Re-entry Dynamics

Question

I've been reading around a lot and I haven't been able to find any clear explanations. Maybe I'm a bit confused or maybe I am just looking in the wrong places.

My question is, what are some basic principles or guidelines to re-entry conditions on a spacecraft? Let's say I have a vehicle in a circular orbit at X km and I perform a retro burn to put my perigee at the tip of Earths atmosphere. Using a blunt-faced capsule design, what are some basic assumptions or relationships I can use to estimate initial velocity, entry angle (corridor bounds), max deceleration, heat flux, etc.? I have found several resources online, however I keep getting wrong numbers from my calculations.

At the very least, could somebody point me in the right direction? Or even explain some concepts I might be lacking? I've dealt with orbital mechanics in the past but this seems like a whole new ball game.

At this point I just need a few numbers to work with (max g load, heat flux and max temp) so that I can use those parameters for material and structural design. Plotting flight profiles will also need to be done however is not a top priority right now.

Here is a worked example for finding max deceleration. According to this FAA PDF, max deceleration can be found using $a_\max = (V^2)*(H*sin(\gamma)) / (2*e)$ where V is initial velocity, H is Earth scale height and $\gamma$ is flight path angle.

So let's say I'm entering Earths atmosphere at 10 km/s and I want a max g load of 8 g's, or 78.48 m/s². Using this equation, I will need a flight path angle of 1.75 degrees which seems very low to me. The Apollo missions had similar velocities and yet their entry corridor sat at around 7 degrees.

Answer 1

There aren't good back-of-the-envelope relations to get you what you want. However all you need is a simple numerical integrator, an atmosphere model with dispersions, and a vehicle model with mass, size, and some aerodynamic coefficients. You can then play with the entry conditions (speed and flight-path angle) to see the trajectories. Ballistic entry is a good place to start, and once you get that working, you can add a small lift-to-drag ratio for control. The dispersions are important, as well as uncertainty in the entry flight-path angle, to assess whether you are too shallow to reliably avoid unintentional skip-out.

Google "Sutton-Graves" for simple formulae to compute the stagnation point convective heating, which gives you an order of magnitude on the peak heating and integrated heat load. Much harder to calculate is the radiative heating, which can be as much as the convective heating for an Earth entry. The codes to get a real prediction of the heating are quite complicated.

Answer 2

This answer is a few years late. You can use the AMAT tool to achieve what you are trying to do.

For example, let us take an example of an Apollo-type entry vehicle entering Earth's atmosphere at 10 km/s and EFPA = -6 deg. For simplicity, let us first consider a scenario where the vehicle has no lifting capability (i.e. L/D = 0). Using AMAT, the entry profile can be simulated as follows.

from AMAT.planet import Planet
from AMAT.vehicle import Vehicle

# Set up the planet and atmosphere model.
planet=Planet("EARTH")    
planet.loadAtmosphereModel(datfile='../../atmdata/Earth/earth-gram-avg.dat',  heightCol=0, tempCol=1, presCol=2, densCol=3)

# Set up the vehicle with L/D = 0
vehicle=Vehicle('Apollo-6', mass=5400.0, beta=400.0, LD=0.0, A=12.0, alpha=0.0, RN=3.0, planetObj=planet)

# Set up entry parameters
vehicle.setInitialState(h0_km=120.0, theta0_deg=0.0,phi0_deg=0.0,
                        v0_kms=10.00, psi0_deg=0.0,gamma0_deg=-6.00, 
                        drange0_km=0.0, heatLoad0=0.0)

# Set up solver
vehicle.setSolverParams(1E-6)

# propogate the entry trajectory, with delta_deg (bank = 0)
vehicle.propogateEntry(t_sec=2400.0, dt=0.1, delta_deg=0)

# plot entry profiles
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rcParams

fig = plt.figure(figsize=(12, 8))
plt.rc('font',family='Times New Roman')
params = {'mathtext.default': 'regular' }          
plt.rcParams.update(params)

plt.subplot(2, 2, 1)
plt.plot(vehicle.v_kmsc, vehicle.h_kmc, 'k-', linewidth=2.0)
plt.xlabel('Speed, km/s',fontsize=16)
plt.ylabel('Altitude, km', fontsize=16)
ax1=plt.gca()

plt.subplot(2, 2, 2)
plt.plot(vehicle.acc_net_g, vehicle.h_kmc, 'b-', linewidth=2.0)
plt.xlabel('Deceleration, Earth g',fontsize=16)
plt.ylabel('Altitude, km', fontsize=16)
ax2=plt.gca()

plt.subplot(2, 2, 3)
plt.plot(vehicle.q_stag_total, vehicle.h_kmc,'r-', linewidth=2.0)
plt.xlabel('Stagnation point heat-rate, '+r'$W/cm^2$',fontsize=16)
plt.ylabel('Altitude, km', fontsize=16)
ax3=plt.gca()

plt.subplot(2, 2, 4)
plt.plot(vehicle.heatload/1.0E3, vehicle.h_kmc, 'm-', linewidth=2.0)
plt.xlabel('Stagnation point heat-load, '+r'$kJ/cm^2$',fontsize=16)
plt.ylabel('Altitude, km', fontsize=16)
ax4=plt.gca()


for ax in [ax1, ax2, ax3, ax4]:
    ax.tick_params(direction='in')
    ax.yaxis.set_ticks_position('both')
    ax.xaxis.set_ticks_position('both')
    ax.tick_params(axis='x',labelsize=16)
    ax.tick_params(axis='y',labelsize=16)

plt.show()

Now, let us consider a lifting entry with L/D = 0.30, and a constant bank angle of 70 deg. In reality, the bank angle is controlled between 0 (full lift-up) and 180 deg. (full lift-down) to control the trajectory and limit the deceleration load. To do this, change two lines above as follows and re-plot the trajectory. We can see the effect of using lift up, as the vehicle flies level for a while and reduces the deceleration compared to having no lift.

# Set up the vehicle with L/D = 0.30
vehicle=Vehicle('Apollo-6', mass=5400.0, beta=400.0, LD=0.30, A=12.0, alpha=0.0, RN=3.0, planetObj=planet)

# propogate the entry trajectory with small lift up
vehicle.propogateEntry(t_sec=2400.0, dt=0.1, delta_deg=70)