Most BAL trough variability comes from transverse motion of absorbing clouds. The timescale of variation constrains the clouds' transverse velocity. That provides a constraint on the cloud's distance from the black hole (via angular momentum conservation plus the fact that a disk wind's terminal velocity is a few times the circular velocity at its launch radius). Uncertainties on the transverse velocity translate directly to uncertainties on the distance; thus, we want to accurately measure transverse velocities of BAL clouds, which requires spectroscopy during a trough variability event.

It is fairly common to see that BAL troughs _have_ varied, but much less common to see them actually vary. That is, either trough changes are very quick so that we almost never see them happening or they are very gradual so that we only notice them after sufficient time has passed. The two possibilities can be discriminated using time series photometry of BAL and non-BAL quasars (e.g., SDSS Stripe 82).

First, a prediction: trough variability will cause BAL quasars to show more color variability than non-BAL quasars, as measured by some variant of the structure function S: if you have N individual measurements i of an object's color c, then S = [\sum_i \sqrt{c_i^2 - sig_i^2}]/N, where sig is the uncertainty on the color.

**Problem:**

*colors aren't like magnitudes; a zero or negative color is perfectly reasonable. What needs to be measured is whether or not the distribution of color measurements differs from that expected from random variations due to noise. So probably best to just use reduced chi^2:*

*\chi_\nu^2=[\sum_i (c_i-c_0)^2 / sig_i^2]/(N-1), where c_0 is the mean color of the object. Now if the c_i values are drawn from Gaussian distributions with variance sig_i^2, then on avg. \chi_\nu^2 = 1, and if there is colour variability, it will be larger. But what matters is the distribution of \chi_\nu^2 for BALs and nonBALs - we expect the distributions to be different. However, a complication is that the \chi_\nu^2 distribution changes as N increases, so you can't intercompare \chi_\nu^2 values from objects with different N.*

**So I'm thinking we should just use the distribution of individual \chi_i^2 = (c_i-c_0)^2/sig_i^2. By plotting the distribution of all those individual values for nonBAL quasars (each with a different c_0 of course), and then the equivalent distribution for BAL quasars, we should see a difference.**

*[In fact, may want to just plot distribution of \chi=(c_i-c_0)/sig_i, to retain maximum information. An object with a few outlying large negative \chi values may not be the same as an object with a few outlying large positive \chi values, but could have an indistinguishable \chi^2 distribution. So that's a reason for using just \chi.]*(Note that a quick look at the Stripe 82 dataset shows that some method for excluding outliers (X sigma from the median color for each quasar?) needs to be implemented before calculating \chi_i^2.)

A refined prediction is that some subset of BAL quasars will show more color variability than non-BALs; any BALs which only show absorption from distant (kpc-scale) gas will show variability consistent with non-BALs, but BALs with pc-scale gas will show more color variability. (One would also have to keep an eye out for objects which were non-BALs during a past spectroscopic epoch but which later developed a BAL trough.)

The \chi or \chi_i^2 distribution won't answer the question of BAL trough variability timescales; to do so, I think you need to look at the distribution of the rate of colour changes r_ij=(c_j-c_i)/(t_j-t_i) where the t values are the rest-frame times of observation. If trough variations are slow, you'll have lots of small values of r_ij. If trough variations are fast, you'll have lots of even smaller values of r_ij, plus a few large values. (One complication is that the uncertainty on r_jk will be correlated with that on both r_ij and r_kl. A crude workaround is to separate each quasar into two datasets: ij in 1, jk in 2, kl in 1, lm in 2, etc., so errors are uncorrelated within each dataset.) One would have to compare the BAL colour change rate distribution to that of non-BALs, as colour changes caused by trough variability will be in addition to any intrinsic colour variability (e.g., quasars usually get slightly bluer when they get brighter). And comparisons should be done in bins of similar rest-frame wavelength

**[<- most important]**, luminosity and black hole mass,

**[and BAL subtype (Hi/Lo/FeLo) and maybe Eddington ratio]**to remove any effects of those parameters on intrinsic variability and to search for any dependency of trough variability on those parameters.

A plot of c_j-c_i vs. t_j-t_i for all i,j pairs would also be useful to see just how fast color changes of a given magnitude can happen.

If that timescale is not too short, then it might be worth monitoring BAL quasars photometrically to catch new cases of trough variability as they start to happen and then trigger spectroscopic followup. It might be possible to demonstrate the feasibility of such an approach using SDSS: establish a threshold of |delta_c/sig_c| > 3 (say), look for quasars whose delta_c/sig_c exceeds that threshold _after_ a spectroscopic epoch in which the quasar appears as a non-BAL, and then look for the subset of objects with a 2nd epoch spectrum after that color change. The 2nd epoch spectrum should show a new BAL, or other unusual effect like a transient increase in reddening or strong emission line variation.

I'll link back to this post y/o post a comment if/when I pick up this idea again. It's worth doing, but I need to finish other projects first.

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