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Astron. Astrophys. 361, 795-802 (2000)

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6. The working of the centrifugal buoyancy deficit mechanism

An accurate treatment of neutron star would of course require a general relativistic analysis (Langlois et al. 1998; Comer et al. 1999), but as a first step towards the estimation of the stress forces needed to maintain equilibrium where the crust constituent is interpenetrated by an independently rotating fluid constituent, it will suffice for our present purpose to work in a Newtonian framework, using a highly idealised two-constituent model in which the corotating crust component (including the protons and electrons, as well as a fraction of the neutrons that is bound into atomic type nuclei) and the neutron superfluid are considered as independent material media having respective mass densities

[EQUATION]

and spatial velocity components [FORMULA] and [FORMULA] ([FORMULA]) where m is the proton mass and [FORMULA] and [FORMULA] are the corresponding baryon number densities. For an approximate description of the kind of scenario envisaged by Alpar et al (1988) in which the rigidly corotating constituent consists not just of the crust lattice but also of the proton superfluid in the core which will be locked to the crust by electromagnetic interactions we adopt a simplified treatment in which it is postulated that the dynamics is governed by Euler type equations of motion of the familiar form

[EQUATION]

[EQUATION]

using [FORMULA] to denote partial differentiation with respect to Newtonian time, where [FORMULA] is the Newtonian gravitational potential, and where [FORMULA], [FORMULA] and [FORMULA], [FORMULA] respectively denote the relevant pressure scalars and force density vectors. In a lowest order approximation in which both components can be considered to obey barotropic equations of state giving their energy densities [FORMULA] and [FORMULA] as functions respectively of [FORMULA] and of [FORMULA], they will be characterised by corresponding chemical potentials

[EQUATION]

from which the associated pressure contributions can be evaluated as

[EQUATION]

This implies that the required gradient terms will be given by

[EQUATION]

(It is to be remarked that in a more detailed analysis the baryon chemical potential [FORMULA] in the component corotating with the crust would be interpretable as the sum of proton and electron contributions, [FORMULA].)

Although adequate for the fluid constituent, a purely barotropic description will not be sufficiently accurate for the crust constituent in which we want to allow for the effects of solidity. The usual way to do this is to replace the isotropic pressure gradient term [FORMULA] by a stress gradient term of the form [FORMULA] where [FORMULA] is the total stress tensor. It will be convenient for our purpose to decompose the latter in the form

[EQUATION]

where the extra anisotropic stress contribution [FORMULA] is a correction term that will be small compared with the dominant isotropic contribution [FORMULA]. This means that while [FORMULA] is to be interpreted as the interaction force density, if any, exerted on the neutron superfluid component by effects such as vortex pinning, on the other hand the term [FORMULA] in (17) will consist, not just of the equal and opposite interaction term [FORMULA] but also of an extra correction term [FORMULA] due to the anisotropic stress correction representing the effect of the solidity property, i.e, we shall have

[EQUATION]

The anisotropic stress contribution [FORMULA] and the associated force density [FORMULA] might also include an allowance for magnetic effects, such as are ultimately responsible for the external braking mechanism and for locking the proton superfluid in the core to the outer crust lattice. However for the equilibrium of the strictly stationary states with which we shall be concerned here such magnetic effects are not important, so it may be considered that the stress force density [FORMULA] arises just from the Coulomb lattice rigidity in the crust, and that it vanishes in the high density core.

Let us now restrict our attention to configurations that are stationary, so that the terms acted on by [FORMULA] will vanish, and let us suppose the motion consists just of a circular motion about the [FORMULA] axis, so that each comoving particle moves with a fixed value of the cylindrical radius [FORMULA]. This means that the velocity gradient terms in the equations of motion will be given by

[EQUATION]

where [FORMULA] is the local angular velocity of the crust constituent and [FORMULA] is the local angular velocity of the superfluid constituent. Under these conditions the Euler Eqs. (17) and (18) can be rewritten in the form

[EQUATION]

and

[EQUATION]

If vortex pinning were effective, it would contribute to [FORMULA] the force density needed to counteract Joukowsky-Magnus type lift force density [FORMULA] that would be exerted on the vortices by the Magnus effect, which would be given by

[EQUATION]

but in the absence of vortex pinning or other coupling forces, the right hand side of (26) will simply vanish, in which case it can be seen that the fluid will satisfy the Taylor-Proudman condition, meaning that its angular velocity [FORMULA] and also the combination [FORMULA] must vary as a function only of the cylindrical radius [FORMULA].

Since the interaction force density [FORMULA] will cancel out of the linear combination of (25) and (26) obtained from the direct sum of (17) and (18), it follows that this combination will take a simple form that is conveniently expressible - independently of whether vortex pinning is actually effective or not - in terms of the "would-be" Joukowsky force density (27) as

[EQUATION]

in which the total pressure P and mass density pressure [FORMULA] are defined in the obvious way as

[EQUATION]

In a systematic calculation by successive approximations, the first stage would be to obtain a zeroth order solution of the stellar equilibrium problem in which the (first order) crust rigidity and differential rotation contributions on the right hand side of (28) would simply be neglected. What we are interested in here is the next stage, which involves the first order equation (from which the zeroth order part has cancelled out) that is obtainable by taking the difference of (25) and (26).

Before going ahead it is necessary to stress that, since only weak interactions are involved, it cannot be taken for granted that the relevant nuclear transitions involved in the "neutron drip" process whereby matter is transferred between the ionic crust material and the interpenetrating neutron superfluid will be very rapid compared with the "secular evolution" timescales on which the state under consideration is significantly modified. If the "neutron drip" process were sufficiently rapid one would obtain not just mechanical equilibrium, such as expressed by Eqs. (25) and (26), but also thermodynamical equilibrium in the rest frame of the crust, in the sense that the energy per baryon of the "normal" matter corotating with the crust, which is just [FORMULA], would be the same as the energy per baryon of the neutron fluid with respect to the crust corotating frame, which has the value [FORMULA]. In practice however, due to the slowness of the relevant nuclear transitions (Haensel 1992), it is necessary to allow for the possibility of a finite deviation,

[EQUATION]

from exact thermodynamic equilibrium. Estimates of the likely values for such a chemical potential excess due to the simple spheroidality adjustment mechanism, discussed above in Sect. 2, have been provided by the recent work of Reisenegger (1995). Significantly larger values are likely to arise from the differential rotation mechanisms considered here due to the resulting tendency for the crust constituent to be convected relative to the neutron fluid constituent.

Including allowance for the possibility of a neutron drip delay contribution

[EQUATION]

representing the force density due to the chemical potential excess (30) if any, the solid stress force density [FORMULA] ultimately responsible for the glitches in which we are interested can be seen to be given by the first order equation obtained by subtracting (26) from (25), which will be expressible in the form

[EQUATION]

The final term in the above equation is what can be interpreted as the extra force needed to compensate for the buoyancy deficit of the crust due to its lack of rotation velocity relative to the neutron superfluid, and is given by

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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