Thesis: Optimized Retroreflector Array for TU Delft Satellites PDF

1 Introduction The first launch of Ariane 6 took place on July 9, 2024, marking a significant milestone in Europe’s space transportation capabilities. From lift-off through the deployment of its eight satellite payload into a 580...

1 Introduction The first launch of Ariane 6 took place on July 9, 2024, marking a significant milestone in Europe’s space transportation capabilities. From lift-off through the deployment of its eight satellite payload into a 580 km orbit, the launch was considered a success. However, during the final phase, when the Vinci engine was expected to ignite for the third time to place the upper stage into a re-entry orbit for safe burn-up in Earth’s atmosphere, the mission encountered an unexpected software malfunction. A sensor registered an excessively high temperature, causing the flight software to shut down the Auxiliary Propulsion Unit (APU) of the upper stage, preventing the third ignition of the Vinci engine. As a result, the deorbit burn was not performed, and the upper stage passivated itself by cutting power to all onboard systems to mitigate the risk of explosions and avoid the generation of space debris. The shutdown of the attitude determination and control system left the upper stage in an unknown attitude. If the upper stage had been equipped with retroreflector arrays, Satellite Laser Ranging (SLR) could have provided an immediate and accurate solution to determine its position and attitude. Fortunately, the upper stage is a large object, and with the help of other systems like cameras, it was later determined to be stable in a low orbit. The Ariane 6 software malfunction is just one instance where SLR could have improved space management by enhancing satellite safety and supporting operational decision- making, thanks to its passive space segment. Delfi-PQ, a satellite launched by TU Delft in 2022, was equipped with four retroreflectors intended for SLR measurements to validate Global Navigation Satellite System (GNSS) data and serve as a demonstration. However, ground stations were unable to retrieve SLR data from Delfi-PQ due to sub- optimal retroreflector selection and array design. Despite this, TU Delft is currently developing two new platforms: Delfi-Twin, the successor of Delfi-PQ, and the Da Vinci satellite, which will participate in the European Space Agency (ESA) program ”Fly Your Satellite.” Both platforms require the implementation of SLR capabilities, leading to the following need: With the development of TU Delft new platforms, there is a need for an optimized design of the retroreflector array to successfully perform SLR measurements. This thesis aims to document the research conducted to address the need for designing an effective and optimized retroreflector array for TU Delft new platforms, while also considering the potential ground stations that could be used for the measurements. TU Delft is currently constructing a laser communi- cation terminal on top of the Aerospace Faculty building, with plans to use it for these measurements. Nevertheless, the International Laser Ranging Service (ILRS) remains the state-of-the-art network for SLR measurements. Based on this need, the following research objectives and related questions were formulated: RO-1: Design a retroreflector array to be mounted on the new TU Delft platforms to obtain Time of Flight (ToF) measurements utilizing a laser ranging terminal from the ILRS network. – RQ-1: Is it technically feasible to obtain SLR measurements using a retroreflector array mounted on the new TU Delft platforms and a laser ranging terminal from the ILRS network? 1  2 RO-2: Design a retroreflector array to be mounted on the new TU Delft platforms to obtain ToF measurements utilizing the laser communication terminal at Delft University. – RQ-2: Is it technically feasible to obtain SLR measurements using a retroreflector array mounted on the new TU Delft platforms and the laser communication terminal at Delft uni- versity? From these research questions, four sub-questions were generated, aimed at exploring specific tech- nical aspects of the proposed designs: RSQ-1: What is the optimal shape and design of the retroreflector array to maximize coverage while facilitating integration with the new TU Delft platforms? RSQ-2: What is the optimal retroreflector radius, and thus the required Optical Cross Section (OCS), to effectively facilitate SLR operations for both the laser communication terminal and the ILRS terminal? RSQ-3: What are the requirements for the laser communication terminal at Delft University of Technology to establish reliable SLR measurements? RSQ-4: How do the Angle of Incidence (AOI) of laser beams and environmental radiation affect the strength and reliability of the reflected signals in SLR operations? The thesis is divided into seven chapters. In chapter 2, a comprehensive literature review is conducted to gather insights on the state-of-the-art technologies related to SLR and the theoretical principles be- hind designing an SLR system. Additionally, post-processing methods used to accurately determine the distances between ground stations and satellites are discussed. Chapter 3 establishes the frame- work that integrates the rationale behind the simulations with the underlying requirements related to the research questions and objectives. This chapter provides a high-level overview of the thesis structure, key components, and the methods employed. In chapter 4, the Probability of Visibility (PoV) simulation is performed to identify the array configuration that maximizes coverage at all elevation angles, based on the satellite attitude. The configuration of the array is finalized in this chapter. Chapter 5 discusses the outcomes of the link budget analysis, which aims to select the optimal retroreflector size for the new TU Delft platforms, considering the configurations of various ground stations. Recommendations are also provided for the Delft laser communication station to ensure successful laser ranging with the selected array. Chapter 6 presents the inspection and potential testing of the ordered retroreflectors, which is crucial for qualifying them for integration into the satellite array. Additionally, the preliminary design of the array holder is carried out. Finally, chapter 7 gathers the outcomes of the research, answers the research questions, and provides recommendations for future work. Supporting graphs generated during the PoV simulation for each satellite attitude and array configuration are compiled in Appendix A.  2 Literature Review This chapter presents a literature review survey conducted to gain insights into the state-of-the-art technologies of SLR, as well as to understand the process behind the design of an SLR system and the post-processing methods used to determine the accurate distances between the ground stations and the satellites. Section 2.1 provides an overview of the history of SLR and the current state-of-the-art systems, de- scribing the basic principles behind SLR measurements. Section 2.2 explains the algorithm used for post-processing SLR data. Furthermore, section 2.3 presents formulations for designing satellite retroreflectors based on the required strength of the returning signal. Section 2.4 gives an overview of the parameters of the considered ground stations. Finally, section 2.5 elaborates on TU Delft satellite legacy, introducing the parameters of interest for the new TU Delft platforms, Delfi-Twin and the Da Vinci satellite. The literature review followed a structured approach guided by the following research questions: Q1) What is SLR, and what are the state-of-the-art techniques? Q2) What are the basic principles of SLR? Q3) What are the different types of applications of SLR? Q4) How is the position of the satellite determined from ToF measurements? Q5) What are the requirements for the space segment to achieve ranging measurements? Q6) What is the design process of a retroreflector array? Q7) What are the common parameters of typical SLR ground stations? Q8) What are the specifications of the new TU Delft platforms? 2.1. Satellite Laser Ranging In this section, the history and state-of-the-art technologies of SLR are discussed, alongside an explo- ration of the basic principles behind SLR measurements. Additionally, a thorough comparison between the implementation of radio ranging measurements and laser ones is undertaken. Finally, several ap- plications of SLR systems in scientific missions are listed. 2.1.1. History of Satellite Laser Ranging In 1964, the first laser range measurements took place at NASA’s Goddard Space Flight Center (GSFC) to Beacon-B, an artificial satellite equipped with optical retroreflectors. These measurements achieved a ten-meter accuracy, marking a significant advancement in technology. However, these initial sys- tems relied on cameras, which provided bias-free angular position measurements but were affected by photographic limitations of 1-2 arcsec and vulnerability to weather conditions. Over time, these pho- tographic constraints were overcome, and advancements in laser technology, coupled with increases 3 2.1. Satellite Laser Ranging 4 in computational power of modern computers and refinement of atmospheric propagation models, pro- pelled SLR measurements to millimeter-level accuracy. Today, a global network of stationary and mobile satellite laser ranging ground stations, the ILRS, cov- ers most of the globe and serves as a crucial interface for SLR measurements to geodetic satellites, GNSS, and commercial satellites. The high level of accuracy of SLR compared to radar ranging measurements has made it an essential tool for geodetic science missions, refining models of Earth’s gravitational field and internal mass redistribution. Additionally, sets of retroreflector arrays were placed on the moon, enabling Lunar Laser Ranging (LLR) measurements. LLR provided insights into Earth- lunar dynamics, including the motion of the center of mass (lunar ephemeris), rotations about the center of mass (librations), internal mass distribution (moments of inertia), and lunar tides. Moreover, Interplanetary Laser Ranging (ILR) measurements have been tested, differing from classi- cal SLR retroreflectors passive systems for the implementation of transponders on the satellites due to the large distances the laser beam has to travel. These measurements aim to complement radar Doppler data to exploit the advantages of both technologies. To date, the only operational implemen- tation of ILR has been on the Lunar Reconnaissance Orbiter (LRO), using the laser altimeter system. However, no mission with dedicated hardware has been created, thus the accuracy has not yet been pushed to the limit. SLR has become an essential technology for satellite ranging and science missions. Increasingly, satellites implement retroreflectors, even if ranging accuracy is not their primary objective, as the im- plementation of passive SLR systems serves as a practical backup for determining satellite positions in cases of satellite control loss or failure of other orbit determination systems. 2.1.2. Radio vs Laser The choice between radio and laser technologies for satellite ranging is pivotal in determining the pre- cision, reliability, and versatility of ranging measurements. Here, we delve into a comprehensive com- parison of the two: Laser: Laser ranging represents a pinnacle of precision, offering unparalleled accuracy for range calculations down to the millimeter level. This precision has made it indispensable for a wide array of scientific and geodetic applications. Moreover, the passive nature of laser payloads makes them highly attractive for Earth orbiting satellites, minimizing the need for additional power and resources. However, laser measurements are not without their challenges. Weather conditions, such as fog or clouds, can obstruct or distort laser signals, impacting the reliability of ranging measure- ments. Furthermore, the absence of Doppler measurements in laser ranging systems precludes the determination of