Are there equations to simulate creation of a solar system?

Sunday, 4 October 2009

Are there equations to simulate creation of a solar system?

I may have to answer this in parts, a bit at a time, so I apologize if I'm not able to answer everything at the moment.

If I wanted to find out how to estimate, say, a planets surface temperature based on a stars temperature and distance, what is the best way to do this?

Ah, I just studied this about a week ago! What you're looking for is the effective temperature of the planet, which can be calculated as

$T=left(frac{L(1-A)}{16 pi sigma D^2} right)^{frac{1}{4}}$
where

$L$ is the star's luminosity,

$A$ is the planet's albedo,

$sigma$ is the Stefan-Boltzmann constant, and

$D$ is the distance from the star.

how does the mass of a star relate to the distance of the frost line, and is there an already existing equation to estimate this?

I wasn't able to find an explicit equation for this, but it's calculable. The frost line is a certain distance from a star in a stellar nebula such that the temperature is about 150 Kelvin.

If you know the mass of the star, you can use the mass-luminosity relation:

$left(frac{L}{L_{odot}} right)=left(frac{M}{M_{odot}} right)^a$
Re-arrange that to solve for the luminosity. The inverse-square law says that the intensity of power drops off as the distance from the source increases:

$I=frac{P}{4 pi r^2}$
You can solve for

$r$ if you can figure out the required intensity:

$r=frac{1}{2}sqrt{frac{P}{pi I}}$
Here,

$P approx L$ . Figure out the necessary intensity to heat the nebula to 150 K and you've found the frost line.

To find the most likely average mass of a potential planet, what information would I need?

I can't help you much there, but I can give you a starter. The density $rho$ of a circumstellar disk at a distance $r$ (as given by Michael Woolfson in On the Origin of Planets) is

$rho (r)=C exp left[-frac{(r-r_{text{peak}})^2}{2 sigma ^2} right]$
where

$sigma$ is one standard deviation and

$r_{text{peak}}$ is the peak density, and

$C$ is a constant.

As promised, here are some more, set forth by Woolfson.

The disk decays over time; its density at time $t$ is approximated by

$rho_t=rho e^{- gamma t}$
where

$gamma$ is a decay constant. If you've taken certain math course, you've probably seen this equation many times before, just in with

$rho=y$ and

$gamma=k$ .

Another equation - really, a couple equations - deal with accretion by a small dust particle in a circumstellar disk. The variables involved are: $m$ (mass of the particle), $t$ (time), $s$ (the radius of the particle), $rho_s$ (the density of the particle), $rho$ (the density of the disk), $f$ (the fraction of the medium that is dust), $T$ (the temperature), and $k$ (Boltzmann's constant, not be confused with the Stefan-Boltzmann constant).

The relevant pair of equations are:

$frac{dm}{dt}=4 pi s^2 rho_s frac{ds}{dt}$

$frac{ds}{dt}=frac{3f rho}{4} left( frac{kT}{4 pi rho_s^3} right)^{1/2}s^{-3/2}$
Using these, you can find the size and mass of a given dust particle at any given time.

AnswerDesk

Sunday, 4 October 2009