Hydrostatic Isothermal disk

The goal is to set up a hydrostatic isothermal rotating disk. Need to solve

as well as

for

and where is the gravitational acceleration due to a point mass at the origin

First we can simplify the equations

as well as

Since omega is a free parameter, we can choose a density field that satisfies the first equation and then integrate the second equation to find as long as

Let's try

where describes the density along the midplane. We can set the height of the disk to be where the density drops to the ambient value - or we can set the background density distribution and replace it with the disk to the height where the pressures balance.

Also, if we don't use softening, then [[latex($f_{z}=-\frac{GMz}{r3}

and

We can then solve the second equation for

If we now consider the density along the midplane

Since we expect and since we have

or

or

Of course this blows up at the origin, so we can either avoid the origin or use some form of softening.

Centrifugal acceleration can only support things from going inward. To keep things from going outward we have to use pressure or gravitational force - Since we want the density to decrease as we go outward, we have to use gravitational force - but this limits how quickly the density can drop. Lets choose

where . If then we expect the equivalent of a BE sphere (but with a point particle instead of self-gravity).. Solving for along the midplane…

Combining all of this we get a solution for the density field

and rotational velocity goes like

Now for softening

First lets consider plummer softening. We can repeat the above procedure but instead of we have where is a softening radius

Let's again try

Now the integrand is which is very similar to before except that and

so

And the integral along the midplane is also modified…

Combining these gives the solution for density

and angular velocity

Spline Softening

If the softening function is compact (only modifies gravity inside of a certain radius) then

So we need the integral of the softened function…

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