Theory of Nuclear Matter for Neutron Stars and ... - Graduate Physics

A.1 Taylor expansion integrationTheintegrandintheYukawaintegrationisexpandtotakeintoaccount thevariationbetweenr and r +∆. For example, for given interval (r i ,r i +δ),f(r) =f(r i )+f ′ (r i )(r −r i )+ 1 2 f′′ (r i )(r −r i ) 2 + 1 3! f(3) (r i )(r −r i ) 3 +···=f i + f i+1 −f i−1(r −r i )+ f i+1 −2f i +f i−1(r −r2∆ 2∆ 2 i ) 2 +··· .(A.4)By doing this, we can eliminate the contamination in the integral between r i and r i+1 whenwe use simple trapezoid rule ( ∫ i+1if(x)dx = 1 2 ∆(f i +f i+1 ) ).A.1.1 1D plane parallel nuclear matterThe study of 1D parallel (semi-infinite) nuclear matter sheds light on how the surface tensionchanges for a given temperature and proton fraction. In the semi-infinite nuclear matter,without loss of generality, we can say g(r) = g(x,y,z) = g(z). Then the original finite rangeintegration becomes˜g(z o ) =∫ ∞−∞∫ ∞ ∫ ∞dx dy−∞ −∞√e − x 2 +y 2 +(z−z o) 2 /adz4πa 2√ x 2 +y 2 +(z −z o ) g(z).2(A.5)Changing variables, x 2 +y 2 = ρ 2 , ∫ dx ∫ dy = 2π ∫ ρdρ give˜g(z o ) = 1 2= 12a= 12a∫ ∞−∞∫ ∞∫ ∞dzg(z)−∞∫ ∞dzg(z)e −|z−zo|/a−∞0e − √ρ 2 +(z−z o) 2 /aa 2√ ρ 2 +(z −z o ) 2 g(z)dzg(z)(−)e − √ρ 2 +(z−z o) 2 /a | 0 ∞(A.6)With the above general equation in 1D plane parallel case, we apply Taylor expansion forsmooth distance dependent function u.ũ(z) = 1 [∫ z∫ ∞]u(z ′ )e (z′ −z)/a dz ′ + u(z ′ )e (z−z′)/a dz ′ ≡ u − (z)+u + (z) (A.7)2a −∞z107