CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN Document Server UCLA/98/TEP/30 Pulsar kicks from neutrino oscillations Alexander Kusenko Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547 Gino Segr`e Department of Physics, University of Pennsylvania, Philadelphia, PA 19104 (November, 1998) Neutrino oscillations can explain the observed motion of pulsars. We show that two different models of neutrino emission from a cooling neutron star are in good quantitative agreement and predict the same order of magnitude for the pulsar kick velocity, consistent with the data. PACS numbers: 97.60.Gb, 14.60.Pq I. INTRODUCTION trino oscillations1. The result obtained in Ref. [6] was in- + correct because the neutrino absorption νen e−p was neglected and also because the different neutrino→ opaci- ties were assumed to be equal to each other. We empha- We recently suggested [1,2] that the observed proper size that in the absence of charged-current interactions motions of pulsars [3] may be the result of neutrino os- the kick from the active neutrino oscillations [1] should cillations in the hot neutron star born in a supernova ex- vanish. plosion. Neutrinos are the only significant cooling agents We will show that, after the charged-current interac- during the first 10 seconds after the onset of the super- tions are included, the two models are, in fact, in good nova, and they carry away most of the energy liberated agreement, as they should be. in the gravitational collapse, 1053erg. A 1% asymme- try in the distribution of neutrinos∼ can account for the measured pulsar velocities 500 km/s. ∼ II. HARD NEUTRINOSPHERES Neutrino oscillations in medium are affected by the po- larization effects due to magnetic field [4]. Consequently, Our calculations [1,2] employed a model with a sharp the strong magnetic field of a neutron star can affect the neutrinosphere, such that the neutrinos of a given type depth at which the neutrino conversions take place. The were assumed trapped inside and free-streaming outside. points of the Mikheev-Smirnov-Wolfenstein resonance lie Several subsequent analyses [5] used the same model to on a surface that is not concentric with the star. For a calculate the kick to the pulsar from the neutrinos whose certain range of neutrino masses, this creates an asym- opacities changed on their passage through matter due metry in the distribution of momenta of the outgoing to oscillations. neutrinos. The ν ν oscillations [1], as well as the µ,τ ↔ e In this model one assumes that the luminosity of the sterile-to-active neutrino conversions [2], can give the pul- emitted neutrinos obeys the Stefan-Boltzmann law and sar a kick consistent with the observation provided that is proportional to T 4. the magnetic field inside the star is of order 1014 1015G. − This model is admittedly simplistic. However, it ap- Other types of neutrino conversions could also produce a pears to work well insofar as predicting the order of mag- similar effect [5]. nitude of the kick. Our explanation [1,2] uses the fact that different neu- trino species have different opacities in nuclear matter. The charged-current interactions are responsible for the III. SOFT NEUTRINOSPHERES difference in the mean free paths of νe and νµ,τ .They are also the cause of the matter-enhanced νe νµ,τ con- Let us now consider the Eddington model for the at- versions. In the case of sterile neutrinos, both↔ charged mosphere which was used by Schinder and Shapiro [7] to and neutral currents contribute to the difference in the opacities. The analyses of Refs. [1,2,5] was based on a simplified model which, as we will see, gave the right order-of-magnitude estimate for the pulsar velocity. 1The active-to-sterile neutrino conversions of Ref. [2] are as- Recently, a different model for the neutrino emission sumed to take place at much higher densities than those dis- was used to calculate the kick [6] due to the active neu- cussed in Ref. [6]. 