For shutterless calibration, IRAC will point to a region of low background ("dark sky") and take a series of dithered observations in all of IRAC's observing modes. These will be combined via the sky dark thread into sky darks, which will be subtracted from all of the data. Previous tests examined the mechanics of how to do this, using a fairly large number of dithered observations. In order to calibrate all of the exposure modes of IRAC, this many dithers will take a prohibitively long amount of time. Here we examine if a similar result can be derived from a much smaller number of dithers. We find that the use of 9 dithers still produces an acceptable result.

2. Test

The requirement set for the darks is that they not contribute to an increase in noise in the calibrated product by more than 10%. From the following one can work out the minimum number of darks:

• The darks are applied to images which have the same exposure time, read noise, etc.
• The darks are always the least background limited data, since they are the darkest sky.
• Therefore, since the other noise sources are the same, a single dark image is always less noisy than the corresponding science data. The limiting case is when the dark and the data are equal.

So we solve for 1.1=sqrt(1^2 + (1/sqrt(n))^2), which gives n >= 5. This is a prime number, and doesn't lend itself well to mapping. Futhermore, it is very small - the way we derive the sky darks requires a certain number of dithers to ensure removal of celestial objects from the data. As a canonical number we consider here the case of 9 dither positions, and address the question, "Is this enough dithers to provide adequate resistance to stellar contamination?".

Assessing this is more problematic than it sounds, since we don't have an example of what a simulated sky image would look like if the stars were perfectly removed, since if we had such an algorithm we would use it. What we can do is run a comparison simulation of the sky which has no stars or galaxies in it. Even so, this still isn't entirely ideal, since the simulation has noise and calibration systematics in it. Five sets of data were assembled:

1. A simulation of the sky containing stars, galaxies, and a background, with 27 dither positions. The 27 dithers were a 3x3 mapping pattern with 100 arcsecond throws, with a 3-position cycling dither at each mapping location. Exposure time was 100 seconds.
2. A subset of the above with just one exposure at each of the 3x3 map points.
3. A simulation of the sky containing just the background, using the same 27 map points.
4. A subset of the above blank sky with 9 points.
5. A second blank sky simulation, with 27 points, to provide an independent baseline set.

   All of the data sets were processed with the previously described sky dark thread. Briefly, this starts with the raw data, applies all of the calibrations (linearity, etc.), detects the objects in the frames, masks them, and then stacks them with outlier rejection. The resulting images are shown in figures 1-3. I then took this output and computed percent differences, i.e. (a/b)/b. In an effort to reference everything to the same image, all of the differences were made relative to the second 27-dither "blank sky" dark. Figure 4 shows the differences between the 27-dither object+sky case vs. the 27-dither sky case for channels 1 through 4. Not unexpectedly, there seems to be a low level of "granularity" in channel 1 (and less so going to loner wavelengths) due to incomplete stellar rejection. Figure 5 illustrates the same case, only now comparing the 9-dither object+sky case to the 27-dither sky case.

   Figure 1 - 9 dither position sky dark, derived from sims with objects.

   Figure 2 - 27 dither position sky dark, derived from sims with objects.

   Figure 3 - 27 dither position sky dark, derived from sims without objects.

   Figure 4 - Error ((star-nostars)/nostars, or fig2-fig3/fig3) for 27 dithers. Channels 1-4, left to right.

   Figure 5 - Error ((star-nostars)/nostars) for 9 dithers. Channels 1-4, left to right.

   Table 1 lists statistics for all of the of the four cases relative to the blank 27-dither case. In all cases the standard deviations are very similar for the sky+objects and blank-sky cases. In the channel 1 case the median values are biased high by approximately 1-2 DN for both the object+star cases.

   Finaly, figure 6 shows the cumulative distribution function of the pixels for channel 1 (normalized to a median of 0). All of the CDFs are similar, confirming that the degree of contamination by stars leaves only a very small imprint on the image. I think that the step functions have to do with the error induced by a single DN (around 1.7%), but this needs further investigation.

   
   Table 1
   The four lines are channels 1 through 4.
   
   9 dither, objects
   no> imstat diff*fits[*,*,1]
   #      MEAN     MIDPT    STDDEV  
       0.02947   0.02914   0.03811  
       0.00652  0.003927    0.0209   
       0.00476  0.004651   0.01191  
       0.02058   0.02053  0.005005 
       
   9dithers, blank sky
   imstat diff*fits[*,*,1]
   #      MEAN     MIDPT    STDDEV  
      0.003149  3.728E-4   0.03874  
     -0.001389  5.822E-5   0.01542  
      0.001178 -1.089E-4   0.01103   
       0.01933   0.01926  0.005071  
    
   27 dithers, object 
       imstat diff*fits[*,*,1]
   #      MEAN     MIDPT    STDDEV   
       0.02624   0.02343   0.02791   
      0.009292  0.008829   0.01367   
      0.002904  0.002159  0.008037   
      0.001433  0.001199  0.005552  
      
   27 dithers, blank sky
   imstat diff*fits[*,*,1]
   #      MEAN     MIDPT    STDDEV   
      0.003979  7.553E-4    0.0279 
      -0.00123 -0.001424   0.01194  
       0.00101  1.728E-4  0.007774  
     -3.958E-5 -1.655E-4  0.004598  
   

   Figure 6 - Cumulative distribution functions for the % differences of the 4 test cases.