BOE Entry Analysis of Apollo Returning From the Moon

Update 4-15-2026:  Note that this article was published long before Artemis-1 and -2 ever flew!  Yet it remains as accurate and useful as ever.  The stagnation heating models I use in the entry spreadsheet now include a plasma radiation model.  The original H. Julian Allen convective model I still use.

Q/A_conv = 1.75E-8 * [(rho, kg/m³)/(Rn, m)]^0.5 * (V, m/s)³

Q/Arad = 27.94 * (rho/rho0)^1.7 * [(V, km/s) / 3.048]^12.5 

(where rho0 refers to scale height model for density built into the atmosphere models)

I typically just scale down the stagnation values for rough estimates of heating rates away from stagnation.  For still-attached flow,  I use stagnation convective/3 and just use the stagnation radiative as it is because the plasma sheath and bow shock are still very close to the surface.  For separated wake zones,  I use stagnation convective/10,  and stagnation radiative/3 because the plasma sheath is more remote from the surface,  as is any shock.  Those are just ballpark guesses on my part.

The accuracy of that spreadsheet entry model (and my Apollo ballistic coefficient and entry angle data) is attested-to by the actual entry time of ~6 minutes from entry interface to end-of-hypersonics at Mach 3,  the end-of-entry altitude (near 120-125,000 feet),  and the peak deceleration gees (Apollo was 10-11 gees).  It does not do skip entries,  just straight-in.

This is now the way I take the plots the spreadsheet creates,  and display them as a representation of what happens during the hypersonics:

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Original article and earlier updates follow:

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I used my Back-of-the-Envelope (BOE) Entry Analysis to model an Apollo command module capsule returning from the moon. Historically, this was a very shallow entry angle below local horizontal, and essentially at Earth escape speed. I used the very best estimates I could find for the Apollo capsule ballistic coefficient (313.5 kg/sq.m) and heat shield radius of curvature (4.59 m). Historically, we also know that Apollo returning from the moon peaked at about 11 gees deceleration during entry.

The BOE model is described in reference 1, and again in the more detailed user’s guide (reference 2). The original Earth entry workbook in the spreadsheet model was set up as a generic example. I modified this to reflect Apollo data, and tweaked the entry angle from the original 1 degree, up to 2 degrees, at which point peak gees matched the historical value. (At 1 degree, peak gees was closer to 5.5, and the heating numbers were a little smaller.)

All the other spreadsheet results are then presented here, for others to compare with actual Apollo entry results, at their leisure. I think that at least the dynamics look pretty close, especially considering just how over-simplified this model really is. As I have always said, the heating model leaves a lot to be desired.

Figures 1-4 present the trajectory profiles of relevant data, annotated to show the Mach 3 point at which model applicability ends. Figure 5 is an excerpted spreadsheet image of the inputs. Figure 6 is a set of excerpts from the entry analysis portion of the spreadsheet showing the relevant calculated values. All figures are below, at the end of this article.

References:

1.“Back-of-the-Envelope Entry Model”, 7-14-12, posted at “http://exrocketman.blogspot.com”
2.“BOE Entry Model User’s Guide”, 1-21-13, posted at “http://exrocketman.blogspot.com”

Figure 1—BOE-Predicted Apollo Deceleration

Figure 2 – BOE-Predicted Apollo Velocity Profile

Figure 3 – BOE-Predicted Apollo Range versus Altitude Profiles

Figure 4 – BOE-Predicted Apollo Heating

Figure 5 – Spreadsheet Inputs for Apollo Entry

Figure 6 – Spreadsheet Results (Excerpts) for Apollo Entry