Context Navigation

← Previous Post
Next Post →

HydroStatic Isothermal Disk

First every distance is softened to avoid trouble at origin

Distance from origin: r=>\sqrt(r^2+\epsilon^2)

Distance from rotation axis: s=>\sqrt(r_s^2+\epsilon^2)

Disk radius: r_D=>\sqrt(r_D^2+\epsilon^2)

Then angular velocity is set to be keplerian and balance force along the 's' direction

\Omega=\sqrt{\frac{G M}{s^3}}

Then the density within the disk is set to be constant along the midplane \rho_0 with a decaying exponential to give vertical hydrostatic equilibrium.

\rho=\rho_0 \exp{ \left [ h \left ( \frac{1}{r}-\frac{1}{s} \right ) \right ]}

where the scale height is given by h=\frac{GM}{c_s^2}

At the outer rim of the disk when s > r_D Omega goes to zero and the profile changes slightly to

\rho=\rho_0 \exp{ \left [ h \left ( \frac{1}{r}-\frac{1}{r_D} \right ) \right ]}

which is just the profile for an envelope about a point mass in hydrostatic equilibrium ie \rho \propto \exp(1/r)

Then for each point \rho => \max(\rho, \rho_B) where \rho_B is the disk background density.

Finally additionally smoothing is done in density and pressure between the disk solution and the original cell values at the outer radius using the disk%thickness to define this smoothing distance - although I did just notice a typo if one uses the hydrostatic disk without using an isothermal EOS…

s(iE)=(gamma7*Disk%pressure-q(iE))*f

should be replaced with

s(iE)=(gamma7*Disk%Pressure/Disk%Density*rho-q(iE))*f

astrobear

Context Navigation

HydroStatic Isothermal Disk

Comments

Download in other formats: