The extension of radiative viscosity to superfluid matter ∗ PI Chun-Mei $^{1}$, YANG Shu-Hua2 ∗∗ , and ZHENG Xiao-Ping 2 1 Department of Physics and Electronics, Hubei University of Education, Wuhan 430205, P.R.China 2 Institute of Astrophysics, Huazhong Normal University, Wuhan 430079, P.R.China, (Dated: Received: , Accepted:) The radiative viscosity of superfluid npe matter is studied, and it is found that to the lowest or- der of δµ/T the ratio of radiative viscosity to bulk viscosity is the same as that of the normal matter. PACS: 97.60.Jd, 21.65.-f, 95.30.Cq As one of the most important transport coefficients, bulk viscosities of simple npe matter, of hyperon matter and even of quark matter, both in normal and superfluid states, have been extensively studied [1–18], for more ref- erences see [19]. In fact, the mechanical energy of density perturbations is not only dissipated to heat via bulk viscosity, but also is radiated away via neutrinos, this was first pointed out by Finiz and Wolf in 1968[1]. However, the damping mech- anism through neutrinos is ignored for several decades, until recently, Sa’d and Schaffner-Bielich [20] named this mechanism the radiative viscosity, and found it is 1.5 times larger than the bulk viscosity to all Urca processes in the lowest order of δµ/T , both in nuclear matter and quark matter. Yang et al. [21] studied non-linear ef- fect of radiative viscosity of npe matter in neutron stars for both direct Urca process and modified Urca process, and found that non-linear effect will decrease the ratio of radiative viscosity to bulk viscosity from 1 . 5 to 0.5 (for direct Urca process) and 0.375 (for modified Urca pro- cess); which means that for small oscillations of neutron star the large fraction of oscillation energy is emitted as neutrinos, but for large enough ones bulk viscous dissi- pation dominates. It’s well known that below certain critical tempera- ture nucleons in neutron star matter will be in superfluid states. The bulk viscosity of superfluid nucleon matter have been studied [4, 5, 17], and it turns out that the superfluidity may strongly reduce bulk viscosity. This paper aims to study the radiative viscosity of superfluid nucleon matter, in other words, we will give the relation- ship between radiative viscosity and bulk viscosity in the superfluid case. Both the bulk viscosity and the radiative viscosity of npe matter are related to direct Urca process ( n → p + e + ν$_{e}$ , p + e → n + ν$_{e}$ ) and modified Urca process ( n + N → p + N + e + ν$_{e}$ , p + N + e → n + N + ν$_{e}$ ) in different conditions. As shown by [22], the direct Urca process is allowed by the momentum conservation when p$_{F}$$_{n}$ < p$_{F}$$_{p}$ + p$_{F}$$_{e}$ , and for pure npe matter where p$_{F}$$_{p}$ = p$_{F}$$_{e}$ , it corresponds to n$_{p}$/n > 1 / 9. This happens if the density is several times larger than the standard nuclear matter density ρ 0 = 2 . 8 × 10 $^{14}$gcm − $^{3}$. In the following, we focus on the direct Urca process. Let us first recall the formulae of radiative viscosity of simple non-superfluid npe matter [7, 20]. Considering a periodic perturbation to the baryon number density It will lead to a deviation from β -equilibrium character- ized by where µ$_{n}$ , µ$_{p}$ and µ$_{e}$ are the chemical potentials of the neutrons, protons and electrons. δµ can be expressed in terms of the variations of two independent variables δn$_{b}$ and δX$_{p}$ , where X$_{p}$ = n$_{p}$/n$_{b}$ is the proton fraction and the coeffi- cient functions C and B are given by To the leading order, the net reaction rate and the incre- ments of neutrino emissivity due to off-equilibrium could be written as where[21, 23] n$_{b}$ ( t ) = n$_{b}$ 0 + R e ( δn$_{b}$e $^{iωt}$) . (1) δµ = µ$_{p}$ + µ$_{e}$ − µ$_{n}$ = δµ$_{p}$ + δµ$_{e}$ − δµ$_{n}$, (2) δµ = C δn$_{b}$ n$_{b}$ + BδX$_{p}$, (3) C = n$_{p}$ ∂µ$_{p}$ ∂n$_{p}$ + n$_{e}$ ∂µ$_{e}$ ∂n$_{e}$ − n$_{n}$ ∂µ$_{n}$ ∂n$_{n}$ , (4) B = n$_{b}$ ( ∂µ$_{p}$ ∂n$_{p}$ + ∂µ$_{e}$ ∂n$_{e}$ + ∂µ$_{n}$ ∂n$_{n}$ ) . (5) Γ$_{ν}$ − Γ$_{ν}$ = − λ 0 ξ 2 δµ , (6) ˙ E loss = S 0 ξ $^{2}$, (7) λ 0 = 714 457 ϵ ( T, 0) , (8) S 0 = 1071 457 ϵ ( T, 0) , (9) $^{∗}$This research is supported by NFSC under Grants No. 11073008. $^{∗∗}$Email: ysh@phy.ccnu.edu.cn Telephone number: 13545359902, 13545119781 arXiv:1001.0382v3 [astro-ph.HE] 11 Sep 2011