in the neutron star cores. In spite of great progress in understanding the neutron-star physics, many properties of this matter are still known with large uncertainty. A comparison of the theoretical cooling models with observations of thermal emission from isolated neutron stars gives a potentially powerful method to explore the internal structure of neutron stars. For a successful modeling of the cooling one needs reliable values of neutrino emissivity in different neutrino reactions.
In this paper we consider the matter of neutron star cores (at densities g cm^{-3}) composed of neutrons (n), protons (p), and electrons (e). It is generally agreed the neutrons and protons can be in superfluid state (as reviewed, e.g., by Lombardo & Schulze 2001). Superfluidity affects the neutrino emission and thus the cooling of neutron stars. According to numerous microscopic calculations, the proton pairing occurs in the singlet (^{1}S_{0}) state of proton pairs. Following Yakovlev et al. (1999) we will call this pairing as pairing A. The neutron pairing occurs either in the ^{1}S_{0} state or in the triplet state (^{3}P_{2}). Neutron pairing A takes place in the matter of subnuclear density ( , where g cm^{-3} is the saturated nuclear matter density), while the ^{3}P_{2} pairing is efficient at higher . We consider the ^{3}P_{2} pairing of two types denoted as B and C. Pairing B occurs in a state of a neutron pair with zero projection of the total angular momentum on the quantization axis, . This pairing has been studied in the majority of papers devoted to ^{3}P_{2}pairing of neutrons. Pairing C occurs in a state with . It has been the subject of some studies (as reviewed, e.g., by Yakovlev et al. 1999). The actual type of neutron pairing (A, B, or C) corresponds to the state with minimum free energy. Pairing C seems to be less realistic than B but cannot be completely ruled out by contemporary microscopic theories. For example, Muzikar et al. (1980) showed that it realizes in matter with strong magnetic field ( G). Amundsen & Østgaard (1985) found that the energetically preferable state of the pair can be a superposition of states with different . The specific feature of pairing C is that it leads to superfluid gaps with nodes at the neutron Fermi surface producing qualitatively different effect on neutrino processes than pairing B (or A).
Note that we do not consider another case: ^{3}P_{2} neutron pairing with . In this case, just as in cases A and B, the superfluid gap does not have any nodes at the Fermi surface (e.g., Amundsen & Østgaard 1985). Therefore, we expect that the results will be similar to those for pairing B or A. On the other hand, the consideration of the pairing is technically much more complicated since the superfluid gap depends not only on the polar angle of neutron momentum at the Fermi sphere (see below) but also on the azimuthal angle .
Let us remind five main neutrino generation mechanisms in the neutron-star cores.
(1) Direct Urca process is the most powerful neutrino process.
It consists of two successive
reactions
(2) Modified Urca process consists of two branches.
Two successive reactions (direct and inverse)
The neutrino emissivity of this process in non-superfluid matter was considered by a number of authors (references can be found in Yakovlev et al. 1999), particularly, by Bahcall & Wolf (1965), Friman & Maxwell (1979), and Yakovlev & Levenfish (1995). The latter authors studied the reduction of the process either by proton superfluidity A, or by neutron superfluidity (A or B). Levenfish & Yakovlev (1996) suggested a simple approximate method to account for the combined effect of the neutron and proton superfluidities. It is based on the similarity relations of the factors which describe the superfluid reduction of the direct and modified Urca processes. These results were used in simulations of the neutron star cooling (as reviewed by Yakovlev et al. 1999,2001). We present a more accurate calculation of the reduction of the modified Urca process by combined action of proton superfluidity A and neutron superfluidity (A, B, or C).
(3) The neutrino-pair bremsstrahlung
at nucleon-nucleon scattering can be of three types:
(4) Neutrino emission due to Cooper pairing
of nucleons (
or p) actually consists of neutrino-pair
(any flavor) emission
(5) Neutrino-pair bremsstrahlung at
electron-electron scattering (Kaminker & Haensel 1999),
The present paper is organized as follows. In Sect. 2 we present general equations for modified Urca process and analyze the reduction factors. In Sect. 3 we consider the reduction factors of nucleon-nucleon bremsstrahlung processes. In Sect. 4 we study the efficiency of various neutrino processes in the cores of neutron stars for different superfluidity types. Analytic fits of the reduction factors of the modified Urca process are given in Appendix.
