where the quantities are as follows:

• f_λ(λ) - Spectrum of object or sky. Sky emission is estimated with the Gemini Observatory IR Sky Background spectra assuming a 3.0 mm water vapor column. The K-band emissivity of telescope in the Gemini model is probably incorrect.
• A_stop(band) - Area of the stop. In the Y, J, and H bands the area is that of Keck's: 76 m². In K and Ks a Lyot stop is used to block stray thermal radiation and the stop area is an effective 72.2 m² (unverified).
• t_exp - Exposure time specified by the user.
• dλ(band) - Size of the resolution element in Å. For a diffraction grating in order m and with a ruling pitch d using a camera with focal length f subtending N_spec pixels of pitch p the following equation holds:
  dλ = d m^-1 p N_spec f^-1 = 6.516 [Å] m^-1 N_spec.
  where m is 6, 5, 4, or 3 in Y, J, H, or K respectively (see MODN06 and DDR) and N_spec is specified by the slit width. Formally the anamorphic magnificaiton changes across the spectrum and field by up to 10%, with a corresponding error in the total counts.
• η(λ) - Efficiency of MOSFIRE including atmospheric transmission, slit loss, glass (windows, collimator, filter, and camera: 70%), grating (65%), and detector quantum efficiency (85%). Typical end-to-end value is around 30%.
• h c - Planck Constant x Speed of light.
• G - Note that the gain measured at PSR is 2.21 e^-/DN.

Note that slit loss is calculated assuming a Gaussian seeing profile so that for a seeing value of FWHM the loss is: 1-erf(1.665 slit width/FWHM) in appropriate units. For typical seeing of 0.75" and slit width of 0.7" (2.9 pix) the loss is about 3% (seems LOW, CHECK!).

The detector noise was measured and reported in the PSR. Read noise is estimated assuming a Fowler 16 readout and is measured to be 4.9 e^- per pixel. The dark current is 0.008 e^-/s/pix and again counting statistics are applied to the dark current. Note that for the dark-current noise to equal the read noise an integration time of 50 minute is required.

Noise calculations often require the number of extraction pixels (N_pix). The number of pixels is equal to the width of the resolution element described above (N_spec) times the number of extraction pixels (N_sptial) which we estimate as 2/3 x (seeing FWHM) where the spatial pixel scale is 0.18" per pixel. For typical seeing of 0.7" and slitwidth this amounts to: N_spec=2.9 and N_spatial=2.6. The number of extracted pixels is typically N_pix=8.

Thus, after converting the signal from erg to DN, the total noise is estimated as follows:

• Detector charge persistence. See PSR.
• Flexure, which should be small with the flexure compensation system.
• Scattered light, which requires on-sky time to determine.
• Detector nonlinearity is not included; however, I do indicate when an observation is near the non-linearity limit.
• Flat fielding.