def HS2011_Omega_SO_3p5PN_pt6(m1, m2, n12U, p1U, p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (+(-(8 / m1 + 9 * m2 / (2 * m1**2)) * dot(n12U, p1U) +
(59 / (4 * m1) + 27 / (2 * m2)) * dot(n12U, p2U)) *
cross(p1U, p2U)[i]) / r12**3
return Omega1
def f_H_SS_2PN(m1,m2, S1U,S2U, nU, q):
S0U = ixp.zerorank1()
for i in range(3):
S0U[i] = (1 + m2/m1)*S1U[i] + (1 + m1/m2)*S2U[i]
global H_SS_2PN
mu = m1*m2 / (m1 + m2)
H_SS_2PN = mu/(m1 + m2) * (3*dot(S0U,nU)**2 - dot(S0U,S0U)) / (2*q**3)
def f_H_NS_3PN(m1, m2, PU, nU, q):
mu = m1 * m2 / (m1 + m2)
eta = m1 * m2 / (m1 + m2)**2
pU = ixp.zerorank1()
for i in range(3):
pU[i] = PU[i] / mu
global H_NS_3PN
H_NS_3PN = mu * (
+div(1, 128) *
(-5 + 35 * eta - 70 * eta**2 + 35 * eta**3) * dot(pU, pU)**4 +
div(1, 16) *
(+(-7 + 42 * eta - 53 * eta**2 - 5 * eta**3) * dot(pU, pU)**3 +
(2 - 3 * eta) * eta**2 * dot(nU, pU)**2 * dot(pU, pU)**2 + 3 *
(1 - eta) * eta**2 * dot(nU, pU)**4 * dot(pU, pU) -
5 * eta**3 * dot(nU, pU)**6) / q +
(+div(1, 16) *
(-27 + 136 * eta + 109 * eta**2) * dot(pU, pU)**2 + div(1, 16) *
(+17 + 30 * eta) * eta * dot(nU, pU)**2 * dot(pU, pU) + div(1, 12) *
(+5 + 43 * eta) * eta * dot(nU, pU)**4) / q**2 +
(+(-div(25, 8) +
(div(1, 64) * sp.pi**2 - div(335, 48)) * eta - div(23, 8) * eta**2)
* dot(pU, pU) +
(-div(85, 16) - div(3, 64) * sp.pi**2 - div(7, 4) * eta) * eta *
dot(nU, pU)**2) / q**3 +
(+div(1, 8) + (div(109, 12) - div(21, 32) * sp.pi**2) * eta) / q**4)
def HS2011_Omega_SO_3p5PN_pt7(m1, m2, n12U, p1U, p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (+(181 * m1 * m2 / 16 + 95 * m2**2 / 4 + 75 * m2**3 /
(8 * m1)) * cross(n12U, p1U)[i] -
(21 * m1**2 / 2 + 473 * m1 * m2 / 16 + 63 * m2**2 / 4) *
cross(n12U, p2U)[i]) / r12**4
return Omega1
def cross(vec1,vec2):
vec1_cross_vec2 = ixp.zerorank1()
LeviCivitaSymbol = ixp.LeviCivitaSymbol_dim3_rank3()
for i in range(3):
for j in range(3):
for k in range(3):
vec1_cross_vec2[i] += LeviCivitaSymbol[i][j][k]*vec1[j]*vec2[k]
return vec1_cross_vec2
def dE_GW_dt_OBKPSS2015_consts(m1, m2, _n12U, S1U, S2U): # _n12U unused.
# define scalars:
m = (m1 + m2)
nu = m1 * m2 / m**2
delta = (m1 - m2) / m
# define vectors:
Stot = ixp.zerorank1()
Sigma = ixp.zerorank1()
l = ixp.zerorank1()
l[2] = sp.sympify(1)
chi1U = ixp.zerorank1()
chi2U = ixp.zerorank1()
chi_s = ixp.zerorank1()
chi_a = ixp.zerorank1()
for i in range(3):
Stot[i] = S1U[i] + S2U[i]
Sigma[i] = (m1 + m2) / m2 * S2U[i] - (m1 + m2) / m1 * S1U[i]
chi1U[i] = S1U[i] / m1**2
chi2U[i] = S2U[i] / m2**2
chi_s[i] = div(1, 2) * (chi1U[i] + chi2U[i])
chi_a[i] = div(1, 2) * (chi1U[i] - chi2U[i])
# define scalars that depend on vectors
s_l = dot(Stot, l) / m**2
# s_n = dot(Stot,n12U)/m**2
sigma_l = dot(Sigma, l) / m**2
# sigma_n = dot(Sigma,n12U)/m**2
return nu, delta, l, chi_a, chi_s, s_l, sigma_l
def HS2011_Omega_SO_3p5PN_pt3(m1,m2, n12U, p1U,p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (+(-9*dot(n12U,p1U)*dot(p1U,p1U)/(16*m1**4)
+dot(p1U,p1U)*dot(n12U,p2U)/(m1**3*m2)
+27*dot(n12U,p1U)*dot(n12U,p2U)**2/(16*m1**2*m2**2)
-dot(n12U,p2U)*dot(p1U,p2U)/(8*m1**2*m2**2)
-5*dot(n12U,p1U)*dot(p2U,p2U)/(16*m1**2*m2**2))*cross(p1U,p2U)[i])/r12**2
return Omega1
def HS2011_Omega_SO_3p5PN_pt5(m1,m2, n12U, p1U,p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (+(+4*dot(n12U,p1U)**2/m1
+13*dot(p1U,p1U)/(2*m1)
+5*dot(n12U,p2U)**2/m2
+53*dot(p2U,p2U)/(8*m2)
-(211/(8*m1) + 22/m2)*dot(n12U,p1U)*dot(n12U,p2U)
-(47/(8*m1) + 5/m2)*dot(p1U,p2U))*cross(n12U,p2U)[i])/r12**3
return Omega1
def HS2011_Omega_SO_3p5PN_pt4(m1,m2, n12U, p1U,p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (+(-3*m2*dot(n12U,p1U)**2/(2*m1**2)
+((-3*m2)/(2*m1**2) + 27*m2**2/(8*m1**3))*dot(p1U,p1U)
+(177/(16*m1) + 11/m2)*dot(n12U,p2U)**2
+(11/(2*m1) + 9*m2/(2*m1**2))*dot(n12U,p1U)*dot(n12U,p2U)
+(23/(4*m1) + 9*m2/(2*m1**2))*dot(p1U,p2U)
-(159/(16*m1) + 37/(8*m2))*dot(p2U,p2U))*cross(n12U,p1U)[i])/r12**3
return Omega1
def f_compute_quantities(mOmega, m1, m2, n12U, S1U, S2U, which_quantity):
if not which_quantity in ('dM_dt', 'dE_GW_dt', 'dE_GW_dt_plus_dM_dt'):
print("which_quantity == " + str(which_quantity) +
" not supported!")
sys.exit(1)
nu, delta, l, chi_a, chi_s, s_l, sigma_l = dE_GW_dt_OBKPSS2015_consts(
m1, m2, n12U, S1U, S2U)
x = (mOmega)**div(2, 3)
# Compute b_5_Mdot:
b_5_Mdot = (-div(1, 4) *
(+(1 - 3 * nu) * dot(chi_s, l) *
(1 + 3 * dot(chi_s, l)**2 + 9 * dot(chi_a, l)**2) +
(1 - nu) * delta * dot(chi_a, l) *
(1 + 3 * dot(chi_a, l)**2 + 9 * dot(chi_s, l)**2)))
if which_quantity == "dM_dt":
return div(32, 5) * nu**2 * x**5 * b_5_Mdot * x**div(5, 2)
b = ixp.zerorank1(DIM=10)
b[2] = -div(1247, 336) - div(35, 12) * nu
b[3] = +4 * sp.pi - 4 * s_l - div(5, 4) * delta * sigma_l
b[4] = (-div(44711, 9072) + div(9271, 504) * nu + div(65, 18) * nu**2 +
