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 ρ^{5/3}. 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 ρ^{4/3}. 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 Y_{e}.
Y_{e} 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 ^{12}C and ^{16}O),
Y_{e} = 0.5, which we use in this tutorial.
Once you find the Chandrasekhar Limit in this tutorial, you may want
to try for other Y_{e}'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 ρ^{4/3}, 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
1.2434979x10^{15} Y_{e}^{4/3} dynes
cm^{2}/g^{4/3},
where Y_{e} is fraction of all nucleons that are protons.
For a C/O white dwarf star, Y_{e} = 0.5, so the value of K is
4.9348247x10^{14} dynes cm^{2}/g^{4/3}.

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.

**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 / ρ^{4/3}).
For the one solar mass star, we get
K = 3.8407278x10^{14} dynes cm^{2}/g^{4/3}
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 = 6.0967753x10^{14} dynes cm^{2}/g^{4/3},
which is clearly above the limit appropriate for degenerate, relativistic
electrons. We have successfully
bracketed the maximum mass between one and two solar masses.

**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 x10^{14} (dynes cm^{2}/g^{4/3}) |
---|---|

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
Y_{e}?