As discussed,
e.g., by Bahcall & Wolf (1965) and Friman & Maxwell (1979),
the general expression for the neutrino emissivity
of modified Urca process
can be written as
(
):
For the proton branch (3) at
one has
The difference of Eqs. (13) and (14) or (15) is the consequence of the fact that is significantly larger than in neutron star matter.
Combining these results one can obtain the neutrino emissivities and in nonsuperfluid matter. The emissivity was calculated by Friman & Maxwell (1979), using the one-pion-exchange approximation for calculating the matrix element, |M|^{2}, and was calculated by Yakovlev & Levenfish (1995) using the same technique.
Now consider the modified Urca process in
the presence of superfluidity of neutrons and protons.
A onset of superfluidity leads to the appearance
of an energy gap
in the momentum dependence of
the particle energy
.
Near the Fermi surface (
),
this dependence can be written as
(e.g., Lifshitz & Pitaevskii 1980)
For further analysis it is convenient
to introduce the dimensionless variables:
We assume that
the neutrino emissivity in superfluid matter
can be calculated from Eqs. (10)-(12)
by replacing
for all
particle species which are in superfluid state.
This assumption is widely used in the literature;
its validity is discussed by Yakovlev et al. (2001).
In this approximation, the neutrino emissivity of the modified
Urca process can be written as
We have composed a code which calculates the reduction factor (22) for proton superfluidity A and neutron superfluidity A, B, or C. The code has been tested by comparing with the analytical asymptotes at large v_{1} and v_{2} and with the results of Yakovlev & Levenfish (1995) who considered superfluidity of either protons or neutrons. The results have also been compared with those calculated from Eq. (22) under simplified assumption , discussed below (see Eq. (41)).
Notice that the results of this section can also be used to describe modified Urca process with muons instead of electrons (see Yakovlev et al. 2001, for details).
In this case Eq. (22) can be simplified.
For pairing A, the dimensionless
energy gap
is angle-independent.
This allows one to decompose the integrals over
the angles and over the dimensionless energies
x_{j}. For the neutron branch of the modified
Urca process we get
Let here and hereafter
refer to neutrons, and
refer to protons.
The reduction of
the proton branch is evidently given by
For example, we outline the derivation of the asymptote of from Eq. (23) in region I; the derivation in other regions is similar. Clearly, the integral (23) can be subdivided into several parts in such a way that any single part contains integrations from to 0 and/or from 0 to +. Now let us introduce the convenient notations for these parts. Let R(2,-1) mean a five-dimensional integral containing the integration from 0 to + over two neutron variables, and over to 0 over a proton variable (in this case, the integration over the third neutron variable is assumed to extend from to 0). Splitting the initial integral (23) into the elementary integrals, we see that the same integral R(2,-1) enters the sum three times. Thus, it is sufficient to calculate the integral once and multiply by 3.
In this way we obtain eight integrals of different types: R(3,+1), R(3,-1), R(2,+1), R(2,-1), R(1,+1), R(1,-1), R(0,+1), and R(0,-1). In the limit of strong superfluidity ( ), each of them is exponentially small. The exponentials are:
Figure 1: Four regions of and where the reduction factors of the neutron and proton branches of modified Urca process can be fitted by different expressions. Inregions I, II, and III at and the factors of neutron branch have different asymptotes. | |
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It is seen that the main contribution into
the asymptote comes from R(2,-1) and R(1,+1).
These terms have the same exponential but
R(2,-1) has a larger pre-exponent.
Therefore, it is
R(2,-1) which gives the main contribution
in region I:
In the same manner in region II we obtain:
The asymptotes themselves contain complicated integrals.
Thus we have calculated the asymptotes of the integrals (29) and (34).
In region I we have:
In region III we have
For region II:
For region III:
The fits of are given in Appendix.
Now consider neutron superfluidity of type B.
According to Eqs. (20) and (18), the dimensionless
energy gap of the neutrons, ,
is angle-dependent.
The asymptotes of
in this case
can be obtained from Eq. (22). As before,
j=1, 2, and 3 enumerates neutrons while
j=4 refers to a proton.
Equation (22) can be written as
Calculations show that
the reduction factor is almost insensitive to
variations of particle Fermi-momenta.
Let us obtain a simplified expression
for
by setting
in Eq. (22) and
integrating over
and
.
The result is
Then, the asymptote in region II
will be written as
We have numerically integrated from Eq. (40) for a dense grid of v_{1} and v_{2}. The calculations have been conducted at and . As mentioned above, the reduction factor is rather insensitive to variations of these parameters. The variations of to the changes of the particle Fermi momenta within reasonable limits ( ) obtained in some test runs are of the order of estimated error of numerical integration. The fits of are given in Appendix.
In this case Eq. (22) can be simplified as
One can easily obtain the asymptotes of the
reduction factor at large values of v_{1} and v_{2}.
For the proton branch of modified Urca process,
the regions where the asymptotes are different can be found
from neutron-branch regions
by replacing
.
For instance, at v_{2} > v_{1}:
The most important feature of this case is that the energy gap vanishes at the poles of the Fermi sphere (see Eqs. (19) and (20)). Equation (40) remains valid in this case. The calculations of have been done at and . As in the previous cases, the reduction factor is rather insensitive to variations of these parameters. The results are approximated by the expressions given in Appendix.
Equation (46) remains valid in this case.
Since
vanishes at the poles of the Fermi sphere, the reduction factor
varies with v_{1} as a power-law
(rather than exponentially).
It is easy to determine its behavior
at large v_{1}. One can see that in this case
the main contribution into integral (46)
comes from the region where
.
Thus, we have
Now consider the superfluid suppression of the neutrino-pair emission in the nucleon-nucleon bremsstrahlung processes (4)-(6). In the absence of superfluidity the emissivities of NN-bremsstrahlung processes in the one-pion-exchange approximation are given, for instance, by Yakovlev et al. (1999).
In analogy to Eq. (21), one can
introduce the superfluid reduction factors
:
The factor
was accurately calculated
by Yakovlev & Levenfish (1995).
For the neutron-proton process we suggest
the similarity relation of the form
Figure 2: Reduction of the neutrino emissivity by neutron and proton superfluidities of type A in the neutron branch of the modified Urca process versus v ( , ) at , 45, 63, and 90. Solid lines show our results, and dashed lines are obtained from similarity criteria, e.g., Yakovlev et al. (1999). | |
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An analysis of
for neutron-neutron bremsstrahlung, Eq. (4),
is more sophisticated (since
no similarity criterion can be formulated).
Let us study the reduction factor
at large v_{1} from Eq. (53).
Now j=1-4 refer to neutrons.
One can see
that the main contribution to
comes
from the range of angles
.
Since the sum of the Fermi momenta
of reacting neutrons must be equal to zero,
the Fermi momenta should concentrate to
the poles of the Fermi sphere: two momenta to one pole
and other two momenta to the other pole.
Now we expand all functions in series over
and
integrate over
,
and
.
In this way we obtain
Figure 3: Same as in Fig. 2 but for . | |
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Figure 4: Same as in Fig. 2 but for . | |
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The results of Sect. 2 allow us to
compare the exact and approximate
reduction factors of modified Urca process.
The comparison is illustrated in Figs. 2-4.
The figures show the dependence of the calculated reduction
factors,
,
,
and
,
on
at several values of (
is the polar angle in the v_{1}-v_{2} plane;
).
Our results (solid lines) are compared with the approximate
reduction factors (dashed lines) constructed
(e.g., Yakovlev et al. 1999) using
the criteria of similarity between the reduction factors
for different neutrino reactions.
The approximate factors
have been used in a number of simulations
of neutron star cooling. One can see that the difference
of the approximate reduction factors from the
exact ones increases with increasing v(but for
in Figs. 2, 3
and
in Fig. 4).
Figure 5: Regions of (of type B) and (of type A) where the different neutrino reactions dominate. | |
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Figure 6: Regions of (of type C) and (of type A) where the different reactions dominate. | |
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Now let us answer the question which neutrino generation mechanism dominates in a superfluid neutron-star core. Taking into account the above results we can calculate the emissivities of all main neutrino processes (Sect. 1) for proton superfluidity A and any neutron superfluidity, A, B, or C. Figures 5 and 6 show which process dominates at different values of and . Figure 5 shows the effect of neutron superfluidity B, while Fig. 6 - the effect of neutron superfluidity C. Both figures are plotted using an equation of state suggested by Prakash et al. (1988) (their model I of symmetry energy with the compression modulus of saturated nuclear matter K = 240 MeV).
