User:Tohline/AxisymmetricConfigurations/SolutionStrategies
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(→Centrally Condensed Initially (n' > 0)) 
(→UniformDensity Initially (n' = 0)) 

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+  [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract Stoeckly (1995)], §II.c, eq. (12)  
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Revision as of 18:03, 11 August 2019
Contents 
Axisymmetric Configurations (Solution Strategies)
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Lagrangian versus Eulerian Representation
In our overarching specification of the set of Principle Governing Equations, we have included a,
When seeking a solution to the set of governing equations that describes a steadystate equilibrium configuration — as has already been suggested in our accompanying discussion of "other forms of the Euler equation" — it is preferable to start from an,
Eulerian Representation
of the Euler Equation,
because steadystate configurations are identified by setting the partial time derivative, rather than the total time derivative, to zero. Notice that if the objective is to find an equilibrium configuration in which the fluid velocity is not zero — consider, for example, a configuration that is rotating — then throughout the configuration, the velocity field must be taken into account, in addition to the gradient in the gravitational potential, when determining the pressure distribution. Specifically, for steadystate flows, the required relationship is,



As we also have mentioned elsewhere, by drawing upon a relevant dot product rule vector identity, this expression can be rewritten in terms of the fluid vorticity, , as,



In certain astrophysically relevant situations — such as the adoption of any one of the simple rotation profiles identified immediately below — the nonlinear velocity term involving the "convective operator" can be rewritten in terms of the gradient of a scalar (centrifugal) potential, that is,



In such cases, the condition required to obtain a steadystate equilibrium configuration is given by the considerably simpler mathematical relation,



In the subsection of this chapter (below) titled, Double Check Vector Identities, we explicitly demonstrate for four separate "simple rotation profiles" that these three separate steadystate balance expressions do indeed generate identical mathematical relations.
Simple Rotation Profile and Centrifugal Potential
"… A necessary and sufficient condition for … to be independent of is that the surfaces of constant pressure coincide with the surfaces of constant density, i.e., that P be a function of ρ only." In this case, a centrifugal potential, , can be defined — see the integral expression provided below — and it "is also a function of only … When exists, the equations of state and of energy conservation may be thought of as determining the form of the Pρ relationship. Hence, by prescribing a Pρ relationship, one avoids the complications of those further equations. This effects a major simplification of the formal problem of constructing rotating configurations. This procedure will, of course, be inadequate for certain objectives …" 
— Drawn from N. R. Lebovitz (1967), ARAA, 5, 465 
Specifying in the Equilibrium Configuration
Equilibrium axisymmetric structures — that is, solutions to the above set of simplified governing equations — can be found for specified angular momentum distributions that display a wide range of variations across both of the spatial coordinates, and . According to the PoincaréWavre theorem, however, the derived structures will be dynamically unstable toward the development shapedistorting, meridionalplane motions unless the angular velocity is uniform on cylinders, that is, unless the angular velocity is independent of . (See the detailed discussion by [T78] — or our accompanying, brief summary — of this and other "axisymmetric instabilities to avoid.") With this in mind, we will focus here on a solution strategy that is designed to construct structures with a
Simple Rotation Profile
which of course means that we will only be examining axisymmetric structures with specific angular momentum distributions of the form .
As has been alluded to immeciately above, after adopting a simple rotation profile, it becomes useful to define an effective potential,
that is written in terms of a centrifugal potential,
The accompanying table provides analytic expressions for that correspond to various prescribed functional forms for or , along with citations to published articles in which equilibrium axisymmetric structures have been constructed using the various tabulated simple rotation profile prescriptions.
Simple Rotation Profiles 








Refs. 
Powerlaw 





d, h 
Uniform rotation 





a, f 
Uniform 





e 
Keplerian 





d 
Uniform specific 





c,g 
jconstant 





a,b 
^{a}Hachisu, I. 1986, ApJS, 61, 479507
(especially §II.c) 
Note that, while adopting a simple rotation profile is necessary in order to ensure that an axisymmetric, barotropic equilibrium configuration is dynamical stability, it is not a sufficient condition. For example, the Solberg/Rayleigh criterion further demands that, for homentropic systems, the specific angular momentum, , must be an increasing function of the radial coordinate, . It is not surprising, therefore, that the above table of example simple rotation profiles does not include references to published investigations in which the powerlaw index, , is negative.
"In order to prevent the RayleighTaylor instability … which arises from an adverse distribution of angular momentum — or, more generally, in order to satisfy the Solberg/Rayleigh criterion — must be a monotonically increasing function of . Aside from this restriction, is free to be any wellbehaved function which we may plausibly expect to have been estabilshed over the history of the star." 
— Drawn from J. P. Ostriker & J. W.K Mark (1968), ApJ, 151, 1084 
Prescribing Based on an Initially NonEquilibrium Spherical Configuration
Each of the simple rotation profiles listed in Table 1 has been defined by specifying the radial distribution of the specific angular momentum, , in the rotationally flattened equilibrium configuration. Here we follow the lead of Stoeckly's (1965) and of Bodenheimer & Ostriker (1973) and, instead, present rotation profiles that are defined by specifying the function, , where is a function describing how the fractional mass enclosed inside varies with .
To better clarify what is meant by the function, , consider a configuration (not necessarily in equilibrium) that is spherically symmetric and that exhibits an — as yet unspecified — massdensity profile, . The mass enclosed within each spherical radius is,
and, if the radius of the configuration is , then the configuration's total mass is,
In contrast, the mass enclosed within each cylindrical radius, , is
where it is understood that the argument of the density function is, .
Example #1: If the configuration has a uniform density, , then its total mass is, , and












Example #2: If the spherically symmetric configuration has a density profile given by the function,



then its total mass is, , and
























UniformDensity Initially (n' = 0)
Drawing directly from §IIc of Stoeckly's (1965) work, … consider a large, gaseous mass, initially a homogeneous sphere of mass and angular momentum rotating as a solid body, and suppose it contracts in such a way that cylindrical surfaces remain cylindrical and each such surface retains its angular momentum. Let , , and denote the initial density, radius, and angular velocity of the [initially unstable configuration], the initial radius of the surface now at radius , and the mass inside this surface. The conditions on the contraction are then



and



By integrating, eliminating between these equations, and eliminating , , and in favor of and , one finds the relation of to the mass distribution to be



Stoeckly (1995), §II.c, eq. (12) 
This is equation (12) of Stoeckly (1965); it also appears, for example, as equation (45) in Ostriker & Mark (1968), as equation (12) in Bodenheimer & Ostriker (1970), and in the sentence that follows equation (3) in Bodenheimer & Ostriker (1973). As Stoeckly points out, the angular momentum distribution implied by this functional form of satisfies the Solberg/Rayleigh stability criterion — that is,
— initially, and also in the final equilibrium configuration because every cylindrical surface conserves specific angular momentum and the surfaces do not reorder themselves.
Centrally Condensed Initially (n' > 0)
In §III.d (pp. 1084  1086) of Ostriker & Mark (1968), we find the following relations:



Ostriker & Mark (1968), §III.d, Eq. (50) 
where,























For a uniformdensity sphere, Ostriker & Mark claim that the abovedefined analytical expression for is retrieved by setting, Let's see …























This implies,

Q. E. D.
In addition, from p. 163 (Table 1) of Bodenheimer & Ostriker (1973) we find the following table of coefficient values.
Coefficients for Expression 

0  +2.5  2.5  
+3.068133  +0.203667  3.271800  +0.801297  
1  +3.825819  +0.857311  4.68313  +0.650981  
+4.887588  +2.345310  7.232898  +0.525816  
2  +6.457897  +6.018111  12.476007  +0.417472  
+8.944150  +18.234305  27.178455  +0.321459  
3  +13.270061  +163.26149  176.53154  +0.235287 
Double Check Vector Identities
Let's plug a few different simple rotation profiles into the Euler equation, using a cylindricalcoordinate base to demonstrate that the three expressions are identical, namely, that





Uniform Rotation
In the case of uniform rotation, we have,
where, is independent of radial position. This also means that,
and,









[A] Hence,






[B} Alternatively,






[C} Or,



This demonstrates that, in the case of uniform angular velocity, all three expressions are identical.
Power Law
In the case of a powerlaw expression, we have,
where, and are both independent of radial position. This also means that,
and,









[D] Hence,






[E} Alternatively,









[F} Or,






This demonstrates that, in the case of powerlaw angular velocity profile, all three expressions are identical.
Uniform v_{φ}
In the case of a uniform (i.e., a flat rotation curve), we have,
where, is independent of radial position. This also means that,
and,






[G] Hence,



[H} Alternatively,






[I} Or,









This demonstrates that, in the case of a constant profile, all three expressions are identical.
jConstant Rotation
In the case of socalled jconstant rotation, we have,
where, , and the characteristic length, , are both independent of radial position. This also means that,
and,












[J] Hence,






[K} Alternatively,















[L} Or,



This demonstrates that, in the case of a jconstant rotation profile, all three expressions are identical.
Technique
To solve the abovespecified set of simplified governing equations we will essentially adopt Technique 3 as presented in our construction of spherically symmetric configurations. Using a barotropic equation of state — in which case can be replaced by — we can combine the two components of the Euler equation shown above back into a single vector equation of the form,
where it is understood that here, as displayed earlier, the gradient represents a twodimensional operator written in cylindrical coordinates that is appropriate for axisymmetric configurations, namely,
This means that, throughout our configuration, the functions () and () must sum to a constant value, call it . That is to say, the statement of hydrostatic balance for axisymmetric configurations reduces to the algebraic expression,
.
This relation must be solved in conjunction with the Poisson equation,
giving us two equations (one algebraic and the other a twodimensional 2^{nd}order elliptic PDE) that relate the three unknown functions, , , and .
See Also
 Part I of Axisymmetric Configurations: Simplified Governing Equations
© 2014  2019 by Joel E. Tohline 