Ballistic Coefficient Study for Earth Entry

It has been suggested that inflatable or extendible heat shields can be used to lower the entry ballistic coefficient,  and thereby lower entry heating,  perhaps to the point of not needing heat protection on a stage or other item returning from low Earth orbit. 

To that end,  I used my spreadsheet version of the old H. Julian Allen and A. J. Eggers 1950’s-vintage entry model,  at fixed entry speed and angle below horizontal,  with a constant entry interface altitude.  I kept the object mass and hypersonic drag coefficient constant,  and used a fixed nose radius to heat shield diameter ratio. 

All I varied was the diameter (and nose radius right with it).  This produced a set of ballistic coefficients β = M/(C_D*A) that decreased dramatically from a near-Apollo value of 300 kg/m²,  down to very low values at very large diameters.  See Figure 1 below for the scope investigated and inputs used (all figures are located at end of this article).

The trajectory model uses a simple scale-height type exponential model of density with altitude.  It presumes a constant angle below horizontal in a 2-D Cartesian modeling set up.  It presumes the drag coefficient (and thus the ballistic coefficient) is constant with speed.  It corresponds to a certain velocity-altitude trend that is doubly exponential.  This is only approximate,  but it really is in the ballpark!  End-of-hypersonics for a blunt object is usually local Mach 3,  which for Earth,  is just about 1 km/s,  but I arbitrarily took this down to 0.7 km/s (about Mach 2.1),  which is well into the range where ribbon chutes can be deployed.

The results I obtained for each of the four ballistic coefficient cases are given in Figures 2 through 5 below.  I expected to see the end of hypersonics altitudes increase,  and the peak stagnation heating rates decrease,  as the ballistic coefficients reduced,  and they did.  I also expected to see peak deceleration gees increase as ballistic coefficients decreased,  but that is not what I got:  peak gees stayed just about the same for all 4 cases.

I then ran stagnation surface temperatures at those peak heating rates,  for a low emissivity and a high-emissivity case.  I did the analysis in US Customary after converting the heating rates,  then converted the temperatures back to metric.  These show a strong decrease as ballistic coefficients get very low,  but are still problematic for anything but high-temperature steels and exotic alloys!   They are reported in Figure 6 below.

I also ran the average pressure exerted upon the heat shield at that observed constant 6.3 gee peak deceleration.  This is nothing but mass times gees times the acceleration of gravity,  then divided by the heat shield blockage area.  These are not as problematic as the stagnation point temperatures,  by far.  They are also reported in Figure 6.

Whether the inflatable or extendible heat shield concepts are survivable,  I leave to others. 

Figure 1 – Inputs Used for Entry Ballistic Coefficient Study

Figure 2 – Entry Trajectory Results for the Highest Ballistic Coefficient

Figure 3 – Entry Trajectory Results for a Lower Ballistic Coefficient

Figure 4 – Entry Trajectory Results for the Next-to-Lowest Ballistic Coefficient

Figure 5 – Entry Trajectory Results for the Lowest Ballistic Coefficient

Update 4-12-2025:  Oops,  found an error converting to degrees C in my data.  Revised Figure 6 replaces the original.  

Figure 6 – Temperature and Pressure Results for the Ballistic Coefficient Study

Update 4-12-2025:

I went ahead and estimated the attached-flow heating rates as stagnation divided by 3,  and the wake zone heating rates as stagnation divided by 10.  This is only an educated guess,  but it is rough ballpark correct. 

From these I computed surface temperatures that equilibriate the convective heating with thermal re-radiation to surroundings at 300 K Earth temperatures.  There is no ablation,  no transpiration cooling,  and no conduction into an interior heat sink.  These temperatures are shown in Figure 7.  Bear in mind that they are very approximate! 

Figure 7 – Temperature Trends Around the Entering Structure

The pictures I’ve seen of inflatable and extendible heat shield concepts seem to fall in the range of 2 to 3 for shield/capsule diameter ratio.  2.5 diameter ratio is about an area ratio near 6.  Factor 6 below typical capsule ballistic coefficients (near 300 kg/m²) would be about 50 kg/m².  Bigger diameter ratio may be too fragile to serve,  since I have not seen any concept images with ratios any bigger than about 3.

At a rather low ballistic coefficient of about 50 kg/m²,  assuming a dark and emissive surface,  we are looking at surface temperatures near 1100 C at stagnation,  near 790 C for attached-flow regions near the rim of the shield,  and near 500 C for all the surfaces immersed on the wake zone behind the heat shield.  If the surfaces are not highly-emissive,  those temperatures will be significantly higher yet!  That is what the plot indicates.

The table just below gives some typical “max service temperatures” for a variety of possible materials of construction.  It would seem that there are no flexible materials one could use to construct inflatable or extendible heat shields for Earth entry from low orbit,  which would not be damaged or destroyed by only one use. 

Carbon cloth might work,  but would suffer both serious oxidation damage,  and heat-induced embrittlement,  preventing any re-use.  It might actually suffer burn-through holes,  if too thin or too-lightweight a weave.