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Propagation of neutrinos in rapidly rotating neutron stars*
Maxim Dvornikov
(1)

1,2

Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, Russia (2)Universidad TИcnica Federico Santa MarМa, Chile

* Based on M. Dvornikov and C. Dib, 0907.1445 [astro-ph]


Neutron stars: an overview
If a degenerate core (or white dwarf) exceeds the Chandrasekhar mass limit (1.4M) it must collapse until neutron degeneracy pressure takes over. Typical parameters: M ~ several M; R ~ 10 km; ~ nuclear; B up to 1015 G These characteristics convert a neutron star a perfect "laboratory" for neutrino physics [G. G. Raffelt, Stars as Laboratories for Fundamental Physics, 1996]


Rotation of a neutron star
Conservation of angular momentum led to the prediction that neutron stars must be rotating very rapidly. The minimum period, P, (or Keplerian angular velocity) of a star is that for which the surface layers are "in orbit" The actual angular velocities of radio pulsars are typically smaller ~102-103 s-1

2 vrot GM <2 R R 4 2 R GM <2 P R


Dirac equation for neutrino mass eigenstates in rotating medium
L=
-
= ,



( i µ µ - f µ µ PL )


(0 )



= ,

m , (r, t = 0) = U a a , (U
a



(r )

=

a =1, 2



)

cos = sin

- sin cos

We study the initial condition for the system of two mixed flavor neutrinos interacting with moving matter [MD, Phys. Lett. B 610, 262 (2005)] The evolution equation for mass eigenstates

i

a 0 = ( p + ma + ( g aa - g aa ) PL ) t 0 +( g ab - g ab ) PL b , a, b = 1, 2
µ

a

µ µ 2 jeµ sin 2 GF (2 je sin - jn ) ( g ab ) = µ (2 jeµ cos 2 - jn jeµ sin 2 2 µ jn ,e = (nn ,e , nn ,e v ), v = ( в r )

)


Neutrino quantum states in a rotating neutron star
For neutrinos with relatively small energies (several eV) a bound state is possible [A. V. Grigoriev, et al., Russ. Phys. J. 50, 845 (2007)]
a (r , , t ) =
( En a )±

n , s =0

(



( ana ) (t )u s

+ a , ns

( ( (r , ) exp[-iEna )+ t ] + bnsa ) (t )u

- a , ns

( (r , ) exp[-iEna ) -t ]

)

2 =-Va ± 4Va n + ma ,

G V1 = F (nn - 2ne sin 2 ), 2 G V2 = F (nn - 2ne cos 2 ) 2

u

± a , ns

Va (r , ) = 2
2

I n -1, s ( a )ei(l -1) iI ( )eil n,s a



,

a = Va r
(a)n

Energy of antineutrino E = -E bound state is not possible!

>0. For an antineutrino a


Wave functions of high energy neutrinos
Neutrino wave functions inside the star
( Va C1in e - a /2 al -1) / 2 F (l - , l , a )ei (l -1) /{(l - 1) ! } u (r , ) = in - a /2 l il iC 2 e a F (l - , l + 1, a )e / l ! in a ,

Neutrino wave functions outside the star

1 out ua , (r , ) = 2

1) C1out H l(-1 ( p r )ei (l -1) out (1) il iC2 H l ( p r )e

Wave functions should be equal at the star surface: ua(in)(R,)=ua(out)(R,) ( 2 E a ) =-Va + 4Va + ma Energy has a continuous value:


Evolution equation for the mass eigenstates
The ordinary differential equations for the coefficients ans
d (a i ans ) (t ) = dt +

{

ns = 0

{


u

+ a , ns

µ (r ) ( g ab

µ

)

( + ub ,ns (r ) d 2r exp i ( En a )+ ( - En b )-

}

a )+

( - En

b )+

)

( t anbs) (t )

u

+ a , ns

µ (r ) ( g ab

µ

)

( - ub ,ns (r ) d 2r exp i ( En

}

)

(b t bns) (t ), a b

In general the transitions (l,s)(l,s±1) are possible If s>>l (neutrino emission from the center of the star)
d i dt a1 /2 s 2= - /2 as a1 m 2 GF GF s ne sin 2 , ne cos 2 = - 2 , = 2 4k 2 2 as

