Expanding explosion from an Asteroid

[Anmoldeep]

An asteroid of mass M explodes into a spherical homogenous cloud in free space. Due to energy received by the explosion, the cloud expands and the expansion is spherically symmetric. At an instant, when the radius of the cloud is R, all of its particles on the surface are observed receding radially away from the center of the cloud with a velocity V. What will the radius of the cloud be, when its expansion ceases?

I got the correct answer by writing the force equation of the differential element just at the edge of the spherical cloud. Also used self-energy of the differential shell element and got a workable differential equation.

What bothers me is, in the solution provided, they derive that the velocity profile is linear with "r", on the basis that at all times, density in the cloud is uniform regardless of radial distance. I wanted to ask that although at the given instance, the density is uniform, it's not necessary that it will remain so throughout the expansion. Moreover, my original answer is independent of the density function, does this mean that there can be different possible profiles of velocity and density variance. Please make me digest the fact that the density of the cloud remains uniform at all times or is it mentioned in the question itself and I am too blind to see it.

 

[PeroK]

If I understand your question: by the shell theorem the outer layer has a gravitational force opposing its expansion independent of the density distribution of the cloud. So, you don't need the assumption of uniform density.

Have you checked whether maintaining uniform density is possible?

 

[jbriggs444]

If I understand your question: by the shell theorem the outer layer has a gravitational force opposing its expansion independent of the density distribution of the cloud. So, you don't need the assumption of uniform density.

Have you checked whether maintaining uniform density is possible?

If the uniform density were to be maintained, the outward velocity profile would need to to be linear with radius. Meanwhile, under this assumption, inward gravitational acceleration would be linear with radius. That would seem to set things up nicely to maintain a linear velocity distribution and, hence, a uniform density distribution.

 

[Anmoldeep]

@jbriggs444 Thanks for the answer. Yeah, that seems understandable. So no compulsion in assuming that.
That aside, I tried my best to solve this using a general radial density variation rho(r) which is, let's say, dependent on the maximum radius of the cloud at any moment.

The density profile was normalized using the fact that
$$ \int _{0}^{R_{o}} \rho _{o} .4\pi r^{2} .dr\ =\ \int _{0}^{R_{max}} \rho _{R_{max}} .4\pi r^{2} .dr\ =\ M$$
which gives constant density in the starting but as the cloud expands density varies.

I also used a general velocity profile, in the everything boils down to simplifying integrals and relating velocity profile to density profile, any insight on how to do this. Just in the general form.

$$\frac{3}{2}\frac{M}{R_{o}^{3}}\int _{0}^{R_{o}} V( r,R_{o})^{2} r^{2} .dr\ -\frac{3}{5}\frac{GM^{2}}{R_{o}} =\ -\ G\left( 16\pi ^{2}\right)\int _{0}^{R_{max}}\left[\int _{0}^{r} \rho _{Rmax} .r^{2} .dr\right] \rho _{Rmax} .r .dr$$

where $V(r,R_{o})$ is the velocity profile when the radius is R_o i.e. at the initial condition and $𝜌_{Rmax}$ is the density profile when expansion ceases. The right-hand side of the above integral is the gravitational self-energy of the sphere at the end.

 

[Anmoldeep]

Sorry for the continuous editing, I have refined the last post and corrected a minor error in the last integral

 

[PeroK]

Does this equation give the answer for the maximum radius $r$?$$\frac{GM}{r} = \frac{GM}{R} - \frac 1 2 V^2$$

 

[Anmoldeep]

Yes

 

[PeroK]

Yes

Do you see why?

 

[Anmoldeep]

Do you see why?

Yes sir. I got it that way, I was just unsure of how to proceed with a general density profile.

 

[PeroK]

Yes sir. I got it that way, I was just unsure of how to proceed with a general density profile.

The shell theorem applies for any spherically symmetric density profile.

 

[Anmoldeep]

Yes, sir, I understand that. However, I no longer want to use shell theorem, I want to mathematically conclude that any density/velocity profile of the cloud normalized to given conditions will end up in the same max radius, not relying on the force equations at the edge, hence I need a way to relate velocity and density profile in the integrals I presented above in a post. I want to work with the bulk and not the external differential shell.

 

[Anmoldeep]

Yes, sir, I understand that. However, I no longer want to use shell theorem, I want to mathematically conclude that any density/velocity profile of the cloud normalized to given conditions will end up in the same max radius, not relying on the force equations at the edge, hence I need a way to relate velocity and density profile in the integrals I presented above in a post. I want to work with the bulk and not the external differential shell.

So explicitly placing, my question is how to relate the generalised version of density and velocity profiles

 

[PeroK]

So explicitly placing, my question is how to relate the generalised version of density and velocity profiles

For the outermost layer they are unrelated. For an interior layer it depends only on how much total mass is enclosed by that layer. It's really the same as a vertical projectile problem.

 

[Anmoldeep]

For the outermost layer they are unrelated. For an interior layer it depends only on how much total mass is enclosed by that layer. It's really the same as a vertical projectile problem.

Thanks, sir, I will give it a try and try to approach the same solution the other way. Thanks a lot for your time.