relative velocity between satellites or other celestial bodies, limiting its utility in certain scenarios where velocity data is crucial. Radar: Radar systems, on the other hand, excel in providing Doppler measurements, enabling precise determination of relative velocity. This capability is particularly valuable in applications such as space debris tracking, where understanding relative motion is essential for collision avoid- ance maneuvers. Moreover, radar systems boast impressive accuracy for relative velocity calcu- lations, achieving precision at the level of millimeters per second. However, the accuracy of radar for range calculations is generally limited to the meter level. This limitation may be acceptable for some applications but can pose challenges in scenarios requiring higher precision. Additionally, radar systems face difficulties in distinguishing between multiple objects, especially in crowded orbital environments, due to their large bandwidth. In conclusion, while laser and radar technologies each offer unique strengths and limitations, their combination can result in a synergistic effect that enhances the capabilities of satellite ranging missions. By leveraging the precision of laser ranging for accurate distance measurements and the Doppler capabilities of radar for relative velocity determination, missions can achieve comprehensive and robust tracking solutions. Understanding the complementary nature of these technologies and integrating them effectively is key to maximizing the success of satellite missions in diverse operational contexts. 2.1. Satellite Laser Ranging 5 2.1.3. Type of Applications Satellite laser ranging serves a multitude of purposes beyond merely determining the position of satel- lites. Its millimeter level accuracy lends itself to various geodetic applications: Maintaining the International Terrestrial Reference Frame (ITRF): SLR contributes to the de- velopment of a Laser Ranging Reference Frame (SLRF), ensuring the precision and consistency of the global reference frame used in geodesy, the ITRF. Precise calibration of radar altimeters: SLR aids in the calibration of radar altimeters and helps separate long-term instrumentation drift from secular changes in ocean topography, ensuring accurate measurements of sea level and oceanic processes. Reference for post-glacial rebound, sea level, and ice volume change: SLR provides a ref- erence system for studying post-glacial rebound, sea level variations, and changes in ice volume, facilitating research into climate change and its impacts on Earth’s surface. Determining temporal mass redistribution: By monitoring temporal mass redistribution within the solid Earth, oceans, and atmosphere, SLR contributes to understanding dynamic processes such as tectonic movements and changes in Earth’s gravitational field. Monitoring atmospheric response to seasonal variations: SLR data helps monitor the re- sponse of the atmosphere to seasonal variations in solar heating, aiding in the study of atmo- spheric dynamics and climate patterns. Testing the theory of general relativity: SLR provides a basis for special tests of the theory of general relativity, contributing to our understanding of fundamental physics and gravitational interactions. Modeling convection in Earth’s mantle: SLR data provides constraints on processes as mantle convection, helping to model the dynamics of Earth’s interior and geological processes. Modeling gravitational fields: SLR contributes to the modeling of the gravitational fields of Earth and other planets, providing valuable insights into planetary dynamics and structure. Space debris orbit determination: SLR could be implemented in the precise determination of the orbits of space debris, crucial for debris mitigation efforts and ensuring the safety of space missions. In essence, SLR serves as a versatile tool for geodetic research, providing precise measurements that contribute to our understanding of Earth’s dynamics, climate, and fundamental physical processes. 2.1.4. Basic Principles Satellite laser ranging involves the emission of short laser pulses from a ground-based station towards an object into orbit or a satellite with mounted retroreflectors. The retroreflectors, typically consisting of an array of corner cube prisms, are designed to reflect incident light back towards its source regardless of the angle of incidence ensuring that the laser pulses return to the ground station with minimum deviation. By measuring the time between the emission of the laser pulse and its reception, also called as ToF, SLR system can determine the distance, d, between the ground station and the satellite. A schematic of the SLR principle is shown in Figure 2.1. 2.1. Satellite Laser Ranging 6 Figure 2.1: Schematic representation of the SLR measurement principle. The distance calculation principle is straightforward as the laser pulses travel at the speed of light, c. Therefore, the distance can be calculated with Equation 2.1. T oF · c d≈ (2.1) 2 However, while Equation 2.1 encapsulates the ranging calculation principle, it only provides a rough approximation of the distance. In fact, corrections due to laser propagation through the atmosphere, discrepancies in clock timing and station biases, along with other induced delays, need to be taken into account as they can severely affect range measurements. The induced time delays depend on a multitude of factors, including atmospheric conditions at the time of the emission of the laser pulse. 2.1.5. SLR System A ground segment and a space segment are identified in an SLR system. An example of a ground segment is the Mini-SLR ground station at Deutsches Zentrum für Luft-und Raumfahrt (DLR) in Stuttgart. An overview is shown in Figure 2.2. Figure 2.2: The Mini-SLR prototype of the DLR institute in Stuttgart. The following components of the ground segment can be identified: Laser source: Generates a train of laser pulses. Telescope: Collects the returning signal. Tracking gimbal and control system: Points the laser towards the satellite. 