Editing White dwarf (section)

=== Mass–radius relationship ===
{{See also|Chandrasekhar's white dwarf equation|Neutron star#Gravity and equation of state}}
The relationship between the mass and radius of white dwarfs can be estimated using the nonrelativistic [[Fermi gas]] equation of state, which gives<ref name="kawaler">{{cite book
 |last1=Kawaler |first1=S. D.
 |chapter=White Dwarf Stars
 |editor1-last=Kawaler |editor1-first=S. D.
 |editor2-last=Novikov |editor2-first=I.
 |editor3-last=Srinivasan |editor3-first=G.
 |date=1997
 |title=Stellar remnants
 |publisher=1997
 |isbn=978-3-540-61520-0
}}</ref>{{rp|25}}

<math display="block"> \frac{R}{R_\odot} \approx 0.012\left ( \frac{M}{M_\odot}\right )^{-1/3} \left (\frac{\mu_e}{2}\right)^{-5/3},</math>

where {{mvar|R}} is the radius, {{mvar|M}} is the mass of the white dwarf, and the subscript <math>\odot</math> indicates relative to the Sun. The [[chemical potential]], <math>\mu_e</math> is a thermodynamic property giving the change in energy as one electron is added or removed; it relates to the composition of the star. Numerical treatment of more complete models have been tested against observational data with good agreement.<ref>{{Cite journal |last1=Bédard |first1=A. |last2=Bergeron |first2=P. |last3=Fontaine |first3=G. |date=October 2017 |title=Measurements of Physical Parameters of White Dwarfs: A Test of the Mass–Radius Relation |journal=The Astrophysical Journal |language=en |volume=848 |issue=1 |pages=11 |doi=10.3847/1538-4357/aa8bb6 |doi-access=free |arxiv=1709.02324 |bibcode=2017ApJ...848...11B |issn=0004-637X}}</ref>

Since this analysis uses the non-relativistic formula {{math|1= ''p''<sup>2</sup> / 2''m''}} for the kinetic energy, it is non-relativistic. When the electron velocity in a white dwarf is close to the [[speed of light]], the kinetic energy formula approaches {{math|1=''pc''}} where {{math|''c''}} is the speed of light, and it can be shown that the Fermi gas model has no stable equilibrium in the [[ultrarelativistic limit]]. In particular, this analysis yields the maximum mass of a white dwarf, which is:<ref name="kawaler"/>

<math display="block">M_{\rm limit} \approx 1.46\left (\frac{\mu_e}{2}\right)^{-2}</math>

The observation of many white dwarf stars implies that either they started with masses similar to the Sun or something dramatic happened to reduce their mass.<ref name="kawaler"/>

[[File:ChandrasekharLimitGraph.svg|thumb|upright=1.2|right|Radius–mass relations for a model white dwarf. {{math|''M''<sub>limit</sub>}} is denoted as ''M''<sub>Ch</sub>.]]

For a more accurate computation of the mass-radius relationship and limiting mass of a white dwarf, one must compute the [[equation of state]] that describes the relationship between density and pressure in the white dwarf material. If the density and pressure are both set equal to functions of the radius from the center of the star, the system of equations consisting of the [[hydrostatic equation]] together with the equation of state can then be solved to find the structure of the white dwarf at equilibrium. In the non-relativistic case, the radius is inversely proportional to the cube root of the mass.<ref name="chandra2" />{{rp|eqn.(80)}} Relativistic corrections will alter the result so that the radius becomes zero at a finite value of the mass. This is the limiting value of the mass – called the ''[[Chandrasekhar limit]]'' – at which the white dwarf can no longer be supported by electron degeneracy pressure. The graph on the right shows the result of such a computation. It shows how radius varies with mass for non-relativistic (blue curve) and relativistic (green curve) models of a white dwarf. Both models treat the white dwarf as a cold [[Fermi gas]] in hydrostatic equilibrium. The average molecular weight per electron, {{math|''μ''<sub>e</sub>}}, has been set equal to 2. Radius is measured in standard solar radii and mass in standard solar masses.<ref name="chandra2" /><ref name="stds">
{{cite web
 |title=Basic symbols
 |url=http://vizier.u-strasbg.fr/doc/catstd-3.2.htx
 |work=Standards for Astronomical Catalogues, Version 2.0
 |access-date=12 January 2007
 |publisher=[[VizieR]]
 |archive-url=https://web.archive.org/web/20170508162629/http://vizier.u-strasbg.fr/doc/catstd-3.2.htx
 |archive-date=8 May 2017
 |url-status=live
}}</ref>

These computations all assume that the white dwarf is non-rotating. If the white dwarf is rotating, the equation of hydrostatic equilibrium must be modified to take into account the [[centrifugal pseudo-force]] arising from working in a [[rotating frame]].<ref>
{{cite web
 |last1=Tohline
 |first1=J. E.
 |author-link=Joel E. Tohline
 |url=http://www.phys.lsu.edu/astro/H_Book.current/H_Book.html
 |title=The Structure, Stability, and Dynamics of Self-Gravitating Systems
 |access-date=30 May 2007
 |archive-url=https://web.archive.org/web/20100627133917/http://www.phys.lsu.edu/astro/H_Book.current/H_Book.html
 |archive-date=27 June 2010
 |url-status=live
}}</ref> For a uniformly rotating white dwarf, the limiting mass increases only slightly. If the star is allowed to rotate nonuniformly, and [[viscosity]] is neglected, then, as was pointed out by [[Fred Hoyle]] in 1947,<ref>
{{cite journal
 |last1=Hoyle |first1=F.
 |date=1947
 |title=Stars, Distribution and Motions of, Note on equilibrium configurations for rotating white dwarfs
 |volume=107 |issue=2
 |pages=231–236
 |journal=Monthly Notices of the Royal Astronomical Society
 |bibcode=1947MNRAS.107..231H
 |doi=10.1093/mnras/107.2.231
 |doi-access=free
}}</ref> there is no limit to the mass for which it is possible for a model white dwarf to be in static equilibrium. Not all of these model stars will be [[dynamics (mechanics)|dynamically]] stable.<ref>
{{cite journal
 |last1=Ostriker |first1=J. P.
 |last2=Bodenheimer |first2=P.
 |date=1968
 |title=Rapidly Rotating Stars. II. Massive White Dwarfs
 |journal=The Astrophysical Journal
 |volume=151 |page=1089
 |bibcode=1968ApJ...151.1089O
 |doi= 10.1086/149507
 |doi-access=free
}}</ref>

Rotating white dwarfs and the estimates of their diameter in terms of the angular velocity of rotation has been treated in the rigorous mathematical literature.<ref>{{cite journal |bibcode=1994CMaPh.166..417C |title=On diameters of uniformly rotating stars |last1=Chanillo |first1=Sagun |last2=Li |first2=Yan Yan |journal=Communications in Mathematical Physics |year=1994 |volume=166 |issue=2 |page=417 |doi=10.1007/BF02112323 |s2cid=8372549 |url=http://projecteuclid.org/euclid.cmp/1104271617 }}</ref> The fine structure of the free boundary of white dwarfs has also been analysed mathematically rigorously.<ref>{{cite journal |bibcode=2012JDE...253..553C |title=A remark on the geometry of uniformly rotating stars |last1=Chanillo |first1=Sagun |last2=Weiss |first2=Georg S. |journal=Journal of Differential Equations |year=2012 |volume=253 |issue=2 |page=553 |doi=10.1016/j.jde.2012.04.011 |arxiv=1109.3046 |s2cid=144301 }}</ref>