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A: IGRINS is optimized for observing stars, and they are the simplest types of objects to observe. Stars are placed in either position A or position B along the slit, and dithered between these two positions between exposures. Typically we take exposures in a sequence called a "quad" where you start at position A, take one exposure, move to position B, take two exposures, and then back to position A for one final exposure, resulting in an ABBA quad. The B positions are subtracted from the A positions such that B acts as the sky, bias, and dark subtraction for A and vice-versa. This way all exposures get signal from your science target, while simultaneously acting . The data reduction pipeline then fits the combined of A-B frames to optimally extract the signal for your science target.
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A: Similar to point sources (see question above), but you must dither your object ON and OFF the slit. A quad would look like ON-OFF-OFF-ON, similar to an ABBA quad but with ~1/√2 times the signal since you are not getting signal during OFF exposures. The pipeline will subtract the OFF frames from the ON frames to account for subtract the sky, dark, and bias.
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Q: I have a star or object of magnitude X and want to get a signal-to-noise S/N of XX. How do I calculate my exposure times?
A: You can look at this chart and equation here for A0 type stars. It is also a good first order estimate for stars of any typical spectral type. You can also calculate your own signal-to-noise S/N estimates with the Exposure Time Calculator (ETC) which can be downloaded here. Note that Park et al. (2014; SPIE Proceedings) found that the S/N estimates from the ETC should be multiplied from the ETC by 0.83. For ABBA quads, set your number of exposures to 4. For ON-OFF-OFF-ON quads, set the number of exposures to 2.
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Q: What S/N and exposure times should I get for an A0 star telluric standard?
The simple answer is you want your telluric standard to have a higher S/N than your science target. Why? When you divide your science target spectrum by the telluric standard, the object with the lowest S/N will dominate the final S/N after applying your telluric correction. The telluric corrected S/N (S/N)corr can be calculated from the following equation (see this website for detailed derivation):
(S/N)corr = [ (S/N)sci-2 + (S/N)std-2]-1/2
where (S/N)sci is the S/N for your science target and (S/N)std is the S/N for your A0 telluric standard. It is clear that the smallest S/N will dominate the sum of (S/N)sci-2 + (S/N)std-2, thus would dominate (S/N)corr.