2 Γ$_{ν}$ − Γ$_{ν}$ = − λ ξ 2 δµ , (20) ˙ E loss = S ξ $^{2}$. (21) R = ( S λ ) ζ. (22) ϵ ( T, 0) = 3 . 3 × 10 − 14 ( x$_{p}$ρ ρ 0 ) 1 / 3 T $^{6}$MeV$^{5}$. (10) 〈 ˙ E diss 〉 = − ζ τ ∫ τ 0 dt ( ∇ · ⃗ v ) 2 , (11) 〈 ˙ E loss 〉 = R τ ∫ τ 0 dt ( ∇ · ⃗ v ) 2 , (12) Γ$_{ν}$ − Γ$_{ν}$ = 4 π (2 π ) $_{8}$T 5   3 ∏ j $_{=1}$∫ d Ω$_{j}$   δ ( P$_{f}$ − P$_{i}$ ) | M$_{fi}$ | 2 3 ∏ j =1 p$_{F}$$_{j}$ m ∗ j + ∞ ∫ 0 dx$_{ν}$x 2 $_{ν}$[ J ( x$_{ν}$ − ξ, v$_{j}$ ) − J ( x$_{ν}$ + ξ, v$_{j}$ )] , (23) ζ = − 2 〈 ˙ E diss 〉 ( υ 0 Δ υ ) 2 ( τ 2 π ) 2 , (13) R = 2 〈 ˙ E loss 〉 ( υ 0 Δ υ ) 2 ( τ 2 π ) 2 , (14) 〈 ˙ E diss 〉 = − ∫ τ 0 (Γ$_{ν}$ − Γ$_{ν}$) δµdt, (15) ϵ ( ξ, v$_{j}$ ) = 4 π (2 π ) $_{8}$T 6   3 ∏ j $_{=1}$∫ d Ω$_{j}$   δ ( P$_{f}$ − P$_{i}$ ) | M$_{fi}$ | 2 3 ∏ j =1 p$_{F}$$_{j}$ m ∗ j + ∞ ∫ 0 dx$_{ν}$x 3 $_{ν}$[ J ( x$_{ν}$ − ξ, v$_{j}$ ) + J ( x$_{ν}$ + ξ, v$_{j}$ )] , (24) J ( x, v$_{j}$ ) = + ∞ ∫ −∞ dx 1 dx 2 dx 3 f ( z 1 ) f ( z 2 ) f ( x 3 ) × δ ( z 1 + z 2 + x 3 − x ) , (25) 〈 ˙ E loss 〉 = ∫ τ 0 ˙ E loss dt. (16) ζ = λC 2 ω 2 + ( λB/n$_{b}$ ) $_{2}$, (17) R = ( S 0 λ 0 ) ζ. (18) R = 3 2 ζ. (19) ε − µ = sign( η ) √ δ 2 + η $^{2}$, (26) ξ = δµ/T , and ϵ ( T, 0) is the neutrino emissivity in equi- librium Now let us consider the effect of nucleon superfluidity on radiative viscosity. Assuming Apparently, For a periodic process, the expansion and contraction of the system will induce not only the dissipation of os- cillation energy to heat, but also the loss of oscillation energy through an increasing of the neutrino emissivity. Bulk viscous coefficient ζ and radiative viscous coefficient R can be defined for the description of these dissipation mechanisms, respectively [20] In the following, we will show how to calculate S /λ in the lowest order of ξ . In non-beta equilibrium, the net reaction rate of direct Urca process with nucleon superfluidity is [24] where ⃗ v is the hydrodynamic velocity associated with the density oscillations, and τ = 2 π/ω is the oscillation period. Using the continuity equation, one obtains and the total neutrino emissivity is[25, 26] where the energy dissipation is and the neutrino emissivity caused by the oscillation is where Here, we only present the results, for detailed calcula- tions see [20] and the subscript j = 1 , 2 , 3 corresponds to n, p, e re- spectively. v$_{j}$ = Δ$_{j}$ T is the gap amplitude and z$_{j}$ = ε$_{j}$ − µ$_{j}$ T (where j = 1 , 2), x 3 = ε$_{e}$ − µ$_{e}$ T . x$_{ν}$ = ε$_{ν}$ T is the dimen- sionless energy of the neutrino, and f ( x ) = (1 + e $^{x}$) − 1 is the Fermi-Dirac functions of nucleons and electrons, p$_{F}$$_{j}$ is the Fermi momentum and m ∗ j is the effective particle mass. d Ω$_{j}$ is the solid angle element in the direction of the particle momentum, and | M$_{fi}$ | 2 is the squared reac- tion amplitude. From Eqs. (8)and(9), one gets This relation only holds for small oscillations. If the per- turbation amplitude is large enough, the non-linear effect must be taken into account and this simple relation is no longer correct [21]. Here, we want to stress that z 1 and z 2 in the above two formula carry all the information about nucleon su- perfluidity. Whether for neutron superfluidity or proton superfluidity, near the Fermi surface we have