[Users] typo in GRHydro thorn guide?

[Erik Schnetter]

Bruno

I find the statement itself weird – the hydrodynamics equations break
down, full stop. What happens to the speed of sound and whether it is
even still defined depends on how to approach the vacuum limit. For
example, one can argue that the temperature will increase if one goes
to vacuum (it does in space), and thus the speed of sound would
increase. I would remove any statement about the speed of sound, and
maybe add the physical reason why the limit is not defined, with the
mean free path length becoming too large (hence we don't have
thermodynamics any more).

Apart from this your calculation looks correct to me. If you take \rho
-> 0 with k = const and \gamma = const then cs -> 0 .

-erik

On Fri, Feb 9, 2024 at 10:16 AM Bruno Giacomazzo
<bruno.giacomazzo at unimib.it> wrote:
>
> The GRHydro thorn guide (https://einsteintoolkit.org/thornguide/EinsteinEvolve/GRHydro/documentation.html) contains the following sentence: "However, in the vacuum limit the continuity equations describing the fluid break down. The speed of sound tends to the speed of light and everything fails (especially the conversion from conserved to primitive variables)."
>
>    A student of mine wrote that the sound speed tends to the speed of light if rho tends to zero citing this, but is this correct?
>
> For example, taking a polytropic EOS P=k*rho^\gamma, the sound speed squared is (assuming c=1)
>
> cs^2= \gamma * k * rho^\gamma / (rho*h)
>
> with
>
> rho*h=rho+rho*eps+P=rho+\gamma*P/(\gamma-1)
>
> So cs^2 goes like rho^\gamma/(rho+\rho^gamma) and if \gamma>1 then cs^2->0 for rho->0 (while it goes to 1 if rho-> infinity).
>
> Am I missing something?
>
> Cheers,
> Bruno
>
> --
>
> Prof. Bruno Giacomazzo
> Department of Physics
> University of Milano-Bicocca
> Piazza della Scienza 3
> 20126 Milano
> Italy
>
> email: bruno.giacomazzo at unimib.it
> phone: (+39) 02 6448 2321
> web: http://www.brunogiacomazzo.org
>
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-- 
Erik Schnetter <schnetter at gmail.com>
http://www.perimeterinstitute.ca/personal/eschnetter/