Semi analytic approximation for the dimensionless stagnation radius

[bolverk]

Almog said:

we were discussing fitting a semi analytic formula to relate the adiabatic index \gamma and the density power law \eta to the position of the stagnation point, so that others will be able to use our results. I tried doing so, but found that each domain in the phase diagram should have its own formula, and this complicates matters. For this reason, I’m not sure this is still worthwhile.

Aleksei said:

That is unfortunate. However, for most physical applications I imagine 1<eta<2 and gamma=5/3 (or perhaps 4/3), maybe we could just discuss this case? I previously estimated that rs~ 4 G M/(nu vw^2), where nu=(4 delta-1)/6, for gamma=5/3.

1. From the discussion about our galactic centre, 1<eta<2 might be the relevant for regular star, but perhaps not for wind emitting stars.
2. Is your previous estimate consistent with the completely analytic solution? gamma=5/3, eta=5/2 and the dimensionless stagnation point is 8/3.
3. I tried using a Pade approximant to get an analytic approximation for the cases gamma=5/3 and 4/3, and 1<eta<2. for the case the coefficients are a=-2.58, b=-2.66 and c=-1.6. For the case the coefficients are a=-3.40, b=-3.84 and c= -1.42.

[alekseygenerozov]

1. Fair enough. Though I would be cautious in describing the galactic center since the bulk of the young stars are in a disk.

2. Yes. It gives exactly 8/3! To be clear my fitting formula is

rs~24/(4 eta-1) G M/vw^2

(Sorry I used delta instead of eta--old habits).

3. Ok it seems like we are in reasonable agreement for gamma=5/3: see the plot below.

[bolverk]

How did you derive rs~24/(4 eta-1) G M/vw^2?

[alekseygenerozov]

For gamma=5/3, r_s= 4 G M/(nu vw^2), where nu is the absolute value of the density power law slope at the stagnation radius. I found that nu(eta)~(4 eta-1)/6, reproduced numerical results for the stagnation radius. I actually didn't specifically try to reproduce the stagnation radius for eta=5/2 (since I was originally interested in eta<2), and I was a bit surprised it worked out exactly.

I was also able to derive it from an analytic ansatz: if you take the integrated form of energy (eq. 16) and set the velocity to 0, you get something like:

c^2/(gamma-1)=1/2+h(r/r_s).

Now if you add a term like h(r/r_s)^2 you an plug the sound speed back into the energy equation, you get an expression for the velocity in the vicinity of the stagnation radius. From this you can the density slope at r_s (nu), which matches the numerically derived result. Of course, this ansatz is not well motivated...

[bolverk]

I'd like to understand your solution better. Can you upload a pdf or a mathematica file with the complete derivation?

[alekseygenerozov]

I have attached a pdf sketching the derivation.

r_s derivation.pdf

[bolverk]

Can you elaborate on the point where you say "After some algebraic manipulation I obtain..."?

[alekseygenerozov]

Ok I have written an updated derivation where I fill in the math. For some reason I can't upload it here, but I will send it as an email attachment...