Three left panels of Fig. 5 illustrate standard neutrino emission at (direct Urca process forbidden) for three values of the internal stellar temperature, , and 10^{9} K, while three right panels correspond to neutrino emission enhanced by direct Urca process at for the same T. Figure 5 is almost the same as obtained earlier by Yakovlev et al. (1999) for another equation of state using the approximate reduction factors of modified Urca process. The selected values of T cover the temperature interval most important for the theory of neutron star cooling. The figures are almost independent of (and of equation of state) as long as does not cross the density threshold of opening direct Urca process. One can see that if the neutrons are superfluid alone and , then the bremsstrahlung due to proton-proton scattering becomes dominant. If protons are superfluid alone and , then the main mechanism is neutrino emission in neutron-neutron bremsstrahlung. Neutrino emission due to Cooper pairing of neutrons always dominates at provided direct Urca process is forbidden. If the direct Urca process is allowed then the Cooper-pairing neutrino emission may dominate provided nucleons of one species are strongly superfluid while nucleons of the other species are moderately superfluid. Finally, in the presence of strong superfluidity of protons and neutrons, all the processes involving nucleons are so strongly reduced that the neutrino-pair emission in electron-electron bremsstrahlung dominates. Figure 6 differs from Fig. 5 mainly by the increase of the efficiency of neutrino emission due to Cooper pairing of neutrons (at ) and direct Urca process (at ).
It is well known that only one neutrino process dominates at a given density in a non-superfluid neutron-star core. It is either direct Urca or modified Urca process. The situation is drastically different in superfluid matter. As seen from Figs. 5 and 6, Cooper-pairing neutrino emission becomes dominant in the presence of a weak neutron superfluidity. With the increase of in the superfluid regime, modified Urca process becomes unimportant and Cooper-pairing neutrino emission dominates.
We have calculated the factors which describe the reduction of the neutrino emissivity in the neutron and proton branches of modified Urca process by superfluidities of neutrons and protons. We have considered singlet-state pairing of protons (pairing A) and either singlet-state or triplet-state pairing of neutrons (A, B or C). The reduction factors are fitted by analytic expressions presented in Appendix to facilitate their use in computer codes.
We have also considered the reduction of neutrino bremsstrahlung due to neutron-neutron and neutron-proton scattering by proton superfluidity A and neutron superfluidity C. We have constructed the approximate reduction factors and fitted them by analytic expressions. We have determined also the dominant neutrino emission mechanisms in a neutron star core at different values of the critical temperatures of the neutron and protons, and , for the cases of neutron superfluidity of type B or C.
Our results combined with those known in the literature (e.g., Yakovlev et al. 2001) allow one to calculate the neutrino emissivity in a neutron-star core in the presence of proton superfluidity A and neutron superfluidity A, B, or C. The results can be useful to study thermal evolution of neutron stars, first of all, cooling of isolated neutron stars. Our cooling simulations based on the present results will be published elsewhere.
Acknowledgements
I am grateful to D. G. Yakovlev for discussions, to M. Ulanov, and K. P. Levenfish for technical assistance, and to anonymous referee for useful remarks. The work was supported partly by RFBR (grants Nos. 02-02-17668 and 00-07-90183).
We have calculated the reduction factors
of the modified Urca process from Eq. (22)
as described in Sect. 2.
Introducing the polar coordinates
(
,
),
in regions I, II, and III we fit
the numerical results by the expression
i | I | II | III | IV |
1 | 0.257798 | -9.495146 | -2.678004 | 0.268730 |
2 | 0.003532 | -1.909172 | 64.33063 | 0.089294 |
3 | 19.57034 | 0.820250 | -2.736549 | 0.002913 |
4 | 0.036350 | 10.17103 | 0.093232 | 1.752838[-5] |
5 | 0.173561 | 5.874262 | 0.380818 | 3.047384[-7] |
6 | 0.039996 | 0.023332 | -0.015405 | 0.022415 |
7 | 0.101014 | 0.003191 | -16.79340 | 0.001835 |
8 | 16.61755 | 201.8576 | 112.4511 | 5.849410[-7] |
9 | 0.063353 | 5.520899 | 517.5343 | 0.001610 |
10 | 0.101188 | 1.257021 | 0.134529 | |
11 | 0.343374 | -2.367854 | -0.174503 | |
12 | -0.135307 | 1.096571 | -0.029008 | |
13 | 2.404372 | 0.481874 | 1.277903 | |
14 | 1.055914 | 487.4290 | -25.70616 | |
15 | 1.086360 | -0.452688 | 558.1592 | |
16 | -257.9342 | 0.328108 | ||
17 | 17.83708 | 0.642631 | ||
18 | 0.260288 |
In the case of neutron and proton superfluidity A for neutron branch of modified Urca process we get the following fits.
