# Celestial Bodies Far-Range Detection with Deep-Space CubeSats

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Absolute and Apparent Magnitude Models

#### 2.1. Magnitude Scale

#### 2.2. Apparent Magnitude of Major Bodies

#### 2.3. Absolute Magnitude of Minor Bodies

#### 2.4. Apparent Magnitude of Minor Bodies

#### 2.5. Magnitude Screening of Solar System Objects

## 3. Radiometric Model

#### 3.1. Irradiance and Flux of Photons

#### 3.2. Emitted Fluxes in a Frequency Range

#### 3.3. Overall Emitted Quantities

#### 3.4. Fluxes

^{2}(this is also known as the solar constant). Similarly, the power flux at the earth’s location from the sun in the visible band is

#### 3.5. Fluxes from an Asteroid

- Let us consider the irradiated power from the sun across all directions. At a given distance, an asteroid receives an incoming power flux from the sun. The overall incident power on the asteroid is given by the power flux from the sun multiplied by the exposed surface of the asteroid, which can be modeled as a half-sphere. Therefore, the power flux in the visible spectrum from the sun at the asteroid location is easily modeled as$${F}_{V}=\frac{1}{4\pi {R}_{a}^{2}}\phantom{\rule{0.166667em}{0ex}}\pi \phantom{\rule{0.166667em}{0ex}}\left(4\pi {r}_{s}^{2}\right)\phantom{\rule{0.166667em}{0ex}}{\int}_{{\lambda}_{L}}^{{\lambda}_{U}}\frac{2h\phantom{\rule{0.166667em}{0ex}}{c}^{2}}{{\lambda}^{5}}\phantom{\rule{0.166667em}{0ex}}\frac{1}{{e}^{\frac{hc}{\lambda {k}_{B}T}}-1}\mathrm{d}\lambda $$Quite a few asteroids have known geometry, and thus they are assumed to be spherical with a diameter given by Equation (3). The albedo of the asteroids ranges between 0.05 and 0.4, and it is usually assumed to be 0.15 for unknown asteroids. Thus, the power received at the asteroid is the incident flux (Equation (16)) multiplied by $\pi {D}^{2}/2$ (the external surface of an half sphere).
- Part of the incident power on the asteroid in the visible band is reflected back into space according to its albedo ${p}_{v}$. An observer placed at a given distance ${R}_{o}$ to the asteroid which sees the asteroid with a phase angle $\alpha $ will receive an incoming power flux of$${F}_{V}=\pi {\int}_{{\lambda}_{L}}^{{\lambda}_{U}}\frac{2\phantom{\rule{0.166667em}{0ex}}h\phantom{\rule{0.166667em}{0ex}}{c}^{2}}{{\lambda}^{5}}\phantom{\rule{0.166667em}{0ex}}\frac{1}{{e}^{\frac{hc}{\lambda {k}_{B}T}}-1}\mathrm{d}\lambda \phantom{\rule{0.166667em}{0ex}}\left(4\pi {r}_{s}^{2}\right)\phantom{\rule{0.166667em}{0ex}}\frac{1}{4\pi {R}_{a}^{2}}\phantom{\rule{0.166667em}{0ex}}{p}_{v}\phantom{\rule{0.166667em}{0ex}}(\pi {D}^{2}/2)\frac{1}{4\pi {R}_{o}^{2}}\frac{\mathrm{cos}\alpha +1}{2}$$$${G}_{V}=\pi {\int}_{{\lambda}_{L}}^{{\lambda}_{U}}\frac{2\phantom{\rule{0.166667em}{0ex}}c}{{\lambda}^{4}}\phantom{\rule{0.166667em}{0ex}}\frac{1}{{e}^{\frac{hc}{\lambda {k}_{B}T}}-1}\mathrm{d}\lambda \phantom{\rule{0.166667em}{0ex}}\left(4\pi {r}_{s}^{2}\right)\phantom{\rule{0.166667em}{0ex}}\frac{1}{4\pi {R}_{a}^{2}}\phantom{\rule{0.166667em}{0ex}}{p}_{v}\phantom{\rule{0.166667em}{0ex}}(\pi {D}^{2}/2)\frac{1}{4\pi {R}_{o}^{2}}\frac{\mathrm{cos}\alpha +1}{2}$$

## 4. Signal to Noise Ratio

#### 4.1. Signal Model

#### 4.2. Noise Model

**Signal shot noise**. The noise due to the source itself (called signal shot noise ${N}_{\mathrm{s}}$) is given by the standard deviation of the source signal itself. Thus,

**Background noise**. The photons coming from the background sky are modeled as a mean cosmic background flux ${G}_{\mathrm{sky}}$ multiplied by the collecting area, the exposure time, and the instrumentation efficiencies. Thus, the noise associated to the sky is modeled as

**Read-out noise**. The read-out noise is a byproduct of the reading of the photons from the detector. This is commonly assumed to have a constant amplitude ${N}_{\mathrm{rn}}$.

