Joel2: Created page with "FORCETOC =Rotationally Flattened White Dwarfs= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffeeee;" |Ostriker
Bodenheimer
& Lynden-Bell
(1966) |} As we have reviewed in an SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|acco..."

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Created page with "__FORCETOC__  =Rotationally Flattened White Dwarfs= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffeeee;" |<b>Ostriker<br />Bodenheimer<br />& Lynden-Bell<br />(1966)</b> |} As we have reviewed in an SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|acco..."

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__FORCETOC__

=Rotationally Flattened White Dwarfs=
{| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like|<b>Ostriker<br />Bodenheimer<br />& Lynden-Bell<br />(1966)</b>]]
|}
As we have reviewed in an [[SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|accompanying discussion]],
[http://adsabs.harvard.edu/abs/1935MNRAS..95..207C Chandrasekhar (1935)] was the first to construct models of spherically symmetric stars using the [[SR#Time-Independent_Problems|barotropic equation of state appropriate for a degenerate electron gas]]. In so doing, he demonstrated that the maximum mass of an isolated, nonrotating white dwarf is <math>M_3 = 1.44 (\mu_e/2)M_\odot</math>. A concise derivation of <math>M_3</math> is presented in Chapter ''XI'' of [[Appendix/References#C67|Chandrasekhar (1967)]].

Something catastrophic should happen if mass is greater than <math>M_3</math>. What will rotation do? Presumably it can increase the limiting mass.
 <br />
 <br />
 <br />
 <br />

==Solution Strategy==

===Our Approach===

When the stated objective is to construct steady-state equilibrium models of rotationally flattened, axisymmetric configurations, the [[AxisymmetricConfigurations/Equilibria#Axisymmetric_Configurations_.28Steady-State_Structures.29|accompanying introductory chapter]] shows how the overarching set of [[PGE#Principal_Governing_Equations|principal governing equations]] can be reduced in form to the following set of three coupled PDEs (expressed either in terms of cylindrical or spherical coordinates):

<table align="center" border="1" cellpadding="10">
<tr><th align="center" colspan="2"><font size="+0">Table 1:   Simplified Set of Three Coupled PDEs</font></th></tr>
<tr>
<th align="center" width="50%">Cylindrical Coordinate Base</th>
<th align="center">Spherical Coordinate Base</th>
</tr>
<tr>
<td align="center">
<font color="#770000">'''Poisson Equation'''</font><br />

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi G \rho </math>
</td>
</tr>
</table>

The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
<br />

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>~{\hat{e}}_\varpi</math>:    </td>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3}
</math>
</td>
</tr>
<tr>
<td align="right"><math>~{\hat{e}}_z</math>:    </td>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math>
</td>
</tr>
</table>

</td>
<td align="center">
<font color="#770000">'''Poisson Equation'''</font><br />

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr]
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi G\rho</math>
</td>
</tr>
</table>

The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
<br />

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>~{\hat{e}}_r</math>:    </td>
<td align="right">
<math>
~0
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr] - \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right"><math>~{\hat{e}}_\theta</math>:    </td>
<td align="right">
<math>
~0
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr] - \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>

This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>j(\varpi)</math>. How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer & Lynden-Bell (1966)] and further detailed and executed by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract J. P. Ostriker & P. Bodenheimer (1968)]?

===Approach Outlined by OBL66===

* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966; hereinafter OBL66)], Phys. Rev. Letters, 17, 816: ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">… 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</font> [Chandrasekhar (1931, ApJ, 74, 81)] <font color="green">mass limit <math>~M_3</math> is increased by only a few percent when uniform rotation is included in the models, …</font>

<font color="green">In this Letter we demonstrate that white-dwarf models with masses considerably greater than <math>~M_3</math> are possible if differential rotation is allowed … models are based on the physical assumption of an axially symmetric, completely degenerate, self-gravitating fluid, in which the effects of viscosity, magnetic fields, meridional circulation, and relativistic terms in the hydrodynamical equations have been neglected.</font>
</td></tr></table>

====Their Equation (4)====

One can immediately appreciate that, independent of the chosen coordinate base, the first expression listed among our trio of governing PDEs derives from the ''differential representation'' of the Poisson equation as [[AxisymmetricConfigurations/PoissonEq#Overview|discussed elsewhere]] and as has been reprinted here as Table 2.

<div align="center">
<table border="1" cellpadding="8" align="center" width="70%">
<tr><th align="center" colspan="2"><font size="+0">Table 2:  Poisson Equation</font></th></tr>
<tr>
<th align="center">Integral Representation</th>
<th align="center">Differential Representation </th>
</tr>
<tr>
<td align="center">
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi(\vec{x})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
</table>
</td>
<td align="center">
{{Math/EQ_Poisson01}}
</td>
</tr>
</table>
</div>

[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] chose, instead, to use the ''integral representation'' of the Poisson equation to evaluate the gravitational potential; specifically, they write,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_g(\vec{x})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (4)
</td>
</tr>
</table>
(Note that, in defining <math>~\Phi_g</math>, [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] have adopted a sign convention for the gravitational potential that is the opposite of ours; that is, <math>~\Phi_g = - \Phi</math>.)

====Their Equations (3) & (5)====

The two relevant components of the Euler equation that are identified, above, result from imposing a ''steady-state'' condition on the,

<div align="center">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\cancel{\frac{\partial \vec{v}}{\partial t} } + (\vec{v} \cdot \nabla)\vec{v}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{\rho} \nabla P - \nabla \Phi \, ,
</math>
</td>
</tr>
</table>
</div>

and adopting a steady-state rotational velocity field in which the angular velocity is either constant or is only a function of the cylindrical-coordinate radius, <math>~\varpi</math>; that is,

<div align="center">
<math>~\vec{v} = \hat{e}_\varphi [v_\varphi] = \hat{e}_\varphi [\varpi \dot\varphi (\varpi)] \, .</math>
</div>

As we have demonstrated in [[AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Solution_Strategies.29|an accompanying discussion]], for any of a number of astrophysically relevant [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple rotation profiles'']] of this form, the [[AxisymmetricConfigurations/PGE#CYLconvectiveOperator|convective operator]] on the left-hand side of this steady-state Euler equation gives (most conveniently written here in a cylindrical-coordinate base),
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\vec{v} \cdot \nabla)\vec{v}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~\hat{e}_\varpi \biggl[\frac{v_\varphi^2}{\varpi} \biggr] = -~\hat{e}_\varpi \biggl[ \varpi {\dot\varphi}^2(\varpi) \biggr] = -~\hat{e}_\varpi \biggl[\frac{j^2(\varpi)}{\varpi^3} \biggr] \, ,</math>
</td>
</tr>
</table>
where, <math>~j \equiv \varpi^2 \dot\varphi</math> is the (radially dependent) specific angular momentum measured relative to the symmetry (rotation) axis. As we have pointed out in an [[AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Solution_Strategies.29|accompanying discussion]], this last expression can be rewritten in terms of the gradient of a scalar (centrifugal) potential; specifically,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~\nabla \Psi \, ,</math>
</td>
</tr>
</table>
if the centrifugal potential is defined such that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi(\varpi)</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~- \int_0^\varpi \frac{j^2(\varpi^')}{(\varpi^')^3} d\varpi^' \, .</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (5)
</td>
</tr>
</table>
(Note that [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] adopted a sign convention for the centrifugal potential that is the opposite of ours; that is, <math>~\Phi_c = - \Psi</math>.) Hence, assuming that our intent is to construct a rotationally flattened equilibrium configuration whose rotation profile is of the form, <math>~\vec{v} = \hat{e}_\varphi [\varpi \dot\varphi (\varpi)] </math>, the ''steady-state'' Euler equation can be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{1}{\rho} \nabla P</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla \Phi - \nabla \Psi \, .
</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (3)
</td>
</tr>
</table>

====Their Adopted Angular-Momentum Distribution====

In what follows, text that has been extracted directly from p. 817 of [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] is presented using a dark green font.

"<font color="darkgreen">The angular-velocity distribution in the model is determined through the specification of a distribution of angular momentum per unit mass <math>~j(m)</math>, where <math>~m</math> is a Lagrangian coordinate equal to the fraction of the total mass interior to a cylindrical surface around the axis of rotation. The specification of <math>~j(m)</math> rather than <math>~\dot\varphi(\varpi)</math> permits the construction of equilibrium models for a given choice of [total] angular momentum <math>~J</math>. The angular-momentum distribution chosen for the computed models is that of a uniformly rotating polytrope of index <math>~\tfrac{3}{2}</math>.</font>"

Later papers refer to models with OBL66's specified angular momentum profile as belonging to an <math>~n^' = \tfrac{3}{2}</math> sequence. It cannot be described by a closed-form analytic expression. But, as a point of reference and drawing from [http://adsabs.harvard.edu/abs/1965ApJ...142..208S Stoeckly's (1965)] work, in an [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_.28n.27_.3D_0.29|accompanying discussion]] we derive the analytic expression for the angular momentum distribution of models that lie along a so-called <math>~n^' = 0</math> sequence.