2.2. Range Model 7 Detector: Detects the returning photons. The detector can be either single-mode or multi-mode. The difference lies in when a returning signal is considered such – either after a single photon hits the detector or when multiple photons do. Optical components: Used to achieve defined beam size and beam divergence. Time of Flight receiver: Records the time of departure and reception of the laser pulses. Meteorological station: Records surface temperature, pressure, and humidity for time of flight corrections. Data storage unit: Used to store the collected measurements. An SLR ground station can be either mono-static or bi-static. In the former, the transmitting and re- ceiving beams follow the same optical path. In these cases, polarized beam splitters need to be im- plemented to distinguish between the transmitting and receiving beams. The latter setup consists of a configuration in which the transmitting and receiving beams follow two different optical paths. The space segment of an SLR system can be either passive or active. As introduced earlier, most of the satellites that support SLR measurements have implemented a passive system consisting of single Cube Corner Retroreflector (CCR) or an Array of CCRs, depending on the required OCS to achieve the desired return signal strength. In Figure 2.3 and Figure 2.4, a Single CCR and an array of CCRs are shown respectively. Figure 2.3: Solid corner cube retroreflector Figure 2.4: Laser retroreflectors array for InSight and an International MGN. When the distances the beam has to travel are too long, as in the case of ILR, an active space segment needs to be implemented. This usually consists of a transponder which generates a returning train of laser pulses. 2.2. Range Model After collecting ToF data from a satellite passing through the Field of View (FoV) of an SLR station, post-processing is required to correct for induced time delays and to aggregate multiple data points, generating an accurate estimate of the range between the station and the satellite. In this section, the range correction models and algorithms implemented as standard by the ILRS network are discussed. 2.2.1. Range Data Point An SLR station typically operates at frequencies ranging from 5 to 10 Hz, while newer systems can reach repetition rates of 100 Hz to kHz. Consequently, during a single satellite pass, thousands of data points are collected. The ILRS network collects the full rate of data points in the Consolidated Range Data (CRD) format , including ToF information, satellite parameters, and weather conditions such as pressure, temperature, and humidity at each epoch of the data point. The time of flight data point, DP, at a given epoch is considered as the point of departure. This DP can be easily converted 2.2. Range Model 8 to a range data point, DPR, in meters using Equation 2.2. ( ) 1 DPT oFi DP Ri = · ·c (2.2) 2 1 · 1012 Where DPT oFi represents the time of flight of a laser pulse emitted at epoch i in picoseconds, and c is the speed of light. The range found needs to be corrected by taking into account the effects of the atmosphere, ∆a , the offset with the satellite center-of-mass, ∆CoM , SLR station range biases, ∆bias , relativistic correction, ∆rel , and random errors, ∆ϵ. Factoring in all the correction factors leads to Equation 2.3. ( ) 1 DPT oFi DP Ri = · · c − ∆a + ∆CoM − ∆bias − ∆rel − ∆ϵ (2.3) 2 1 · 1012 These correction factors can be expressed in time or range values. To convert between units, it is sufficient to multiply by the speed of light. The main correction values and their uncertainties are sum- marized in Table 2.1. These values are indicative, as the corrections depend on the specific station and satellite. Table 2.1: Main range corrections’ values and uncertainties Correction Range Value Time Value Uncertainty Atmospheric Propagation 2-8.5 m 6.67-28.35 ns 1-16 mm Center of Mass ∼250 mm ∼0.83 ns 1-5 mm Station Biases 0.001-300 m 0.1-1000 ns few mm Relativistic 0.1-1 mm few ps few mm The result of Equation 2.3 is considered as the observed range, and it is used together with predicted range values in the algorithm to calculate the residuals as part of the SLR data analysis process. 2.2.2. Correction Factors The correction factors to be taken into account when retrieving the observed range from the ToF of the laser pulses are discussed in this subsection. Atmospheric Delay When passing through the atmosphere, the train of laser pulses emitted by an SLR ground station is affected by atmospheric propagation delay. The main two effects that cause this delay are both associated to the change in the index of refraction, n, within each atmospheric layer. The first effect is the velocity change due to refractive index variation. The speed of light in a medium is determined by the refractive index, n, of that medium. As light passes through different layers of the atmosphere with varying refractive indices, its velocity changes according to Equation 2.4, where v is the velocity of the light in the medium. c v= (2.4) n The other effect is the curvature of light path. In fact, the laser pulses passing through the atmosphere are curved due to the refraction and do not follow a straight path. The atmospheric delay is the largest correction factor of the range model, it can reach several meters as shown in Figure 2.5. 