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Orbital Transfer DeltaV Comparison for Different Orbits

  1. Oct 1, 2011 #1
    1. The problem statement, all variables and given/known data
    Assuming planar orbits, calculate the Hohmann ∆V required to transfer from a low-Earth
    circular orbit with radius of 1.03 Earth radii to the orbit of the Moon (assume 60 Earth
    radii). Compare this to the 3-maneuver strategy of (1) escaping to infinity on a parabolic
    orbit, (2) maneuvering from infinity to approach the Moon’s radius on a tangential path,
    and (3) circularizing at the Moon orbital radius. Use at least 5 significant digits in these
    calculations. Locate the crossover point (the ratio r2/r1) where the escape option becomes
    more ∆V efficient.

    The first parts are solved and cheked to be right.The bold part of the question is the place where i am stuck, finding an iteration to find the crossover point.
    2. Relevant equations
    Equations are right, the iteration doesn't work.
    3. The attempt at a solution

    This is a MATLAB ITERATION.

    mu = 3.986004418000000e+005;%gravitaional parameter
    r1=1.03*6378;%first point
    r2=r1*1.5:1:382680;%final points

    for i=1:length(r2);

    vesc1=(2*mu/r1)^.5;%escape from leo
    vesc2=(2*mu./r2(i)).^.5;%escape from moon
    del1=vesc1+vesc2;%left side of deltaone
    at=(r1+r2(i))/2;%transfer semimajor axis
    et=-mu/2/at;%transfer orbit energy

    %%%%%%%%%%%%%%%%%%%%%%%%%
    vt1=(2*(et+mu/r1))^.5;%transfer velocity at periapsis1
    vt2=(2*(et+mu./r2(i))).^.5;%transfer velocity at apoapsis
    v2=(mu./r2(i))^.5;%circular moon orbit velocity

    %%%%%%%%%%%%%%%%%%%%%%%%%
    del2=vt1-vt2+2*v2;%rhs
    del3=del1-del2;%lhs
    if del3<0.00001
    ratio=r2(i)/r1;%finds the ratio
    else
    i=i+1;
    end
    end
     
    Last edited: Oct 1, 2011
  2. jcsd
  3. Oct 1, 2011 #2
    Ok, i found the right iteration and solved the question. I moved the variables that vary to the right side and then tried to iterate. this is the solution. the ratio comes out to be 11.938. I will not delete the post since it might help someone.

    mu = 3.986004418000000e+005;%gravitaional parameter
    r1=1.03*6378;%first point
    r2=r1*1.5;%final points
    vesc1=(2*mu/r1)^.5;
    del1=vesc1;
    for i=1:1000000;
    r2(i+1)=r2(i)+1;
    vesc2(i)=(2*mu./r2(i)).^.5;%escape from moon
    %left side of deltaone
    at=(r1+r2(i))/2;%transfer semimajor axis
    et=-mu/2/at;%transfer orbit energy
    %%%%%%%%%%%%%%%%%%%%%%%%%
    vt1=(2*(et+mu/r1))^.5;%transfer velocity at periapsis1
    vt2=(2*(et+mu./r2(i))).^.5;%transfer velocity at apoapsis
    v2(i)=(mu./r2(i))^.5;%circular moon orbit velocity
    %%%%%%%%%%%%%%%%%%%%%%%%%
    del2=vt1-vt2+2*v2(i)-vesc2(i);%rhs

    if del2-del1<0.00000000000001
    ratio=r2(i)/r1;%finds the ratio
    else
    i=i+1;
    end
    end
     
    Last edited: Oct 1, 2011
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