In region I:
A=p_{1} v_{1}^{2} + p_{2} v_{2}^{2} + p_{3} v_{1}^{2} v_{2}^{2} + p_{4} v_{1}^{6} + p_{5} v_{2}^{6}, | |
B=1 + p_{6} v_{1}^{2} +p_{7} v_{2} ^{2} +p_{8} v_{1}^{4} , | |
C=1+ p_{9} v_{2}^{4}. | (A6) |
In the case of proton superfluidity A and neutron
superfluidity B for neutron branch of modified Urca
process we get the following fits.
In region I:
i | I | II | III | IV |
1 | -0.719681 | -6.475443 | 0.316041 | 0.565001 |
2 | -0.024591 | -1.186294 | -289.2964 | 0.087929 |
3 | 0.297357 | 0.591347 | 2480.961 | 0.006756 |
4 | 1.260056 | 6.953996 | -268.8219 | 1.667194[-4] |
5 | 0.100466 | 3.366945 | 1984.115 | 3.782805[-6] |
6 | 0.148464 | -9.172994 | 3503.094 | 0.173165 |
7 | 0.253881 | -2.675793 | 0.331551 | 1.769413[-5] |
8 | 140.3699 | 1.053679 | -0.265977 | 7.710124[-8] |
9 | 0.132615 | 10.38526 | 1098.324 | 0.001695 |
10 | 0.280765 | 7.138369 | 65528.01 | |
11 | 0.375796 | 0.024500 | ||
12 | -0.096843 | 0.120536 | ||
13 | 3.100942 | 89.79866 | ||
14 | 0.275434 | 5719.134 | ||
15 | 0.330574 | 285.8473 | ||
16 | 0.402111 | |||
17 | 16657.19 |
A=p_{1} v^{2}_{1} + p_{2} v^{2}_{2} +p_{3} v^{2}_{1} v^{2}_{2} + p_{4} v^{6}_{1}, | |
B= | |
C=1 + p_{9} v^{4}_{2}. | (A10) |
In the case of proton superfluidity A and neutron superfluidity B for proton branch of modified Urca process we get the following fits.
In region I:
i | I | II | III | IV |
1 | 0.288203 | 0.398261 | 0.387542 | 0.272730 |
2 | -0.124974 | -0.054952 | -195.5462 | 0.165858 |
3 | 17.39273 | -0.084964 | 3032.985 | 0.005903 |
4 | 0.083392 | -0.036240 | -189.0452 | 2.555386[-5] |
5 | 0.059046 | -0.168712 | 3052.617 | 2.593057[-7] |
6 | 0.028084 | -0.704750 | 442.6031 | 0.023930 |
7 | -0.019990 | -0.066981 | 0.041901 | 0.006180 |
8 | 28.37210 | 1.223731 | -0.022201 | 1.289532[-5] |
9 | 0.244471 | 0.363094 | 5608.168 | 0.005368 |
10 | -0.610470 | -0.357641 | -10761.76 | |
11 | 0.023288 | 0.869196 | 0.064643 | |
12 | 0.475196 | -0.364248 | 0.296253 | |
13 | -0.180420 | 2.668230 | 106.3387 | |
14 | 25.51325 | -0.765093 | -75.36126 | |
15 | 0.281721 | -4.198753 | 84.65801 | |
16 | -0.080480 | 0.530223 | ||
17 | -0.191637 | -86.76801 |
In region IV:
A=p_{1} v^{2}_{2} + p_{2} v^{2}_{1} + p_{3} v^{2}_{1} v^{2}_{2} + p_{4} v^{6}_{2} + p_{5} v^{6}_{1}, | |
B=1+ p_{6} v^{2}_{2} + p_{7} v^{2}_{1} + p_{8} v^{4}_{2}, | |
C=1 + p_{9} v^{4}_{1}. | (A13) |
In the case of proton superfluidity A and neutron superfluidity C for neutron branch of modified Urca process we get the following fits.
