Subject:
Re: wind modeling of the C IV line in the B star beta Cru
From:
Stan Owocki
Date:
Thu, 29 Nov 2007 00:48:57 -0500
To:
David Cohen
CC:
Fullerton Alex , Emma Wollman , Gagne Marc , Townsend Rich , ud-Doula Asif , Anilmis Nurdan , Madura Tom , Russell Chris , Oksala Mary
David,
This is good stuff. Here are a few quick comments.
If you want a slightly better model for the base velocity, ie. in the sonic region, you could use something like:
v(r) = vs e^(r-Rs)/H ; r == Rs
Here H = vs2/g is the atmospheric scale height for surface gravity g.
The above is continuous in v, but has a discontinuity in dv/dr at the sonic point, which will lead to a jump in tau.
Clearly one issue is that your line absorption troughs are narrower than observed. There are lots of possible effects that could broaden these, most obviously a micro-turbulence velocity vturb. Those who fit observed line profiles routinely use quite high vturb, i.e. 100 km/s or more, which is about 10 times the typical ion thermal speed vth ~ 10 km/s, and even several times the sound, vs~25 kim/s.
In fact, the need to include such high vturb is the main reason for using SEI codes instead of the pure Soboelev approach youre using. In your case, you're ignoring the source function, so this would just be a simple exact integration, pure absorption code.
Another thing that might broaden the absorption troughs a bit is integration over the finite disk. This can be done using either the SEI code, or still semi-analytically, i.e. in Mathematica and/or Maple, using the Sobolev approximation but with the optical depth now given by tau ~ kappa rho vht/(dvz/dz), where z is the coordinate along a ray with a fixed inpact parameter p, with then an integration over 0 < p < R to get the total flux over the finite disk of radius R.
All this is stuff that is covered in my class, but unfortunately I don't have any good web notes available. But I do have a Mathematica notebook that computes Sobolev P-"Cygni profiles. Did I already give that to you??
So another thing to keep in mind here is that in principle the small but non-zero emission component could be partly filling in the absorption trough. But certainly this can't explain the apparent redward shift you note. This might indeed be a wavelength calibration issue, coupled with some vturb ~ 100 km/s to broaden the overall trough.
Anyway, I think you've shown that marginally optically thick line profile can give a rough fit to the observed profile, especially if you include some lowering of the ionization fraction with height. As Emma noted, if you parameterize this as a power of velocity, then it mimics a different beta. Actually, if you imagine this ion fraction as set by some balance between photoionization vs radiative recombination, then it should scale with ne/W, where W is the dilution factor. Since ne/W ~ 1/v, that tends to make the lower ion more abundant in the inner wind, and if you identify that as CIV, then this goes in the direction of mimicking a high beta. But that seems a bit artificial.
As for where to go from here, I guess there are several options. Let me think a bit more about it and we can discuss it when we meet on Monday.
Stan
On Nov 28, 2007, at 3:25 PM, David Cohen wrote:
> Alex, Stan, et al.,
>
> At your suggestion, I've been doing a few experiments with the IUE C IV wind lines of the early B star beta Cru.
>
> Recall that the issue -- from my perspective, in the context of narrow (hwhm ~ 150 km/s) x-ray lines -- is what the wind terminal velocity of this star actually is. Blue edge velocities (Prinja 1989) max out at 420 km/s in C IV (Si IV similar; Si III has no wind signature). V_esc ~ 1000 km/s for this B0.5 III star, and CAK theory (using Abbott's 1982 alpha, k, delta tabulation) predicts v_inf ~ 2000 km/s.
> A related and interesting-in-its-own-right issue is the mass-loss rate. CAK theory via Abbott says 1e-8. M-dot*q for C IV is 1e-11. By the way, if I take the X-ray emission measure and assume it comes from a smooth wind with an x-ray filling factor of ~unity above 1.5 Rstar then M-dot = 1e-9 is required.
>
> So, a new student - Emma Wollman, who's a junior - has been making basic models of the absorption component of the C IV doublet, using Sobolev theory - and has some preliminary results we'd like to share with you.
> First off, here's the complex, as we show it in the beta Cru manuscript:
>
> http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/civ_nov07.png
>
> This is (all? most? - I should really know this) of the ~10 IUE spectra co-added. Mike Kuhn, my old student and coauthor on the beta Cru Chandra paper, looked for variability and found nothing significant (at least in terms of coherent variability; the stack of spectra look only to be affected by statistical noise, but that's not a rigorous statement at this point). Emma can chime in and remind me if she's investigated this to any extent.
