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EqLPZonal.m
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%% EqLPZonal
% Desc: Computes long-period effects up to first order for Jn (n > 2) for Equinoctial elements.
% Equinoctial elements: [a l+g+h tan(i/2)*cos(h) tan(i/2)*sin(h) e*cos(g+h) e*sin(g+h)];
% Reference:
% B. Mahajan, S. R. Vadali and K. T. Alfriend, �Analytic Solution For Satellite
% Relative Motion: The Complete Zonal Gravitational Problem,� 26th AAS/AIAA
% Space Flight Mechanics Meeting, Napa, CA, February 2016.
% Author: Bharat Mahajan (https://github.com/princemahajan)
function [Eqlp, Dlp] = EqLPZonal(Xm,degree,mu,Re,Jcoeff,InverseOn, JacobianOn,tol)
O2Msign = 1;
if InverseOn == true
O2Msign = -1;
end
% for each zonal
narr = 3:1:degree;
J2 = Jcoeff(2);
a = Xm(1); Lambda = Xm(2); p1 = Xm(3); p2 = Xm(4); q1 = Xm(5); q2 = Xm(6);
DelXm = Del2Eqn(Xm,mu,true);
l = DelXm(1); g = DelXm(2); h = DelXm(3);L = DelXm(4);G =DelXm(5); H= DelXm(6);
e = sqrt(q1^2 + q2^2);
eta = sqrt(1 - e^2);
i = 2*atan(sqrt(p1^2 + p2^2));
si = sin(i);
ci = cos(i);
ch = cos(h);
sh = sin(h);
si2 = sin(i/2);
ci2 = cos(i/2);
sgh = sin(g+h); cgh = cos(g+h);
sg = sin(g); cg = cos(g);
% J2 Equinoctial First Order Long-Period Effects (Using Maple)
J = J2;
R__e = Re;
% First order Long-Period effects due to higher degree zonals
Jn_LPX1 = zeros(6,1);
Dlpn = zeros(6);
for ctr = 1:length(narr)
n = narr(ctr);
Jn = Jcoeff(n);
% SMA
anlp1 = 0;
% Mean Longitude
[lW1W2] = getW1W2(@MlongTerm,n,Jn,J2,a,e,i,eta,g,mu,Re);
Lambdanlp1 = sin(i)/sqrt(mu*a)*lW1W2;
% p1,p2
[pW1W2] = getW1W2(@p1p2Term,n,Jn,J2,a,e,i,eta,g,mu,Re);
[pW1W4] = getW1W4(@dummy,n,Jn,J2,a,e,i,eta,g,mu,Re);
p1nlp1 = cos(i)/(2*G*(cos(i/2))^2)*(sin(h)*pW1W2 - cos(h)*pW1W4);
p2nlp1 = cos(i)/(2*G*(cos(i/2))^2)*(-cos(h)*pW1W2 - sin(h)*pW1W4);
% q1,q2
[qW1W2] = getW1W2(@q1q2Term1,n,Jn,J2,a,e,i,eta,g,mu,Re);
[qW3W2] = getW3W2(@q1q2Term2,n,Jn,J2,a,e,i,eta,g,mu,Re);
[qW3W4] = getW3W4(@dummy,n,Jn,J2,a,e,i,eta,g,mu,Re);
q1nlp1 = sin(i)*sin(g+h)/G*(qW1W2 + eta^2*qW3W2) + sin(i)*eta/L*cos(g+h)*qW3W4;
q2nlp1 = -sin(i)*cos(g+h)/G*(qW1W2 + eta^2*qW3W2) + sin(i)*eta/L*sin(g+h)*qW3W4;
Jn_LPX1 = Jn_LPX1 + J2*[anlp1; Lambdanlp1; p1nlp1; p2nlp1; q1nlp1; q2nlp1];
if JacobianOn ~= true
continue;
end
% SMA LP partials
dadx = [0,0,0,0,0,0];
% Mean Longitude partials
dMLda = (-1/2/a)*Lambdanlp1 -(2*n-5)/2/a*Lambdanlp1;
dMLdML = 0;
W1W2t23 = getW1W2(@MLp12termW12,n,Jn,J2,a,e,i,eta,g,mu,Re);
W1W4t4 = getW1W4(@MLp12termW14,n,Jn,J2,a,e,i,eta,g,mu,Re);
dMLdp1 = 1/sqrt(mu*a)*(cos(i)*lW1W2*(1+ci)*ch + W1W2t23*(1+ci)*ch + W1W4t4*(1+ci)*sh);
dMLdp2 = 1/sqrt(mu*a)*(cos(i)*lW1W2*(1+ci)*sh + W1W2t23*(1+ci)*sh - W1W4t4*(1+ci)*ch);
mldq12t1 = getW3W2(@MLq12termW321,n,Jn,J2,a,e,i,eta,g,mu,Re);
mldq12t2 = getW3W2(@MLq12termW322,n,Jn,J2,a,e,i,eta,g,mu,Re);
mldq12t3 = getW3W4(@MLq12termW343,n,Jn,J2,a,e,i,eta,g,mu,Re);
dMLdq1 = si/sqrt(mu*a)*(mldq12t1*q1 + mldq12t2*cgh + mldq12t3*(-sgh));
dMLdq2 = si/sqrt(mu*a)*(mldq12t1*q2 + mldq12t2*sgh + mldq12t3*(cgh));
% p1
dp1da = -(2*n - 4)/2/a*p1nlp1;
dp1dML = 0;
t32 = e*getW3W2(@p12p12term32,n,Jn,J2,a,e,i,eta,g,mu,Re);
t34 = e*getW3W4(@p12p12term34,n,Jn,J2,a,e,i,eta,g,mu,Re);
t3b4 = e*getW3bW4(@p12p12term3b4,n,Jn,J2,a,e,i,eta,g,mu,Re);
t3b2 = e*getW3bW2(@p12p12term3b2,n,Jn,J2,a,e,i,eta,g,mu,Re);
t3bnjp = e*getW3bW2njp(@p12p12term3b2njp,n,Jn,J2,a,e,i,eta,g,mu,Re);
t3bjstar = e*getW3jstar(@p12p12termjstar,(n-1)/2,n,Jn,J2,a,e,i,eta,g,mu,Re);
dp1dp1 = 1/eta/sqrt(mu*a)*(sh*t32*ch + ch*t34*ch + sh*t3b4*sh + ch*t3b2*sh + ch*t3bnjp*sh + (sh*sg+ch*cg)*t3bjstar*sh);
dp1dp2 = 1/eta/sqrt(mu*a)*(sh*t32*sh + ch*t34*sh - sh*t3b4*ch - ch*t3b2*ch - ch*t3bnjp*ch - (sh*sg+ch*cg)*t3bjstar*ch);
t32sc = getW3W2(@p1q1Term32shcgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t34cc = getW3W4(@p1q1Term34chcgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t34ss = getW3W4(@p1q1Term34shsgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t32njpcs = getW3W2njp(@p1q1Term32njpchsgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
dp1dq1 = 1/eta^2*p1nlp1*q1 + ci/eta/sqrt(mu*a)/(1+ci)*(sh*t32sc*cgh + ch*t34cc*cgh + sh*t34ss*sgh + ch*t32njpcs*sgh);
dp1dq2 = 1/eta^2*p1nlp1*q2 + ci/eta/sqrt(mu*a)/(1+ci)*(sh*t32sc*sgh + ch*t34cc*sgh - sh*t34ss*cgh - ch*t32njpcs*cgh);
% p2
dp2da = -(2*n - 4)/2/a*p2nlp1;
dp2dML = 0;
dp2dp1 = 1/eta/sqrt(mu*a)*(-ch*t32*ch + sh*t34*ch - ch*t3b4*sh + sh*t3b2*sh + sh*t3bnjp*sh - (ch*sg-sh*cg)*t3bjstar*sh);
dp2dp2 = 1/eta/sqrt(mu*a)*(-ch*t32*sh + sh*t34*sh + ch*t3b4*ch - sh*t3b2*ch - sh*t3bnjp*ch + (ch*sg-sh*cg)*t3bjstar*ch);
dp2dq1 = 1/eta^2*p2nlp1*q1 + ci/eta/sqrt(mu*a)/(1+ci)*(-ch*t32sc*cgh + sh*t34cc*cgh - ch*t34ss*sgh + sh*t32njpcs*sgh);
dp2dq2 = 1/eta^2*p2nlp1*q2 + ci/eta/sqrt(mu*a)/(1+ci)*(-ch*t32sc*sgh + sh*t34cc*sgh + ch*t34ss*cgh - sh*t32njpcs*cgh);
% q1
dq1da = -(2*n-4)/2/a*q1nlp1;
dq1dML = 0;
t12sghch = getW3W2(@q1p1Term32sghch,n,Jn,J2,a,e,i,eta,g,mu,Re);
t14cghch = getW3W4(@q1p1Term34cghch,n,Jn,J2,a,e,i,eta,g,mu,Re);
t14sghsh = getW3W4(@q1p1Term34sghsh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t32njpcghsh = getW3W2(@q1p1Term32njpcghsh,n,Jn,J2,a,e,i,eta,g,mu,Re);
dq1dp1 = 1/eta/sqrt(mu*a)*(t12sghch*sgh*ch + eta^2*t14cghch*cgh*ch + t14sghsh*sgh*sh ...
