Subject: Re: ArcTan[x]/x From: Stan Owocki Date: Fri, 6 Jan 2006 09:15:58 -0500 To: David Cohen David, Sorry, but I ended up going to bed earlier than expected. I just suddenly ran out of gas. However, I did manage to get a better feeling for what's going on. Namely, in the inner wind, the density is higher and therefore it's easier for the blobs to become optically thick, and so give a reduction in the overall optical depth. This is especially true if the smooth wind is already very optically thick, i.e. tau_* >> 1. Indeed, you can see from my arctan(x)/x plot that there is already a modest (ca. 20%) reduction in the overall tau_eff* even for x=h*tau_* = 1. This works out to require a porosity length of only H= R_*/tau_*. That is indeed much smaller than our original estimate that you need H = R_1, which is R_* TIMES \tau_*, in order to have a significant effect on the radius of optical depth unity. This is because this radius of optical depth unity depends on conditions in the OUTER wind, where the density is lower, requiring a much bigger porosity length for the clumps to be optically thick. Indeed, this is the same reason that the arctan(x)/x curve flattens out. In effect, it's easier to get rid of the big optical depth in the interior than the marginal optical depth in the outer wind. The overall upshot is that a modest porosity length can indeed make a very thick wind less optically thick, but it can't actually make the wind optically thin, ie. transparent. For example, for our standard case that tau_*=10, getting arctan(x)/x=1/tau_*=0.1 requires x=15. But because x~sqrt(H) this solves to an H/R_* = 152/10 = 22.5 In fact, rather remarkably, it turns out that the value of H/R_* needed to give tau_eff*=1 scales almost (but not exactly) LINEARLY (!) with tau_*, very nearly as 2.5(tau_*-1). Another point is that all this is assuming that the porosity length is a fixed value in the wind, whereas more realistically it might vary with radius, perhaps, for example, increasing with r. I've not yet tried to examine a case like that. Anyway, all this now makes me think that the porosity issue is somewhat more subtle than we thought. We can discuss this further on the phone today if you like, but if you're willing to hold off in the paper submission, perhaps it would be best to talk about it all further at the AAS meeting next week. It does seem that we may need to bite the bullet and make a more concerted effort to understand better the Feldmeier et al. papers on this, to establish whether indeed this porosity effect could make the wind transparent enough to explain the relatively symmetric profiles. I'll be at home for the next hour or so, so give me a call if you like. Or we can talk this afternoon, or wait till AAS. Stan On Jan 5, 2006, at 11:28 PM, David Cohen wrote: > Stan, > > Thanks - it's interesting that it levels off (i.e. you can't realistically get an order of magnitude reduction in tau_star). > > But to get even a factor of 3 reduction, h would have to be 0.4 for tau_star=10. That means a porosity length of only 0.16 R_1... which is about Rstar. Still seems big. The bigger you make tau_star to allow for a smaller h to get a significant reduction in optical depth, the bigger tau_star and thus R_1 is, and so the small h still implies an H of order Rstar, right? > > About discussing Feldmeier and Oskinova and vice versa - it's true that the paper short-changes them. There's a brief mention at the top of p. 13 in the current pdf file of the manuscript. I do think we really need to tackle their work and figure out why they find such a big effect, if it's somehow that their model doesn't conserve mass or what. I guess we felt that it was a whole can of worms that we didn't want to open in this particular paper. How do you feel about that? > > The new paper - which I haven't looked at - is at: http://arxiv.org/ abs/astro-ph/0511019 > > OK, now I've read the abstract and can see it's a conference proc. > > I will bring it, and their earlier two papers to the meeting. > > David > > Stan Owocki wrote: