[Webnucleo-public] Polytrope Tool vs Standard Solar Model (BP2004)

[Bradley S. Meyer]

On Thu, December 13, 2007 8:11 am, Lonnie Cumberland wrote:

> [snip] since I have only just started using the Polytrope Tool, (which I
think is
> VERY good by the way), I was not sure of the out data from the tool as it
> did not seem to compare to the tables in the BP2004 data file.
>
> I am wondering if someone might be able to tell me if I am missing some
> scaling or something?
>

No, Lonnie, you are not missing something.  Remember a polytrope is only
an approximation to a real stellar model.  To construct a polytrope, you
assume the pressure P is a constant K time the density rho raised to the
(1+1/n) power.  K and n are constant throughout the model.  This
particular relationship allows you to solve for the structure of the star
via the Lane-Emden equation.

For a real star such as the Sun, the relationship between the pressure and
density will vary depending on the equation of state, the detailed
relationship between the pressure, density, temperature, and chemical
composition.  Since you have the BP2004 data, try plotting the logarithm
of the Pressure versus the logarithm of the density.  Unlike for the
polytrope, where this plot will be a straight line, you will find a more
complicated plot.  If you can extract those data, try using the curve
fitting tool at

http://www.webnucleo.org/home/online_tools/curve_fit/0.3/

to fit log rho vs. log P to a line.  The slope of the fitted line will
give an estimate for the "overall" n of the BP2004 model.  It should be
reasonably close to n = 3.

Without a detailed equation of state, it is not possible to obtain the
temperature T from a polytrope.  You can use the usual gas law P = N k T,
where N is the number density and k is Boltzmann's constant.  N would be
rho times Avogadro's number divided by the average mass number of nuclear
species present.  A polytrope also doesn't follow energy generation and
abundance changes.  You have to use much more complicated models (like
BP2004) to get that information.

What's remarkable is how well polytropes work.  You can see why people
used them to study stellar structure in the days before fast computers. 
Best wishes.

Brad Meyer


-- 
Bradley S. Meyer
Department of Physics and Astronomy
Clemson University
Clemson, SC 29634-0978
mbradle at clemson.edu
http://nucleo.ces.clemson.edu