[
]
M
(1 / g - 2) 1 + ψk13 Mσ(0.5 / g + 1) / 3 + k12
H=
(C25)
σ
where
M 1
1
A = C21,
b1 = - + + C21φ ,
b2 = - C21φ,
3 St
3
G
1
M=
σd =
Xd .
,
1+ R
∆G1
The derivatives of β and g can be found from the following equations
a +a a
dβ
= β′ = 5 1 4
(C26)
dσ
a3 + a2a4
dg
= g′ = a1 - a2β′
(C27)
dσ
where
(σ + φ)β(β + 2)
α 21
a1 =
1 -
m
m
α 21(σ + φ)
[2σ(β + 1) + 2φ]
a2 =
m2
M 2ρ21(g - 1)
Aψk13 (2 - 1 / g)βM 2
a3 =
-
- k21σ +
3gH
g
ST
Mψk13 (2 - 1 / g)βM 2
2ρ21 M
βM 2
[
]
(0.5 / g + 1)Mσ d k13 + σ / 2 + σ d k13
+ 2 β +
a4 =
σg3
g
ST
3gH
βM 2 σd k13 (1 - 2g)
a5 = k21(β + 2) +
+ P
σg
σ2g
ψk13 (2 - 1 / g) β
Mσd k13 (1 - 2g)
M
- A (2σ + φ) + σ + φ - 1 / ST + (0.5 / g + 1)
P=
- 1
σH
σg
3
3
[
]
m = β σ(β + 2) + 2φ .
The problem has now been reduced to a simple numerical quadrature of eq C23 using the auxiliary rela-
tions of eq C24C26. A FORTRAN program to carry out the integration is listed in Appendix E as
PFTSYN.FOR.
This approximation is inferior to method 1 since the variables in eq C23 are not strictly separable.
Nevertheless, predictions for modest times compare quite well to those of method 1.
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