## 2. **Breadboard Design and Implementation**
### Limiting Magnitude of the Sensor and Visual Magnitude of Debris  
The sensor's limiting magnitude sets its detectability of space debris. In other words, the sensor can only detect objects that reflect enough light, otherwise, they will be under the \(SNR=5\) threshold for reliable detection. Therefore, different scenarios can be simulated to evaluate the detectability of the camera. Simulations were done considering a spherical debris in a circular orbit of 800 km altitude. After that, the distance between the sensor and the object (range) was changed. Also, the same pessimistic solar-phase angle of 120 deg was considered for all the scenarios. Taking into consideration the limiting magnitude of 10.2 for a fixed source, like a star, at an \(SNR=5\) and an integration time of 1 s, one would expect to detect the signal of a piece of space debris with the same magnitude. However, space debris are fast-moving objects, and their signal is not integrated over the same pixels but rather spread over its streak. The longer the integration, the longer the streak. For example, a piece of debris orbiting the Earth at an 800 km altitude would have an orbital velocity of \(v_{orb}=7.452\) km/s, an orbital period of \(P_{orb}=6052\) s, and an angular velocity relative to the Earth center of \(\omega_{orb}=214.1\) arcsec/s. Its visibility, for an observer on Earth, occurs during an Earth-centered angle of \(\psi=54.617\) deg \(=196621\) arcsec. For an observer on the surface of the Earth, and not at its centre, the average sky motion of such object is \(\delta=705.7\) arcsec/s. Since the average pixel scale of our detector is s \(=9.79\) arcsec/pix, during a 1 s integration, the debris will trail over 72.1 pixels on the detector, spreading its signal on the streak. While the signal of a fixed point-source is accumulated on 16 pixels, the debris signal will be spread over 342 pixels. Therefore, the maximum magnitude of a piece of space debris on a circular orbit at an altitude of 800 km, detectable at an \(SNR=5\) with our equipment, for a 1 s exposure is \(m=8.6\). For a 3 s exposure, the signal would be spread over 994 pixels, and for 5 s spread over 1651 pixels, resulting in magnitudes of 9.2 and 9.5 respectively. Analogously, for debris in circular orbits at 1000 km altitude, we get a maximum magnitude of about 8.7, and for circular orbits at 2000 km altitude, we get about 8.9, both for a 1 s exposure. The higher the orbit, the closer to the fixed-star maximum magnitude we get. This apparently counter-intuitive output does not evidently result in a detection of smaller debris at higher altitudes. In order not to leave a streak on the image, a piece of debris would need to be moving at a rate lower than 1 arcsec/s relative to the Earth's centre, which would mean orbiting at an altitude higher than 200,000 km, an orbit of no immediate interest, and for such distance, only a debris with a diameter larger than about 20 m would be measured with magnitude 10.2 for a 1 s exposure. Furthermore, considering a scenario where the Sun elevation is \(\alpha_{sun}=-20\) deg (already after astronomical twilight) and the elevation of a debris is \(\alpha_{debris}=40\) deg, both with equal azimuths, from (12) we get a sky separation angle between the two of \(\theta=60\) deg and a phase angle of \(\phi=120\) deg considering an observer on the surface of the Earth. In addition, the range is also a function of the satellite elevation and is given by [26]:  
\[R=r_{E}\left(\sqrt{\left(\frac{H+r_{E}}{r_{E}}\right)^{2}-\left(\cos\alpha_{ debris}\right)^{2}}-\sin\alpha_{debris}\right) \tag{14}\]  
In (14), \(r_{E}\) is the Earth radius and \(H\) the object's altitude. A piece of space debris exhibiting a magnitude of 8.6, moving on a circular orbit at an altitude of 800 km, being observed with such configuration, is at 1159 km from the observer, using (14).