Contributions to from various neutrino emission mechanisms (except for the Cooper-pair
mechanism, which will be considered later in this section) versus
are plotted in Figures 61
, 62
and 63
.
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We start with the crust of a very young neutron star, with a temperature (age
1 year), Figure 61
. For density
, the contribution
is dominant. However, with
increasing density, electrons become degenerate, and positrons disappear in the matter, so that
is
strongly suppressed at
. We also notice that
is never important in the inner crust,
because of the strong electron degeneracy.
from the plasmon decay gives the dominant contribution
to
from
down to the bottom of the crust. We notice also that
behaves
differently than the other contributions. Namely, at
its density dependence is
very weak, and
scales approximately with the magnetic field
as
. Finally,
one notices jumps of
and
, which result from jumps in
and
in the
ground-state matter. As we will see, this feature is even more pronounced at lower temperatures
.
Let us now consider the case of a colder crust at , Figure 62
. Except for
, which is
just scaled down due to the decrease of temperature, there is a dramatic change in the overall landscape.
For a magnetic field
,
dominates in the lowest-density region. On the contrary,
is of marginal importance, and is influenced by
(increases with
). Moreover,
contribution of
is negligible. Neglecting the effect of magnetic fields, one concludes that
dominates in the outer crust, while
dominates in the inner crust. Let us notice
that
reaches its maximum near
and then decreases by four orders of
magnitude when the density falls below
; this characteristic behavior is due to the
factor, Equation (265
). On the contrary,
rises steadily with increasing
density.
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Finally, in Figure 63 we consider an even colder crust at
. Pair and photoneutrino
constributions have disappeared completely, while
dominates for
, whereas
at higher densities,
is the main source of neutrino emission. At the magnetic field
, characteristic of magnetars, synchrotron radiation dominates in the density interval near
, but then at
,
becomes the strongest neutrino radiation
mechanism.
Two general remarks are in order. First, as we have already mentioned, jumps in and
are due to specific factors involving
and
and reflect the jumps in
and
in the layered
crust. For the other mechanisms, the electron chemical potential
with its smooth dependence on
plays the role of the crucial plasma parameter, and therefore no jumps are seen. Secondly,
were the magnetic field
,
would be overall dominant for
and
.
The Cooper-pair mechanism of neutrino radiation differs fundamentally from the other mechanisms of
neutrino cooling, discussed above, and therefore we consider it separately. depends sensitively on the
interplay between temperature
and the
pairing gap
of the dripped neutrons. The gap itself
depends on
, rising from zero at
to the asymptotic value
for
(see Section 8.2.2). As we already discussed in Section 8.2.1, the dependence of
on the free neutron
density,
, is very poorly understood, and this introduces a large uncertainty in the calculated values of
. Notice that the relation
, needed to get
, depends on the model of the inner
crust.
Figure 64 refers to
, a selected model of neutron superfluidity, and a selected model of the
inner crust. In the BCS theory (Section 8.2.1), the maximum of
, denoted by
, corresponds
to the maximum of
, given by
(Section 8.2.2). In the case presented in
Figure 64
,
is significantly larger than
. The Cooper-pair mechanism is efficient
only within a narrow range of temperature below
, namely for
, and is strongly
damped outside this region. In view of this,
usually has two maxima, around
and
, which
are the two solutions of
(remembering the bell shape of the pairing gap as a function of
density). Only the lower-density maximum can be seen in Figure 64
. Because
, the localization
of the peaks (at
1012 g cm–3 and at
1014 g cm–3) does not change much with decreasing
temperature. However, the heights of the peaks decrease very fast. For the selected superfluidity
model, and at
,
in the peak region dominates over all other neutrino emission
mechanisms. Let us notice that a proper inclusion of the in-medium modification of the weak
interactions could significantly decrease the maximum value of
by about two orders of
magnitude [368, 245].
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