Object Spectrum

Sensitivity calculations can be performed with reference to theoretical spectra given in analytical form, to synthetic spectra derived from numerical simulations and fitted by some mathematical function, or to templates taken from the literature. Two common analytical forms are the flat spectrum

$$F_{\lambda} \equiv F_{\lambda,0}~~~\mathrm{[J \: / \: m \: m^2 \: s]}~,$$

and the constant reduced brightness spectrum

$$\lambda~F_{\lambda} \equiv F_0~~~\mathrm{[J \: / \: m^2 \: s]}~.$$

When integrated over a wavelength interval centred on $\lambda_0$ and of width $\Delta\lambda$

$$\left[\lambda_0-\frac{\Delta\lambda}{2},\lambda_0+\frac{\Delta\lambda}{2}\right]~,$$

these two spectra yield²

$$F_{\Delta\lambda}=\Delta\lambda~F_{\lambda,0}~,$$

and

$$F_{\Delta\lambda}=\ln\left(\frac{\lambda_0+\Delta\lambda/2}{\lambda_0-\Delta\lambda/2}\right)~F_0~.$$

To a first approximation, one can assume that the response of the system and the spectrum of the source are both flat, so as to integrate a constant function over a given range. This will preclude the perception of possible details where the response and the spectrum show some structure such as rapid increases or decreases, but at the same time will allow simple computations.