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A MATLAB Script for Propagating Interplanetary Trajectories from Earth to Mars

A MATLAB Script for Propagating Interplanetary Trajectories from Earth to Mars

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Numerically integrate the orbital equations of motion of an Earth to Mars interplanetary trajectory.

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 emu req j2 j4 jdtdb_wrk

global lgrav mgrav ccoef scoef

% initialize acceleration vector

agrav = zeros(3, 1);

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 * emu / 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 * emu / 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 * emu / (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
    
    jdate = jdtdb_wrk + t / 86400.0d0;
    
    jd1 = fix(jdate);
    
    jd2 = jdate - jd1;
    
    gst0 = gast4 (jd1, jd2, 1);
    
    pmt = mod(gst0, 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 = emu / 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

agrav = agrav - emu * y(1:3) / r3;

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

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

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