<p>Suppose an asteroid explodes, into a spherically symmetrical cloud, of radius <em><strong>R</strong></em>, mass <em><strong>M</strong></em>, and surface receding velocity <em><strong>V</strong></em> (normal to the edge of the cloud), initially the density is radially uniform.</p>
<p>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.</p>
<p>I am unable to derive a relation between the velocity profile and the density profile at any time.</p>
<p>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
<span class="math-container">$$R_{MAX} =\frac{2GMR}{2GM-V^{2} R}$$</span></p>
<p>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 <em><strong>pulsate</strong></em> with time.</p>

CC BY-SA 4.0
2021-09-04T07:59:18.513Z
Have a look at the [Sedov-Taylor theory](https://en.m.wikipedia.org/wiki/Taylor%E2%80%93von_Neumann%E2%80%93Sedov_blast_wave).
Prallax
CC BY-SA 4.0
2021-09-04T08:27:47.950Z
@Prallax thanks. One more thing, in the self-similar solution section of this theory, they have related pressure, density, and volume. Here, besides gravitational pressure, what kind of pressure forces exist. I can't just equate gravitational forces and pressures of any elemental shell, because this is not hydrostatic equilibrium. I can't use Bernoulli's equation either since this is not a steady-state flow. Nor is there any mention of temperature in the original question I am dealing with.
Anmoldeep
CC BY-SA 4.0
2021-09-04T10:12:56.060Z
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.
Prallax
CC BY-SA 4.0
2021-09-04T10:14:10.753Z
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.
Prallax
CC BY-SA 4.0
2021-09-04T10:19:19.663Z
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
Prallax
CC BY-SA 4.0
2021-09-04T12:47:49.960Z
Ah yes, sorry for that, it actually is a problem from a physics/astronomy textbook for olympaids. They have only asked for the max radius when expansion of the cloud ceases, which can be found without messing with density profiles. Which had me wondering if the same answer could be achieved with other density and velocity. Because keeping the density uniform with time, the velocity profile initially has to be linear. I tried using a quadratic velocity profile but the density profile with that at end of expansion is a pain to deal with, so I thought whether some generalisation could be made
Anmoldeep
CC BY-SA 4.0
2021-09-04T20:27:21.397Z
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 $v_0<v_{esc}$.
AtmosphericPrisonEscape
CC BY-SA 4.0
2021-09-04T20:49:02.527Z
@AtmosphericPrisonEscape Got it captain, thanks a tonne!
Anmoldeep
<gravity><asteroids><explosion>
Density and Velocity profiles of a self gravitating explosion
Anmoldeep
# Density and Velocity profiles of a self gravitating explosion

(Asked by: username_0 on 2021-09-04)

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.

**Tags:** gravity, asteroids, explosion
> **Comments:**
> **username_1**: Have a look at the [Sedov-Taylor theory](https://en.m.wikipedia.org/wiki/Taylor%E2%80%93von_Neumann%E2%80%93Sedov_blast_wave).  
> **username_0**: @Prallax thanks. One more thing, in the self-similar solution section of this theory, they have related pressure, density, and volume. Here, besides gravitational pressure, what kind of pressure forces exist. I can't just equate gravitational forces and pressures of any elemental shell, because this is not hydrostatic equilibrium. I can't use Bernoulli's equation either since this is not a steady-state flow. Nor is there any mention of temperature in the original question I am dealing with.  
> **username_1**: 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.  
> **username_1**: 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.  
> **username_1**: 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  
> **username_0**: Ah yes, sorry for that, it actually is a problem from a physics/astronomy textbook for olympaids. They have only asked for the max radius when expansion of the cloud ceases, which can be found without messing with density profiles. Which had me wondering if the same answer could be achieved with other density and velocity. Because keeping the density uniform with time, the velocity profile initially has to be linear. I tried using a quadratic velocity profile but the density profile with that at end of expansion is a pain to deal with, so I thought whether some generalisation could be made  
> **username_2**: 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 $v_0<v_{esc}$.  
> **username_0**: @AtmosphericPrisonEscape Got it captain, thanks a tonne!  

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## 0 answers

