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White Dwarf

A white dwarf star is one that supports itself against gravitational collapse by electron degeneracy pressure. Chandrasekhar showed that there is a limit to how massive a white dwarf star can be, and this tutorial shows how to use the Polytrope Tool to compute this limiting mass, appropriately named the Chandrasekhar Limit. For low-mass white dwarf stars, the degree of electron degeneracy is low and the degenerate electrons are non-relativistic (i.e., they move at speeds much less than the speed of light). This means their pressure P is proportional to the mass density ρ raised to the 5/3 power: P = K ρ5/3. When the pressure in the star follows this pressure law, the star is appropriately modeled by an n = 3/2 polytrope. As the white dwarf star becomes larger in mass, however, the degree of the electron degeneracy increases, and the most energetic electrons become relativistic (travel near the speed of light). In this case, the pressure follows the equation P = K ρ4/3. This is an n = 3 polytrope.

The idea of the Chandrasekhar Limit is that there is mass at which the pressure at the center of the n = 3 polytrope is higher than the fully degenerate, relativistic electrons can provide. Gravity would win and the star would collapse to a neutron star or black hole.

This tutorial will use a standard model polytropic white dwarf with n = 3. You will see that as you increase the polytrope's mass, the pressure constant K will increase. Eventually it will exceed the number that fully degenerate, relativistic electrons can provide, and you will have found the Chandrasekhar Limit. Importantly, this Limit depends on the composition of the star, which is characterized for our purposes by the number Ye. Ye is the fraction of all nucleons (neutrons and protons) that are protons. Since most white dwarf stars are Carbon and Oxygen white dwarfs (comprised of 12C and 16O), Ye = 0.5, which we use in this tutorial. Once you find the Chandrasekhar Limit in this tutorial, you may want to try for other Ye's.

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 3, the default value, in the polytropic index box.

Click "Calculate Polytrope" and see the calculation results in a separate window. Close this window to unfreeze the Polytrope Tool and click on "Physical Polytrope" in the Task List or click "Next" the required number of times to get to the "Physical Polytrope" panel.

Physical Polytrope

Recall that for n = 3, the pressure P is given by P = K ρ4/3, where K is a constant and ρ is the density. For the n = 3 polytrope only, K is independent of the central density. For relativistic electrons, the value of K is given by 1.2434979x1015 Ye4/3 dynes cm2/g4/3, where Ye is fraction of all nucleons that are protons. For a C/O white dwarf star, Ye = 0.5, so the value of K is 4.9348247x1014 dynes cm2/g4/3.

We would expect the electrons at the center of a white dwarf to be highly degenerate and relativistic, with the electrons becoming less relativistic farther from the center. The most extreme possible configuration for a white dwarf is that all the degenerate electrons are relativistic. Thus, any n = 3 polytrope that has a K greater than the number above cannot be supported by electron degeneracy pressure and therefore cannot be a white dwarf.

We now want to use the Polytrope Tool to estimate the Chandrasekhar limit. In the field marked "Central Density" you may put any value since the K for an n = 3 polytrope is independent of the central density. For simplicity, enter 1 in the central density box. In the field marked "Mass," type in 1, that is, one solar mass. Click on "Generate Physical Polytrope," wait for the calculation to finish and close the calculation window. The last two panels in the tool should unfreeze. Click "Next" twice or go to "Physical Table" on the Task List.

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Physical Table

Check the "Pressure" and "Mass Density" checkboxes on the tool and click "View HTML Table". You may use any point to calculate the value of K (recall that K = P / ρ4/3). For the one solar mass star, we get K = 3.8407278x1014 dynes cm2/g4/3 which is below the Chandrasekhar limit for K. Now go back to "Physical Polytrope", leave the central density at 1 and change the mass to 2 solar masses. Proceed through the steps and view the same mass density and pressure table. You will find K = 6.0967753x1014 dynes cm2/g4/3, which is clearly above the limit appropriate for degenerate, relativistic electrons. We have successfully bracketed the maximum mass between one and two solar masses.

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Finding the Limit

Performing the procedure outlined in the last section with different mass values will help us "zero-in" on the Chandrasekhar limit. Doing so, you can generate this table in increments of tenths of solar mass:

Mass (Solar Mass) K x1014 (dynes cm2/g4/3)
1.0 3.8407278
1.1 4.0926882
1.2 4.3371160
1.3 4.5748383
1.4 4.8065361
1.5 5.0327771
1.6 5.2540417
1.7 5.4707409
1.8 5.6832294
1.9 5.8918171
2.0 6.0967753

Notice this gives us a finer bracket of (1.4,1.5) solar masses. Use the procedure above to refine your estimate of the Chandrasekhar limit. Confirm that your result is independent of your choice of central density. What happens if you try a different Ye?

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