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 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. $\endgroup$ – Magic Octopus Urn Apr 1 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 at 23:42
  • $\begingroup$ 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. $\endgroup$ – JCRM Apr 3 at 16:20

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