# Absolute Radiometric Calibration of TESS-W and SQM Night Sky Brightness Sensors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}), a negative logarithmic unit system for filter-weighted radiances commonly used in astronomy and astrophysics. However, the exact definition of the device-specific magnitude scales for these kinds of sensors and their relationship with the actual incident radiance have not been thoroughly addressed in the literature. Whilst this lack of definition does not completely prevent the use of these sensors for detecting qualitative overall trends in the evolution of the anthropogenic night sky brightness, it makes the quantitative comparison of their measurements against the predictions of different atmospheric light propagation models difficult. In order to fill this void, the purpose of this paper is threefold: (i) to develop a formal optoelectronic model for this kind of sensors, (ii) to describe their absolute radiometric calibration procedure, and (iii) to show how to define a rigorous photometric absolute (AB) magnitude system that allows to assign a definite physical meaning to their mag/arcsec

^{2}readings in terms of the incident spectral radiance. The sensor modelling and its absolute calibration procedure, as well as the definition of the AB magnitude system, are described in Section 2. In Section 3 we present the particular results obtained for the TESS-W and SQM (model USB enabled data-logging light meter, SQM-LU-DL) devices. Additional remarks are included in Section 4, and overall conclusions drawn in Section 5. Some formal mathematical steps leading to the key modelling equations are described in Appendix A.

## 2. Materials and Methods

#### 2.1. Detector Modelling

^{−2}sr

^{−1}nm

^{−1}) of the incident light field along the direction specified by the angular vector $\mathsf{\omega}=\left(\theta ,\text{}\varphi \right)$, such that ${L}_{\lambda}\text{}\left(\mathsf{\omega}\right)\text{}\mathrm{d}\lambda $ is the radiance (Wm

^{−2}sr

^{−1}) contained within the spectral interval $[\lambda ,\text{}\lambda +\mathrm{d}\lambda ]$. The two-dimensional differential factor ${\mathrm{d}}^{2}\mathsf{\omega}$ is the elementary solid angle (sr) around the direction $\mathsf{\omega}$ (in case of using spherical coordinates with the Z axis along the central ray of the field-of-view ${\mathrm{d}}^{2}\mathsf{\omega}=\mathrm{sin}\theta \text{}\mathrm{d}\theta \text{}\mathrm{d}\varphi $), and $P\left(\mathsf{\omega}\right)$ is the weighting function describing the field-of-view of the device (units sr

^{−1}), normalized such that ${\int}_{\Omega}P\left(\mathsf{\omega}\right)\text{}{\mathrm{d}}^{2}\mathsf{\omega}}=1$, where Ω stands for the angular half-space subtended by the forward-facing hemisphere. $T\left(\lambda \right)$ is the photometric band of the device, that is, the normalized spectral transmittance of the whole setup including the spectral sensitivity of the irradiance-to-frequency converter, the spectral transmittance of the protective glass, filters, optical collector, and any other wavelength-dependent factor. $T\left(\lambda \right)$ is a unitless function normalized to 1 at its maximum. The constant K, with units Hz/(Wm

^{−2}sr

^{−1}), provides the absolute link between the converter output frequency and the spectrally weighted and field-of-view averaged incident radiance. The dark frequency${f}_{D}$ accounts for the output of the converter under complete darkness conditions.

^{−2}sr

^{−1}Hz

^{−1}) and the dark frequency ${f}_{D}$ (Hz) have been determined by calibration. A complete characterization of the radiometric properties of the detector also requires the precise measurement of the function $T\left(\lambda \right)$ characterizing the device’s photometric band.

#### 2.2. Radiometric Calibration

#### 2.3. Formalizing the Absolute (AB) Astronomical Magnitudes Units System

^{2}, ${m}_{T}$, is defined as:

^{−26}Wm

^{−2}Hz

^{−2}). The associated reference source for the (radiance) scale of AB magnitudes per square arcsecond is the one producing this irradiance at the entrance plane of the measuring device under normal incidence per square arcsecond (i.e., $\Delta {\omega}_{0}$ = 1 arcsec

^{2}= 2.3504 × 10

^{−11}sr) of solid angle extent. Hence the spectral radiance of the reference source is ${L}_{0}\left(\nu \right)={E}_{0}\left(\nu \right)/\Delta {\omega}_{0}$ (per unit frequency interval) or, equivalently, ${L}_{0}\left(\lambda \right)=\left(c/{\lambda}^{2}\right){E}_{0}\left(\nu \right)/\Delta {\omega}_{0}$ (per unit wavelength interval), where $c$ is the speed of light in vacuum. Note that, whereas ${L}_{0}\left(\nu \right)$ is constant, ${L}_{0}\left(\lambda \right)$ turns out to be wavelength-dependent due to the relationship $d\nu =\left(-c/{\lambda}^{2}\right)d\lambda $ between the frequency and wavelength differential intervals.