>
> Note that these data are just downloaded directly from MAST. Just the "Download ASCII table of small aperture fluxes & wavelengths" option near the bottom of, for example, this page:
>
> http://archive.stsci.edu/cgi-bin/mastpreview?mission=iue&dataid=SWP05201
>
> There are some details here: http://archive.stsci.edu/iue/preview/mx_preview_info.html but it's not clear to me how well these data are calibrated.
>
> In any case, Emma co-added at least some substantial fraction of these IUE spectra, and wrote a little module in Matlab to compare simple line profile models to the data.
> The basis of these models is the following:
>
> We assume a smooth, spherically symmetric outflow, described by a modified beta-law:
>
> v(r) = v_s + v_inf(1 - Rstar/r)^beta
>
> We calculate the Sobolev optical depth as:
>
> tau = kappa*rho*v_th/(dv/dr)
>
> and get an expression like:
>
> tau(v/v_inf) \propto (v/v_inf)^(1/beta - 2), based on mass continuity to eliminate rho. This could be cast as a function of r/Rstar, but since we're in wavelength space already with the data, it seems to me to make more sense to just keep everything in terms of the velocity.
>
> One aside: the optical depth expression doesn't formally (in these notes) take the sound speed modification to the beta law into account, but Emma *does* take it into account in her modeling code.
>
> Emma does not (yet) do formal, statistical fitting, but she can choose parameters by hand (nicely displayed in the Matlab interface, as you'll see in a moment), calculate a model, normalized to the continuum level in the data, as:
>
> I/Io = exp^(-tau), with tau coming from the expression above - tau \propto v^(1/beta - 2) - but accounting for the doublet. In other words, tau really equals tau_blue + tau_red.
>
> The free parameters of Emma's model are:
>
> normalization (can cast this in terms of the maximum optical depth; or in terms of mass-loss rate, eventually, given various assumptions)
> beta
> v_inf
> v_s
>
> Note/aside/question-for-Alex: Reading Prinja (1989), I wonder about his equation (2) (p. 726), which parameterizes tau(v). It seems like what Raman is doing here is fitting tau from the data (parameterizing its dependence on v/v_edge using two fitting parameters, tau_o and gamma). In any case... my question is - what's the justification for this, physically? What *physical* quantities are really being parameterized here? Emma is just assuming she knows tau(v) from continuity and sobolev in the context of smoothness and spherical symmetry. Does Raman's eqn. (2) basically parameterize our lack of knowledge of...the ionization fraction? Of deviations from a monotonic, smooth velocity law and/or density profile?
>
> In any case, once Emma chooses her four parameter values, everything is completely specified and she can plot the model on the data, as follows:
>
> Here's a model that assumes v_inf = 2000 (the expected CAK value) and beta=0.8:
>
> http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/2000_8.png
>
> Note that you can see, in the graphical interface, at least some of the parameter choices. Correct me if I'm wrong Emma, but basically Emma's procedure is to choose values of v_inf and beta ahead of time, and then optimize the model fit by eye, varying the normalization (controlling the maximum absorption or optical depth) and v_s (which she denotes "v_surface") - though within reason (10, 20, 30 km/s).
> I've suggested that she focus on the blue wing of the blue line in assessing when her fit is good (enough), rather than the absolute depth of the core of the line.
>
> OK - to be honest, there are a couple of other parameters:
>
> The relative strength of the red and blue components is a free parameter (it's obvious that the ratio of their strengths is *not* the ratio of gf values, which is 2, with the blue line being stronger). Emma can twiddle this between 1 and 2.
>
> And how Emma defines the continuum level is another free parameter, though not so easy to control. Basically, she chooses a wavelength range over which to normalize the data. This part of the exercise is frustratingly imprecise, at least partly because of all the photospheric lines in the data.
