# Exploring the effects of *q* - a radially-varying filling factor

For the purposes of this exercise, we will fit the Fe XVII line at 15.014 over the wavelength range [14.87:15.13] in the MEG data only and assume a terminal velocity of 2250 km/s - the *baseline* model (with the weak background subtracted).

Here is the baseline model fit, shown on the main page and other sub-pages.

And here's the fit when we allow *q* to be a free parameter:

Note that allowing this additional free parameter only modestly increases the confidence limit range on the other parameters (like taustar, *u*_{o}, and the normalization). And in this case, the slightly negative value of *q*, which puts more emission farther from the star and thus would tend to lead to broader lines, leads to larger values of *u*_{o} which correspond to more emission near the star (compensating, in some sense, for the negative *q* value).

The value of C is nearly the same for the best-fit model as for the baseline model, where *q* is held constant (obviously; as the best-fit q is nearly equal to zero -- we will repeat this exercise for other lines, and see if values of *q* ~ 0 are generally preferred).

Note that the filling factor is assumed to scale as *r ^{-q}*. The emissivity scales as density squared times the filling factor.

Here are the 68%, 90%, and 95% confidence contours when *q* is allowed to be a free paramter at each taustar-*u*_{o} point:

For comparison, here are the confidence contours for the model fit in which the value of *q* was held fixed at *q* = 0 (which we've already shown on the page about parameter uncertainty estimation).

As we've shown above, by looking at the one-dimensional uncertainties, or, more correctly, confidence limits, the confidence regions, like the best-fit model, shifts location a little bit when we allow *q* to be a free parameter. But the more important effect is that the confidence regions get bigger. This is because at each point in the two-dimensional parameter space we plot here, *q* is allowed to be a free parameter (in the top plot; it's fixed at zero in the bottom plot). At each point on the grid we plot, *q* can, in principle, have a different value. We don't show what the value is in these plots, but we could easily include *q* as one of the dimensions in a similar confidence contour plot.

## Conclusions

For this one line, allowing *q* to be a free parameter has only a small effect on the confidence limits on the more important parameters. To some extent, this is because values near *q*=0 are preferred.

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*last modified*: 25 April 2008