What is the significance of Eq. 15 in the expansion of HII regions?

[matteo86bo]

I'm trying to understand what Eq. 15 of this paper (http://www.google.com/url?sa=t&sour...NnQXLAvA&sig2=IRuyhaREKgGjFDLIn77YHg&cad=rja") means:

$\frac{d}{dt}(M_{sh}\dot r)=A_{sh}\rho(c^2+u(u-\dot r))$

This equation should describe the evolution of a HII regions expanding, in particular the evolution of the momentum of the expanding shell. $A_{sh}$ is the shell's area and $M_{sh}$ its mass. What does the two terms represents?
The first is the pressure and the second, they say in the article, is the thrust caused by the exhaust of ionized gas at a velocity u relative to the cloud. Well, I don't get this last part, do you any idea what this means?

 

[Astronuc]

I'm trying to understand what Eq. 15 of this paper means:

On the Role of Massive Stars in the Support and Destruction of Giant Molecular Clouds
http://iopscience.iop.org/0004-637X/566/1/302/15209.text.html

$\frac{d}{dt}(M_{sh}\dot r)=A_{sh}\rho(c^2+u(u-\dot r))$

This equation should describe the evolution of a HII regions expanding, in particular the evolution of the momentum of the expanding shell. $A_{sh}$ is the shell's area and $M_{sh}$ its mass. What does the two terms represents?
The first is the pressure and the second, they say in the article, is the thrust caused by the exhaust of ionized gas at a velocity u relative to the cloud. Well, I don't get this last part, do you any idea what this means?

The u² term would be the momentum flux due to the mass traveling inward at a radial velocity u, relative to the shell if the shell was not moving. The second term $-u \dot r$ is the correction for the fact that mass of the shell is moving in the opposite (radially outward) from the direction of u.

It's the same principle as a rocket moving at velocity u with an exhaust velocity u_e. The thrust is proportional to u_e - u, because the mass of the propellant is initially moving at u until it is exhausted at u_e in the oppositive direction.

 

[matteo86bo]

The u² term would be the momentum flux due to the mass traveling inward at a radial velocity u, relative to the shell if the shell was not moving. The second term $-u \dot r$ is the correction for the fact that mass of the shell is moving in the opposite (radially outward) from the direction of u.

It's the same principle as a rocket moving at velocity u with an exhaust velocity u_e. The thrust is proportional to u_e - u, because the mass of the propellant is initially moving at u until it is exhausted at u_e in the oppositive direction.

Thanks, that helped a lot! But there's still something I don't understand. Let's say the the shell in not moving, why the pressure of mass traveling inward is positive?

 

[Astronuc]

Thanks, that helped a lot! But there's still something I don't understand. Let's say the the shell in not moving, why the pressure of mass traveling inward is positive?

A cloud had a temperature and pressure. What is pressure?

u is just the local relative velocity. It would only produce a thrust in the direction of the velocity vector. What heating might there be in such a cloud? What is a characteristic temperature of such a cloud?

At ~293 K (~0.0253 eV), neutrons have a mean speed of about 2.2 km/s.

If the inward velocity, u, is not positive, or it is zero, then the gas at the inner surface of the shell is not providing thrust against the cloud. u is taken as positive such that it would provide thrust to the cloud. Isn't u a variable?

 

[Ken G]

There are actually a few aspects of that expression I don't understand, in fact I don't think they are right. It sounds like the general simplification is the "Stromgren sphere" type of simplification, where we imagine a very sharp boundary between the ionized gas and the non-ionized gas. That boundary is also a surface of force imbalance, because the ionized gas is both hotter and higher density (it just got ionized, so H turns into p+e). The high pressure inside the HII region presses on the essentially zero-pressure outside region, piling up a shell of mass that is moving supersonically relative to the external gas. The rate of change of momentum of that shell is the left-hand side, where note both its mass and velocity can change.

The right-hand side analyzes the momentum deposition into the shell. As the ionization front eats into that shell, a jetlike flow at speed u moves back into the HII region, and the recoil from that jet pushes on the shell (the second term on RHS). Also, there is the pressure from the HII region (the first term on the RHS). It all makes sense, but there seem to be some discrepancies.

First look at the pressure term, the first term on the RHS. That term basically comes from particles in the HII region bouncing off the shell, where c is the sound speed in the HII region. But the recoil of those particles will not be c, it will be more like c-rdot, just as is there in the second term. Also, the rate they encounter the shell will also be like c-rdot, not c. Presumably c >> u is implied, so they choose to worry about u-rdot but not c-rdot. Maybe that's OK, though I'm a little surprised that c >> u, so I would have thought you'd need both effects or neither.

A potentially more serious problem, it seems to me, is that they are assuming the same density in the HII region as in the jetlike flow coming off from the shell. What's the justification for that? The shell should be at higher density since it is being compressed. What's more, the u seems hard to connect with anything you'd actually see in the HII region, as it would quickly interact presumably (astronuc's question about u being a variable). Why wouldn't they want to characterize the recoil by the ionization rate instead? i would think you'd have an ionization front eating into the shell, controlled by how many ionizing photons are leaking out of the expanding HII region. If you knew that, then you'd know the rate that ionization was eating into the shell, and you know the recoil due to each ionization because the electron pretty much ends up moving backward into the HII region at something like c. That term would be quite independent of the density of the HII region.

Put differently, imagine there was no HII region at all, just a hollow void surrounded by a shell. You'd still have the ionizing photons leaking into the shell, and it would still cause ionization and recoil, but there rho=0. So I don't understand the second term on the RHS, it doesn't look right to me. I like the paper anyway though.