D. Cohen

Fits to selected lines in the sigma Ori AB Chandra MEG spectrum

*Spot-checking Gaussian fits to individual lines

In XSPEC v.11, using a custom model "lgauss" written by M. Leutenegger. Nothing fancy, but defines the model in wavelength space and using sensible units (sigma is in mA, for example).  I first fit the nearby continuum with a n=2 powerlaw (flat in lambda-F_lambda space).  Then I fit the line itself with a lgauss + pow model and froze the two pow parameters (n and normalization) at the values found from fitting just the continuum.  I find confidence limits on parameters of interest using the "error" or "steppar" commands in XSPEC (which implement the delta-Chi-sq formalism to assess confidence limits).  I use the C statistic as my fit statistic.  I fit the -1 and +1 order spectra simultaneously but not co-added.  When I plot the data and my fits, though, I do coadd the two orders. 

In general, because the source is weak and soft, the continuum is very, very weak; often barely detectable, especially if you're careful to avoid wavelengths where weak lines are expected.  I often find that the best-fit powerlaw normalization is unbelievably low, so I have generally used the 68% confidence upper limit.  Even that's pretty low.  Let me know if you want to see plots of the continuum fits. 

Fe XVII at 15.014 - this is one of the strongest lines in the spectrum, unblended, with a well-behaved surrounding continuum.  

I use a continuum normalization of 4.5e-5 cts/s/A
and fit the line over the range (14.96:15.08)

fit parameters: 

centroid: 15.014 (frozen)
sigma (mA): 11.1 (9.0 : 12.6 at the 68% confidence level) - corrsponds to 222^{+31}_{-22} km/s
norm (ph/s/cm2): 7.92e-5 (6.51e-5 : 8.38e-5 at the 68% confidence level)

C-stat = 43.64 for 46 bins which implies a rejection probability of 88% (based on the "goodness nosim" command in XSPEC, which does a Monte Carlo simulation of fake data sets which it then refits with the same model to derive an empirical distribution of the C statistic; "88%" means 88% of these fits to MC fake datasets gave a fit statistic better than 43.46, implying that only 12% of the time would we get a fit statistic as bad as we found just due to random fluctuations.  I'd interpret this as saying the fit's not great, but can't be rejected outright.  When you look at it, you'll note some pretty big statistical fluctuations on the red side of this particular line. 

Finally, I also tried letting the centroid be a free parameter to see if that would change/improve the fit.  It didn't.  Nor did fitting the data over a narrower wavelength range. 

A plot of the data over the fit range with the best-fit model is here: 

http://astro.swarthmore.edu/~cohen/projects/sigOri/fevii_15014_lgauss_fixedcentroid.png

Moving on to another line now: 

Ne X at 12.134 (this is a doublet; the wavelength I list here is the oscillator-strength weighted mean of the two lines; I fit these very closesly spaced lines as if they are a single line at this wavelength) - there's also an Fe XVII line in APEC at 12.124; this could in fact be affecting the analysis I report on below, but for now, I don't include this line specifically. 

I use a continuum normalization of 2.2e-5 cts/s/A - again, this is the 68% confidence upper limit; my best-fit value for the continuum level is nearly zero.  I fit the continuum on 12.02:12.08; 12.18:12.24. 

For fitting the line itself (with the fixed continuum level under it), used a range of (12.08:12.18). 

fit parameters: 

1st, with the centroid of the Gaussian frozen at the lab rest wavelength: 

centroid: 12.134 (frozen)
sigma (mA): 9.9 (7.2 : 11.3 at the 68% confidence level) - corrsponds to 245^{+35}_{-67} km/s
norm (ph/s/cm2): 1.94e-5 (1.65e-5 : 2.14e-5 at the 68% confidence level)

C-stat = 30.61 for 38 bins which implies a rejection probability of 45% - so the fit is formally good.  But the fit does not *look* good: 

http://astro.swarthmore.edu/~cohen/projects/sigOri/nex_12134_lgauss_fixedcentroid.png

The relatively low S/N (compared to the Fe XVII line at 15.014) is responsible for this. 

However, we can explore whether the fit improves if we let the centroid of the Gaussian be a free parameter.  It does: 

2nd, with the centroid of the Gaussian thawed: 

centroid: 12.129 (12.1270 : 12.1303 at the 68% confidence level)
sigma (mA): 7.0 (4.6 : 8.9 at the 68% confidence level) - corrsponds to 173^{+47}_{-59} km/s
norm (ph/s/cm2): 1.95e-5 (1.58e-5 : 2.10e-5 at the 68% confidence level)

C-stat = 20.23 (a very significant improvement over the fit with the fixed centroid) for 38 bins which implies a rejection probability of 3% - so the fit is formally very good. Too good.  In my experience, this what happens when S/N is poor.  Here's the best-fit model overplotted on the data: 

http://astro.swarthmore.edu/~cohen/projects/sigOri/nex_12134_lgauss.png

I think it's likely that the blue shift here is due to blending with the Fe XVII line.  I haven't bothered to fit a model with both lines, though.  

Finally, I'll note that I refit the line over a narrower range of wavelength bins and there was very little change in the best-fit parameters of the confidence limit ranges on these parameters. 

Trying a third line:

O VIII 16.006 (a quite weak line; I'm trying this one because Steve mentioned that he wasn't able to put reasonable constraints on the width of the line). 

Once again, the continuum is very weak.  I use a 68% confidence upper limit normalization of 4.5e-5 cts/s/A based on fitting on the intervale 15.85:15.95 and 16.04:16.06 (there's a Fe XVIII line at 16.071; it's weak, but it's in the data and so I wanted to avoid it). 

With a Gaussian centroid frozen at the lab rest wavelength of 16.006, and the powerlaw continuum model fixed underneath the line model, I find: 

centroid: 16.006(frozen)
sigma (mA): 10.1 (7.5 : 13.2 at the 68% confidence level) - corrsponds to 189^{+58}_{-49} km/s
norm (ph/s/cm2): 2.87e-5 (2.35e-5 : 3.36e-5 at the 68% confidence level)

C-stat = 19.15 for 26 bins which implies a rejection probability of 14% - so the fit is formally good.  

Note that I obtained the confidence limits on the line width, sigma both using the "error" command in XSPEC and by using the "steppar" command.  The results from these two methods were in good agreement. 

The best-fit model overplotted on the data is here: 

http://astro.swarthmore.edu/~cohen/projects/sigOri/oviii_16006_lgauss_fixedcentroid.png

Finally, I then thwawed the centroid of the Gaussian, but the best-fit turns out to be very, very close to 16.006, so this didn't affect the fit or confidence limits. 

Note that I am *not* using the background spectrum (which formally would invalidate the C statistic).  But in my experience, subtracting a background spectrum from the data does not generally affect the fit results for O stars.  However, it might be a useful exercise to test this for sig Ori A.  At the very least, plotting and visually inspecting the background spectrum could be useful for convincing ourselves that it's not important.