**Quantization noise**. The quantization noise is due to the analog-to-digital conversion which has a finite number of bits to represent the signal. The noise can be modeled as

**Photo response non-uniformity**. The photo response non-uniformity models the different response of each pixel to the number of incoming photons. Indeed, given the same amount of photons, the number of collected electrons is not the same for every pixel due to small imperfections. Thus, this non-uniformity can be modeled as

**Dark current**. The dark current noise is generated by the thermal energy of the CCD. The mean level of the dark current can be easily subtracted from each image, but the dark current shot noise (DCSN) will be present. This is modeled as

**Fixed pattern noise**. The fixed pattern noise (FPN) is an offset which is constant in time but variable in space (over the detector). It is made up of three components: per pixel FPN, per-column FPN, and per-row FPN. These noise sources can be modeled as having maximum noise values of ${N}_{\mathrm{pfpn}}$, ${N}_{\mathrm{cfpn}}$, and ${N}_{\mathrm{rfpn}}$, in units of $\sqrt{{\mathrm{e}}^{-}}$.

#### 4.3. SNR

## 5. Celestial Bodies Far-Range Detection

#### 5.1. The M–ARGO Mission

_{2}point to the asteroids. The baseline trajectories for the five selected asteroids are shown in Figure 3. The trajectories are shown in the geocentric solar ecliptic (GSE) frame that originates in the earth; the x axis points toward the sun; the z axis is parallel to the ecliptic north pole; and its y axis completes the right-handed frame. Table 2 reports the orbital elements of the five target asteroids together with their absolute magnitude H and estimated diameter D.

#### 5.2. Apparent Magnitude Assessment

#### 5.3. Planets Apparent Magnitude

#### 5.4. Asteroids Apparent Magnitude

#### 5.5. M–ARGO Targets Detection

**Apparent Magnitude.**The apparent magnitude of the M–ARGO targets (computed with the model in Section 2.4) is shown in Figure 9 for the baseline trajectories. As it can be seen from Figure 9, the apparent magnitude of the M–ARGO targets in deep space is the range 10–30, and they become brighter only when the spacecraft is approaching the targets. Note that typical navcams for CubeSats are revisited versions of the star trackers. Noting that the limit magnitude of common star trackers for CubeSats is in the range 5–7.5 (see Table 3), the M–ARGO targets are not visible for the vast majority of the spacecraft trajectory, and they become visible in the star tracker only during the final approach phase. Table 3 reports the characteristics of current COTS star trackers for CubeSats in terms of the field of view (FOV), camera limit magnitude, acquisition rate, and nominal sun exclusion angle. For CubeSats, the FOV spans between 10 × 12 deg to 15 × 20 deg, the faintest detectable magnitude is 7.5, and the sun exclusion angle spans between 90 and 22 deg according to the proper baffle.

**Signal-to-noise ratio.**The SNR of the asteroids can be evaluated with different exposures to assess their detectability with larger distances. Figure 10 and Figure 11 show the relative geometry and the visibility of the M–ARGO targets for each trajectory. The distance to the target (${R}_{a}$), its phase angle ($\alpha $), its apparent magnitude (V), and signal-to-noise ratio (SNR) as seen from M–ARGO are shown in each plot. In terms of apparent magnitude, the detection range is when the sensor limit magnitude is reached (six for common star trackers). Regarding the SNR, a value of five commonly corresponds to a possibility of detection.

#### 5.6. Detectability Envelope

#### 5.7. Sensitivity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**M–ARGO baseline trajectories to the five selected asteroids in the GSE frame. The earth is at (0, 0), the sun is always at (−1, 0) direction, the M–ARGO departure is from the sun–earth L2 point and the asteroid position at arrival is at the final location of the trajectory. (

**a**) 2000 SG344; (

**b**) 2010 UE51; (

**c**) 2011 MD; (

**d**) 2012 UV136; (

**e**) 2014 YD.

**Figure 6.**Planets app. magnitude during M–ARGO trajectories (30 deg sun angle). (

**a**) M–ARGO to UV136; (

**b**) M–ARGO to UV136 (30 deg sun angle).

**Figure 7.**Asteroids’ apparent magnitude for M–ARGO to UV136. (

**a**) Asteroids’ apparent magnitude; (

**b**) asteroids versus apparent magnitude.

**Figure 8.**Asteroids’ apparent magnitude for M–ARGO to UV136. (

**a**) Asteroids’ apparent magnitude; (

**b**) asteroids versus apparent magnitude.

**Figure 9.**Asteroids apparent magnitude during the M–ARGO baseline trajectories. (

**a**) 2000 SG344; (

**b**) 2010 UE51; (

**c**) 2011 MD; (

**d**) 2012 UV136; (

**e**) 2014 YD.

**Figure 10.**Distance to the asteroid (${R}_{a}$), asteroid phase angle ($\alpha $), asteroid apparent magnitude (V), and asteroid SNR for different exposure windows during the M–ARGO arrival at the asteroid (

**a**) 2000 SG344 baseline trajectory and (

**b**) 2010 UE51 baseline trajectory. The detection epoch (evidenced with a dot) is determined when the apparent magnitude equals 6 (V = 6) or the signal-to-noise ratio equals 5 (SNR = 5).