====Their Adopted Barotropic Equation of State====

Because they were interested in constructing equilibrium models of rotationally flattened white dwarfs, [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] chose a barotropic equation of state that describes a ''zero-temperature Fermi (degenerate electron) gas''. As has been documented in our [[SR#Barotropic_Structure|accompanying discussion of barotropic equations of state]], the set of key relations that define this equation of state is,

<div align="center">
{{Math/EQ_ZTFG01}}
Reference (original): [http://adsabs.harvard.edu/abs/1935MNRAS..95..207C S. Chandraskehar (1935)]<p></p>
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (2)
</div>

As we also have [[SSC/Structure/WhiteDwarfs#ChandrasekharMass|reviewed elsewhere]], the ''Chandrasekhar limiting mass'' that is associated with this equation of state in ''nonrotating'' stars is given by the expression,
<div align="center">
<math>
\frac{M_\mathrm{Ch}}{M_\odot} = \frac{5.742}{\mu_e^2} .
</math><p></p>
[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 816, Eq. (1)
</div>

===Further Implementation by OB68===

If we define an effective potential,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi_\mathrm{eff}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\Phi + \Psi \, ,</math>
</td>
</tr>
</table>
and recall that, for a [[SR#Barotropic_Structure|barotropic equation of state]], we can make the substitution, <math>~\nabla P \rightarrow \rho\nabla H</math>, where <math>~H</math> is the fluid enthalpy, then OBL66's hydrostatic balance equation — their equation (3) — can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla [H + \Phi_\mathrm{eff} ]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 </math>
</td>
<td align="center">    <math>~\Rightarrow</math>    </td>
<td align="right">
<math>~H + \Phi_\mathrm{eff}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_B \, , </math>
</td>
</tr>
</table>
where, <math>~C_B</math> is a constant (i.e., independent of position). Rotationally flattened, steady-state (equilibrium) configurations can be constructed by finding spatial density distributions that simultaneously satisfy the Poisson equation and this deceptively simple algebraic relation. [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract Ostriker & Bodenheimer (1968; hereinafter, OB68)] used this "self-consistent field" technique to obtain models of rotationally flattened white dwarfs; it is a technique of choice that we [[AxisymmetricConfigurations/SolutionStrategies#Technique|broadly promote]] as well.

In the specific case of a zero-temperature Fermi (degenerate electron) gas — see our [[SR#Barotropic_Structure|related discussion of barotropic structures]] — to within an additive constant, the enthalpy associated with <math>~P_\mathrm{deg}</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[ (1 + \chi^2 )^{1 / 2} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl\{ \biggl[1 + \biggl(\frac{\rho}{B_\mathrm{F}}\biggr)^{2/3} \biggr]^{1 / 2} \biggr\} \, .
</math>
</td>
</tr>

<tr>
<td align="center" colspan="6">[https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68], p. 1090, Eq. (4)</td>
</tr>
</table>
Note that, using this expression, the enthalpy at the surface <math>~(\rho = 0)</math> is <math>~H_s = 8A_\mathrm{F}/B_\mathrm{F}</math>. ([[SR#Barotropic_Structure|Our tabulated expression for the enthalpy]] has been shifted by this constant value so that the enthalpy naturally goes to zero at the surface.) If we use <math>~\Phi_\mathrm{eff,s}</math> to denote the surface value of the effective potential, the constant in the algebraic hydrostatic-balance expression must be,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~C_B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~H_s + \Phi_\mathrm{eff,s}</math>
</td>
</tr>
</table>
Then, at every other spatial location, <math>~\vec{x}</math>, we must have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H(\vec{x}) - H_s + \Phi_\mathrm{eff}(\vec{x}) - \Phi_\mathrm{eff,s}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{8A_\mathrm{F}}{B_\mathrm{F}} \Biggl[ \biggl\{ \biggl[1 + \biggl(\frac{\rho(\vec{x})}{B_\mathrm{F}}\biggr)^{2/3} \biggr]^{1 / 2} \biggr\} - 1 \Biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}</math>
</td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{\rho(\vec{x})}{B_\mathrm{F}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Biggl[ \biggl\{ \frac{B_\mathrm{F}}{8A_\mathrm{F}} \biggl[- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}\biggr] + 1 \biggr\}^2 - 1 \Biggr]^{3/2} \, .
</math>
</td>
</tr>

<tr>
<td align="center" colspan="3">[https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68], p. 1090, Eq. (5)</td>
</tr>
</table>
(Note that, as with OBL66, a different sign convention was adopted by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68] for the effective potential than we have used; that is, <math>~\mathfrak{B} = - \Phi_\mathrm{eff}</math>.)

=See Also=
* Our discussion of [[Apps/RotatingWhiteDwarfs|Rotating White Dwarfs: Example Equilibria]]
* [[ThreeDimensionalConfigurations/BinaryFission|Fission Theory of Binary Star Formation]]

{{ SGFfooter }}

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