This page will briefly describe how to use this tool to calculate the extreme case of an n = 0 polytrope. The density of such a polytrope is constant throughout the radial distance from the star's center. We will use the plot tool to investigate the behavior of this polytrope.
This tutorial assumes some familiarity with the polytrope tool. There is a beginner tutorial available for first time users. While using the tool, click on any blue link to learn more about a particular subject.
Polytropic Index
After clicking the 'Launch Polytrope Tool' button located on the main tool page, the polytrope tool will open in a separate window. Enter a polytropic index of "0."
Once your polytropic index is entered properly, click "Calculate Polytrope" and see the result:
Notice that the ratio of average density to central density (RC) is one, which indicates a uniform mass density along the radial distance from the star's center.
Close the Polytrope Calculation window to unfreeze the Polytrope Tool and click "Next" on the Tool to plot the results.
Plot Results
In this panel, you can plot quantities related to the scaled polytrope. We will plot scaled mass density vs. distance variable. To do this, select "Mass Density" in the drop-down menu for the y-axis.
Once the mass density is selected in the drop down menu, click "View Plot" to obtain this result:
Notice that it apears that nothing has been plotted. In fact, there is a graph there, but we need to adjust the axes to see it. To do so, increase the maximum y-axis value from "Default" to 1.5 and the maximum x-axis value from "Default" to 3.0. The fields you edit are illustrated below.
This is the result of the axes change if you click on "View Plot":
Notice that the mass density is indeed one all the way up to the root of the solution to the Lane-Emden equation (2.44949) where it drops sharply to zero.
Another interesting graph to plot is the solution to the Lane-Emden equation (Phi in the drop-down menu) vs. scaled distance. The n = 0 polytrope has an analytic solution for φ as a function of the scaled distance ξ. The solution happens to be parabolic: φ(ξ) = 1 - ξ2/6. Click "Next" to view a table of data.
Table
In this panel, you may select a table of polytropic data to investigate. To generate the table of data that produced the plot in the last selection, for example, check the boxes for Mass Density and Distance Variable. You may then generate an html or ascii table of the data.
Please click "Next" to generate a physical polytrope.
Physical Polytrope
In this panel, we can enter physical parameters to investigate a polytropic star. Using the defaults, which are the central density (150 g/cc) and mass of the Sun (one solar mass), we can see how close an n = 0 polytrope gets to the measured solar radius (6.9599x1010 cm). After you click on "Generate Physical Polytrope," notice that the agreement is not very good, but still within an order of magnitude. Of course, the Sun's mass density is far from constant throughout the star and is, in fact, physically closer to an n = 3 polytrope, as discussed in the beginner tutorial. Feel free to investigate this polytrope further by using the other panels.