**Figure 11.**Distance to the asteroid (${R}_{a}$), asteroid phase angle ($\alpha $), asteroid apparent Magnitude (V), and asteroid SNR for different exposure windows during the M–ARGO arrival at the asteroid (

**a**) 2011 MD baseline trajectory and (

**b**) 2012 UV136 baseline trajectory. The detection epoch (evidenced with a dot) is determined when the apparent magnitude equals 6 (V = 6) or the signal-to-noise ratio equals 5 (SNR = 5).

**Figure 12.**Asteroid app mag function of the relative distance (${r}_{o}$) and asteroid phase angle ($\alpha $). (

**a**) 2000 SG344; (

**b**) 2010 UE51; (

**c**) 2011 MD; (

**d**) 2012 UV136; (

**e**) 2014 YD.

**Figure 13.**Zoom on the asteroid apparent magnitude in approach conditions. (

**a**) 2000 SG344; (

**b**) 2010 UE51; (

**c**) 2011 MD; (

**d**) 2012 UV136; (

**e**) 2014 YD.

**Figure 15.**SNR sensitivity analysis in the LP, RP, and HP cases for the five selected targets. (

**a**) 2000 SG344; (

**b**) 2010 UE51; (

**c**) 2011 MD; (

**d**) 2012 UV136; (

**e**) 2014 YD.

Size | Wet Mass | Prop. Mass | Departure (SE L_{2}) | Transfer | CPO |
---|---|---|---|---|---|

12 U | 22.6 kg | 2.8 kg | 2023–2024 | ≤3 years | ≤6 months |

**Table 2.**Orbital elements (ecliptic J2000) and properties for the 5 selected asteroids (see https://www.minorplanetcenter.net/iau/mpc.html (accessed on 19 January 2023)).

Name | a [AU] | e [-] | i [deg] | ω [deg] | Ω [deg] | H [-] | D [m] |
---|---|---|---|---|---|---|---|

