# Space Debris In-Orbit Detection with Commercial LiDAR Sensors
## 2 Materials and Methods
### Maximum Distance from LiDAR to Object at Orbital Velocities  
The initial focus is on determining the effective range distance from which we can reliably detect an object with the orbital velocity of the sensor. As all objects that orbit the Earth must maintain a specific relationship between altitude and orbital velocity, it is possible to calculate the relative velocity between two objects that share the same orbital parameters and are separated by a radial distance \(R\), which represents the sensor's detection range. This maximum distance between objects is characterized by Equation (1):  
\[r_{object}-r_{lidar}=R \tag{1}\]  
where \(r_{object}\) is the distance from the object to the center of the Earth and \(r_{lidar}\) is the distance from the LiDAR sensor onboard the satellite to the center of the Earth. Figure 1: Commercial Automotive LiDAR Sensors. Equation (2) shows the relationship between the orbital velocity _V(r)_ and the distance r from the centre of Earth, where \(G\) is the [PERSON]'s gravitational constant and \(M\) is the Earth's mass. Figure 2 illustrates how the orbital velocity _V(r)_ varies inversely with distance from the Earth's centre, according to the principle that the orbital velocity decreases as an object moves far away from Earth's gravitational influence. \[V(r)=\sqrt{\frac{GM}{r}} \tag{2}\]  
For example, with an orbit altitude of \(r\) = 840 km, the sensor velocity is around 7437.8654 m/s, and the relative velocity between the object and the sensor, located at \(R\) = 50 m far away from the sensor (_R_ is the sensor range), is 0.26 m/s or around 1 km/h (expressed with the common units of speed of a car). In order to be able to detect the object, the velocity of the detection system, defined by the signal integration speed, must be equal or greater than the relative velocity between the sensor and the object, so that the signal integration speed of all commercial LiDARs is higher than the relative low velocity between the object and the sensor with a range distance between them, enabling the commercial automotive sensor to detect the object even at orbital velocities. We consider the LiDAR sensor rotating around the Earth in a circular orbit, and the object rotating with the same orbit parameters as the sensor, but at an altitude equal to the sensor's altitude plus the sensor's range of view. The simplest form of a LiDAR sensor (1D) can only detect objects directly perpendicular to its motion within its field of view. Consequently, the object becomes visible to the sensor only when it passes directly in front of it. With a relative velocity of 1 km/h between the LiDAR and the object (assuming that the LiDAR has a range of vision of 50 m), this means that the sensor is moving 1 km/h faster than the object. Therefore, if the object is located 1 km away, it will take one hour for the sensor to detect it. Following the same reasoning, imagine that two objects orbit the Earth at different altitudes characterized by the angular velocity. * Object 1--LiDAR: orbiting at an altitude of 840 km. * Object 2--debris: orbiting slightly higher, at 840.05 km. The relative angular velocity between the sensor and object is crucial to understand the total orbit scanning time, in order to detect all the objects placed on the target orbit. The objects move at slightly different angular velocities because of their altitude difference. The relative angular velocity is defined in Equation (3):  
\[\Delta\omega=(\varpi_{lidar}-\varpi_{object}) \tag{3}\]  
Figure 2: Orbital velocity _V(r)_ as a function of the distance r to the Earth’s surface (LEO orbits). To calculate the time required for object 1 (the LiDAR sensor) to catch up to object 2, placed at the maximum possible angular distance from the sensor, which is 360 degrees, we calculate the time needed for object 1 to cover this distance at the relative angular speed. Thus, the total orbit scanning time is defined in Equation (4).