Goodman Red Camera Exposure Times and Signal-to-Noise Calculations with the 400l/mm Disperser

Introduction:

This document is designed to be a guide for astronomers to determine a reasonable exposure time (EXPTIME) for a desired signal-to-noise (SN) value based upon real data obtained during "grey" and "bright" time. The data were obtained during SOAR engineering time in 28-29 Nov 2018 and 18-19 Dec 2018. On this page we present observations of several stars that vary in Spectral Type from DA2 to K0V and from V~11.4 to V~17.7.

Data:

The data were obtained using the Goodman Spectrograph at the SOAR Telescope during Engineering time in Nov and Dec 2018. The spectrograph was configured to use the 400l/mm grism in the 400M1 and 400M2 preset modes with the 1" slit, 2x2 pixel binning, and a gain of 1.48 e/ADU and a read noise of 3.89e (344 kHz ATTN 3 readout mode). The objects were acquired on the slit using the Goodman Acquisition camera at the parallactic angle.

The 2D data were reduced using standard IRAF procedures for overscan-correction, trimming, bias-subtraction, and flat-fielding (i.e., ccdproc). These reduced 2D data are linked to in the table below. After basic reductions were completed, we extracted 1D spectra from the 2D images using the IRAF (twod.apextract) package. Wavelength calibration spectra using HgAr+Ne lamps were extracted for each object spectrum using the same extraction parameters and applied to the objects.

Analysis:

After the extraction and wavelength correction of these data, we are left with 1D wavelength-calibrated spectra that are in units of total counts (CNTS) for a given exposure time (EXPTIME). To the 0^th-order, in a photon-dominated (e.g., Poisson statistics) regime, the S/N can be given as the square root of the CNTS
SN_λ = √ (CNTS_λ)
and the CNTS_λ are related to the flux (F_λ) of the object and the exposure time (EXPTIME). Thus, the S/N_λ at a given wavelength can be given by:
SN_λ = √(F_λ * EXPTIME).

If we have an observation of an object of a known spectral type, magnitude (M_V1), and exposure time (EXP₁) with a measured signal-to-noise (SN₁), we can use this information to estimate the expoure time (EXP₂) or the signal-to-noise (SN₂) of a second object if we know something about its flux (F₂; or M_V2 and spectral type). The ratio of the SN for each object reduces to
SN₂/SN₁ = √(F₂ * EXP₂) / √(F₁ * EXP₁)
which can be simplified to
SN₂ = SN₁ * √(2.5^{M_V1 - M_V2} * EXP₂/EXP₁) or
EXP₂ = EXP₁ * (SN₂ / SN₁)² / 2.5^{M_V1 - M_V2}.
Because of the flux at a given wavelength is dependent upon the spectral type, it is recommended that the above relation be used for objects of similar spectral types. Using the data presented in the table below, one can select an object of a given spectral type and measure the resulting SN at the desired wavelengths. From there, it is possible to determine the necessary exposure time to reach a desired SN.

We note that total counts in the spectra for the same object obtained using the same settings, can show variations of ~50%. Part of this is due to the differences in Lunar phase, sky brightness, and Lunar distance. Some of the observations were also affected by variable clouds and/or poor sky conditions, so we urge observers to use caution in the calculations or to consult with an instrument support scientist if there are questions.

Last updated 2020-07-20 [sdp]