2.2. Range Model 9 Figure 2.5: Increase in range due to atmospheric delay as a function of elevation for a 7-day arc of 11 SLR stations with 15◦ as cut-off elevation. Therefore, there is a need of very accurate atmospheric models to correct for the propagation delay, mainly at low elevation angles. An often used atmospheric correction is the Marini-Murray model de- veloped in 1972 and further improved by Mendes et al. with the implementation of newly derived mapping functions for optical wavelengths. A high accuracy zenith delay prediction at optical wave- lengths currently implemented by the ILRS analysis group defines the atmospheric propagation delay experienced by a laser signal in the zenith direction with Equation 2.5. ∫ ra ∫ ra dzatm = dzh + dznh = 10−6 · Nh · dz + Nnh · dz (2.5) rs rs In this equation, the atmospheric propagation delay, dzatm , is divided into a hydrostatic, dzh , and a non- hydrostatic, dznh , component. The former is also defined as the dry component due to the fact that its refractivity is the results of the dry gases in the troposphere, although it contains the non-dipole component of water vapour refractivity. While, the latter is defined as the wet component. The total group refractivity of moist air is denoted by N = (n − 1) · 106 , where n is the total refractive index of moist air. Furthermore, rs and ra are the geocentric radius of the laser station and the geocentric radius of the top of the neutral atmosphere respectively. Direct equation to calculate the hydrostatic and non-hydrostatic components in the zenith direction are provided in Mendes and Pavlis model analysis. The hydrostatic component, dzh , is given by Equation 2.6. fh (λ) dzh = 0.002416579 · · Ps (2.6) fs (ϕ, H) Where Ps is the surface barometric pressure and fh (λ) is the modified group refractivity index of the dry air component, also defined as the dispersion equation, and can be calculated with Equation 2.7. [ ] (k0 + σ 2 ) (k2 + σ 2 ) fh (λ) = 10−2 · k1∗ · + k ∗ · · CCO2 (2.7) (k0 − σ 2 )2 3 (k2 − σ 2 )2 Where k0 = 238.0185 µm−2 , k1∗ = 19990.975 µm−2 , k2 = 57.362 µm−2 and k3∗ = 579.55174 µm−2. σ = λ−1 is the wave number with λ in micrometers, and CCO2 = 1 + 0.534 · 10−6 · (xc − 450). Where xc is the carbon dioxide content in parts per million (ppm), following International Association of Geodesy (IAG) recommendations xc = 375 ppm. Leading to CCO2 = 0.99995995. The denominator function in Equation 2.6, f (ϕ, H), is given by Equation 2.8. f (ϕ, H) = 1 − 0.00266 · cos(2ϕ) − 0.00028 · H (2.8) Where ϕ is the geodetic latitude of the SLR station, and H is the geodetic height in kilometers. The non-hydrostatic component, dznh , can be computed with Equation 2.9. es dznh = 10−4 · (5.316 · fnh (λ) − 3.759 · fh (λ)) · (2.9) fs (ϕ, H) 2.2. Range Model 10 Where es is the water vapour pressure at the surface which can be calculated with Equation 2.10. [ ( )10 ] 6.1078 · 273.3+T 7.5·T · RH es = (2.10) 100 RH represents the relative humidity as a percentage and T is the temperature in degree Celsius. Fur- thermore, the dispersion formula for the non-hydrostatic component is given by Equation 2.11. fnh (λ) = 0.0031101 · (ω0 + 3 · ω1 · σ 2 + 5 · ω2 · σ 4 + 7 · ω3 · σ 6 ) (2.11) Where ω0 = 295.235, ω1 = 2.6422 µm2 , ω2 = −0.032380 µm4 , ω3 = 0.004028 µm6. With the aforementioned equations it is possible to compute the atmospheric propagation delay in the zenith direction, dzatm. However, in order to compute the delay in the direction of ranging, datm , a Mapping Function (MF) needs to be implemented to map the delay to the elevation angle, e, at which the laser beam is being fired. Due to the fact that the refraction of water vapour at visible wavelengths is small compared to the total refractivity, a single MF can be applied to the atmospheric propagation delay in the zenith direction as shown in Equation 2.12. However, if the wavelength of the laser is not in the visible range, different MFs need to be developed and applied to the hydrostatic and non-hydrostatic components. datm = dzatm · m(e) (2.12) The MF developed by Mendes is based on a truncated form of the Marini continued fraction in terms of 1/ sin(e), normalised to unity. The MF can be calculated with Equation 2.13. a1 1+ a2 1+ 1+a 3 m(e) = a1 (2.13) sin(e) + a2 sin(e)+ sin(e)+a 3 The coefficients, ai , can be found using the formulation in Equation 2.14 suggested by Mendes in its improved mapping function model called FCULa. The values of the sub-coefficients can be found in Table 2.2. ai = ai0 + ai1 · Ts + ai2 · cos(ϕ) + ai3 · H (2.14) Where Ts is the temperature at the station in degree Celsius, ϕ is the altitude of the station, and H is the height of the station in meters. Table 2.2: Improved mapping function sub-coefficients of the FCULa model. aij Value a10 1.21008 · 10−3 a11 1.7295 · 10−6 a12 3.191 · 10−5 a13 −1.8478 · 10−8 a20 3.04965 · 10−3 a21 2.346 · 10−6 a22 −1.035 · 10−4 a23 −1.856 · 10−8 a30 6.8777 · 10−2 a31 1.972 · 10−5 a32 −3.458 · 10−3 a33 1.06 · 10−7 The atmosphere propagation model described in this section can be utilized to derive the atmospheric range correction to be applied to SLR ToF measurements. This model tends to degrade from low to high latitudes of the stations due to higher seasonal variations of surface temperature. However, it achieves high accuracy up to low elevation angles, with a standard deviation of 1 mm, 4 mm, and 16 mm for 15◦ , 10◦ , and 6◦ elevation angles, respectively. Nonetheless, if the laser wavelength lies outside 2.2. Range Model 11 the visible range, the mapping function developed in this section needs to be revised to individually map the hydrostatic and non-hydrostatic components of the propagation delay. This may require the introduction of more complex models, such as the Vienna Mapping Function for optical wavelengths (VMF30). However, if the elevation angles at which the measurements are performed do not go below 20◦ , the impact of the mapping function might not be significant, relieving the necessity of developing new mapping functions. Center of Mass Correction Another effect that induces a time delay, which needs to be corrected in the range model, is the offset between the front face of the CCR that is hit by the laser beam wavefront and the actual Center of Mass (CoM) of the satellite. The center of mass correction is unique per satellite due to different shapes and configurations of CCR arrays. To retrieve the CoM signature of a satellite, its response function needs to be analyzed. As a consequence of having multiple CCRs within the array, the response signal is broadened. This characteristic is shown schematically in Figure 2.6. In fact, the CCR which is perpendicular to the approaching laser beam will reflect the signal before the CCRs which have an offset in incidence angle, resulting in a time delay in the reflected signal. Figure 2.6: Broadening signature effect of a spherical satellite with a CCRs array on the reflected laser pulse. Moreover, a phase delay can also be observed due to the same principle, and if individual reflector responses overlap, destructive interference could be witnessed. To compute the time delay per CCR, the sketch shown in Figure 2.7 is used to geometrically explain the parameters involved in the CoM characterization. 2.2. Range Model 12 Figure 2.7: Sketch of typical geodetic satellite such as LAGEOS, defining parameters required to discuss satellite impulse response In the geometrical sketch shown in Figure 2.7, two CCRs are shown, and it can be denoted that the time delay between CCR1 with incidence angle θi = 0, and CCR2 with θi > 0 is caused by the distance δR − nL, which is the distance between the reflection points, P , of the two reflectors in the direction of the laser beam wavefront. The distance between the CCR front surface and the point of reflection, ∆R, can be computed with Equation 2.15. √ ( )2 sin(θi ) ∆R = n · L · 1 − (2.15) n The time delay related to CCR2 can be calculated with Equation 2.16. 2 ∆T = {Rs − [Rs − ∆R] · cos(θi )} (2.16) c Where Rs is the radius of the satellite from its center of mass to the reflector surface. Each retroreflec- tor has its own time delay in the laser response depending on its incident angle and its distance to the satellite CoM. The total response function of the reflected laser beam can be computed by summing up all responses of the individual CCRs and applying the respective delays. Nevertheless, if the satel- lite is not a perfect sphere with equally distributed CCRs, the total response function varies with the orientation of the satellite. Therefore, the convoluted response is computed for a multitude of orienta- tions, and an average response is retrieved. The CoM correction, ∆CoM , is equal to the centroid of the averaged total response function. For LAGEOS, the standard value found for the CoM correction is 251 mm. This correction is applied to all ToF measurements performed on LAGEOS. However, as explained before, different satellite orientations affect the response function. Therefore, utilizing a standard value retrieved from an average could lead to small errors. On this topic, researchers are developing new methodologies to apply CoM corrections based on the instantaneous responses of the satellite. However, the implementation requires modifying the processing of SLR data centers, making it impractical for the gained accuracy of approximately 1 mm. Station Biases Range bias and time bias due to station-dependent errors must be included in the range model cor- rection budget. These errors can originate from various sources, each identified individually per SLR station. They may stem from non-linearities in interval counters, inaccuracies within sensors for baro- metric pressure, temperature, and relative humidity, as well as uncertainties in the ITRF position of the ground station and its relative velocity due to tectonic plate movement. Additionally, laser settings can contribute to time errors. For instance, if the emitted laser pulse is long, multiple photons within the laser pulse will be emitted. However, upon the return of a photon from the 2.2. Range Model 13 laser pulse, it is impossible to specify which photon in the pulse it was, leading to time uncertainty. Moreover, having a high repetition rate will result in multiple pulses flying simultaneously towards the target satellite, making it unfeasible to match the returning pulse to a specific emitted pulse. These biases typically range within a few millimeters and are specifically characterized for each SLR station. Relativistic Correction When light propagates within a reference frame containing multiple masses, space and time are warped, deviating the light path and introducing relativistic time delays. Equation 2.17 encapsulates the effect of gravitational masses on the travel time of the laser. ( ) | x⃗2 (t2 ) − x⃗1 (t1 ) | ∑ 2 · G · MJ rj1 + rj2 + ρ t 2 − t1 = + · ln (2.17) c c3 rj1 + rj2 − ρ J The sum of all bodies, J, with mass Mj centered at xj is considered. While, rj1 =| x⃗1 − x⃗j |, rj2 =| x⃗2 − x⃗j |, and ρ =| x⃗2 − x⃗1 |. In practice, ρ is the range between ground station and satellite before the relativistic correction is applied. For near-Earth satellites, analyses are done in the geocentric frame of reference, and the only body to be considered in the relativistic correction model is the Earth. 2.2.3. Normal Point Algorithm After collecting the full-rate ToF data from an SLR station, data centers utilize the Herstmonceux al- gorithm to post-process the data and generate normal points. This procedure aims to reduce the size of the data package for further analyses. With the normal point algorithm, the size of the data package can be decreased from thousands of data points to hundreds of normal points. For example, the full-rate data of LAGEOS-1 from station 8834 in September 2019 consisted of 140,718 points, and after implementing the algorithm, only 311 normal points were obtained. The Herstmonceux algo- rithm requires as input the full-rate data of the SLR station and the orbit predictions of the observed satellite. Data of orbit predictions can be found per satellite on the ILRS website in the Consolidated Prediction Format (CPF). Predictions are obtained using orbit propagation models, which predict the future satellite position and velocity based on its current state. The Herstmonceux algorithm consists of two stages. The first stage involves screening the full-rate data with respect to predicted ranges to remove outliers. The second stage involves forming normal points using the accepted data points from stage one. The steps to be taken within these stages are displayed in the flow diagram shown in Figure 2.8. Furthermore, each step implemented in the algorithm is described below. Figure 2.8: Flow diagram of the Herstmonceux algorithm 2.2. Range Model 14 Stage 1: Full-rate Data