# Orbital Transfer DeltaV Comparison for Different Orbits

1. Oct 1, 2011

### leiden2

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. Oct 1, 2011

### leiden2

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