3 J ( x$_{ν}$ − ξ, v$_{j}$ ) − J ( x$_{ν}$ + ξ, v$_{j}$ ) = − 2 ξ∂J ( x$_{ν}$, v$_{j}$ ) /∂x$_{ν}$, (27) J ( x$_{ν}$ − ξ, v$_{j}$ )+ J ( x$_{ν}$ + ξ, v$_{j}$ ) − 2 J ( x$_{ν}$, v$_{j}$ ) = ξ $^{2}$∂$^{2}$J ( x$_{ν}$, v$_{j}$ ) /∂x 2 $_{ν}$, (28) ΔΓ = Γ$_{ν}$( ξ, v$_{j}$ ) − Γ$_{ν}$( ξ, v$_{j}$ ) = 4 π (2 π ) $_{8}$T 5   3 ∏ j $_{=1}$∫ d Ω$_{j}$   δ ( P$_{f}$ − P$_{i}$ ) | M$_{f i}$ | 2 3 ∏ j =1 p$_{F}$$_{j}$ m ∗ j × ( − 4 ξ ) + ∞ ∫ 0 dx$_{ν}$x$_{ν}$J ( x$_{ν}$, v$_{j}$ ) , (29) ˙ E loss = ϵ ( ξ, v$_{j}$ ) − ϵ (0 , v$_{j}$ ) = 4 π (2 π ) $_{8}$T 6   3 ∏ j $_{=1}$∫ d Ω$_{j}$   δ ( P$_{f}$ − P$_{i}$ ) | M$_{f i}$ | 2 3 ∏ j =1 p$_{F}$$_{j}$ m ∗ j × 6 ξ 2 + ∞ ∫ 0 dx$_{ν}$x$_{ν}$J ( x$_{ν}$, v$_{j}$ ) . (30) S /λ = 3 2 , (31) where η = v F ( p − p F ), v F and p F are the Fermi velocity and Fermi momentum, respectively; and δ 2 = Δ $^{2}$F ( ϑ ), Δ is the energy gap and F ( ϑ ) describes the dependence of the gap on the angle ϑ between the quantization axis and the particle momentum. For different types of nu- cleon superfluidity, the expression of F ( ϑ ) is completely different, which can be seen in [24–26]. In the case of ξ ≪ 1, to the lowest order we have which is the same as S 0 /λ 0 . This means that the nucleon superfluidity doesn’t change the value of R /ζ . In summary, we have studied the radiative viscous co- efficient of superfluid npe matter, and find that for di- rect Urca process, the ratio of radiative viscosity to bulk viscosity in the lowest order of ξ yielded by Sa’d and Schaffner-Bielich [20] for normal nucleons could be ex- tended to the superfluid case. In fact, it can be seen from our calculations that this is correct for modified Urca process, too. Thus, to the lowest order of ξ , the relation factor of 3 2 between the radiative viscosity and the bulk viscosity of npe matter is a generic one. The following are the extensive discussions to our re- sult: then the net reaction rate is and the neutrino emissivity due to the departure from β -equilibrium is First, Although we have S /λ = S 0 /λ 0 in the low- est order, the viscosities in the superfluid case are dif- ferent from these in the non-superfluid case. The bulk and radiative viscous coefficients could be easily ex- pressed as ζ = ζ 0 R Γ and R = R 0 R$_{ϵ}$ , where ζ 0 and R 0 are the viscosities in the non-superfluid case, R Γ = ΔΓ( ξ, v$_{j}$ ) / ΔΓ( ξ ) and R$_{ϵ}$ = ˙ E loss ( ξ, v$_{j}$ ) / ˙ E loss ( ξ ) are the reduction factors caused by superfluidity. Of course, our result shows that to the lowest order of ξ , R$_{ϵ}$ equals to R Γ (note that R Γ has been calculated numerically by Haensel et al. [4, 5]). Second, our result satisfies the general relationship ∂ ˙ E loss /∂δµ = 3(Γ$_{ν}$ − Γ$_{ν}$), which has been found by Flores- Tuli´ an and Reisenegger [27]. They found that this rela- tionship holds both in the case of normal nucleons and in the case of superfluid nucleons, and both in linear case and for precise solutions. Finally, our result based on the fact that to the low- est order of ξ , the existence of superfluidity don’t change the relationship between the neutrino emissivity due to the departure from β -equilibrium and energy dissipation in the form of heat. Nevertheless, it is not correct for higher order calculations. As shown in [21, 23], for nor- mal matter ΔΓ and ˙ E loss can be solved analytically as polynomials of ξ , while for superfluid matter they must be solved numerically. As a result, we expect that in non-linear regime, the ratio of radiative viscosity to bulk viscosity of superfluid matter no longer equals to that of normal matter. However, the calculation in the non- linear case is far more complicated and we will consider it in our further study. Note that, the leading order of ΔΓ is the first order of ξ , while for ˙ E loss it is the second order of ξ . Unlike normal matter, Eqs.(29) and (30) haven’t exact analytical solutions. However, one can find that ΔΓ δµ is in proportion to ˙ E loss , thus we can easily obtain [1] Finzi A and Wolf R A 1968 Astrophys. J. 153 835 [2] Sawyer R F 1989 Phys. Rev. D 39 3804 [3] Haensel P and Schaeffer R 1992 Phys. Rev. D 45 4708 [9] Gupta V K, Wadhwa A, Singh S and Anand J D 1997 Pramana - J. Phys. 49 443 [10] Lindblom L and Owen B J 2002 Phys. Rev. D 65 063006 [11] Zheng X P, Yang S H and Li J R 2002 Phys. Lett. B 548 29 [12] Zheng X P, Liu X W, Kang M and Yang S H 2004 Phys. Rev. C 70 015803 [4] Haensel P, Levenfish K P and Yakovlev D G 2000 Astron. Astrophys. 357 1157 [5] Haensel P, Levenfish K P and Yakovlev D G 2001 Astron. Astrophys. 372 130 [6] Wang Q D and Lu T 1984 Phys. Lett. B 148 211 [7] Sawyer R F 1989 Phys. Lett. B 233 412 [8] Madsen J 1992 Phys. Rev. D 46 3290 [13] Zheng X P, Kang M, Liu X W and Yang S H 2005 Phys. Rev. C 72 025809 [14] Pan N N, Zheng X P, and Li J R 2006 Mon. Not. Roy.