1 Astrophysical Constraints on Dense Matter in Neutron Stars
29
age therefore depends strongly on how long neutrino emission dominates. The
temperature tool is blunt because there are many processes that can enhance neutrino
production, and many processes that can suppress it, and thus mere measurement of
the temperature with age of neutron stars would not allow us to distinguish easily
between the multiple candidate effects.
1.5.1 The URCA Processes
To start, we note that as the star cools down from its high-temperature birth the
processes p + e − → n + ν e and n → p + e − + ¯
ν e can produce neutrinos efficiently.
In this context these are known as the URCA processes, named thus by George
Gamow after the Urca casino in Rio de Janeiro because the URCA processes are a
perfect sink for energy just as the casino is a perfect sink for money! Phase space
considerations indicate that the URCA emissivity (energy per volume per time)
scales as T 6 .
As the temperature drops below the Fermi temperature, however, we can see that
these processes become impossible unless the proton to neutron ratio is sufficiently
high. The derivation of the critical ratio is similar to what we presented in Sect. 1.2:
the momenta of the particles, which are dominated by Fermi momenta, must satisfy
the triangle inequality p n ≤ p p + p e . The Fermi momenta of all three particles
are determined by their respective number densities, so this means n
1/3
n ≤ n
1/3
p +
n
1/3
e . Charge neutrality means n e = n p , so n
1/3
n
≤ 2n
1/3
p and thus the criterion
for the URCA process to be possible is n n ≤ 8n p . This is on the low side for many
traditional equations of state but can be achieved in some cases at high density. Note
that if muons are also present (these are higher-mass analogs to electrons with the
same electric charge), and thus electrons have a number density n e = xn p with
x ≤ 1, the criterion becomes n n ≤ (1 + x 1/3 ) 3 n p , so even higher proton fractions
would be required.
If the URCA process is suppressed, bystander particles can soak up the extra
momentum, e.g., n+n → n+p+e − + ¯
ν e . This is usually called the modified URCA
process to distinguish it from the direct URCA (sometimes DURCA) processes
described above. Given that the neutrons are degenerate, only a fraction T /T F
of them can interact. Thus the modified URCA process is suppressed by a factor
(T /T F ) 2 (one factor is for the initial neutron and one is for the final) compared to
the direct URCA process. Given that in a neutron star core T F ∼ 10 12 K and T can
be ∼10 9 K, the suppression factor can easily be a million.
The huge difference between the direct and modified URCA rates means that
if enough protons are present, or if there are any other channels for neutrino
production, then cooling can be enhanced dramatically. We now consider such
channels.
29
age therefore depends strongly on how long neutrino emission dominates. The
temperature tool is blunt because there are many processes that can enhance neutrino
production, and many processes that can suppress it, and thus mere measurement of
the temperature with age of neutron stars would not allow us to distinguish easily
between the multiple candidate effects.
1.5.1 The URCA Processes
To start, we note that as the star cools down from its high-temperature birth the
processes p + e − → n + ν e and n → p + e − + ¯
ν e can produce neutrinos efficiently.
In this context these are known as the URCA processes, named thus by George
Gamow after the Urca casino in Rio de Janeiro because the URCA processes are a
perfect sink for energy just as the casino is a perfect sink for money! Phase space
considerations indicate that the URCA emissivity (energy per volume per time)
scales as T 6 .
As the temperature drops below the Fermi temperature, however, we can see that
these processes become impossible unless the proton to neutron ratio is sufficiently
high. The derivation of the critical ratio is similar to what we presented in Sect. 1.2:
the momenta of the particles, which are dominated by Fermi momenta, must satisfy
the triangle inequality p n ≤ p p + p e . The Fermi momenta of all three particles
are determined by their respective number densities, so this means n
1/3
n ≤ n
1/3
p +
n
1/3
e . Charge neutrality means n e = n p , so n
1/3
n
≤ 2n
1/3
p and thus the criterion
for the URCA process to be possible is n n ≤ 8n p . This is on the low side for many
traditional equations of state but can be achieved in some cases at high density. Note
that if muons are also present (these are higher-mass analogs to electrons with the
same electric charge), and thus electrons have a number density n e = xn p with
x ≤ 1, the criterion becomes n n ≤ (1 + x 1/3 ) 3 n p , so even higher proton fractions
would be required.
If the URCA process is suppressed, bystander particles can soak up the extra
momentum, e.g., n+n → n+p+e − + ¯
ν e . This is usually called the modified URCA
process to distinguish it from the direct URCA (sometimes DURCA) processes
described above. Given that the neutrons are degenerate, only a fraction T /T F
of them can interact. Thus the modified URCA process is suppressed by a factor
(T /T F ) 2 (one factor is for the initial neutron and one is for the final) compared to
the direct URCA process. Given that in a neutron star core T F ∼ 10 12 K and T can
be ∼10 9 K, the suppression factor can easily be a million.
The huge difference between the direct and modified URCA rates means that
if enough protons are present, or if there are any other channels for neutrino
production, then cooling can be enhanced dramatically. We now consider such
channels.