#### 2.4. Experimental Calibration Setup

_{max}= 68.2° and the photodiode effective field-of-view function is ${F}_{p}=\pi {\mathrm{sin}}^{2}\left({\theta}_{max}\right)=2.71$ sr.

## 3. Results

#### 3.1. TESS-W Detectors

#### 3.2. SQM Detectors

## 4. Discussion

^{−2}sr

^{−1}Hz

^{−1}), that corresponds to the in-band detected radiance per Hz, is smaller for the TESS-W, what amounts to a higher sensitivity in absolute terms. The reproducibility of the calibration constants within each type of detector is fairly good, with constants G differing in the range 1–5%, and absolute AB zero points differing 0.03–0.04 mag

_{AB}/arcsec

^{2}. It must be noted, however, that these results are provided here as an example of application of the calibration procedure described in this paper, and that more extensive tests with larger samples should be performed before establishing the reproducibility of both families of instruments.

^{2}using some variant of the Johnson V photometric system, with zero points ${ZP}_{m}$ provided by the manufacturer. These measurements can be directly converted to the AB system by adding to them a magnitude correction term $\Delta ={ZP}_{AB}-{ZP}_{m}$.

## 5. Conclusions

^{−2}sr

^{−1}) or, equivalently, in AB mag/arcsec

^{2}with explicit reference to the precise definition of the specific photometric band of the measuring device.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

^{−2}), and ${{E}_{\lambda}}^{\prime}$ is the spectral irradiance on its surface.

^{−2}nm

^{−1}) incident on an infinitesimal patch ${\mathrm{d}}^{2}{x}^{\prime}$ of the surface around ${x}^{\prime}$, due to a small bundle of rays of spectral radiance ${{L}_{\lambda}}^{\prime}\left({x}^{\prime},{\mathsf{\omega}}^{\prime}\right)$ and solid angle spread ${\mathrm{d}}^{2}{\mathsf{\omega}}^{\prime}$ incident along the direction ${\mathsf{\omega}}^{\prime}$, is

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**Figure 1.**Normalized spectral sensitivity of the Telescope Encoder and Sky Sensor (TESS-W) night sky brightness meters (

**a**) stars3, and (

**b**) stars222

**Figure 2.**Normalized spectral sensitivity of the of the SQM USB enabled data-logging light meters (SQM-LU-DL) (

**a**) #2370, and (

**b**) #2747.

Constant | TESS-W Stars3 | TESS-W Stars222 | σ | Units |
---|---|---|---|---|

$G$ | 1.22·10^{−6} | 1.16·10^{−6} | 0.06·10^{−6} | Wm^{−2}sr^{−1}Hz^{−1} |

${L}_{r,AB}$ | 521.8 | 516.9 | 7.8 | Wm^{−2}sr^{−1} |

${ZP}_{AB}$ | 21.58 | 21.62 | 0.06 | mag_{AB}/arcsec^{2} |

Constant | Serial#2370 | Serial#2747 | σ | Units |
---|---|---|---|---|

$G$ | 1.51·10^{−6} | 1.49·10^{−6} | 0.08·10^{−6} | Wm^{−2}sr^{−1}Hz^{−1} |

${L}_{r,AB}$ | 433.9 | 415.4 | 7.9 | Wm^{−2}sr^{−1} |

${ZP}_{AB}$ | 21.15 | 21.12 | 0.06 | mag_{AB}/arcsec^{2} |

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## Share and Cite

**MDPI and ACS Style**

Bará, S.; Tapia, C.E.; Zamorano, J.
Absolute Radiometric Calibration of TESS-W and SQM Night Sky Brightness Sensors. *Sensors* **2019**, *19*, 1336.
https://doi.org/10.3390/s19061336

**AMA Style**

Bará S, Tapia CE, Zamorano J.
Absolute Radiometric Calibration of TESS-W and SQM Night Sky Brightness Sensors. *Sensors*. 2019; 19(6):1336.
https://doi.org/10.3390/s19061336

**Chicago/Turabian Style**

Bará, Salvador, Carlos E. Tapia, and Jaime Zamorano.
2019. "Absolute Radiometric Calibration of TESS-W and SQM Night Sky Brightness Sensors" *Sensors* 19, no. 6: 1336.
https://doi.org/10.3390/s19061336