I have a simulation that describes a vehicle traveling at very high speeds (near or even above orbital) up out of the atmosphere and into space. I'd like to chart the rate of heating ($\dot{q}_{conv}$ and $\dot{q}_{rad}$) of the vehicle's nose cone versus time. Input parameters that I would like to provide are the atmospheric density, atmospheric temperature, atmospheric composition (that is, not necessarily Earth's atmosphere), nose cone angle, and nose tip radius, and airspeed. (If I forgot anything important, feel free to mention that in your answer.)

The equations should work for airspeeds in the 1 to 15 km/s range.

I'm fine with assuming the shape below, but if it's more convenient to provide an answer for a better shape that's fine too.

enter image description here enter image description here

(from nose cone design)

I'm hoping that there are some formulas that I can use that resemble these

formulas for stagnation heating (slide 8)

I would also like to be able to reference a credible source that describes the equations that I end up using, and their limitations.

I'm willing to assume a non-ablative nosecone for now.

enter image description here

I'm also more interested in minimizing drag to get out of the atmosphere with minimal loss of speed, so this problem is different from the usual reentry problem where the goal is to slow down the vehicle without it burning up.

I don't want to blow up the scope of this question too much, but any additional advice from someone with expertise in this area would be very much appreciated. For example, what other information would an expert working on this problem feel is important to chart to assess the engineering feasibility of launching a vehicle with a specified shape along a certain proposed trajectory?

Update - just want to share a bit of what I discovered - it might help other people to get started. I found an relevant article called "Projectile Nosetip Thermal Management for Railgun Launch to Space" which, in turn, referenced a book called.

J. D. Anderson, Jr, Hypersonic and High Temperature Gas Dynamics. New York: McGraw-Hill, 1989, pp. 289–291.

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    $\begingroup$ ntrs.nasa.gov/api/citations/19850008666/downloads/… See page 1027 $\endgroup$ Commented Apr 22 at 20:25
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    $\begingroup$ You also need to account for shockwaves generated by the nose cone. Can you give us a more specific question as heating also has to do with Mach number and mass as well as exposed surface area of the nose cone. The nose of the cone would experience significantly more heating than the tail. $\endgroup$ Commented Apr 28 at 2:28
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    $\begingroup$ The Mach number (or speed) is an input parameter in the question. But I'd like the formulas to work in the range of 1 to 15 km/s. I'm not sure why mass would change q, so if you could explain that maybe I can answer. Exposed surface are is ... the entire curved side of the cone? Not quite sure what you asking here as that seems like too obvious an answer. The proposed shape is that the entire vehicle is a cone with a blunt nosetip, but if you have a source with data for a different shape I'll take it. $\endgroup$
    – phil1008
    Commented May 12 at 7:21
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    $\begingroup$ @LawnHollanderLawn Maybe the Sprint Missile will illustrate the kind of heating that I'm interesting in finding formulas to describe. $\endgroup$
    – phil1008
    Commented May 12 at 7:48
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    $\begingroup$ I found an online copy of Hypersonic and High-Temperature Gas Dynamics, 2nd Edition at IIT Bombay. $\endgroup$
    – uhoh
    Commented May 13 at 22:09


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