# Apollo Spacecraft Re-entry Program

1. Jan 3, 2014

### roldy

I'm working on an Apollo re-entry program and would like to somehow check my results I got for re-entry. I'm using 3 DOF equations to calculate the trajectory of the spacecraft.

Equations of motion:

$\dot{V} = -\frac{D}{m} - g\sin{\theta}$

$\dot{\phi} = -\frac{V\cos{\theta}}{R+h}$

$\dot{\theta} = \frac{L}{mV} - \frac{g\cos{\theta}}{V} + cdot{\phi}$

$\dot{h} = V\sin{\theta}$

Drag and lift is calculated using Newtonian Theory:

$C_d = 2\sin^3{\alpha}$

$C_l = 2\sin^2{\alpha}\cos{\alpha}$

$D = 1/2C_d \rho V^2 S$

$L = 1/2C_l \rho V^2 S$

Density and gravity are calculated as follows:

$\rho = \rho_0e^{\frac{h_0-h}{H}}$

$g = g_0(\frac{R}{R+h})^2$

Constants:

S = 12.0687 m2 (surface area of the capsule)
R = 6371 km (radius of the Earth)
g0 = 9.80665 m/s2 (gravity at 60 km)
ρ = 2.7649*10-4 kg/m3 (density at 60 km)
m = 5500 kg (mass of capsule)
h0 = 60 km (end of sensible atmosphere)
H = 7 km (scaled height)

Inputs:

V = 7162.8 m/s (velocity of capsule)
$\phi$ = 10.556° (angular displacement, range)
$\theta$ = -2.0 (flight path angle)
h = 80 km (re-entry altitude)
$\alpha$ = 53° (attack angle)

These are some values that I picked up from a pdf on a similar project I found online. The project goes through the optimization of the initial conditions to avoid overshoot and undershoot. First they picked some initial values and ran the simulation to get the reference trajectory. I am only concerned with the plots to this reference trajectory since I will be utilizing my own optimization scheme later.

On page 7 of the attached pdf shows some plots for the reference trajectory. My plots are close in shape by some of the numbers are off. I'm not sure why.

Is there any other way to check my results? Perhaps with another simulation? Attached is a zip file containing MATLAB code for the simulation as well as the pdf.

#### Attached Files:

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• ###### AM_LQR.pdf
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Last edited: Jan 4, 2014
2. Jan 3, 2014

### SteamKing

Staff Emeritus
The report states that the air density of 2.7649*10^-4 kg/m^3 is the figure for the reference altitude of 60,000 m, not the surface of the earth. (See p. 5)

3. Jan 4, 2014

### roldy

My mistake. This has no effect on the programming though.