2000 SG344 | 0.9775 | 0.0669 | 0.1121 | 275.3026 | 191.9599 | 24.7 | 39.4 |

2010 UE51 | 1.0552 | 0.0597 | 0.6239 | 47.2479 | 32.2993 | 28.3 | 7.5 |

2011 MD | 1.0562 | 0.0371 | 2.4455 | 5.9818 | 271.5986 | 28 | 8.6 |

2012 UV136 | 1.0073 | 0.1392 | 2.2134 | 288.6071 | 209.9001 | 25.5 | 27.3 |

2014 YD | 1.0721 | 0.0866 | 1.7357 | 34.1161 | 117.6401 | 24.3 | 47.4 |

Name | FOV | Limit Mag | Rate | Sun Excl. |
---|---|---|---|---|

Blue C. NST | 10 × 12 deg | 7.5 | 5 Hz | 45 deg |

Clyd. ST-200 | - | - | 5 Hz | 45/30 deg |

Clyd. ST-400 | - | - | 5 Hz | 40 deg |

Ku Leuv. ST | - | 6 | 10 Hz | 40 deg |

TY NST-4 | 15 × 12 deg | 5.8 | 10 Hz | 25 deg |

MAI-SS | - | 6 | 4 Hz | 90/45 deg |

Sincl. ST-16 | 15 × 20 deg | - | 2 Hz | 34 deg |

Tyvak (IRM) | 16.8 × 12.6 deg | 6 | 6 Hz | - |

Parameter | Value | Unit |
---|---|---|

Field of View | 16 × 10 | deg^{2} |

Image Size | 2048 × 1280 | pixels |

Focal Length | 40 | mm |

F-number | 3.2 | - |

Aperture | 12.5 | mm |

Bit depth | 12 | bits/pix |

HAS2 | FaintStar | CMV4000 | LCMS | ||
---|---|---|---|---|---|

Parameter | Unit | Value | Value | Value | Value |

Sensor Format | pix | 1024 × 1024 | 1024 × 1024 | 2048 × 2048 | 512 × 512 |

Pixel Size | μm | 18 | 10 | 5.5 | 25 |

ADC Res | bits | 12 | 12 | 12 | 12 |

QE × FF | – | 0.45 | 0.49 | 0.60 | 0.40 |

FWC | ke^{−} | 100 | 80 | 13.5 | 75 |

Quantization Noise | e^{−} | 7 | 6 | 3 | 6 |

Fixed-Pattern Noise | e^{−} | 115 | 44 | 13 | 20 |

Dark Signal | e^{−}/s | 190 | 174 | 125 | 1000 |

DSNU | e^{−}/s | 275 | 50 | 40 | 100 |

Read-Out Noise | e^{−} | – | 40 | 13 | 60 |

PRNU | – | 0.018 | – | 0.010 | 0.015 |

Parameter | Unit | Value | Parameter | Unit | Value |
---|---|---|---|---|---|

Field of View | deg | 16 × 10 | QE × FF | – | 0.50 |

Focal length | mm | 40 | FWC | ke^{−} | 100 |

F-number | – | 3.2 | Quantization Noise | e^{−} | 7 |

Aperture Diameter | mm | 12.5 | Fixed-Pattern Noise | e^{−} | 100 |

Sun Exclusion Angle | deg | 35 | Dark Signal | e^{−}/s | 200 |

Sensor Size | pix | 2048 × 1280 | DSNU | e^{−}/s | 100 |

Pixel Size | μm | 5.5 | Read-Out Noise | e^{−} | 100 |

Plate Scale | arcsec/pix | 28.1250 | PRNU | – | 0.02 |

ADC Res | bits | 12 | Overall Noise Margin | % | 20 |

Asteroid | Traject. | Arrival | Det. Epoch | Det. Epoch | Det. Epoch | Det. Epoch |
---|---|---|---|---|---|---|

App Mag | SNR—1 s | SNR—10 s | SNR—100 s | |||

[MJD] | [MJD] | [MJD] | [MJD] | [MJD] | ||

2000 SG344 | Baseline | 9545 | 9518 | 9518 (+0) | 9512 (+06) | 9511 (+07) |

Backup | 9785 | 9769 | 9769 (+0) | 9759 (+10) | 9757 (+12) | |

2010 UE51 | Baseline | 9241 | 9240 | 9240 (+0) | 9233 (+07) | 9226 (+14) |

Backup | 9887 | 9885 | 9885 (+0) | 9878 (+07) | 9871 (+14) | |

2011 MD | Baseline | 9410 | 9402 | 9402 (+0) | 9388 (+14) | 9381 (+21) |

Backup | 9612 | 9610 | 9610 (+0) | 9603 (+07) | 9596 (+14) | |

2012 UV136 | Baseline | 9517 | 9510 | 9510 (+0) | 9502 (+08) | 9495 (+15) |

Backup | 9939 | 9930 | 9930 (+0) | 9923 (+07) | 9909 (+21) | |

2014 YD | Baseline | 9329 | 9322 | 9322 (+0) | 9308 (+14) | 9296 (+26) |

Backup | 9625 | 9617 | 9617 (+0) | 9602 (+15) | 9588 (+29) |

**Table 8.**Settings for SNR sensitivity analysis: (1) low performance (LP), (2) reference performance (RP), and (3) high performance (HP).

Parameter | Unit | LP | RP | HP |
---|---|---|---|---|

F-number | – | 4.0 | 3.2 | 2.6 |

QE × FF | – | 0.45 | 0.50 | 0.55 |

Quantization Noise | e^{−} | 10 | 7 | 4 |

Aperture Diameter | mm | 10 | 12.5 | 15 |

Fixed-Pattern Noise | e^{−} | 125 | 100 | 75 |

Sun Exclusion Angle | deg | 40 | 35 | 30 |

Dark Signal | e^{−}/s | 250 | 200 | 150 |

Sensor Size | pix | 1280 × 1280 | 2048 × 1280 | 2048 × 2048 |

DSNU | e^{−}/s | 150 | 100 | 50 |

Read-Out Noise | e^{−} | 150 | 100 | 50 |

PRNU | – | 0.025 | 0.020 | 0.015 |

FWC | ke^{−} | 100 | 100 | 100 |

ADC Res | bits | 12 | 12 | 12 |

Overall Noise Margin | % | 25 | 20 | 15 |

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Franzese, V.; Topputo, F.
Celestial Bodies Far-Range Detection with Deep-Space CubeSats. *Sensors* **2023**, *23*, 4544.
https://doi.org/10.3390/s23094544

**AMA Style**

Franzese V, Topputo F.
Celestial Bodies Far-Range Detection with Deep-Space CubeSats. *Sensors*. 2023; 23(9):4544.
https://doi.org/10.3390/s23094544

**Chicago/Turabian Style**

Franzese, Vittorio, and Francesco Topputo.
2023. "Celestial Bodies Far-Range Detection with Deep-Space CubeSats" *Sensors* 23, no. 9: 4544.
https://doi.org/10.3390/s23094544