Dwarf Star

White Dwarfs

The material remaining after the ejection of a planetary nebula forms the core remnant of the star. It is mostly carbon or oxygen, and its temperature is not high enough for further nuclear fusion to take place. The gas pressure is not high enough to support the star against gravitational collapse.

Stars supported by electron degeneracy pressure are characterized by their small size, as they need to collapse to a high density before the degeneracy pressure halts the collapse. These remnants of stellar cores are hot and are observed on the HR diagram as white dwarfs.

Electron degeneracy

This collapse would continue forever if not for an additional source of pressure when a high enough density is reached. This pressure arises from electron degeneracy.

Properties of white dwarfs

A degenerate gas, such as a white dwarf, has low radiation opacity due to the difficulty of electron transitions to higher energy states. Degenerate gases, like metals, exhibit high thermal conductivity, limiting temperature differences. White dwarfs, unlike normal stars, maintain a nearly constant internal temperature. The outermost layer cools from significantly. White dwarfs feature strong magnetic fields. As they radiate energy without a new energy source, they cool, eventually becoming too cool to observe. Despite their current temperatures, the small size of white dwarfs makes them challenging to detect directly, often identified in binary systems by their gravitational influence on visible companions.

Relativistic effects

The treatment of degeneracy pressure requires adjustments to account for the special theory of relativity, as first recognized by S. Chandrasekhar. Chandrasekhar demonstrated that these corrections lead to a reduction in degeneracy pressure, establishing an upper limit on the mass sustained by electron degeneracy pressure. The modification stems from the limitation that electrons cannot exceed the speed of light. While \(p_x=(h/2π)(n_e)^{1/3}\) still valid, the assumption that \(v_x = p_x/m_e\) is no longer valid, and the correct relativistic expression must be employed.

To find the maximum degeneracy pressure, we take \(v_x = c\). This gives:

\[P_{max}=(h/2π)c(n_e)^{4/3}\]

A more detailed calculation gives approximately 0.8 times this. Using this, the expression analogous to equation \(P= 2(\frac { h}{2\pi})^2 \ \frac {{[\frac ZA\ \frac{\rho}{m_p}]}^{5/3}}{m_e\ }\) is \(P= 0.8(\frac { h}{2\pi}) \ c\ {[\frac ZA\ \frac{\rho}{m_p}]}^{4/3}\)

If we use the same assumptions to find the mass–radius relation

\(GRM^{1/3}= 2\frac { h^2}{4\pi ^2 m_e} {{[\frac Z{4Am_p}\ ]}^{5/3}} \), we find that the radius drops out, and we simply have an expression for the mass

\[M=0.05(\frac {hc}{2\pi G})^{3/2} (\frac {Z}{Am_p})^2\]

This mass corresponds to the maximum pressure, so it is the maximum mass that can be supported by electron degeneracy pressure. A more accurate calculation, which takes into account variations in pressure and density with distance from the center of the star. The resulting mass, called the Chandrasekhar limit. A star whose nuclear processes have stopped, and whose mass is greater than Chandrasekhar limit, will continue to collapse beyond the white dwarf phase.