1 describe the emission of a single neutrino species. We We will now include the absorptions of neutrinos. will generalize it to include several types of neutrinos. Some of the electron neutrinos are absorbed on their In the diffusion approximation, the distribution func- passage through the atmosphere thanks to the charged- tions f are taken in the form [7]: current process eq + eq ξ @f νen e−p : (6) fνi fν¯i f + ; (1) → Λ @m ≈ ≈ i 2 2 The cross section for this reaction is σ =1:8G Eν,where eq F where f is the distribution function in equilibrium, Λi Eν is the neutrino energy. The total momentum trans- denote the respective opacities, m is the column mass fered to the neutron star by the passing neutrinos de- density, m = ρdx, ξ =cosα,andαis the normal angle pends on the energy. of the neutrino velocity to the surface. At the surface, Both numerical and analytical calculations show that one imposes theR same boundary condition for all the dis- the muon and tau neutrinos leaving the core have much tribution functions, namely higher mean energies than the electron neutrinos [8,9]. Below the point of MSW resonance the electron neutrinos f (m; ξ)=0; for ξ<0; νi (2) have the mean energies 10 MeV, while the muon and f (m; ξ)=2feq; for ξ>0: νi tau neutrinos have energies≈ 25 MeV. The origin of the kick in≈ this description is that the However, the differences in Λi produce unequal distribu- tions for different neutrino types. neutrinos spend more time as energetic electron neutri- Generalizing the discussion of Refs. [6,7] to include six nos on one side of the star than on the other side, hence flavors, three neutrinos and three antineutrinos, one can creating the asymmetry. Although the temperature pro- write the energy flux as file remains unchanged in Eddington approximation, the unequal numbers of neutrino absorptions push the star, 1 3 so that the total momentum is conserved. =2 ∞ 3 ( + ) (3) F π E dE ξdξ fνi fν¯i ; Below the resonance Eνe <Eντ,µ .Abovethereso- 0 1 i=1 nance, this relation is inverted. The energy deposition Z Z− X into the nuclear matter depends on the distance the elec- (0) 2 2 We will assume that Λi =Λi (E =E0 ). tron neutrino has traveled with a higher energy. This We use the expressions for fνi from equation (1). distance is affected by the direction of the magnetic field Changing the order of differentiation with respect to m relative to the neutrino momentum. and integration over and , and using the fact that E ξ We assume that the resonant conversion νe ντ takes eq is isotropic, we arrive at the result similar to that of ↔ f place at the point r = r0 + δ(φ); δ(φ)=δ0cos φ.The Ref. [7]: position of the resonance depends on the magnetic field B inside the star [1]: 2π3 3 2 @T 2 F = E2 : (4) 9 0 (0) @m eµeB dNe "i=1 Λ # δ = ; (7) X i 0 2π2 dr The basic assumption of the model is that flux F is conserved. In other words, the neutrino absorptions where Ne = YeNn is the electron density and µe is the + electron chemical potential. νen e−p are neglected. Since the sum in brackets, as well→ as the flux F are treated [7] as constants with Below the resonance the τ neutrinos are more ener- respect to m,onecansolveforT2: getic than the electron neutrinos. The oscillations ex- change the neutrino flavors, so that above the resonance 1 3 − 1=2 the electron neutrinos are more energetic than the τ neu- 2 9 2 2 30 T (m)= E− Fm+ F (5) trinos. The number of neutrino absorptions in the layer 2π3 0 (0) 7π5 "i=1 Λi # of thickness 2δ(φ) around r0 depends on the angle φ be- X tween the neutrino momentum and the direction of the Swapping the two flavors in equation (5) leaves the magnetic field. Each occurrence of the neutrino absorp- temperature unchanged in the Eddington approximation. tion transfers the momentum Eνe to the star. The dif- Hence, neutrino oscillations do not alter the temperature ference in the numbers of collisions per electron neutrino 2 profile in this approximation . between the directions φ and φ is − ∆k =E =2δ(φ)N [σ(E ) σ(E )] (8) e νe n 1 − 2 µ eB =1:8G2[E2 E2] e h cos φ, (9) 2 F 1 2 2 Ne It was also assumed in our earlier calculations [1,2] that − Ye π neutrino oscillations do not have a significant effect on the 1 temperature profile. We then took the spectral temperature where hNe =[d(ln Ne)=dr]− . of free-streaming neutrinos to be equal to matter temperature We use Ye 0:1, E1 25 MeV, E2 10 MeV, at their respective neutrinospheres. µ 50 MeV,≈ and h ≈6 km. After integrating≈ over e ≈ Ne ≈ 2 angles and taking into account that only one neutrino Incidentally, the Stefan-Boltzmann relation between species undergoes the conversion, we obtain the final re- the luminosity and temperature is, of course, present in sult for the asymmetry in the momentum deposited by both models. In the diffusion approximation, equation the neutrinos: (5) implies that F T 4 at m =0. In both models the∝ kick is a manifestation of the un- ∆k B =0:01 ; (10) equal neutrino opacities which are caused by the charged- k 2 1014G × current interactions. The two models are in good quantitative agreement.