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.

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:

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:

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

Accuracy of the Model

Appraisal of the Model

Page last updated: 10/1/2015