Trapping of neutrinos in extremely compact stars

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by  Zdenek Stuchlik

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TRAPPING OF NEUTRINOS IN EXTREMELY COMPACT STARS∗
ZDENEˇKSTUCHL´IK,MARTINURBANEC,GABRIELTO¨RO¨K,STANISLAVHLED´IK
ANDJANHLAD´IK
Institute of Physics, Faculty of Philosophy and Science, Silesian UniversityinOpava,
Bezruˇcovo n´am. 13, Opava 746 01, Czech Republic
7 [email protected]
0
0
2 Trapping of neutrinos in extremely compact stars containing trapped null geodesics is
studied.Wecalculatedtheratioofproducedtotrappedneutrinosinthesimplestmodel
n ofuniformdensitystars.Thisgivestheupperlimitontrappingcoefficientsinrealobjects.
a
J

1 1. Introduction
1
Trapped null geodesics are concentrated around the stable circular geodesic.2,3 We

2
suppose that neutrinos have zero mass and can move along null geodesics in the
v
situationswhentheneutrinomeanfreepathexceedsthestarradiusR.Considering
9
0 the internal Schwarzschild spacetimes, we use the effective potential related to the
3 impact parameter to calculate the ratios of trapped to produced neutrinos.
1
0
7
2. Internal Schwarzschild spacetime, effective potential and impact
0
/ parameter
h
p The line element of internal Schwarzschildspacetimes of uniform density1 reads
-
o
ds2 =−e2Φ(r)dt2+e2Ψ(r)dr2+r2(dθ2+sin2θdφ2). (1)
r
t
s
Thetemporalandradialcomponentsofthemetrictensoraregivenbytheformulae
a
:
v (−g )1/2 =eΦ = 3Y − 1Y(r), (g )1/2 =eΨ =1/Y(r), (2)
i tt 2 1 2 rr
X
and in geometric units c=G=1 there is
r
a
r2 1/2 R2 1/2 1 2M
Y(r)= 1− , Y ≡Y(R)= 1− , = , (3)
a2 1 a2 a2 R3
(cid:18) (cid:19) (cid:18) (cid:19)
where R is the radius andM is the mass ofthe object. The parametera represents
the curvature of the internal Schwarzschild spacetime.
Due totheexistenceoftwoKillingvectorfields,thetemporal∂/∂tone,andthe
azimuthal ∂/∂φ one, two conserved components of the 4-momentum must exist:
E =−p (energy), L=p (axial angular momentum). (4)
t φ
Becausethemotionplaneiscentral,onecansetθ =π/2=const,choosingtheequa-
torial plane. The motion along null-geodesics is independent of energy (frequency)
∗ThisresearchhasbeensupportedbyCzechgrantsMSM4781305903 andLC06014(M.U.).
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40
35
30 l2
c(e)
25
Veff 20
15
10
5
rb(e) R
0
0 0.5 1 1.5 2 2.5 3
r
Fig. 1. Behavior ofthe effective potential forR=2.5GM/c2. Neutrinos produced inthe region
r <r<Raretrappedifℓ>l andescapewhen ℓ<l .
b(e) c(e) c(e)
and can conveniently be described in terms of the impact parameter ℓ=L/E.The
radial motion is restricted by an effective potential related to ℓ by
4a2[1−Y2(r)]
Vint = forr ≤R
ℓ2 ≤V = eff [3Y1−Y(r)]2 (5)
eff r3
Veeffxt = r−2M forr >R,
Veiffnt (Veeffxt) is the effective potential of the null-geodetical motion in the internal
(external) Schwarzschildspacetime4(see Fig. 1).
3. Escaped to produced neutrinos ratio
We assume that neutrinos are locally produced by isotropically emitting sources.
Thenescaped-to-produced-neutrinosratiodepends onageometricalargumentonly
and it is determined through directional escaping angle Ψ given by
e
33/2 3 2M 1/2 1 2M r 2 1/2
sinΨ (r,R)= 1− − 1− . (6)
e
r 2 R 2 R R
" (cid:18) (cid:19) (cid:18) (cid:16) (cid:17) (cid:19) #
Let N , N and N denote the number of produced, escaped and trapped neu-
p e b
trinos per unit time of an external static observerat infinity. In order to determine
the global correction factors
N (R) N (R)
E(R)≡ e , B(R)≡ b =1−E(R), (7)
N (R) N (R)
p p
it is necessary to introduce the local correction factor for escaping neutrinos at a
given radius r ∈(r ,R). The escaping solid angle is given by
b(e)
Ψe2π
Ω (Ψ )= sinΨdΨdφ=2π(1−cosΨ ) (8)
e e e
Z Z
0 0
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3
1
0.8
βmax MR
0.6
b(c) 2.3
=
β0.4
2.5
0.2
2.8
2.9
0 2.95
0.2 0.4 0.6 0.8 1
r/R
Fig.2. Behaviorofthecoefficientoflocaltrappingβ(r,R)(left)forseveralvaluesofR/M andof
thetotalcoefficientoftrappingB(R)(right).QLandXLdeterminetrappingofinternal(external)5
neutrinos;BL=QL+XL
and the escaping (trapping) correction factors ǫ(r,R) (β(r,R)) are given by
dN (r) 2Ω(ψ (r,R))
ǫ(r,R)=1−β(r,R)= e = e =1−cosψ (r,R). (9)
e
dN (r) 4π
p
Thecoefficientβ(r,R)determineslocalefficiencyoftheneutrinotrapping(seeFig.2
for behavior of local (left) and global (right) trapping coefficients). The global es-
caping (trapping) coefficients are given by integrating the local production rate
andescaping(trapping)coefficientsthroughthestar,anditstrappingzone,respec-
tively.5 We assume local production rate uniformly distributed through the star
from the point of view of local observersbut it is not uniform as viewed by distant
static observers.5
4. Conclusions
Wehaveshowninthesimplestcaseofuniformdensitystarsthattrappingofneutri-
nos plays a significant role in cooling mechanism. The local effect given by β(r,R)
is 10% for R/M = 2.95 while the global effect described by B(R) is 10% for
R/M =2.87.We expectthis trappingefficiencycouldbe reachedinsomerealstars
with specific but realistic equations of state.
References
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424–434, 1916.
2. Z. Stuchl´ık, S. Hled´ık, J. Sˇolt´es, and E. Østgaard. Phys. Rev. D, 64(4):044004
(17 pages), 2001.
3. M. A. Abramowicz, J. Miller, and Z. Stuchl´ık. Phys. Rev. D, 47(4):1440–1447, 1993.
4. C.W.Misner,K.S.Thorne,andJ.A.Wheeler.Gravitation.Freeman,SanFrancisco,
1973.
5. Z.Stuchl´ık,GabrielT¨or¨okandStanislavHled´ık.InS.Hled´ıkandZ.Stuchl´ık,editors,
Proceedings of RAGtime 4/5: Workshops on black holes and neutron stars, Opava,
14–16/13–15 October 2002/03, Opava,2004. Silesian Universityin Opava.