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I saw the question Could escape velocity be achieved in the atmosphere? and thought it probably could've been asked better.

I'm sure there's an equation to calculate the average heat generated by an object traveling in the atmosphere. More specifically, I am asking for an equation simplified based on the following variables:

  • Speed of Object
  • Height of Atmosphere (or density of atmosphere)

Which results in:

  • Heat/Energy Applied to Object

Taking this even further I would like my rocket to be made out of something which has a melting point of 1,650 °C (3,000 °F) (atmospheric re-entry for the average space shuttle mission).

Can anyone provide me a graph for how fast I am able to go at altitude $X$ without melting?

You may make any assumptions you would like, I do not mind an ideal world here. I would love to see this done with equations if possible.

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    $\begingroup$ Slightly related What aspects of reentry heating 'scale as the 8th power'? $\endgroup$
    – uhoh
    Apr 1 '19 at 23:09
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    $\begingroup$ @uhoh is there a meta exchange post on those... LaTeX style equations? Markup wise? I wanted to use some earlier. $\endgroup$ Apr 1 '19 at 23:33
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    $\begingroup$ MathJax basic tutorial and quick reference I just type "mathjax stackexchange" into a search engine to find it each time. $\endgroup$
    – uhoh
    Apr 1 '19 at 23:42
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    $\begingroup$ I suspect this is too broad. I'd be very surprised if there were "an equation" - shape of the object would influence the amount of heat it had to handle; materials would have impacts on what it could handle and cooling system would dictate how it would handle it. $\endgroup$
    – user20636
    Apr 3 '19 at 16:20
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This is just silly enough to be fun. Here is the requested plot:

Max velocity for given equilibrium temperature

NASA Aerothermodynamics Course Lecture Notes has all of the equations I used:

  • Sutton-Graves Convective Heating: $$\dot{q}_{conv}=k(\frac{\rho}{R_n})^\frac{1}{2}V^3$$ where $k=1.7415*10^{-4}$ for Earth.

  • Martin Radiative Heating: $$\dot{q}_{rad}=\rho^{1.6}R_n^{1.0}V^{8.5}$$

  • Total heating rate is the sum of convective and radiative. With radiative equilibrium assumed: $$\dot{q}_{rerad}=\dot{q}_{conv}+\dot{q}_{rad}=\epsilon\sigma T^4, \sigma=5.67*10^{-8}$$

  • I assumed a blackbody radiator ($\epsilon=1$) and a nose radius of 1 meter ($1650 °C=1923 K$).

  • Solve for velocity V.

I used Wikipedia's International Standard Atmosphere and the mean solar activity model for extended altitudes from Braeunig Rocket & Space Technology to find the density ($\rho$) at various altitudes.

Important: these are just engineering models, not proper "physics-like" equations, they are moderately accurate when used properly and highly inaccurate when used like this, but hey, we're having some fun here :). I've noted the approximate region of validity for these models in the image below.

It is also important to consider that this is all steady-state modeling. It would take some time to reach the max temperature from ambient temperature and there's not much time be had at some of these speeds!

I made an excel file that will take your inputs of nose radius and temperature limits to automatically calculate the max velocity for this steady-state temperature and configuration. It is available on GitHub: excel file usage

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