Screening 1. Use high precision predictions to generate prediction residuals P R = observation - prediction. Both observation and prediction need to include all the correction factors of the range model; 2. Identify large outliers defining a suitable range window and remove them; 3. Solve for a set of parameters (preferably orbital) to remove the systematic trends of the prediction residuals. The resulting fitted function is the trend function f , with value f (P R) at the epoch of the residual P R; 4. Compute fit residuals F R = P R − f (P R); 5. Remove outliers by computing the Root Mean Square (RMS) of the fit residuals and utilizing a rejection level of nxRMS. The value of n is recommended to be taken as n=2.5 for systems using single photon detection and n=3 for systems using multi-photon detection. Now iterate step 3, 4 and 5 until reaching convergence. The outliers removed in step 5 should not be considered in step 3 of next iteration. However, for step 4 and 5 of each iteration all data points should be utilized even the one removed in previous iterations. Stage 2: Normal Point Formation 1. Take the accepted fit residuals, F R, and subdivide them into fixed intervals. These intervals are called bins and their size is specifically defined per satellite through recommendations provided by the ILRS; 2. Compute the mean value F Ri and the mean epoch of the accepted fit residuals within each bin i; 3. Match the particular observation Oi with its fit residual F Ri , whose observation epoch ti is nearest to the mean epoch of the accepted fit residuals in bin i; 4. Compute the normal point for each bin as: N Pi = Oi − F Ri + F Ri ; 5. Compute √ the RMS of the accepted fit residuals in each bin i from their mean value with ∑ RM Si = n1i · i (F Rj − F Ri )2. Where ni is the number of accepted fit residuals within bin i. Further analysis of the data distribution can be performed by calculating the skewness and kurtosis parameters. These parameters are useful for describing some of the features of the data distribution. When computing these parameters, only the fit residuals that have been retained within the nxRMS screening in steps 3, 4, and 5 are considered. If the data have insignificant skewness, then the mean value of the retained fit residuals will be located at the peak of the distribution of the data. Conversely, if the data are significantly skewed, then the mean will be offset from the peak, in the direction of the skewness. The difference of the mean from the peak is another useful indicator of the distribution of the data to characterize satellite signatures. 2.2.4. Precision of Normal Points The precision of ToF measurements of an SLR station is given by the total variance of the system which is computed summing up the individual variances of the involved SLR segments, as shown in Equation 2.18. σT2 ot = σLaser 2 2 + σDetector + σT2 imer +... + σSpace 2 (2.18) Therefore, to maximize precision of ToF measurements the total variance needs to be minimize. Follow- ing the aforementioned algorithm for the generation of normal points from full-rate data, an improved range precision can be achieved per normal point, ∆RN P , as shown by Equation 2.19. σT ot ∆RN P = √ (2.19) n Where n is the number of data points used to generate each normal point. This number could vary per normal point, nevertheless increasing the laser pulse frequency generating more data points leads to a faster desired normal point precision. Furthermore, the resulting normal point will represent a shorter orbital arc length obtaining a higher resolution orbit. 2.3. Retroreflectors Design 15 2.3. Retroreflectors Design In this section, the design of the space segment is described with a focus on defining the required OCS of the retroreflector array to generate a strong enough returning pulse. Furthermore, various types of retroreflectors are investigated, along with the effect of radiation on their reflectivity. 2.3.1. The Link Budget Equation To assess if the returning signal has enough strength to be collected by the ground station detector the mean signal flux in the receiver needs to be computed. The radar link equation is used to compute the mean number of photo-electrons, npe , reaching the ranging detector, as shown in Equation 2.20. ( )2 1 npe = Nt · GT · σ · · Ar · ηt · ηr · ηd · Ta2 · Tc2 (2.20) 4 · π · R2 From this equation, it can be seen how the strength of the returning signal decreases by a factor of R4. Each of the variables in the radar link equation are individually investigated below. Emitted photons of laser, Nt : The number of emitted photons within the laser pulse is given by Equation 2.21. Ep · λ Nt = (2.21) h·c Where Ep is the pulse energy which is the average power divided by pulse length, λ is the laser wavelength, h is the Planck’s constant, and c is the speed of light in vacuum. Slant range, R: The slant range to the target is the distance between the SLR ground segment and the target object in space. R is calculated utilizing the geometric coherence, as shown in Equation 2.22. (π ) R = −(RE + hs ) · cos − αe √[ 2 (π )]2 (2.22) + (RE + hs ) · cos − αe + 2 · RE · (hsat − hs ) + h2sat − h2s 2 Where RE is the Earth radius, hs is the station altitude above sea level, hsat is the satellite altitude above sea level, and αe is the elevation angle. Transmitter gain, GT : The transmitter gain, GT , describes how well the laser energy is converted in a specific direction. Modern SLR systems produce a quasi-gaussian spatial and temporal laser profiles. Therefore, the formulation in Equation 2.23, applicable to Gaussian beams, can be used to calculate the transmitter gain. ( ) ( ∆θ )2 8 −2· θ p GT = · e d (2.23) θd2 Where θd represents the far-field beam divergence half-angle between the beam center and the 1/e2 intensity point. The exponential term denotes the static beam pointing loss, where ∆θp represents the resulting beam pointing error from the center of the beam. However, it is essential to note that this expression does not account for the radial truncation of the beam Gaussian profile