Aerobraking and reentry

I have added a simple atmosphere model to my orbit simulator and done some aerobraking and reentry simulations. Originally I just wanted to see how hard it would be to aerobrake into LEO when returning from the Moon. Then, to test the accuracy of the model I tried three other cases; reentry from an X-prize flight, from LEO and directly from Lunar return. I was gratified to see the double hump profile that Apollo actually experienced during it's reentry, so I guess the simple model is accurate enough. Each graph below shows the second by second G load over 300 seconds. As you can see, aerobraking into LEO when returning from the Moon is not too hard, with a peak G load just under 3, the lowest of the four simulations. You better have good navigation though because position errors of 1 km or velocity errors of 1 meter per second can completely change the result.

Seems good. I noticed that the re-entry from LEO was pretty hairy, though. Is that just straight aerobraking? If so, can it be lowered to more comfortable levels?

...

You should have seen the 14 G reetnry from the 600km orbit before I lowered it to 200 km! A bit of a surprise give that Lunar return stays under 8 Gees.

The vehicle's motion under the influence of gravity is calculated as described in that other thread, but using Earth's gravity instead of the Sun's. Then I added deceleration as density*speed^2*ballistic coefficient, where atmospheric density is modeled as 0.5*EXP(-Altitude/8.42) and ballistic coefficient is assumed to be 2.
Yes, it is all drag, no lift. You could compute the magnitude of the drag, then simply apply only 20% of it against the velocity and the other 80% perpendicular to the velocity and call that part lift. That would be L/D of 4. I don't know how valid that is, but it would be fun to play with.

Here is an excel spreadsheet with the first 400 lines:
(EDIT) New corrected code in new location
http://home.austin.rr.com/campbelp/astro/reentry.xls (/EDIT)
Each line represents 1 second of time. Just copy the last line as often as needed to extend the simulation beyond 400 seconds. It starts at 100 km altitude (Y=Earth's radius + 100) and 0 velocity. Starting at X=Re+400 and X velocity 7.63 gives a nice orbit that does not reenter, but you will need over 5,000 lines to compute one complete orbit. Use a slightly slower speed to simulate LEO reentry. I used 15,000 lines for the Lunar return, simulation.

Couldn't you find the equation with respect to time and plug it into something like Maple or even MATLAB?

I don't know.
The method I am using is like a fractal calculation, where an arbitrary starting value is fed into a non-linear equation and the result is used as another input to the same equation. If there is an analytic equation that could be used instead, I don't know it. As far as I know any orbital mechanics problem involving more than two point masses has never been solved, in general. I think even NASA uses numerical methods, although far more advanced than I am using.

Well, hell, we need to get that guy from Numb3rs to have a crack at it. I mean, if he can solve the nastiest stats problems in an hour, he oughta be able to get this one in a couple days!

Besides, why are you using more than two point masses? Isn't everything besides the ship and the earth negligible? And if you're using the moon, isn't the ship itself of negligible mass? Oh, right: even infinitely small mass will screw up the equation..... Or would it?

can you input a range of posibilities and have the computer figure out the optimum? What about a 3-d graph, which also has all the factors displayed?

I am only considering Earth's gravity, but atmospheric drag is the complicating factor for which I don't have an analytical soultion. Gravitational acceleration depends only on position but drag depends on position and speed. I have just been entering different starting values and seeing what I get. Excel can calculate thousands of values in a fraction of a second. Enter different values on one line and see where it goes. Actually it is quite fun to play with.

Oh Oh Oh

Just as I am typing this the TV is on and I see that the PBS show American Experience will have a program about Apollo 8 tomorrow night. I'll have to tape it, my kids and Halloween will never let me watch it live.

I found an error in my simulation. Got a sign wrong in the drag function (do I get partial credit?). Here is the corrected Excel file:
http://home.austin.rr.com/campbelp/astro/reentry.xls
And here are new Gee graphs. These cover 600 seconds of time. The LEO entry is much gentler because I found a better entry angle, not because of fixing the math. Note though that the LEO entry does show a slight double hump profile now.

Yes, Excel is a lot of fun to play with for iteration problems. And it's also a lot quicker than building your own for loop.

Those new values are all completely acceptable. The two minutes or so of 3g on return from LEO is somewhat annoying, but definitely tolerable for all but the most fragile cargo and passengers.

I added a column to calculate heating by assuming heating equals, or at least is proportional to, the difference of the velocity squared. For example slowing from 1 to 0 has a heat load of 1 and slowing from 5 to 4 has a heat load of 5*5-4*4=9.

There is good and bad news.
Good:
LEO entry has a peak heat load only about 5 times suborbital.
Bad:
Aerobraking into LEO from Lunar return has a peak heat load 10 times suborbital, twice that of LEO reentry.

Where'd that come from? I'll admit to not being in Thermo yet, so please elaborate.

Pete is assuming (correctly) that the vast majority of kinetic energy is converted to heat in aerobraking manuevers.

Kinetic energy being mass times velocity squared...

Actually kinetic energy is one half the mass times the velocity squared. And yes, I assume it all changes to heat.