+ eta^2*t32njpcghsh*cgh*sh);
dq1dp2 = 1/eta/sqrt(mu*a)*(t12sghch*sgh*sh + eta^2*t14cghch*cgh*sh - t14sghsh*sgh*ch ...
- eta^2*t32njpcghsh*cgh*ch);
t32sc = getW3W2(@q1q1Term32sghcgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t322sc = getW3bW2(@q1q1Term322sghcgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t34cc = getW3W4(@q1q1Term34cghcgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t324cc = getW3bW4(@q1q1Term324cghcgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t32cs = getW3W2(@q1q1Term32cghsgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t34ss = getW3W4(@q1q1Term34sghsgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t3b4ss = getW3bW4(@q1q1Term3b4sghsgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t3b2cs = getW3bW2(@q1q1Term3b2cghsgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
t3b2njpcs = getW3bW2njp(@q1q1Term3b2njpcghsgh,n,Jn,J2,a,e,i,eta,g,mu,Re);
dq1dq1 = 1/eta/sqrt(mu*a)*(sgh*t32sc*cgh + sgh*t322sc*cgh + cgh*t34cc*cgh + cgh*t324cc*cgh ...
+ cgh*t32cs*sgh + sgh*t34ss*sgh + sgh*t3b4ss*sgh + cgh*t3b2cs*sgh + cgh*t3b2njpcs*sgh);
dq1dq2 = 1/eta/sqrt(mu*a)*(sgh*t32sc*sgh + sgh*t322sc*sgh + cgh*t34cc*sgh + cgh*t324cc*sgh ...
- cgh*t32cs*cgh - sgh*t34ss*cgh - sgh*t3b4ss*cgh - cgh*t3b2cs*cgh - cgh*t3b2njpcs*cgh);
% q2
dq2da = -(2*n-4)/2/a*q2nlp1;
dq2dML = 0;
dq2dp1 = 1/eta/sqrt(mu*a)*(-t12sghch*cgh*ch + eta^2*t14cghch*sgh*ch - t14sghsh*cgh*sh ...
+ eta^2*t32njpcghsh*sgh*sh);
dq2dp2 = 1/eta/sqrt(mu*a)*(-t12sghch*cgh*sh + eta^2*t14cghch*sgh*sh + t14sghsh*cgh*ch ...
- eta^2*t32njpcghsh*sgh*ch);
dq2dq1 = 1/eta/sqrt(mu*a)*(-cgh*t32sc*cgh -cgh*t322sc*cgh + sgh*t34cc*cgh + sgh*t324cc*cgh ...
+ sgh*t32cs*sgh - cgh*t34ss*sgh - cgh*t3b4ss*sgh + sgh*t3b2cs*sgh + sgh*t3b2njpcs*sgh);
dq2dq2 = 1/eta/sqrt(mu*a)*(-cgh*t32sc*sgh -cgh*t322sc*sgh + sgh*t34cc*sgh + sgh*t324cc*sgh ...
- sgh*t32cs*cgh + cgh*t34ss*cgh + cgh*t3b4ss*cgh - sgh*t3b2cs*cgh - sgh*t3b2njpcs*cgh);
DLPn = [dadx;
dMLda, dMLdML, dMLdp1, dMLdp2, dMLdq1, dMLdq2;
dp1da, dp1dML, dp1dp1, dp1dp2, dp1dq1, dp1dq2;
dp2da, dp2dML, dp2dp1, dp2dp2, dp2dq1, dp2dq2;
dq1da, dq1dML, dq1dp1, dq1dp2, dq1dq1, dq1dq2;
dq2da, dq2dML, dq2dp1, dq2dp2, dq2dq1, dq2dq2];
Dlpn = Dlpn + DLPn;
end
% Long-period effects
EqLPDelX1 = O2Msign*Jn_LPX1;
EqLPDelX2 = O2Msign*0;
Eqlp = EqLPDelX1 + EqLPDelX2;
% Dlp matrix
Dlp = O2Msign*J2*Dlpn;
end
% Intermediate term for Mean Longitude
function func = MlongTerm(k,j,n,i,e,eta)
func = -(2*n-5)/eta -eta*k/(1+eta) + 10*cos(i)*(1 - cos(i))/(eta*(1 - 5*(cos(i))^2)) ...