In region I:
i | I | II | III | IV |
1 | 0.897393 | -3.471368 | 0.322115 | 0.175090 |
2 | -0.045357 | -0.133540 | -15.05047 | 0.088159 |
3 | 0.309724 | 0.143230 | 112.9733 | 3.055763[-3] |
4 | -0.739962 | 3.634659 | -13.79012 | 3.984607[-7] |
5 | 0.222597 | 0.496579 | 128.3156 | 5.591497[-8] |
6 | 0.032104 | 0.030609 | 39.82789 | 0.046496 |
7 | -0.054011 | 0.005056 | 0.164614 | 1.452790[-5] |
8 | 61.73448 | 0.438608 | 49.07699 | 4.505614[-8] |
9 | 0.195679 | -2.970431 | -3.145006 | 1.779724[-3] |
10 | -0.001851 | 0.284703 | 5132.076 | 2.136809[-4] |
11 | 0.482581 | 0.898355 | 0.018737 | 5.365717[-4] |
12 | -0.001637 | -0.036420 | 0.100223 | |
13 | -0.685659 | 0.407393 | 4.055407 | |
14 | 1.528415 | -0.058942 | 390.6242 | |
15 | -0.053834 | 0.605413 | 6.594365 | |
16 | -0.452426 | 2.851209 | 175.7396 | |
17 | -0.053502 | -0.800218 | 441.3965 | |
18 | 1.497718 | |||
19 | 1.476375 |
A=p_{1} v_{1}^{2} + p_{2} v_{2}^{2} + p_{3} v_{1}^{2} v_{2}^{2} + p_{4} v_{1}^{6}, | |
B= | |
C=1 + p_{9} v_{2}^{4} + p_{10} v_{2}^{2} + p_{11} v_{1}^{2} v_{2}^{2}. | (A16) |
In the case of proton superfluidity A and neutron superfluidity C for the proton branch of modified Urca process at we get the following fits
In region I:
i | I | II | III | IV |
1 | 0.049947 | -4.985248 | 0.100241 | 0.272905 |
2 | -0.029006 | -0.025984 | 0.005432 | 0.058684 |
3 | 3872.363 | -0.007404 | -0.748377 | 2.053694[-3] |
4 | 0.250385 | 5.294455 | 0.050631 | 1.800867[-7] |
5 | -0.245758 | -0.201654 | 0.007900 | 1.911708[-8] |
6 | 0.018241 | 0.184431 | -0.032915 | 0.052786 |
7 | 0.090256 | -0.139729 | -0.000768 | 2.043824[-5] |
8 | 108.8302 | 0.415562 | 0.044312 | 4.458912[-8] |
9 | 1.007326 | 2.692073 | -0.697892 | 1.101541[-3] |
10 | 0.061586 | -0.385832 | 0.032534 | 3.312811[-4] |
11 | 0.797695 | 1.055347 | 0.080109 | 2.682799[-4] |
12 | 175.5965 | 0.013667 | 0.031994 | |
13 | 9.306619 | -0.509106 | 8.724039 | |
14 | -0.551550 | -0.267675 | 2.982355 | |
15 | 1.203014 | 0.034585 | -0.062076 | |
16 | 0.096598 | |||
17 | -0.441039 |
A=p_{1} v_{2}^{2} + p_{2} v_{1}^{2} + p_{3} v_{2}^{2} v_{1}^{2} + p_{4} v_{2}^{6}, | |
B= | |
C=1 + p_{9} v_{1}^{4} + p_{10} v_{1}^{2} + p_{11} v_{2}^{2} v_{1}^{2}. | (A20) |
Process | I | II | III |
, | 20 | 23 | 25 |
, | <10 | <15 | <20 |
, | 15 | 15 | 25 |
, | <20 | <20 | <50 |
, | 19 | 21 | 26 |
, | <15 | <26 | <20 |
, | 22 | 13 | 13 |
, | <20 | <5 | <3 |
, | 23 | 16 | 13 |
, | <13 | <5 | <2 |
In Table 6 we give maximum values of of our fit expressions in regions I, II, and III, and maximum fit errors at in these regions. These maximum errors occur at , where the reduction factors are very small (and are thus unimportant for calculation of the total neutrino emissivity). At our fit expressions are not reliable and we recommend to set the corresponding reduction factors equal to zero in computer codes.