User:Tohline/SSC/Structure/WhiteDwarfs
From VisTrailsWiki
(→See Also) 
(→The Nauenberg MassRadius Relationship) 

Line 120:  Line 120:  
</math>  </math>  
</div>  </div>  
  where,  +  <span id="ChandrasekharMass">where,</span> 
<div align="center">  <div align="center">  
<math>  <math> 
Current revision as of 09:34, 9 August 2019
Contents 
White Dwarfs
 Tiled Menu  Tables of Content  Banner Video  Tohline Home Page  
MassRadius Relationships
The following summaries are drawn from Appendix A of Even & Tohline (2009).
Chandrasekhar mass
Chandrasekhar (1935) was the first to construct models of spherically symmetric stars using the barotropic equation of state appropriate for a degenerate electron gas, namely,
In so doing, he demonstrated that the maximum mass of an isolated, nonrotating white dwarf is , where is the number of nucleons per electron and, hence, depends on the chemical composition of the white dwarf. A concise derivation of M_{Ch} (although, at the time, it was referred to as M_{3}) is presented in Chapter XI of Chandrasekhar (1967), where we also find the expressions for the characteristic Fermi pressure, , and the characteristic Fermi density, . The derived analytic expression for the limiting mass is,
,
where the coefficient,
,
represents a structural property of n = 3 polytropes (γ = 4 / 3 gasses) whose numerical value can be found in Chapter IV, Table 4 of Chandrasekhar (1967). We note as well that Chandrasekhar (1967) identified a characteristic radius, , for white dwarfs given by the expression,
The Nauenberg MassRadius Relationship
Nauenberg (1972) derived an analytic approximation for the massradius relationship exhibited by isolated, spherical white dwarfs that obey the zerotemperature whitedwarf equation of state. Specifically, he offered an expression of the form,
where,
n 


N_{0} 


R_{0} 


is the atomic mass unit, is the mean molecular weight of the gas, and ζ and ν are two adjustable parameters in Nauenberg's analytic approximation, both of which are expected to be of order unity. By assuming that the average particle mass denoted by Chandrasekhar (1967) as (μ_{e}m_{p}) is identical to the average particle mass specified by Nauenberg (1972) as and, following Nauenberg's lead, by setting ν = 1 and,
,
in the above expression for N_{0}, we see that,
Hence, the denominator in the above expression for n becomes the Chandrasekhar mass. Furthermore, the above expressions for R_{0} and R become, respectively,
and,
Finally, by adopting appropriate values of and , we obtain essentially the identical approximate, analytic massradius relationship for zerotemperature white dwarfs presented in Eqs. (27) and (28) of Nauenberg (1972):
where,
Eggleton MassRadius Relationship
Verbunt & Rappaport (1988) introduced the following approximate, analytic expression for the massradius relationship of a "completely degenerate star composed of pure helium" (i.e., μ_{e} = 2), attributing the expression's origin to Eggleton (private communication):
where M_{p} is a constant whose numerical value is . This "Eggleton" massradius relationship has been used widely by researchers when modeling the evolution of semidetached binary star systems in which the donor is a zerotemperature white dwarf. Since the Nauenberg (1972) massradius relationship discussed above is retrieved from this last expression in the limit , it seems clear that Eggleton's contribution was the insertion of the term in square brackets involving the ratio M / M_{p} which, as Marsh, Nelemans & Steeghs (2004) phrase it, "allows for the change to be a constant density configuration at low masses (Zapolsky & Salpeter 1969)."
Highlights from Discussion by Shapiro & Teukolsky (1983)
Here we interleave our own derivations and discussions with the presentation found in [ST83].
In our accompanying discussion, we have shown that the equilibrium radius of an isolated polytrope is given, quite generally, by the expression,



Inverting this provides the following expression for the total mass in terms of the equilibrium radius:









As is shown by the following boxedin equation table, this expression matches equation (3.3.11) from [ST83], except for the sign of the exponent on , which is demonstratively correct in our expression.
Equations extracted^{†} from §3.3 (p. 63) and §2.3 (p. 27) of Shapiro & Teukolsky (1983)
"Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects"
(New York: John Wiley & Sons)  


(Eq. 3.3.12) 

(Eq. 2.3.23) 

^{†}Each equation has been retyped here exactly as it appears in the original publication. 
Given that (see equation 3.3.12 of [ST83]; see the boxedin equation table) in the relativistic limit, — that is, — and acknowledging as we have above that, for isolated polytropes,
,
this polytropic expression for the mass becomes,



Separately, [ST83] show that the effective polytropic constant for a relativistic electron gas is (see their equation 2.3.23, reprinted above in the boxedin equation table),



Together, then, the [ST83] analysis gives,



Given that the definitions of the characteristic Fermi pressure, , and the characteristic Fermi density, , are,
we have,












which matches the expression presented above for the Chandrasekhar mass if we set .
See Also
 Edmund C. Stoner (1930), The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 7, Volume 9, Issue 60, p. 944963: The Equilibrium of Dense Stars
 S. Chandrasekhar (1931), ApJ, 74, p. 81: The Maximum Mass of Ideal White Dwarfs
 J. P. Ostriker, P. Bodenheimer & D. LyndenBell (1966), Phys. Rev. Letters, 17, 816: Equilibrium Models of Differentially Rotating ZeroTemperature Stars
… work by Roxburgh (1965, Z. Astrophys., 62, 134), Anand (1965, Proc. Natl. Acad. Sci. U.S., 54, 23), and James (1964, ApJ, 140, 552) shows that the [Chandrasekhar (1931, ApJ, 74, 81)] mass limit is increased by only a few percent when uniform rotation is included in the models, … In this Letter we demonstrate that whitedwarf models with masses considerably greater than are possible if differential rotation is allowed … models are based on the physical assumption of an axially symmetric, completely degenerate, selfgravitating fluid, in which the effects of viscosity, magnetic fields, meridional circulation, and relativistic terms in the hydrodynamical equations have been neglected. 
© 2014  2019 by Joel E. Tohline 