(+div(287, 96) + div(1, 24) * nu) * dot(chi_s, l)**2 -
(+div(89, 96) + div(7, 24) * nu) * dot(chi_s, chi_s) +
(+div(287, 96) - 12 * nu) * dot(chi_a, l)**2 +
(-div(89, 96) + 4 * nu) * dot(chi_a, chi_a) +
div(287, 48) * delta * dot(chi_s, l) * dot(chi_a, l) -
div(89, 48) * delta * dot(chi_s, chi_a))
b[5] = (-div(8191, 672) * sp.pi - div(9, 2) * s_l -
div(13, 16) * delta * sigma_l + nu *
(-div(583, 24) * sp.pi + div(272, 9) * s_l +
div(43, 4) * delta * sigma_l))
if which_quantity == "dE_GW_dt_plus_dM_dt":
b[5] += b_5_Mdot
b[6] = (+div(6643739519, 69854400) + div(16, 3) * sp.pi**2 -
div(1712, 105) * gamma_EulerMascheroni -
div(856, 105) * sp.log(16 * x) +
(-div(134543, 7776) + div(41, 48) * sp.pi**2) * nu -
div(94403, 3024) * nu**2 - div(775, 324) * nu**3 -
16 * sp.pi * s_l - div(31, 6) * sp.pi * delta * sigma_l)
b[7] = (
+(+div(476645, 6804) + div(6172, 189) * nu - div(2810, 27) * nu**2)
* s_l +
(+div(9535, 336) + div(1849, 126) * nu - div(1501, 36) * nu**2) *
delta * sigma_l + (-div(16285, 504) + div(214745, 1728) * nu +
div(193385, 3024) * nu**2) * sp.pi)
b[8] = (+(-div(3485, 96) * sp.pi + div(13879, 72) * sp.pi * nu) * s_l +
(-div(7163, 672) * sp.pi + div(130583, 2016) * sp.pi * nu) *
delta * sigma_l)
b_sum = sp.sympify(1)
for k in range(9):
b_sum += b[k] * x**div(k, 2)
return div(32, 5) * nu**2 * x**5 * b_sum
def HS2011_Omega_SO_3p5PN_pt2(m1,m2, n12U, p1U,p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (+(-3*dot(n12U,p1U)*dot(n12U,p2U)*dot(p1U,p1U)/(2*m1**3*m2)
-15*dot(n12U,p1U)**2*dot(n12U,p2U)**2/(4*m1**2*m2**2)
+3*dot(p1U,p1U)*dot(n12U,p2U)**2/(4*m1**2*m2**2)
-dot(p1U,p1U)*dot(p1U,p2U)/(2*m1**3*m2)
+dot(p1U,p2U)**2/(2*m1**2*m2**2)
+3*dot(n12U,p1U)**2*dot(p2U,p2U)/(4*m1**2*m2**2)
-dot(p1U,p1U)*dot(p2U,p2U)/(4*m1**2*m2**2)
-3*dot(n12U,p1U)*dot(n12U,p2U)*dot(p2U,p2U)/(2*m1*m2**3)
-dot(p1U,p2U)*dot(p2U,p2U)/(2*m1*m2**3))*cross(n12U,p2U)[i])/r12**2
return Omega1
def f_H_SO_3p5PN(m1,m2, n12U,n21U, S1U, S2U, p1U,p2U, r12):
Omega1_3p5PNU = ixp.zerorank1()
Omega2_3p5PNU = ixp.zerorank1()
for i in range(3):
Omega1_3p5PNU[i] = HS2011_Omega_SO_3p5PN_pt1(m1,m2, n12U, p1U,p2U, r12)[i]
Omega1_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt2(m1,m2, n12U, p1U,p2U, r12)[i]
Omega1_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt3(m1,m2, n12U, p1U,p2U, r12)[i]
Omega1_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt4(m1,m2, n12U, p1U,p2U, r12)[i]
Omega1_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt5(m1,m2, n12U, p1U,p2U, r12)[i]
Omega1_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt6(m1,m2, n12U, p1U,p2U, r12)[i]
Omega1_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt7(m1,m2, n12U, p1U,p2U, r12)[i]
Omega2_3p5PNU[i] = HS2011_Omega_SO_3p5PN_pt1(m2,m1, n21U, p2U,p1U, r12)[i]