+ (n-2*j)/eta*(-cos(i)/2/(cos(i/2))^2);
end
% Intermediate term for Mean Longitude partials wrt p1p2 term2-term3
function func = MLp12termW12(k,j,n,i,e,eta)
ci = cos(i); si=sin(i);
func = (-(2*n-5)/eta -eta*k/(1+eta) + 10*ci*(1-ci)/(eta*(1-5*ci^2)) ...
+ (n-2*j)/eta*(-ci/(1+ci)))*(-10*si^2*ci/(1-5*ci^2) + (n-2*j-1)*ci) ...
+ si^2/eta/(1-5*ci^2)*((-10+20*ci) - 100*ci^2*(1-ci)/(1-5*ci^2)) ...
+ si/eta/(1+ci)*(si - ci*tan(i/2));
end
function func = MLp12termW14(k,j,n,i,e,eta)
ci = cos(i);
func = (-(2*n-5)/eta -eta*k/(1+eta) + 10*ci*(1-ci)/(eta*(1-5*ci^2)) ...
+ (n-2*j)/eta*(-ci/(1+ci)));
end
% Intermediate term for Mean Longitude partials wrt q1q2 term2-term3
function func = MLq12termW321(k,j,n,i,e,eta)
ci = cos(i); si=sin(i);
func = (-(2*n-5)/eta -eta*k/(1+eta) + 10*ci*(1-ci)/(eta*(1-5*ci^2)) ...
+ (n-2*j)/eta*(-ci/(1+ci)))*(2*n-5)*e/eta^2 + e*(-(2*n-5)/eta^3 +k/eta/(1+eta)^2 ...
+ 10*ci*(1-ci)/(eta^3*(1-5*ci^2) + (n-2*j)/eta^3*(-ci/(1+ci))));
end
function func = MLq12termW322(k,j,n,i,e,eta)
ci = cos(i); si=sin(i);
func = (-(2*n-5)/eta -eta*k/(1+eta) + 10*ci*(1-ci)/(eta*(1-5*ci^2)) ...
+ (n-2*j)/eta*(-ci/(1+ci)))*k;
end
function func = MLq12termW343(k,j,n,i,e,eta)
ci = cos(i); si=sin(i);
func = (-(2*n-5)/eta -eta*k/(1+eta) + 10*ci*(1-ci)/(eta*(1-5*ci^2)) ...
+ (n-2*j)/eta*(-ci/(1+ci)));
end
% Intermediate terms for p12 partials wrt p12
function term = p12p12term32(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = tan(i/2)*((n-2*j - 10*(si)^2/(1 - 5*ci^2))*(-1 -10*ci^2/(1-5*ci^2)*(1+ci)) ...
- (n-2*j-1)*10*ci^2/(1-5*ci^2)*(1+ci) + 80*ci^2/(1-5*ci^2)^2*(1+ci))...
+ (n-2*j)*(n-2*j-1)*ci^2;
end
function term = p12p12term34(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = tan(i/2)*(1 + 10*ci^2/(1-5*ci^2)*(1+ci)) - (n-2*j-1)*ci^2;
end
function term = p12p12term3b4(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = (-1 + (n-2*j - 10*(si)^2/(1 - 5*ci^2)))*ci;
end
function term = p12p12term3b2(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = (-(n-2*j - 10*(si)^2/(1 - 5*ci^2)))*ci;
end
function term = p12p12term3b2njp(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = 1*ci;
end
function term = p12p12termjstar(k,j,n,i,e,eta)
if mod(n,2) == 0
term = 0;
else
si = sin(i);ci = cos(i);
term = ci*10*si/(1-5*ci^2)*getGamma(0,j,n)*nchoosek(k, (k-1)/2);
end
end
function term = p1q1Term32shcgh(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = (n-2*j - 10*(si)^2/(1 - 5*ci^2))*((2*n-5)*e^2/eta^2 + k);
end
function term = p1q1Term34chcgh(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = (-(2*n-5)*e^2/eta^2 - k);
end
function term = p1q1Term34shsgh(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = -(n-2*j - 10*(si)^2/(1 - 5*ci^2));
end
function term = p1q1Term32njpchsgh(k,j,n,i,e,eta)
si = sin(i);ci = cos(i);
term = -1;
end
% intermediate terms for q1 q2 partials
function term = q1p1Term32sghch(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
termi = (e*(2*n-5) - 5*e*sin(2*i)*tan(i/2)/(1-5*ci^2) + e*ci*(n-2*j)/(1+ci));
term = termi*(ci*(1+ci)*e + e*(-10*si^2*ci/(1 - 5*ci^2) + (n-2*j-1)*ci)*(1+ci)) ...
