Calibration is the final step in obtaining the binary star visibilities that will be used in orbit fitting. When observing an object with an interferometer, the visibility returned by the instrument is always diminished from the intrinsic visibility of the object due to the effects of the atmosphere, instrumental vibrations, optical aberrations, etc. McAlister (2005). The standard method of compensating for these effects is observation of a calibrator star. An ideal calibrator is an unresolved, non-variable, single star. The standard form of linear calibration is given by Boden (2007):
Vtgt
Vcal
= Vtgt,corr
Vcal,corr
(3.10)
whereVtgt and Vcal are the respective intrinsic visibilities of the target and calibrator
and Vtgt,corr and Vcal,corr are defined in equation 3.7. The intrinsic visibility of the
calibrator is calculated by the visibility equation for a single star:
Vcal = 2J1(πΘcalB λ ) πΘcalB λ (3.11)
where J1 is the first-order Bessel function, Θcal is the angular diameter of the cal-
ibrator, and B and λ are the baseline and wavelength of observation. Finally, the value of highest interest, the intrinsic visibility of the target, can be calculated using equations 3.7, 3.10, and 3.11. Before 3.7 can be substituted into 3.10, the components of the system must be designated as either brighter or dimmer. This leads to two possible outcomes: the target is brighter or the calibrator is brighter. In the case of
the target being brighter than the calibrator, the intrinsic visibility of the target is given by: Vtgt =Vcal βVtgt,obs Vcal,obs . (3.12)
In the case of the calibrator begin brighter, the intrinsic visibility of the target is given by:
Vtgt =Vcal
Vtgt,obs
βVcal,obs
. (3.13)
Vtgt can now be used in orbit fitting to determine the orbital elements of a binary
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– 4 –
Side-Lobe Interference
In Chapter 2, it was mentioned that it is preferable to observe SFPs when they are well-separated. In Chapter 3, it was stated that for SFPs with very small separa- tions, the fringe-fitting to individual packets is insufficient in determining the visibil- ities of the components. The reason for both of these is the existence of side-lobes. Polychromatic interferometric observations produce both a central fringe packet and side-lobes extending out infinitely on either side. When observing SFPs, each central fringe packet from which we derive a visibility will interact with the side-lobes of the other packet, leading to a distortion of the intrinsic visibility of the packet. The inter- action manifests itself as either constructive or destructive “interference”, leading to an enhancement or diminution of the amplitude of the packet. A qualitative example of this effect is shown in Figure 4.1. Because this project depends on the accurate calculation of the visibilities of two fringe packets, it is imperative that the effect of “side-lobe interference” is taken into account.
Extensive modeling has been conducted to calculate the quantitative effect of side- lobe interference and determine ways to correct for its effect. Two of the main effects that are evident in modeling are the change in amplitude and change in position of the packets. The change in amplitude should be obvious from Figure 4.1. The position change is less intuitive, but can occur if, for example, the central fringe of the LAFP lies on the null between the first and second side-lobes of the HAFP while the third
Figure. 4.1: Side-Lobe Interference Effect. This is a qualitative look at how side- lobes affect the amplitudes of the fringe packets. On each of the plots, two individual fringe packets, the larger of which is twice the amplitude of the smaller, are plotted on the bottom half. From left to right, the separation between these individual packets is decreased. The top half of each plot shows the combined interferogram of the individual packets below. It can clearly be seen that the amplitude of the LAFP changes significantly with varying values of separation.
fringe of the LAFP’s central packet lies on the peak of the HAFP’s first side-lobe. In that case, the peak of the LAFP would be identified at the position of the third fringe of the LAFP, rather than its actual position at the central fringe. Because the position of the fringe packets can change, the separation between them will also change from its intrinsic value. The percent error in the visibilities of both packets and the separation between them is calculated as a function of intrinsic separation.
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The modeling was conducted for SFPs with intrinsic visibility ratios of 1, 1.5, 2, and 3 and is presented in Figures 4.2, 4.3, 4.4, and 4.5. All models are calculated using λ
= 2.1329 µm and ∆λ = 0.350 µm. As expected, the error in each case is substantial at small true separations between the packets. At small values of intrinsic separation, the secondary packet is interfering with either the central packet or the first (and largest) side-lobe of the primary packet, so the error should be large. Also, the errors in the separation and the visibility of the LAFP increase substantially with increasing ratio. For a ratio of 3, the error inVLAFP can exceed 60%, while the error in separation
can reach almost 40%. Even the best case scenario (VHAFP
VLAFP = 1) results in an error
of up to 10% in separation and up to 20% in VHAFP and VLAFP. More important,
however, than the individual visibilities is the ratio between them (VRatio = VVHAFPLAFP).
Looking at equations 3.12 and 3.13, only the ratio of the observed visibilities is needed to calculate the calibrated visibility of the target. Therefore, if a correction can be applied to the visibility ratio, the errors on the individual visibilities can be ignored. The change in the ratio as a function of separation is given in Figures 4.6 and 4.7 for intrinsic ratios of 1, 1.5, 2, and 3. For a ratio of 1, the two fringe packets’ side-lobes affect each other equally, so the ratio never deviates from 1. As the intrinsic ratio increases, the side-lobes of the HAFP become larger in size relative to the LAFP while the side-lobes of the LAFP become smaller relative to the HAFP. This leads to the ratio oscillating sinusoidally about the intrinsic ratio as a function of separation.
Figure. 4.2: Side-lobe Interference Modeling for a Visibility Ratio of 1. These plots are constructed by comparing the parameters of the two individual fringe functions to the parameters of the combined fringe function. The upper left plot shows the percent error in the separation as a function of the intrinsic separation of the fringe packets. The upper right plot shows the percent error of the visibility of the HAFP, while the lower plot shows the same for the LAFP.
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Figure. 4.3: Side-lobe Interference Modeling for a Visibility Ratio of 1.5. For more information, see Figure 4.2.
Figure. 4.4: Side-lobe Interference Modeling for a Visibility Ratio of 2. For more information, see Figure 4.2.
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Figure. 4.5: Side-lobe Interference Modeling for a Visibility Ratio of 3. For more information, see Figure 4.2.
Figure. 4.6: Visibility ratio as a function of separation. The top plot represents a ratio of 1, while the bottom represents a ratio of 1.5.
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Figure. 4.7: Visibility ratio as a function of separation (cont’d). The top plot repre- sents a ratio of 2, while the bottom represents a ratio of 3.