Instrumental visibilities are obtained for a data set by determining the instrumental visibility of a packet in each individual scan, then averaging all of those values. It should be noted that because of the changing baseline and atmospheric effects, vis- ibilities cannot be determined by coadding the individual fringe packets from each scan and deriving a visibility value from the coadded data (Benson et al. 1995). In order to obtain visibilities from each scan, fringe-fitting is performed on both packets.
Fringe-fitting is performed in one of two ways, depending on the separation between the packets. If the fringe packets are relatively far apart, the two packets should be fit separately. When the packets are close together, such that they are overlapping, the two packets should be fit simultaneously. In this section, only separately fitting the packets is discussed, as simultaneous fitting will be addressed in detail in Chapter 4.
When the fringe packets are fit separately, a five-parameter fit of equation 3.3 is applied to each packet with the goal being the derivation of the visibility amplitude,
V, of each packet. The peak of the packet is considered the center, and the fit is performed on 50 total points, 25 on each side of the center. The five parameters of the fit are V, ∆σ, vg, σ0, and φ. Although the values of λ and ∆λ are nominally
known from the manufacturer’s testing of CHARA’sK-band filter, the parameter ∆σ
is kept as a free parameter in the solution because of a small bias that is introduced by the transmissive and reflective properties of the atmosphere, mirror surfaces, and optical windows as well as the detector spectral response across the K passband. Previous attempts to determine ∆λ have produced varied results: ∆λ = 0.350 µm from Bowsher (2010) vs. ∆λ = 0.622 µm from Farrington et al. (2010). Also, the group velocity,vg is not constant during a scan. The fringe scanning mirror is attached
to a moving cart that acts to equalize the optical path length of the two telescopes as a target moves across the sky. The cart’s movement introduces an acceleration term into the movement of the fringe scanning mirror, so the velocity must be treated as
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a free parameter as well.
The five-parameter fit is applied after initial “guess” values are assigned to the five parameters. The initial values of ∆σ(equation 2.3) andσ0 (= 2λπ) are determined
using the standard values ofλ= 2.1329µm and ∆λ= 0.350µm. The initial value ofvg
is determined by taking the distance that the dither mirror traveled over one scan and dividing by the time taken to travel that distance. The phase parameter φ is always initially set to zero. Three different fringe fits, centered on the highest amplitude individual fringe in the packet and on its two nearest neighbors, are performed on each packet. The initial value of V is the peak of the individual fringe that is being used in the fit. The fit with the smallest standard deviation of its residuals is accepted as the best fringe fit. An example fringe fit is given in Figure 3.9.
In general, the standard deviations of the fitted values for both vg and λ among
all non-rejected scans in a data set are both less than 3%. The standard deviation of ∆λ is larger, at roughly 15%. This shows that even within a single data set, there is a large uncertainty in the value of ∆λ. This is most likely due to atmospheric seeing changing the shape of the fringe packets. If the timescale for atmospheric seeing is smaller than the timescale of the scanning of a fringe packet (0.1 sec), the fringe packets can be stretched or compressed on the time axis. If seeing moves a fringe packet in the direction of the motion of the dither mirror, the fringe packet will be stretched. Similarly, if seeing moves the fringe packet opposite the direction of motion of the dither mirror, the fringe packet will be compressed.
Figure. 3.9: Single packet fringe fit. The diamonds represent the data points while the dashed line just connects them, showing the general trend of the data. The solid line is the best-fit solution to the data.
The result of the fringe fitting is a pair of visibilities for each data scan. The locations of the packets on the time axis allows the designation of each packet as either the “left” or “right” packet. The final two visibilities derived for each data set are the average of the left-packet visibilities and the average of the right-packet visibilites for all scans that are not weighted zero. All of the scans not weighted zero are weighted equally in the average. These averaged results represent the observed instrumental visibilities of the target and calibrator for a single data set and will now be designated asVobs. These values must undergo a few additional corrections before
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