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Analytical expressions for BE paper
Been thinking about the equations we were deriving the other day, and their origin.
Starting from the Navier Stokes equation,
\rho (\frac{Dv}{Dt}) = -\triangledown P - \rho g
We can solve for P in hydrostatic equilibrium (dv/dt=0), assuming P=P(z). A volume integral of the above equation then gives,
x y P(z) \hat{z} = - \int_V \rho (z) g dxdydz \hat{z}
or
P= -\frac{1}{A} \int \rho(z) g dxdydz = -\int \rho(z) g dz
This says the Pressure at some depth is a function of the instantaneous weight of all the material on top of it.
From here, it seems we were thinking of extrapolating this equation now for the dynamic case,
P(z,t)= - \int \rho(z,t) g dz
where the bounds went from Rbe (radius of BE sphere) to Rbe + R(t), where R(t) depends on whether we are in either a subsonic or supersonic regime. (This integral would actually be in spherical coordinates, left in cartesian for simplicity).
Does this seem okay to people still?? I am not entirely convinced, and am having a hard time finding any discussion of the physics here. If it is okay, I can try to crank out some numbers for the expected P(Rbe,t), although I am not sure what function I would use for rho(r,t). Thanks
- Posted: 11 years ago (Updated: 11 years ago)
- Author: Erica Kaminski
- Categories: (none)
Comments
(sorry about all of the super and subscripting above…) boy do I wish I could edit comments… - hang on - I can use sqlite to dump my comment and reformat it here…
The basic idea was to calculate at a given time - how much of the ambient material would have had time to fall onto the BE sphere. If we assume the ambient is in free fall towards the BE sphere… then given a time t we can estimate how much material will have fallen onto the sphere…
tff = torbit/2 = sqrt(pi2 R3 / (8*G*M))
if we invert this and solve for the volume of gas we get Mamb=rhoamb * 32 * G * M * t2 / (3 * pi)
Then if all of this mass has collected at the edge of the BE sphere… this can be converted into a pressure… P=G*M*Mamb / (4 * Pi a4) where a is the size of the BE sphere
which gives P=rhoamb*M2/a4*t2 * 8*G2/(3*pi2)
or something like that… Realistically there will be some ram pressure as well - and all of the mass won't be sitting at the edge of the BE sphere - but will probably be extended over some scale height… which would depend on the sound speed… So that could be added to the derivation…
(so after posting above comment - I deleted my previous poorly formatted comment)