Can a Self-Gravitating Gas Ball Simulate Stellar Formation Dynamics?

[Glurth]

I have a simulation I'm trying out (for fun). A self-gravitating ball of gas, in deep space. (The sim uses a fixed-time-step for each iteration.)

I'd like to use Boyles Ideal gas law, the force of gravity, and energy as internal heat. (I don't want to touch enthalpy unless I don't realize it.) This sim will ignore relativity, and gradient of internal pressure (assumes constant homogeneous pressure within the ball or gas).PV=nRT (R being the ideal gas constant)

Since I'm using a single type of matter, and I'm not really concerned with the scale of each unit, I'm simplifying this to use the total mass of the gas ball (M):

PV=mass * T

I'd like to define Temperature using internal heat (H)

T = mass * internal heat

PV= mass^2 * internal heat
p= mass^2 * internal heat/Volume

Given a ball of Radius (R) the Volume is (4/3)*pi*R^3
P= mass^2 * internal heat/((4/3)*pi*R^3)

Pressure is the Force time the surface area(S).
FS=P

Given a ball of Radius (R) the surface area(S) is 4*pi*R^2
F=P/4*pi*R^2

F= mass^2 * internal heat/((4/3)*pi*R^3 *4*pi*R^2 )
F= mass^2 * internal heat * Constant/R^5

OK, so I'm pretty confident in the above, but please point out any mistakes.

Now, how do I compute force or pressure, due to gravity?

F=GmM/r^2

Obviously, I use the mass of the gas-ball for capital M, and I suspect, the radius for r, but what do I use for lowercase m?

Goal: I'd like to be able to add the force due to the gas pressure and the force due to gravity, and use this force sum to accelerate the Radius.

I will keep track of the radius velocity(change in radius per unit time), adjusting each iteration by the acceleration of the radius.

F=ma

Fsum = m * (acceleration of radius)

But I'm not sure if I should be using the entire mass of the gas ball here for the variable "m", though I'm not sure what else would be applicable. Perhaps some fraction of the mass, related to the current radius and its velocity?

I will use the radius velocity to determine the distance traveled, over the period of the time the force is applied.
I like to extract the WORK applied (by gravity pulling in), or used (by gas pressure pushing out)
W=Fs
Work= Fsum + distance of Radius change
The work done/used can then simply be added to the internal heat.

Then we run another iteration using the new radius, radius velocity, and internal heat.

 

[russ_watters]

How big of a ball and how dense? My recollection from prior discussion is that it starts as a pure ballistic collapse before transitioning at a certain density to gas behavior.

 

[Oudeis Eimi]

The OP has many misconceptions. Pressure as force times area? Temperature as mass times "internal heat" (whatever that is)? Homogeneous pressure in a self-gravitating cloud? Besides, how many particles are you simulating, OP? Unless you have a lot, temperature and pressure aren't going to be properly defined.

 

[Glurth]

>Russ
How big and dense? What happens at various sizes, masses, and temperatures, is what I'd like the sim to show me. Let's assume it is dense enough to be modeled as a gas.

>Oudeis
P=F*area
oops! good catch, thank you, I'll use...
F=P*area (http://hyperphysics.phy-astr.gsu.edu/hbase/press.html)

Internal heat is a measure of energy inside a given body. In particular, the energy you detect when measuring temperature. I thought this made sense because if I cut the ball of gas in thirds, each third would have the same tempurature, but only one third the total "internal energy". Units would be equivalent to Calories. Would you recommend a different expression or term?

Homogeneous pressure in a self-gravitating cloud? Yes, it's just a simple sim, so I'd like to ignore the pressure gradients. Perhaps we should consider the average pressure?
Besides, how many particles are you simulating, OP? Enough to assume it behaves as a gas. (no "particles", just a ball of gas with inital variables: Radius(changes during sim), mass, internal energy(changes during sim), gasConstant

 

[Khashishi]

I like to extract the WORK applied (by gravity pulling in), or used (by gas pressure pushing out)
W=Fs
Work= Fsum + distance of Radius change
The work done/used can then simply be added to the internal heat.

It sounds like you are treating the compression as reversible. In reality, there is viscosity in the infalling gases which will convert some kinetic energy to heat irreversibly. But this is a more difficult problem.

T = mass * internal heat

That's not right. Temperature doesn't scale with mass. Try:
T = U/N
where U is internal heat and N is the number of particles.

PV=mass * T

It's not at all clear why you want to equate particle number with mass. You explained that you were working with a single species, but still... why?

p= mass^2 * internal heat/Volume

That's not right. You should have P = (N/V) kT

But I'm not sure if I should be using the entire mass of the gas ball here for the variable "m", though I'm not sure what else would be applicable. Perhaps some fraction of the mass, related to the current radius and its velocity?

The problem is that you want to treat the ball of gas as a single "object".
It sounds like you need to treat the ball of gas as a set of concentric shells. The mass m will depend on your choice of a thickness for the outer shell.
The issue arises because you want to treat the pressure as constant within the ball. In reality, the pressure will decrease with radius, so it is possible to calculate a force on each layer. Since you treat the pressure as constant, you will have to pick some m which will be on the order of magnitude of M. I don't think there is a right answer here, so try m = M/2.

 

[NFuller]

How big and dense? What happens at various sizes, masses, and temperatures, is what I'd like the sim to show me. Let's assume it is dense enough to be modeled as a gas.

I think that russ is concerned about whether the conditions are sufficient for collapse to begin. The gas will only condense if it is sufficiently dense such that the gravitational potential energy is larger than the internal kinetic energy. This occurs when the average density exceeds the Jeans density
$$\rho_{J}=\frac{3}{4\pi M^{2}}\left(\frac{3k_{B}T}{2Gm}\right)^{3}$$
where $M$ is the total mass of the gas cloud, $m$ is the particle mass, and $T$ is temperature.

Now, how do I compute force or pressure, due to gravity?

From what I remember from a stellar astrophysics class I once took, the pressure and the acceleration of the gas is governed by the differential equation
$$-\frac{d^{2}r}{dt^{2}}=\frac{1}{\rho(r)}\frac{dP}{dr}+\frac{G}{r^{2}}\int_{0}^{r}\rho(r')4\pi r'^{2}dr'$$
Although it is straightforward to deduce equilibrium conditions from this equation, using it to calculate the dynamics of the collapsing gas will be difficult. I assume there are specialized numerical algorithms for dealing with this type of problem but I have never used them.