Here's a question for Steve Thorsett (UCSC), who I am hoping is a lurker here at Hogg's Research:
After a nice talk by Gruzinov here about pulsar emission (Gruzinov has recently discovered a force-free electrodynamics "solution" for a rotating dipole, with rotation and dipole axes aligned), he and I talked a bit about the Eddington limit. No pulsars appear to emit above the Eddington limit (even when you consider total nebular emission and/or implied spin-down luminosity). The Crab and one other are basically at the Eddington limit, and all others are below. Is this a coincidence? Part of me says "yes", because gravity doesn't enter in any of the usual considerations about the emission mechanism. But part of me says "no", because (a) nothing emits above Eddington, and (b) if the pulsar is above Eddington, the emission might significantly distort the plasma in the magnetosphere and "break it" or even distort the outer layers of the NS, and change the moments of inertia.
Interestingly, if people are right about spin-down, the Crab was hugely super-Eddington in the past, and it is a coincidence that we see the Crab only now just as it has passed into the sub-Eddington phase! That sounds like the kind of argument a cosmologist would make.
Of course I am talking about the total nebular emission, not the pulsed emission (which, I understand, is a small fraction of the total—either nebular or spin-down—luminosity).
Hey Hogg -- I guess I'm not sure what you are getting at with your point (b). Gravity isn't a particularly important contributor to the force equations for charged particles in the magnetosphere, so the Eddington luminosity isn't particularly relevant to the analysis of the stability of the particle flows in the strong B field region.
ReplyDeleteI think that the part of you that argues for coincidence is probably right. Pulsars don't spend very much of their time at superEddington luminosities (under a thousand years for the Crab), and there aren't many very young pulsars around.
One might think a bit harder about this. If you assume a constant B field for any particular pulsar, then the age at which it happens to be at the Eddington luminosity varies inversely with the strength of B (if I've done that right in my head), so a pulsar with a tenth the B field of the Crab will be at L_edd when it is about 10k years old (but still spinning faster than the Crab is now), and vice versa. For the observed field distribution and birthrate, one could then (with a little time) estimate the number of expected superEddington pulsars, which would be an overestimate if pulsars aren't all born spinning fast. Would be pretty straightforward to do the calculation correctly (with the age/field population convolution) if it is still interesting. --steve
Steve! I knew it.
ReplyDeleteYeah, gravity doesn't enter the equations, except in the equation of the neutron star itself. So as long as the radiation "emerges" or is "generated" far from the star surface, Eddington won't matter, I guess.
Hogg -- Of course at the surface of the star the force balance equation is particularly interesting. What is the work function? On my father's knee I learned that neutron stars have crystalline iron crusts, which if true means you need to do a little condensed matter physics before you can write down the forces. And then in the region between the surface and the light cylinder, it is still not particularly useful to consider the Eddington limit, since its derivation depends on the assumption of an isotropic radiation field (clearly far from correct), non-relativistic and classical limits (also wildly wrong for plasma in this region). L_edd only become useful when talking about the region outside the light cylinder, so I can only imagine it becoming relevant for the emission question if there were some way that it could shut down a return current closing a circuit created by the pulsar wind. But then you would introduce significant charge separation between the pulsar and ISM, which would introduce an additional correction to L_edd.... --s
ReplyDeleteLeaving aside the non-sphericity of the radiation field, I agree that the best mechanism would be that the radiation pressure "messes up" the plasma (eg, current sheets, etc) in the pulsar magnetosphere. But of course the EM forces probably far, far exceed the gravitational forces, so the "effective" Eddington limit would be much, much higher.
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