+ e*(10*si^2*e*(5*ci^2 -2*ci +1)/(1-5*ci^2)^2 + e*si*ci*(n-2*j)*tan(i/2)/(1+ci))*(1+ci) ...
+ eta^2*ci*k*(1+ci) + (-10*si^2*ci/(1 - 5*ci^2) + (n-2*j-1)*ci)*k*(1+ci);
end
function term = q1p1Term34cghch(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
term = ci*(1+ci) + (1+ci)*(-10*si^2*ci/(1 - 5*ci^2) + (n-2*j-1)*ci);
end
function term = q1p1Term34sghsh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
termi = (e*(2*n-5) - 5*e*sin(2*i)*tan(i/2)/(1-5*ci^2) + e*ci*(n-2*j)/(1+ci));
term = e*termi*(1+ci) + eta^2*k*(1+ci);
end
function term = q1p1Term32njpcghsh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
term = -(1+ci);
end
function term = q1q1Term32sghcgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
t1 = ((2*n-5) - 5*sin(2*i)*tan(i/2)/(1-5*ci^2) + ci*(n-2*j)/(1+ci));
term = t1*(si*e^2/eta^2*e + (2*n-5)*e^2/eta^2*si + e*si + e*si) + si*e/eta^2*k -2*e*k*si + (2*n-5)*e*si*k;
end
function term = q1q1Term322sghcgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
t1 = ((2*n-5) - 5*sin(2*i)*tan(i/2)/(1-5*ci^2) + ci*(n-2*j)/(1+ci));
term = (t1*(e*si) + (eta^2*k*si))*(k-1);
end
function term = q1q1Term34cghcgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
term = si*(-e) + si*(2*n-5)*e;
end
function term = q1q1Term324cghcgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
term = si*eta^2*(k-1);
end
function term = q1q1Term32cghsgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
t1 = e*((2*n-5) - 5*sin(2*i)*tan(i/2)/(1-5*ci^2) + ci*(n-2*j)/(1+ci));
term = -t1*si;
end
function term = q1q1Term34sghsgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
t1 = e*((2*n-5) - 5*sin(2*i)*tan(i/2)/(1-5*ci^2) + ci*(n-2*j)/(1+ci));
term = -si*e*t1;
end
function term = q1q1Term3b4sghsgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
term = -si*(eta^2*k - eta^2);
end
function term = q1q1Term3b2cghsgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
term = -si*(eta^2*k);
end
function term = q1q1Term3b2njpcghsgh(k,j,n,i,e,eta)
si = sin(i); ci = cos(i);
term = -si*(-eta^2);
end
% Intermediate term for p1,p2
function func = p1p2Term(k,j,n,i,e,eta)
func = n-2*j - 10*(sin(i))^2/(1 - 5*(cos(i))^2);
end
% Intermediate term-1 for q1q2
function func = q1q2Term1(k,j,n,i,e,eta)
func = e*(2*n-5) - 5*e*sin(2*i)*tan(i/2)/(1-5*(cos(i))^2) + e*cos(i)*(n-2*j)/2/((cos(i/2))^2);
end
% Intermediate term-2 for q1q2
function func = q1q2Term2(k,j,n,i,e,eta)
func = k;
end
% dummy
function func = dummy(k,j,n,i,e,eta)
func = 1;
end
function [W1FW2] = getW1W2(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor(n/2)
blp = betalp(j,n,i);
sumk1 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
alp = getAlpha(k,n,e);
% compute W2, W4
[W2] = getW2(n,j,k,g);
sumk1 = sumk1 + alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W1FW2 = deln*sumj1;
end
function [W1FW2] = getW1W2jk(func,jf,k0,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor((n-jf)/2)
blp = betalpjf(j,n,i,jf);
sumk1 = 0;
% innner k loop
for k1 = (k0/2):(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
alp = getAlpha(k,n,e);
% compute W2, W4
[W2] = getW2(n,j,k,g);
sumk1 = sumk1 + alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W1FW2 = deln*sumj1;
end
function [W1FW4] = getW1W4(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj2 = 0;
% outer j loop
for j = 0:floor(n/2)
blp = betalp(j,n,i);
sumk2 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
alp = getAlpha(k,n,e);
% compute W2, W4
[W4] = getW4(n,j,k,g);
sumk2 = sumk2 + alp/2^k*func(k,j,n,i,e,eta)*W4;
end
if isinf(blp) == 0
sumj2 = sumj2 + blp*sumk2;
elseif sumk2 ~= 0
disp('Inclination singularity');
end
end
W1FW4 = deln*sumj2;
end
function [W3FW2] = getW3W2(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor(n/2)
blp = betalp(j,n,i);
sumk1 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
if k < 1
continue;
end
alp = getAlpha1(k,n,e);
% compute W2, W4
W2 = getW2(n,j,k,g);
sumk1 = sumk1 + (n-k)/k*alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W3FW2 = deln*sumj1;
end
function [W3FW2] = getW3bW2(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor((n-2)/2)
blp = betalp2(j,n,i);
sumk1 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
if k < 1
continue;
end
alp = getAlpha1(k,n,e);
% compute W2, W4
W2 = getW2(n,j,k,g);
sumk1 = sumk1 + (n-k)/k*alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W3FW2 = deln*sumj1;
end
function [W3FW2] = getW3ebW2(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor((n-2)/2)
blp = betalp2(j,n,i);
sumk1 = 0;
% innner k loop
for k1 = 1:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
if k < 2
continue;
end
alp = nchoosek(n-1,k-1)*e^(k-2);
% compute W2, W4
W2 = getW2(n,j,k,g);
sumk1 = sumk1 + (n-k)/k*alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W3FW2 = deln*sumj1;
end
function [W3FW2] = getW32W4(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor((n-2)/2)
blp = betalp2(j,n,i);
sumk1 = 0;
% innner k loop
for k1 = 1:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
if k < 2
continue;
end
alp = nchoosek(n-1,k-1)*(k-1)*e^(k-2);
% compute W2, W4
W2 = getW4(n,j,k,g);
sumk1 = sumk1 + (n-k)/k*alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W3FW2 = deln*sumj1;
end
function [W3FW2] = getW3jstar(func,jstar,n,Jn,J2,a,e,i,eta,g,mu,Re)
if mod(n,2) == 0
W3FW2 = 0;
else
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
blp = betalp(jstar,n,i);
sumk1 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
if k < 1
continue;
end
alp = getAlpha1(k,n,e);
% get the customized function
sumk1 = sumk1 + (n-k)/k*alp/2^k*func(k,jstar,n,i,e,eta);
end
W3FW2 = deln*blp*sumk1;
end
end
function [W3FW2] = getW3W2njp(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor((n)/2)
blp = betalp(j,n,i);
sumk1 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
if k < 1
continue;
end
alp = getAlpha1(k,n,e);
% compute W2, W4
W2 = getW2njp(n,j,k,g);
sumk1 = sumk1 + (n-k)/k*alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W3FW2 = deln*sumj1;
end
function [W3FW2] = getW3bW2njp(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj1 = 0;
% outer j loop
for j = 0:floor((n-2)/2)
blp = betalp2(j,n,i);
sumk1 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end
if k < 1
continue;
end
alp = getAlpha1(k,n,e);
% compute W2, W4
W2 = getW2njp(n,j,k,g);
sumk1 = sumk1 + (n-k)/k*alp/2^k*func(k,j,n,i,e,eta)*W2;
end
if isinf(blp) == 0
sumj1 = sumj1 + blp*sumk1;
elseif sumk1 ~= 0
disp('Inclination singularity');
end
end
W3FW2 = deln*sumj1;
end
function [W3FW4] = getW3W4(func,n,Jn,J2,a,e,i,eta,g,mu,Re)
deln = deltalp(n,Jn,J2,a,eta,mu,Re);
sumj2 = 0;
% outer j loop
for j = 0:floor(n/2)
blp = betalp(j,n,i);
sumk2 = 0;
% innner k loop
for k1 = 0:(floor(n/2)-1)
if mod(n,2) == 0
% even harmonic
k = 2*k1;
else
% odd harmonic
k = 2*k1 + 1;
end