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# Self Gravity

(This is a skeleton while a new version of this page is under construction.)

## Physics

### Poisson Equation of Self Gravity

The electric field generated by a point chargeis

On the other hand, the gravitational acceleration by a point mass of is

The Poisson equation for electrostatic potential

is

Analytically we can get the Poisson equation for the gravity

where

is the mass density. This equation describes how the potential is determined by the mass density distribution.

For example, consider the uniform density distribution

Use spherical coordinates, the Poisson equation for the density distribution Equation (1) can be written as

Or

Solving Equation (3) with the density Equation (2), we obtain the solution for the potential for the uniform density sphere:

### Free-fall Time

The free-fall time is the characteristic time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse. Using Gauss's theorem, the gravitational acceleration

is given bywhere

is the mass insideSo the equation of motion for a gas molecule under a control of the self-gravity can be written as

Equation (7) can be written as the form of the conservation of mechanical energy

Suppose the gas molecule is initially located at

. So the total energyNow consider gas collapse. So

. Equation (8) gives

Let

From the initial state we have

. So Equation (10) reduces toSolve Equation (12) we have

Let

or . we have the free-fall timewhere

is the average density of the gas.

## Implementation

### Uniform Density Cloud

In this section we consider the problem of the collapse of a pressureless (

), uniform, initially motionless cloud which has an analytic solution.

The definition of

in Equation (11) can also be written asFrom Equations (13) and (14) we have

Equation (17) describes the time it takes for the cloud to collapse to a density

.

### Boundary Conditions

## Present Considerations

### Some notes on periodic BC's and particles

With periodic boundary conditions, we would like the total potential to be unchanged under the conversion of gas to particle. This requires adjusting the particle potential so that it has the same constant of integration as well as being periodic. When hypre solves for the potential, the constant of integration is a free parameter and I believe hypre adjusts it so that

. To calculate the potential of a point charge, we need the Greens function corresponding towhere D is the dimenesion of the problem. For 3D,

and for 2D,

and for 1D,

Now with periodic BC's, the solution to the gas potential is given by

, so the solution to the potential would actually be given by where is the mass of the particle. Note this is not the same as using mirror versions of the particle… An easier way to solve this however, is to use fourier transforms.

with . Now is just the fourier transform of a delta function times the mass, which is , so and

This can be discretized as

This still requires summation over many wave numbers - although the higher wave numbers have less impact because of the

dependence. However in 2D this is lessened because there are more wave vectors in each anuli, and in 3D there is equal power in each shell. This then, becomes an order N^{6 operation }

Solutions:

- Don't ever use sink potential?
- Don't use sink potential to update gas potential
- Solve for gas potential after accretion?

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