density versus radius
log density versus radius
log pressure versus radius
log temperature versus radius
mass fraction versus radius
temperature versus radius
equation of hydrostatic equilibrium
derivation of Lane-Embden equation part 1
derivation of Lane-Embden equation part 2
derivation of Lane-Embden equation part 3
derivation of Lane-Embden equation part 4
derivation of Lane-Embden equation part 5
n = 3 polytropic model
various polytropic models
Stellar Models
One of the main tasks of an astrophysicist is to use physics theory and the results of empirical measurements to
make physical models which make predictions about the nature of stars. The predictions of these models can then
be tested by further measurements or observations and refined with the end result of understanding more about
stars and what makes them work. The graphs below show how density, pressure, temperature and mass are
predicted to vary as we move outwards from the centre of the Sun (radius  = 0) to the observable surface of the Sun
(radius = 1 solar radius). This model was what we call an n = 3 polytropic model.
The model behind these computations is explained below.
The model requires us to input a value for n. The resulting pair of first-order differential equations can only be solved
analytically only for n = 0, for n = 1, and n = 5. Alternatively, the resulting pair of first-order differential equations can
be solved with the help of a computer to obtain approximate numerical solutions. I used the Euler method in MS Excel
(which is not the best approach, as i shall explain later, but is quite sufficient for our purposes here). The
spreadsheet can be
downloaded here. The general approach is to increment the dimensionless radius (Greek letter
xsi) by a set amount (the step size) in a series of steps starting from an initial value of zero. We use the initial
conditions theta = 1 (maximum density) and phi = 0 (zero mass contained within a sphere of radius 0) at the centre of
the star. For each step, the dimensionless radius (xsi) is incremented and fed into the two first-order differential
equations and phi calculated at this radius using the theta from the previous step (or initial condition) and theta
calculated for this radius using phi from the previous step (or initial condition). By this iterative process an
approximation to the solution is obtained. The solutions are plotted below:
Above: the polytrope solutions for 9from left to right) n = 1, n = 1.5, n = 3, and n = 5. Notice that the n = 5 solution
extends to infinity and is not physically realistic for a star and so must be rejected. We can use these solutions to
obtain predictions for how the density, temperature, pressure and mass vary with radial distance from the centre
within the star, and plot the results using more familiar units rather than these dimensionless variables. We can feed
in empirical data on the radius and mass of the star being modelled to obtain the additional variables. This has been
done for the Sun, modeled with an n = 3 polytrope as its equation of state, below:
This solution was used to generate the graphs at the top of this page.

Accuracy of the Model

Appraisal of the Model

Page last updated: 10/1/2015