 # Reentry of a Capsule

Subject Mathematics Mathematics - Other

## Question

Reentry of a Capsule
In this project, you’ll model the trajectory of a satellite/space capsule “reentering” the Earth’s atmosphere, and predicting the point of impact. The main effects you should take into account are:
• Aerodynamic drag (i.e. friction caused by air)
• Gravity
Are there other aspects that appear relevant? The key steps of the project are:

(1) Do some research on the modeling of air drag. There are different models for different regimes. Which one would you choose?

(2) The drag will depend on the height of the satellite, as air becomes less dense with in- creasing height. Find a (simple) model for the air density depending on the height. How does this affect the model for air drag?

(3) Choose parameters: What are reasonable values for mass, diameter, drag coefficient, . . . ? Assuming that the capsule is spherical might be helpful. You could use data from the Apollo-missions.

(4) Write down the equations of motion for the capsule. Specify a coordinate system.

(5) Find a transformation that allows you to describe the trajectory of the satellite in terms of latitude, longitude and height above ground.

(6) Plot the trajectory of the satellite on a world map, and the height of the satellite, as a function of time. Find initial conditions that make the satellite crash somewhere in the pacific.

(7) Outlook: What are flaws of this model? How could it be improved? Do you trust the prediction?
Upon completion, you will give a short (5-10 min) presentation of your model and your findings.

## Solution Preview

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clear;clc;
AU=1.495978707*10^8; %in km
a=2.2871731 *AU; %in km (Semi-major axis)
e=0.07160901;% Eccentricity
i=5.16052;%in deg (Inclination)
w=31.51952;%in deg (Argument of perihelion)
omega=105.70810;%in deg (Longitude of the ascending Node)
mu=1.32712428*10^11;%in km^3/s^2
ta=2.445151603388958e+02; %in deg
k=1;
for ta=ta:1:ta+360
%Distance to the Sun from Asteroid
r=a*(1-(e^2))/(1+(e*cosd(ta)));
%Position of asteroid in Heliocentric cartesian coordinates
x(k)=r*((cosd(omega)*cosd(w+ta))- (sind(omega)*sind(w+ta)*cosd(i)));
y(k)=r*((sind(omega)*cosd(w+ta))+(cosd(omega)*sind(w+ta)*cosd(i)));
z(k)=r*(sind(i)*sind(w+ta));
R(k,:)=[x(k)/AU y(k)/AU];
k=k+1;
end
plot(R(:,1), R(:,2),'-*','LineWidth',2)...

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