>
> So, looking back at this model overplotted on the data (again): http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/2000_8.png there are a couple of things to note:
>
> 1. The lines (in the data) look a bit red shifted. Or, to be more precise, there's absorption up to about 100 km/s redward of the laboratory rest wavelength of each line. Within the context of our model, there's nothing we can do to reproduce this. But if this represents a wavelength calibration error, then we should "slide the spectrum over" and that will change our assessment of how well a given model reproduces the data on the blue wing. Or, maybe the lines are subject to some sort of broadening mechanism(s) that's symmetric (thermal, instrumental, rotational - vsini = 35 km/s for this star, though). Alex, the spectral resolution of these data is... 30 km/s? Is that FWHM? What's your take on the wavelength calibration? Have a look at the Si III line in the same data: http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/SiIII.png - the lab rest wavelength is indicated with a dotted line. The line seems a bit red shifted. But it also seems broad (given the vsini; but, I think, also given the appearance of other (photospheric) lines in the IUE spectrum). What's going on here?
>
> 2. The core of the model line(s) are narrower and deeper than the data. We haven't accounted for instrumental broadening. Or any other form of broadening. There's a distressingly strong interplay between our assumed value of v_s and the behavior of this low-velocity core of the line, which has some effect on the level of the line wing. Stan, I'd asked you about choices for parameterizing the velocity law when the velocity is low to keep the density from blowing up. Perhaps that's an issue here. But maybe if we convolved our models with the instrument broadening function, this problem would go away (because the details of the narrowness and the depth at low velocity would get smeared out).
>
> 3. The tentative science result: The model predicts visible absorption at high velocity that's not seen in the data. But...
>
> 3b. Maybe our continuum placement should be different/higher. Clearly, the model doesn't return to the continuum at v > v_inf (e.g. 1538 A). But the continuum does seem to match pretty well on the red side of the spectral window.
>
> 4. As mentioned above, the red and blue components are almost the same depth. Is this an effect of clumping? Or is it possible that the lines are more saturated than they look (e.g. the zero point of the y-axis isn't in the right place). Alex: IUE calibration wisdom? And yes, I recognize that these ascii spectral files are not "real" data products.
>
> OK, here are some more models:
>
> I asked Emma to look at varying beta. If the wind accelerates slower, then all other things being equal, we should get less absorption at high velocities. Here's v_inf=2000 (still), but beta=1.5:
>
> http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/2000_15.png
>
> Looking at this, I think it's plausibly a good fit to the far blue wing, in the sense that between ~1540 and 1545, the model and data are pretty indistinguishable. Conclusion: We can "hide" the high velocity part of the wind, so that it wouldn't be detectable in the IUE data if we let beta=1.5.
> Now, we haven't (yet) parameterized the possible variation in the C IV ionization fraction. Emma noted that if we take the ionization fraction to be a power law in v/v_inf, then, mathematically, it's indistinguishable from modifying the value of beta. Treating the ionization fraction as a power law in r/Rstar might make more physical sense. We haven't tried it yet, but we could. Let us know if you think that would be worth it.
>
> So, again to summarize: v_inf = 2000 seems perhaps consistent with the data, even if beta=0.8 as long as the ionization fraction of C IV falls off as v-0.58 (by my calculation).
>
> Now, we can look at some other models. Let's turn the situation around: Are the data consistent with v_inf=400? Here's the beta=0.8, v_inf=400 model:
>
> http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/400_8.png
>
> Not bad, with again the problem of the red-shifted absorption and of the narrow, deep core. The shapes of the blue wings are pretty well reproduced. I think Stan suggested that the shape of the blue wing and edge varied characteristically with the maximum optical depth (e.g. optically thin lines have shallow rises back to the continuum and saturated lines have steep ones). Certainly this is true (we see it in suites of models we made), but I wonder if we're just not getting much discrimination here because of the uncertainties/problems I've already enumerated regarding the model line cores and the noisy continuum in the data.
>
> Here's v_inf=400, beta=1.5:
>
> http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/400_15.png
>
> The model lines are too narrow, I'd say.
>
> Here are two intermediate cases:
>
> http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/1000_8.png
> http://astro.swarthmore.edu/~cohen/projects/betacru/UVlines/1000_15.png
>
> parameters can be gleaned from the file names and/or interface shown in the images.
>
> OK - this has gotten to be a pretty lengthy memo.
> I'd be very appreciative of any impressions and suggestions any of you could provide; especially IUE calibration gut-feelings from Alex. And also suggestions for where to go from here, in addition to thoughts about what the things we're showing you here mean.
>
> Emma, feel free to add your own thoughts and, in case I've left out something important or misrepresented something you did, please do let us know.
>
> Thanks,
> David