Omega2_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt2(m2,m1, n21U, p2U,p1U, r12)[i]
Omega2_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt3(m2,m1, n21U, p2U,p1U, r12)[i]
Omega2_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt4(m2,m1, n21U, p2U,p1U, r12)[i]
Omega2_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt5(m2,m1, n21U, p2U,p1U, r12)[i]
Omega2_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt6(m2,m1, n21U, p2U,p1U, r12)[i]
Omega2_3p5PNU[i]+= HS2011_Omega_SO_3p5PN_pt7(m2,m1, n21U, p2U,p1U, r12)[i]
global H_SO_3p5PN
H_SO_3p5PN = dot(Omega1_3p5PNU,S1U) + dot(Omega2_3p5PNU,S2U)
def f_H_SSS_3PN_pt(m1,m2, nU, S1U,S2U, p1U,p2U, r):
p2_minus_m2_over_4m1_p1 = ixp.zerorank1()
for i in range(3):
p2_minus_m2_over_4m1_p1[i] = p2U[i] - m2/(4*m1)*p1U[i]
H_SSS_3PN_pt = (+div(3,2)*(+dot(S1U,S1U)*dot(S2U,cross(nU,p1U))
+dot(S1U,nU)*dot(S2U,cross(S1U,p1U))
-5*dot(S1U,nU)**2*dot(S2U,cross(nU,p1U))
+dot(nU,cross(S1U,S2U))*(+dot(S1U,p1U)
-5*dot(S1U,nU)*dot(p1U,nU)))
-3*m1/(2*m2)*( +dot(S1U,S1U) *dot(S2U,cross(nU,p2U))
+2*dot(S1U,nU) *dot(S2U,cross(S1U,p2U))
-5*dot(S1U,nU)**2*dot(S2U,cross(nU,p2U)))
-dot(cross(S1U,nU),p2_minus_m2_over_4m1_p1)*(dot(S1U,S1U) - 5*dot(S1U,nU)**2))/(m1**2*r**4)
return H_SSS_3PN_pt
def f_Omega_SO_2p5PN(m1, m2, n12U, p1U, p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (+(+(-div(11, 2) * m2 - 5 * m2 ** 2 / m1) * cross(n12U, p1U)[i]
+ (6 * m1 + div(15, 2) * m2) * cross(n12U, p2U)[i]) / r12 ** 3
+ (+(-div(5, 8) * m2 * dot(p1U, p1U) / m1 ** 3
- div(3, 4) * dot(p1U, p2U) / m1 ** 2
+ div(3, 4) * dot(p2U, p2U) / (m1 * m2)
- div(3, 4) * dot(n12U, p1U) * dot(n12U, p2U) / m1 ** 2
- div(3, 2) * dot(n12U, p2U) ** 2 / (m1 * m2)) * cross(n12U, p1U)[i]
+ (dot(p1U, p2U) / (m1 * m2) + 3 * dot(n12U, p1U) * dot(n12U, p2U) / (m1 * m2)) *
cross(n12U, p2U)[i]
+ (div(3, 4) * dot(n12U, p1U) / m1 ** 2 - 2 * dot(n12U, p2U) / (m1 * m2)) * cross(p1U, p2U)[
i]) / r12 ** 2)
return Omega1
def HS2011_Omega_SO_3p5PN_pt1(m1,m2, n12U, p1U,p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = ((+7*m2*dot(p1U,p1U)**2/(16*m1**5)
+9*dot(n12U,p1U)*dot(n12U,p2U)*dot(p1U,p1U)/(16*m1**4)
+3*dot(p1U,p1U)*dot(n12U,p2U)**2/(4*m1**3*m2)
+45*dot(n12U,p1U)*dot(n12U,p2U)**3/(16*m1**2*m2**2)
+9*dot(p1U,p1U)*dot(p1U,p2U)/(16*m1**4)
-3*dot(n12U,p2U)**2*dot(p1U,p2U)/(16*m1**2*m2**2)
-3*dot(p1U,p1U)*dot(p2U,p2U)/(16*m1**3*m2)
-15*dot(n12U,p1U)*dot(n12U,p2U)*dot(p2U,p2U)/(16*m1**2*m2**2)
+3*dot(n12U,p2U)**2*dot(p2U,p2U)/(4*m1*m2**3)
-3*dot(p1U,p2U)*dot(p2U,p2U)/(16*m1**2*m2**2)
-3*dot(p2U,p2U)**2/(16*m1*m2**3))*cross(n12U,p1U)[i])/r12**2
return Omega1
def f_MOmega(m1,m2, chi1U,chi2U, r):
a = ixp.zerorank1(DIM=10)
MOmega__a_2_thru_a_4(m1,m2, chi1U[0],chi1U[1],chi1U[2], chi2U[0],chi2U[1],chi2U[2])
a[2] = a_2
a[3] = a_3
a[4] = a_4
MOmega__a_5_thru_a_6(m1,m2, chi1U[0],chi1U[1],chi1U[2], chi2U[0],chi2U[1],chi2U[2])
a[5] = a_5
a[6] = a_6
MOmega__a_7( m1,m2, chi1U[0],chi1U[1],chi1U[2], chi2U[0],chi2U[1],chi2U[2])
a[7] = a_7
global MOmega
MOmega = 1 # Term prior to the sum in parentheses
for k in range(8):
MOmega += a[k]/r**div(k,2)
MOmega *= 1/r**div(3,2)
def f_H_Newt__H_NS_1PN__H_NS_2PN(m1,m2, PU, nU, q):
mu = m1*m2 / (m1+m2)
eta = m1*m2 / (m1+m2)**2
pU = ixp.zerorank1()
for i in range(3):
pU[i] = PU[i]/mu
global H_Newt, H_NS_1PN, H_NS_2PN
H_Newt = mu*(+div(1,2)*dot(pU,pU) - 1/q)
H_NS_1PN = mu*(+div(1,8)*(3*eta-1)*dot(pU,pU)**2
-div(1,2)*((3+eta)*dot(pU,pU) + eta*dot(nU,pU)**2)/q
+div(1,2)/q**2)
H_NS_2PN = mu*(+div(1,16)*(1 - 5*eta + 5*eta**2)*dot(pU,pU)**3
+div(1,8)*(+(5 - 20*eta - 3*eta**2)*dot(pU,pU)**2
-2*eta**2*dot(nU,pU)**2*dot(pU,pU)
-3*eta**2*dot(nU,pU)**4)/q
+div(1,2)*((5+8*eta)*dot(pU,pU) + 3*eta*dot(nU,pU)**2)/q**2
-div(1,4)*(1+3*eta)/q**3)
def f_p_t(m1, m2, chi1U, chi2U, r):
q = m2 / m1 # It is assumed that q >= 1, so m2 >= m1.
a = ixp.zerorank1(DIM=10)
p_t__a_2_thru_a_4(m1, m2, chi1U[0], chi1U[1], chi1U[2], chi2U[0], chi2U[1],
chi2U[2])
a[2] = a_2
a[3] = a_3
a[4] = a_4
p_t__a_5_thru_a_6(m1, m2, chi1U[0], chi1U[1], chi1U[2], chi2U[0], chi2U[1],
chi2U[2])
a[5] = a_5
a[6] = a_6
p_t__a_7(m1, m2, chi1U[0], chi1U[1], chi1U[2], chi2U[0], chi2U[1],
chi2U[2])
a[7] = a_7
global p_t
p_t = 1 # Term prior to the sum in parentheses
for k in range(8):
p_t += a[k] / r**div(k, 2)
p_t *= q / (1 + q)**2 * 1 / r**div(1, 2)
# Some references use r, others use q to represent the
# distance between the two point masses. This is rather
# confusing since q is also used to represent the
# mass ratio m2/m1. However, q is the canonical position
# variable name in Hamiltonian mechanics, so both are
# well justified. It should be obvious which is which
# throughout NRPyPN.
r, q = sp.symbols('r q', real=True)
chi1U = ixp.declarerank1('chi1U')
chi2U = ixp.declarerank1('chi2U')
# Euler-Mascheroni gamma constant:
gamma_EulerMascheroni = sp.EulerGamma
# Derived quantities used in Damour et al papers:
n12U = ixp.zerorank1()
n21U = ixp.zerorank1()
p1U = ixp.zerorank1()
p2U = ixp.zerorank1()
for i in range(3):
n12U[i] = +nU[i]
n21U[i] = -nU[i]
p1U[i] = +pU[i]
p2U[i] = -pU[i]
# Step 2.a: Define dot and cross product of vectors
def dot(vec1,vec2):
vec1_dot_vec2 = sp.sympify(0)
for i in range(3):
vec1_dot_vec2 += vec1[i]*vec2[i]
return vec1_dot_vec2
def f_Omega1(m1, m2, n12U, p1U, p2U, r12):
Omega1 = ixp.zerorank1()
for i in range(3):
Omega1[i] = (div(3, 2) * m2 / m1 * cross(n12U, p1U)[i] -
2 * cross(n12U, p2U)[i]) / r12**2
return Omega1
