Another way in which it is theoretically possible to correct for side-lobe interference is by employing simultaneous fitting of the two fringe packets for every data scan. In Chapter 3, it is explained that the single-packet method of fringe-fitting is applied to each packet individually to derive a value ofVobsfor each. The method of simultaneous
fitting alters the fringe-fitting by adding another term into equation 3.3 to represent a second packet. Thus, the equation for simultaneous fitting of the packets is a seven-parameter fit:
V(t) = VL sin(π∆σvgt) (π∆σvgt) cos(2πσ0vgt+φ)+VR sin(π∆σvg(t−t0)) (π∆σvg(t−t0)) cos(2πσ0vg(t−t0)+φ) (4.1) whereVL and VR represent the respective visibility amplitudes for the packets on the
left and right side of the scan,t0 represents the separation between the packets in the
time domain, and all other parameters are the same as in equation 3.3. Before the fit is applied, the initial values of VL and VR are the peaks of the respective packets,
while the initial value fort0 is the separation of those peaks in the time domain. Two
markers are placed on each scan, one 25 points to the left of the left fringe packet’s peak and the other 25 points to the right of the right fringe packet’s peak, and the fit is performed on every point between these two markers.
The overall quality of the resulting simultaneous fits is a mixed bag. About half of the fitting attempts fit the data reasonably well, while the other half are very poor. Examples of both types are shown in Figure 4.9. The main concern with the poor fits is that they do not fit the separation very well. In these fits, the position of the HAFP is generally consistent with the data, but the position of the LAFP is very different. Again, the separation is a crucial parameter in these fits because it determines the nature of the interference between the central packets and the side-lobes.
Another way to examine the effectiveness of this method is to compare the original visibility ratio from data reduction to the ratio obtained from the best-fit parameters. Given in Table 4.2 is the comparison for two different targets on one night each. From
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Figure. 4.9: Example simultaneous fits. These fits are performed on individual data scans for HD 157482. Diamonds represent the individual data points, the solid line represents the fit, and the dashed line represents the general trend of the data. The left plot is an example of a good fit, where the peak of the lower amplitude fringe packet in the scan matches the peak from the fit. The right plot is an example of a bad fit, where the peak of the LAFP lies at 95 ms in the data, but at 75 ms from the fit.
the information in this table, the ratio is increased in every case by the simultaneous fitting. It was hoped that the ratio would decrease at the peak of the sinusoidal variation and increase at the trough. Thus, the simultaneous fitting method seems to shift the sinusoidal variation caused by the side-lobe, rather than solving it. Figure 4.10 shows the comparison of the the original and corrected ratios for two nights. It looks like the fitting routine settles on a state of destructive interference in every case, such that the individual visibilities of the fits are almost always lower than their original values.
Piston error may be another factor that contributes to the unexpected results seen with simultaneous fitting. The fitting method works with individual data scans
Figure. 4.10: Original vs. corrected visibility ratio for simultaneous fitting. The dia- monds/solid line represents the original ratio and the triangles/dotted line represents the corrected ratio. For two different targets, it seems that simultaneous fitting is not solving for the sinusoidal variation, but rather just shifting it.
rather than the resulting average values for a data set of a few hundred scans. Piston error causes one of the packets to move relative to the other between successive scans. At very small separations, the shapes of the fringe packets can change significantly from this effect, even in high signal-to-noise data. The shapes may change such that the two packets blend together in some scans but are both visible in others. This can essentially fool the fitting program into identifying a false LAFP.
The combination of poor fits, a lack of solving for the sinusoidal variation in the visibility ratio, and the strong possibly that piston error corrupts the results have lead to the conclusion that simultaneous fitting is not a favorable way to reduce data on SSFPs.
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Table 4.2. Original and corrected visibilities and ratios from simultaneous fitting for HD 35411 and HD 157482
Original Corrected
Epoch (MJD) ρµm VHAFP VLAFP VRatio ρµm VHAFP VLAFP VRatio
HD 35411 54721.46828 13.2 0.181 0.068 2.64 13.3 0.173 0.059 2.93 54721.47467 16.2 0.191 0.061 3.15 16.7 0.190 0.041 4.63 54721.48104 23.8 0.184 0.057 3.19 21.4 0.178 0.043 4.13 54721.48662 19.0 0.188 0.064 2.91 19.0 0.179 0.043 4.16 54721.49233 15.7 0.190 0.077 2.47 15.4 0.169 0.065 2.60 54721.49820 15.1 0.191 0.085 2.24 14.9 0.164 0.070 2.34 54721.50391 15.7 0.192 0.083 2.33 15.7 0.187 0.059 3.16 HD 157482 54662.27239 24.7 0.163 0.061 2.67 24.9 0.163 0.045 3.62 54662.27566 22.5 0.152 0.059 2.57 22.7 0.148 0.041 3.61 54662.27978 22.0 0.149 0.065 2.29 22.0 0.148 0.048 3.08 54662.28312 23.2 0.164 0.070 2.34 23.0 0.160 0.049 3.27 54662.28616 21.2 0.153 0.065 2.35 21.1 0.147 0.045 3.27 54662.28944 19.2 0.163 0.075 2.17 19.0 0.161 0.061 2.64 54662.29268 17.2 0.163 0.070 2.33 17.0 0.162 0.051 3.18 54662.29596 17.7 0.161 0.072 2.24 17.5 0.162 0.055 2.95 54662.29921 16.8 0.147 0.066 2.23 16.4 0.142 0.053 2.68 54662.30251 17.1 0.154 0.066 2.33 16.9 0.151 0.057 2.65