Code covered by the BSD License

# The Gravity Perturbed Hohmann Transfer

### David Eagle (view profile)

28 Feb 2013 (Updated )

MATLAB script for solving the Hohmann transfer problem perturbed by non-spherical Earth gravity.

gravity (t, y)
```function agrav = gravity (t, y)

% first order equations of orbital motion

% N degree and M order gravitational acceleration

% input

%  t = simulation time (seconds)
%  y = state vector

% output

%  agrav = ECI gravitational acceleration
%          vector (km/sec/sec)

% Orbital Mechanics with MATLAB

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

global mu req omega j2 j4 gst0

global lgrav mgrav ccoef scoef

% initialize acceleration vector

agrav = zeros(1:3);

r2 = y(1) * y(1) + y(2) * y(2) + y(3) * y(3);

r1 = sqrt(r2);

r3 = r2 * r1;

if (lgrav == 0 && mgrav == 0)

% Keplerian motion

elseif (lgrav == 2 && mgrav == 0)

% j2 only

r5 = r2 * r3;

d1 = -1.5 * j2 * req * req * mu / r5;

d2 = 1 - 5 * y(3) * y(3) / r2;

agrav(1) = agrav(1) + y(1) * d1 * d2;

agrav(2) = agrav(2) + y(2) * d1 * d2;

agrav(3) = agrav(3) + y(3) * d1 * (d2 + 2);

elseif (lgrav == 4 && mgrav == 0)

% j2 and j4 only

r5 = r2 * r3;

d1 = -1.5d0 * j2 * req * req * mu / r5;

d2 = 1.0d0 - 5.0d0 * y(3) * y(3) / r2;

aj2(1) = y(1) * d1 * d2;

aj2(2) = y(2) * d1 * d2;

aj2(3) = y(3) * d1 * (d2 + 2.0d0);

r4 = r2 * r2;

r7 = r2 * r5;

d3 = 15.0d0 * j4 * req^4 * mu / (8.0d0 * r7);

aj4(1) = y(1) * d3 * (1.0d0 - 14.0d0 * y(3) * y(3) ...
/ r2 + 21.0d0 * y(3)^4 / r4);

aj4(2) = y(2) * d3 * (1.0d0 - 14.0d0 * y(3) * y(3) ...
/ r2 + 21.0d0 * y(3)^4 / r4);

aj4(3) = y(3) * d3 * (5.0d0 - 70.0d0 * y(3) * y(3) ...
/ (3.0d0 * r2) + 21.0d0 * y(3)^4 / r4);

for i = 1:1:3

agrav(i) = aj2(i) + aj4(i);

end

else

% user-defined degree and order gravity model

sr2 = y(1) * y(1) + y(2) * y(2);

sr1 = sqrt(sr2);

sphi = y(3) / r1;

phi = asin(sphi);

% right ascension of greenwich

pmt = mod(gst0 + omega * t, 2.0 * pi);

% east longitude of the spacecraft

lamda = mod(atan3(y(2), y(1)) - pmt, 2.0 * pi);

im = mgrav;

if (mgrav < lgrav)
im = mgrav + 1;
end

p = legend(lgrav, im, sphi);

[cn, sn, tn] = angles(mgrav, lamda, phi);

e1 = 0;
e2 = 0;
e3 = 0;

d1 = req / r1;
d2 = 1;

for il = 1:1:lgrav
f1 = 0;
f2 = 0;
f3 = 0;

il1 = il + 1;

for im = 0:1:il
if (im > mgrav)
break;
end

im2 = im + 2;
im1 = im + 1;

d3 = ccoef(il1, im1) * cn(im1) + scoef(il1, im1) * sn(im1);

f1 = f1 + d3 * p(il1, im1);

if (im2 <= il1)
f2 = f2 + d3 * (p(il1, im2) - tn(im1) * p(il1, im1));
end

if (im2 > il1)
f2 = f2 - d3 * tn(im1) * p(il1, im1);
end

if (im ~= 0)
f3 = f3 + im * (scoef(il1, im1) * cn(im1) - ccoef(il1, im1) * sn(im1)) * p(il1, im1);
end
end

d2 = d2 * d1;

e1 = e1 + d2 * il1 * f1;

e2 = e2 + d2 * f2;

e3 = e3 + d2 * f3;
end

d3 = mu / r1;

e1 = -e1 * d3 / r1;

e2 = e2 * d3;

e3 = e3 * d3;

d1 = e1 / r1;

d2 = y(3) / (r2 * sr1) * e2;

d3 = e3 / sr2;

agrav(1) = agrav(1) + (d1 - d2) * y(1) - d3 * y(2);

agrav(2) = agrav(2) + (d1 - d2) * y(2) + d3 * y(1);

agrav(3) = agrav(3) + d1 * y(3) + sr1 * e2 / r2;

end

% complete gravity acceleration vector

for i = 1:1:3
agrav(i) = agrav(i) - mu * y(i) / r3;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [cn, sn, tn] = angles (m, a, b)

% multiple angles

cn = zeros(21, 1);
sn = zeros(21, 1);
tn = zeros(21, 1);

cn(1) = 1;
sn(1) = 0;
tn(1) = 0;

if (m == 0)
return;
end

cn(2) = cos(a);
sn(2) = sin(a);
tn(2) = tan(b);

if (m == 1)
return;
end

for i = 2:1:m
cn(i + 1) = 2 * cn(2) * cn(i) - cn(i - 1);
sn(i + 1) = 2 * cn(2) * sn(i) - sn(i - 1);
tn(i + 1) = tn(2) + tn(i);
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function p = legend (n, m, x)

% Legendre polynomials

p(1, 1) = 1;
p(2, 1) = x;

for i = 2:1:n
p(i + 1, 1) = ((2 * i - 1) * x * p(i, 1) -(i - 1) * p(i - 1, 1)) / i;
end

if (m == 0)
return;
end

y = sqrt(1 - x * x);

p(2, 2) = y;

if (m == 1)
% null
else
for i = 2:1:m
p(i + 1, i + 1) = (2 * i - 1) * y * p(i, i);
end
end

for i = 2:1:n

i1 = i - 1;

for j = 1:1:i1

if (j > m)
break;
end

p(i + 1, j + 1) = (2 * i - 1) * y * p(i, j);

if (i - 2 >= j)
p(i + 1, j + 1) = p(i + 1, j + 1) + p(i - 1, j + 1);
end
end
end
```