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

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7. Estimation of the centrifugal buoyancy deficit force density

The solidity property of the crust implies that, in a stationary state, its rotation must be rigid, i.e.

[EQUATION]

where [FORMULA] is a uniform angular velocity value (the one that is actually observable from outside), so the formula (33) for the buoyancy deficit force density can be immediately simplified to the form

[EQUATION]

If the superfluid were macroscopically irrotational, i.e. if there were no vortices present, then [FORMULA] would have a uniform value so the right hand side of (35) would also vanish, i.e. the effective buoyancy deficit force density [FORMULA] would be zero.

What we actually anticipate in the context of the pulsar slowdown problem is that [FORMULA] will be approximately uniform (representing rigid rather than irrotational motion) with a value equal to that of the crust component at a rather earlier stage, perhaps just after the previous glitch, and that the velocity difference will therefore be small compared with the total angular velocity

[EQUATION]

Thus, by neglecting corrections of quadratic order in this velocity difference, we see that (35) can be conveniently approximated by the simpler formula

[EQUATION]

which will be accurate to linear order in the difference [FORMULA]. It is to be remarked that, to the same order of accuracy, the neutron drip delay force contribution (31) will be given by the approximation

[EQUATION]

It is to be observed that the formula (37) for the buoyancy deficit force density closely ressembles the Joukowsky formula (27) for the lift force density [FORMULA] that would be exerted on the vortices by the Magnus effect if they are pinned to the crust: this Joukowsky force density is evidently related to the buoyancy deficit force density by the simple proportionality relation

[EQUATION]

It follows that, in terms of the effective (centrifugally adjusted) gravitational potential

[EQUATION]

the basic stellar equilibrium Eq. (28) will reduce to the form

[EQUATION]

in which the zeroth order terms are grouped on the left and the first order terms are on the right (while the final second order term on the right of (28) has been neglected). Since the left hand side consists just of the small difference left over after the approximate cancellation of the dominant zeroth order terms, this equation does not provide any utilisable information about the solid force density [FORMULA] in which we are interested: on the contrary, after [FORMULA] has been evaluated by other means, (41) can be used to calculate the corresponding first order adjustments to the zeroth order pressure and density distributions.

The equation that does supply the relevant information about the solid stress force density [FORMULA] in which we are interested is the first order equilibrium condition (32), whose terms can be instructively regrouped in the form

[EQUATION]

in which it can be seen from (37) that the right hand side will always be approximately proportional to the first order difference [FORMULA] whether or not pinning is effective. (This shows incidentally that differential rotation would be impossible if both the rigidity force [FORMULA] and the chemical delay contribution [FORMULA] were negligible.)

In particular, the relation (42) shows, by (39) that in the pinned case, i.e. when the superfluid is submitted to a force density

[EQUATION]

on the crust, the stress force density [FORMULA] necessary for equilibrium will be given by the simple formula

[EQUATION]

in which the term on the right is just the Joukowsky-Magnus contribution, as is assumed in the conventional presentation of the vortex pinning theory of pulsar glitches.

The formula (44) is potentially misleading in that it gives the false impression that if the pinning were ineffective, so that instead of being given by (43) the force exerted on the superfluid by the crust were simply zero,

[EQUATION]

then the term on the right of the stress force density formula would similarly disappear, whereas in fact substitution of (45) in (32) leads to the replacement of (44) by the formula

[EQUATION]

in which, instead of the Joukowsky-Magnus contribution [FORMULA], the right hand side is now given by the oppositely directed buoyancy deficit force contribution [FORMULA].

Our reasonning so far does not make it obvious whether or not the crust will develop a sufficiently non-uniform chemical potential excess [FORMULA] to provide a significant chemical excess force [FORMULA]. If it is a good approximation to suppose that chemical excess force in the crust vanishes,

[EQUATION]

(as seems to have been implicitly asumed in most previous works but which needs to be confirmed or infirmed quantatively) then, in the case where pinning would not be effective (as has been advocated by Jones (1991) contrarily to earlier works), it follows from (46) that there will still be a solid stress force density given by

[EQUATION]

in which the centrifugal buoyancy deficit force density on the right is given by Eq. (37). This formula can be seen to differ from the (alternative) well known formula - for the stress due to pinning -, deduced from (44) by the same assumption (47),

[EQUATION]

with the Joukowsky-Magnus term on the right hand side given by (27), only by having the opposite sign and by having a proportionality factor given by the density [FORMULA] of the corotating crust component instead of the density [FORMULA] of the differentially rotating neutron superfluid component.

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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