The rotation causes almost no effect on the neutrinos emitted from the center of the star


Neutrinos with big angular momentum
We can use the conventional Schrodinger equation based approach to describe neutrino flavor oscillations in moving matter [A. Grigoriev, et al., Phys. Lett. B 535, 187 (2002)]
sin 2 - cos 2 1 0 GF - ne (1 - v) 2 0 -1
2 H12 P ( x, y ) = 2 si n 2 H12 + H11 2

m 2 cos 2 H= 4 E sin 2

(

2 2 H12 + H11 y

)
0.0124 0.0124 0.0123 0.0123 = ± 0.05 = ± 0.1 =0

G cos 2 - F ne (1 - x), H11 = 4 E 2 H12 = 4 E
F
outside neutrinosphere

m

2

m

2

sin 2



0.0122 0.0122

j

µ ,

= Fµ , je , F =



0.0121

dx dy P ( x , y )

0.0121 0.012 0

0.05

0.1

0.15

R

0.2

0.25

0.3

0.35


Neutron star spin-down
Main mechanisms of the neutron star spin-down at the latest stages of the evolution (thousands of years) are magnetic dipole radiation, for radio pulsars, and gravitational waves radiation (D. R. Lorimer and M. Kramer, Handbook of Pulsar Astronomy, 2004) Neutrinos may significantly contribute to the neutron star spin-down at the initial stages of the evolution (10 sec)


Collisions with moving matter
We have demonstrated that rotation causes very small effect on flavor dynamics of neutrinos. Therefore neutrinos keep their initial flavors Matrix element for the reaction (k1) f(p1) (k2) f(p2) [L. B. Okun', Leptons and Quarks, 1990]
M= GF f ( p2 ) g L µ (1 - 5 ) + g R µ (1 + 5 ) f ( p1 ) (k2 ) µ (1 - 5 ) (k1 ) 2 p1, 2 = ( E1,2 , p1,2 ), k1,2 = (1,2 , k 1,2 )

Differential cross section
d 1 = d 128 M
2 2



F 1 E1 E2

2

,

2 E2 = E12 - 212 cos + 2 - 2 E12 v f sin sin ,

2 =

E11 , sin ) + 1 (1 - cos ) E1 (1 - v f sin

F = 1 + (2 - 1 cos - Ev f sin sin ) / E2 1


Angular momentum carried away by neutrinos
Angular momentum per unit time

Lz =



k r sin n f (r ) d 3r,

k = 2 sin sin J (r )

Assuming that ne,p << nn as well as vf and 1/E1 are small parameters we express the final result in the form

d d d

E R nn J0 Lz 0.1 L0 10 MeV 10 km 1038 cm -3 1043 cm -2s -1
Neutrino luminosity reaches 1052 erg/s during t ~ 1 s [T. Totani, et al., Astrophys. J. 496, 216 (1998)] giving the neutrino flux at the neutron star surface 1043 cm-2s-1. Finally we obtain that neutrinos can carry away ~10% of the initial angular momentum.

3

3

M M s

-1


Discussion
Rotation of the neutron star causes small effect on flavor oscillations of neutrinos The trapping of neutrinos by rotating matter is possible. Antineutrinos cannot be trapped. Neutrinos can carry away an essential fraction of the initial angular momentum [see also K. Mikaelian, Astrophys. J. 214, L22 (1977); R. Epstein, Astrophys. J. 219, L39 (1978)]. An effort to infer initial angular velocities of neutron stars was made by E. van der Swaluw and Y. Wu [Astrophys. J. 555, L49 (2001)]. It was revealed that there should be a significant uncertainty in the results. Thus there exists a possibility that neutrino emission can contribute to the spin-down of a neutron star.


Acknowledgements
Work was supported by Conicyt (Chile), Programa Bicentenario PSD-91-2006 I am thankful to C. O. Dib, L. B. Leinson, J. Maalampi, A. Reisenegger, I. Schmidt and A. I. Studenikin for helpful discussions.