Wavelength Sensitivity Calibration

The observed spectrum of an object ( $S(\lambda)$) is represented as,

$$S(\lambda) = S_{o} (\lambda) T_a(\lambda) I(\lambda),$$ (44)

where $S_o (\lambda)$ is the spectrum of the object, $T_a(\lambda)$ is the atmospheric transmittance, and $I(\lambda)$ is the effective instrumental response including the diffraction efficiencies of gratings and transmittance of optics. In order to extract the spectrum of the object, $T_a(\lambda)$ and $I(\lambda)$ must be corrected.

The observed spectrum of a standard star ( $S_s(\lambda)$) is represented as,

$$S_s(\lambda) = S_{*} (\lambda) T_{a,s}(\lambda) I_{s}(\lambda),$$ (45)

where $S_{*}$ is the spectrum of the standard star. The spectrum of the standard star is obtained at the same airmass as the objects. Thus one can assume that $T_a(\lambda) = T_{a,s}(\lambda)$ and $I(\lambda) = I_s(\lambda)$.

A-type stars are usually used as standards because they show almost featureless infrared spectra except for hydrogen absorption lines. One can thus assume that the continuum spectra of standard stars can be represented by a black-body spectra $B_{T_{eff}}(\lambda)$ after correcting the features of hydrogen absorption lines. Thus one can obtain the product of $T_a(\lambda)$ and $I_{s} (\lambda)$ from normalized $S_s(\lambda) / B_{T_{eff}}(\lambda)$.

The spectrum of an object can then be obtained from the following relation

$$S_o(\lambda) = {S (\lambda)} \cdot \langle \frac{B_{T_{eff}}}{S_s (\lambda)} \rangle .$$ (46)

The details of the procedure are explained below.