Aerocapture: Exploring Aerodynamics, Heating & High Altitude Modeling

[MattRob]

Aerocapture: Aerodynamics OR Aerodynamic Heating OR High-Altitude Atmospheric Model

I apologize for the many topics covered at once here, but, this is why:

I'm trying to design a vehicle to make aerocaptures from Earth to Mars, and from Mars to Earth (I do these kinds of things quiet a lot as a hobby).

I'm trying to find out if a heatshield would be necessary, or if heating would be such that the planned LI-900 HRSI tiles would be insufficient.

For that, I'm piecing together a spreadsheet to calculate the vehicle's trajectory every second. I realize there's a small amount of error due to that, but it's an acceptable amount.

So, I need to know a few things.

1. For Aerodynamic drag at these high mach numbers (25+ for Mars, 33+ for Earth), what equation do I use? Could I simply use dynamic pressure and multiply it by the exposed (windward) surface area to get a close approximation? (Obviously it fails to take into account waveforms and such, but I only need a decent approximation, here.)

2. How do I calculate Aerodynamic heating?
My guess was it comes from the discrepancy in conservation of momentum and conservation of energy.
(If a 100,000 kilo spacecraft drops from 7,500 to 7,000 m/s, and the air that dragged it is only 166,667 kg and was accelerated to only 300 m/s (about mach 1 for Mars), that conserves momentum, but 355 GJ of energy has gone missing - where? Into heat?)
Based off that, I decided loss of energy into other forms such as light, and the heat absorbed by the air were negligible compared to the heat dumped on the vehicle, so I was just going to model the heat dumped onto the vehicle in Watts using that method (loss of kinetic energy not accounted for in movement of air).

3. And finally, where can I find a formula for high-altitude air density? Haha, seems like quiet a thing to ask for for Mars, but perhaps at least someone knows of how I could obtain this for Earth? For Mars would be great, too, of course, even better, but I'm calling that a Long shot.
Here's charts. It would be great if I knew how to reverse engineering the functions...
I've currently taken Calc 1, so perhaps I could plot a number of points then solve for, say, a fourth-order equation to approximate the function?
(y = ax^4 + bx^3 + cx^2 + dx + f)
http://blogs.agi.com/agi/wp-content/uploads/2010/11/MarsGRAM-density-graph1.jpg

 

[mfb]

1. For Aerodynamic drag at these high mach numbers (25+ for Mars, 33+ for Earth), what equation do I use? Could I simply use dynamic pressure and multiply it by the exposed (windward) surface area to get a close approximation? (Obviously it fails to take into account waveforms and such, but I only need a decent approximation, here.)

The shape of the cone is important - a formula without any geometry inside cannot be good.

I think you can get some ideas from others.

The lost energy will heat air and spacecraft - the distribution (air<-> spacecraft and within the spacecraft ) is the interesting thing.
To get an approximate density based on the charts, I would try to get several different formulas for different regions (focus on h=0km to h=100km, the other parts are negligible). The basic law for pressure is an exponential function, where the prefactor in the exponent depends on temperature and chemical composition (it is variable, but not so much). Density depends on pressure and the two variables, too.
For earth, you can find temperature profiles to get a better approximation. For mars, it might be tricky.
$$p=ce^{ax^2+bx}$$
$$\rho=gp(2ax+b)$$
Where a,b,c are some constants and g is the local gravitational acceleration.

 

[cjl]

At hypersonic speed, perhaps the simplest approximation you could use (while still maintaining some semblance of accuracy) would be the Newtonian approximation, as described on page 15 here:

http://www.engr.sjsu.edu/nikos/courses/ae264/pdf/Hypersonic.pdf

It's not ideal, especially for a blunt body, but it'll at least give you a place to start.

As for the heating? If you just assume all of the heat goes into the object, you'll find it nearly impossible to have enough protection. Fortunately the majority goes into heating the air rather than the vehicle, but determining that proportion is not easy.