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Suppose an asteroid explodes, into a spherically symmetrical cloud, of radius R, mass M, and surface receding velocity V (normal to the edge of the cloud), initially the density is radially uniform.

Can anything be explicitly stated with respect to the velocity profile with time, and the density profile. Although initially, the density profile is uniform, I suspect it is bound to change with time and result in a radially varying density profile.

I am unable to derive a relation between the velocity profile and the density profile at any time.

Also the max radius (radius when expansion ceases) according to me is independent of density and velocity profiles (initial or at any time t) and is $$R_{MAX} =\frac{2GMR}{2GM-V^{2} R}$$

EDIT: I only seem to find information of a spherically symmetric collapse of such a cloud, not regarding an expansion, so another supplement question would be, is such an explosion bound to pulsate with time.

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    $\begingroup$ Have a look at the Sedov-Taylor theory. $\endgroup$
    – Prallax
    Sep 4, 2021 at 7:59
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    $\begingroup$ I don't know whether the details of your problem can really be fit into the Sedov-Taylor theory, because you may have to consider gravity, you may have a different equation of state and you may or may not have to consider the pressure and density of the interstellar medium around your asteroid. The link was merely a hint towards a possible direction you can investigate. $\endgroup$
    – Prallax
    Sep 4, 2021 at 10:12
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    $\begingroup$ About the equation you can use to relate pressure, density and temperature, if you can neglect viscosity, then Euler's equations are the way to go (like it is done in Sedov-Taylor). Otherwise you may try with Navier-Stokes equations, but the situation is more complicated. $\endgroup$
    – Prallax
    Sep 4, 2021 at 10:14
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    $\begingroup$ I'm treating this like this is some kind of research problem you are doing. If this question comes from an exercise/homework, then you probably don't need to mess with Sedov-Taylor or Navier-Stokes, but you need to give more details to frame your problem $\endgroup$
    – Prallax
    Sep 4, 2021 at 10:19
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    $\begingroup$ Sedov-Taylor is the entirely wrong path to go down here. You asteroid splits up into a number of uncoupled, ballistically flying particles, so no continuum exists, hence all fluid-dynamic theory is invalid. You want to compute the trajectory of a single particle with some initial kinetic energy. The formula you quoted looks a lot like a ballistic particle in a gravitational field just flying 'up' and reaching some maximum radius, given that is has $v_0<v_{esc}$. $\endgroup$ Sep 4, 2021 at 20:27

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