The magazine of the Melbourne PC User Group Saturn V - Apollo 11 Part 8 - Re-entry Ken Holmes

The Command Module is at 112 km altitude with a speed of close to 11 km/sec, and it is entering the earth's atmosphere at an angle of 4 degrees to the horizontal. The splashdown area is 18 degrees around the earth or 2000 km ahead so, in Figure 11, we place the CM 9 deg. to the right of the centreline and thus the velocity vector points to the left and upwards by 5 deg. (9 - 4).

The earth's rotation is carrying the atmosphere forward at 0.4 km/sec so we have 10.6 km/sec of velocity to get rid of before we can deploy parachutes. Earth gravity acceleration, g, is close to 0.01 km/sec/sec so we need about 1060 seconds of 1g deceleration or, more to the point, 260 secs at 4g, which the heat shield and our astronauts are designed to tolerate. The CM is a pure drag machine, meant to go base first, and the 10 cm of heat shield is a good insulator designed to ablate, or vaporise, carrying of heat into the plasma of superheated air streaming around it.

The CM is deliberately loaded to place its centre of gravity off its central axis with, I understand, the effect that it adopts a small angle of incidence between its axis and the direction of travel. Perhaps this confers a greater degree of stability, but it also produces a small lateral force component at right angles which could provide a crude means of steering. After the correction burn 24000 km before, the exact trajectory was closely monitored and Houston could preprogram a series of small thruster firings to rotate the CM about its axis to direct this side force. There were several ships scattered around the retrieval area 1500 km SW of Hawaii and steering right or left, or lifting or lowering the range, would minimise the error in approaching the most promising of these. I do not have any information on exact details in this regard.

Nor do I have good information on the density of the upper atmosphere or on the drag characteristics of the CM at various speeds and air densities. We will assume that the density follows an exponential relationship, in Listing 8, but don't quote me on the equation used. We will assume the drag is proportional to density times velocity squared as in subsonic flow - drawing a very long bow. We are in effect assuming a constant drag coefficient which is combined with whatever is the coefficient required to give the density, into a single coefficient, dc. This was adjusted by trial and error to 1.17 to give a range of 2000 km. Out of this we can only get a general impression of the sort of decelerations and times endured by the astronauts. Since it is all so inaccurate we will ignore the 0.4 km/sec motion of the atmosphere to materially simplify the program.

The resulting graph gives a double hump in the deceleration; it rises to about 4g as the thin upper atmosphere is penetrated at high velocity and then eases off before rising again to over 5g in the lower atmosphere. The altitude trace associates the easing with a levelling out at about 70 km. This may be an artefact due to my hairy assumptions but a hazy memory suggests that I have read about a similar effect in reality. The higher peak would be hard on the astronauts and the heat shield and it would be useful if it could be reduced by judicious rolling to direct the lateral force up or down. I have tried this in a modified program but with limited success. In view of its importance, NASA would certainly have exploited any possibilities very thoroughly.

You may note on the graph that, as the velocity decreases from 11 km/s, the Specific Energy decreases from near zero, or escape energy, towards -60, its value on the surface. (For those who worry about these things, Specific Energy units are the same as velocity squared, or km.km/sec/sec in this program). At the end, the drag is still a little over 1g; ie. it is counteracting gravity, plus a bit which is still slowing Apollo. In the acceleration polygons along the bottom, the resultant acceleration vector is from the top of the vertical 1g to the end of the drag vector.

It only remains to deploy parachutes and plop into the Pacific. You will all be aware of the rescue caravan of divers, flotation collars, helicopters and aircraft carriers.

Conclusion

I would apologise to those who found the series incomprehensible, but they are no longer reading. Even if you have no interest in programming, I hope that the text and graphs may have stirred old memories and perhaps quantified the details of this most significant venture. 

An understanding of orbital mechanics is useful general knowledge since the Shuttle, the Space Station, interplanetary probes and the burn up of spent satellites will continue to regularly feature in the news. I hope the programs have stirred the interest of younger programmers; perusing, criticising and improving on others' programs is a valid way to diversify the range of your own efforts. 

Reprinted from the June 2001 issue of PC Update, the magazine of Melbourne PC User Group, Australia

[About Melbourne PC User Group]