due to a limiting aperture, nor does it consider the central obscuration of the beam, possibly resulting from a secondary mirror in a Cassegrain telescope. These effects lead to a transfer of energy from the central lobe to the outer rings of the profile. Telescope receive area, Ar : To compute the effective receiver area, the radiation lost due to blockage of a secondary mirror, if any, and the spillover at the spatial filter, if any, and at the detector need to be taken into account. Equation 2.24 computes the receiver area including the aforementioned losses. ( ) k · Rd Ar = Ap · (1 − γ 2 ) · ηd · γ · (2.24) 2·F Where Ap = π · rp2 is the area of the primary receiver, γ is the receiver obscuration ratio and ( ) (1 − γ 2 ) is the fraction lost due to blockage of the secondary receiver. The term ηd · γ · k·R 2·F d is 2.3. Retroreflectors Design 16 the fraction of light intercepted by a detector of radius Rd. Alternately, Rd could be the radius of a spatial filter in the focal plane of the receiver. F is the F-number of the receiving telescope and k = 2π/λ. Satellite optical cross-section, σ: The optical cross section of the satellite is the only variable related to the space segment which has an impact on the returning signal. Therefore, this term will be the driving factor for the design of the retroreflector array. An in depth analysis of σ is performed in the next section. Transmitter optical throughput efficiency & receiver optical throughput efficiency, ηt , ηr : The parameters ηt and ηr represent the total transmission efficiency of the transmitter and receiver optics, respectively. The total throughput optical efficiency, ηt,r , is the product of the transmission efficiencies of all elements ηi in the beam path, as shown in Equation 2.25. ∏ n ηt,r = ηi (2.25) i Quantum efficiency of detector, ηd : The quantum efficiency refers to the effectiveness of a de- tector in converting an incident photon into an electron. This measure is expressed in probabilistic terms and depends on the type of detector implemented. Two-way atmospheric attenuation factor, Ta2 : The attenuation of light in the visible and the infrared wavelength range occurs due to absorption and scattering caused by air molecules, solid, and liquid particles. The one-way atmospheric attenuation factor, Ta (λ, V, hr , θzen ), is dependent on the laser wavelength, λ, the visibility, V , the relative height of the SLR ground station above sea level, hr , and the zenith angle, θzen. The zenith angle is the complement of the elevation angle, αe. The two-way attenuation factor is computed as the square of the one-way attenuation factor. The attenuation factor can be computed assuming the laser path to be a straight line ignoring the refractive bending. Figure 2.9 shows a plot displaying the one-way attenuation factor against laser wavelength at sea level and clear visibility V = 60 km, at different zenith angles. Figure 2.9: One-way attenuation factor against laser wavelength at sea level and clear visibility V = 60 km at θzen = 0◦ , 50◦ , 70◦. Two-way cirrus cloud attenuation factor, Tc2 : Atmospheric losses arise from the presence of cirrus clouds, which are sub-visible clouds that are overhead 50% of the time in most locations, thereby attenuating the signal strength. For wavelengths between 0.3 and 12 μm, no significant 2.3. Retroreflectors Design 17 dependence has been found on the cirrus cloud attenuation factor. From experiments, it has been determined that the one-way cirrus cloud attenuation factor, Tc , can be computed using Equation 2.26. Tc = e−0.14·(t·sec(θzen )) 2 (2.26) Where t is the cirrus cloud thickness. From a global study on cirrus clouds thickness, it has been found that the average cirrus clouds thickness is 1.341 km. With the aforementioned variables, it is possible to solve the radar link equation to obtain the mean number of photo-electrons reaching the SLR detector. This number significantly varies with respect to weather conditions. For instance, considering the worst and best case scenarios for laser ranging of LAGEOS using MOBLAS-6, the maximum number of received photo-electrons could vary between 612 and 0.05 per pulse due to variations in weather conditions, as well as changes in orbital and hardware settings. Therefore, unless there are other compelling reasons, an SLR station with minimal cloud cover and high atmospheric transparency will result in more detected data and higher accuracy than any other station. 2.3.2. Required Mean Number of Returning Photo-electrons The required mean number of returning photo-electrons, npe , to achieve ranging measurements is determined by the detector settings, specifically by defining the threshold for the minimum number of photo-electrons, nt , needed for signal detection. This threshold helps decrease the probability of detecting a signal when no signal is present, also known as the probability of false alarm. False alarms are a consequence of background noise entering the detector FoV. Therefore, it is important to set a threshold high enough to minimize their occurrence. Furthermore, npe is also related to the probability of detection, which is the detector capability to convert an arriving photon into an electric impulse. To retrieve the required npe , it is essential to first define the background photo-electron rate, Λ, as de- scribed by Equation 2.27. ηd Λ= · N l · Ω r · Ar · η r (2.27) h·v Where Nl is the background spectral radiance, Ωr is the receiver FoV in steradians, and v is the laser frequency. The background spectral radiance, Nl , is computed for both daylight and nighttime, depend- ing on when ranging measurements are performed. Usually, due to the high background photo-electron rate during daylight, ranging measurements are performed at night. During daylight, the solar spectral irradiance, Iλ , can be retrieved from Figure 2.10 for specific wave- length ranges passing through the narrow band-pass filter. It is appropriate to consider the wavelength of the emitted laser pulse. The most important parameter that determines the solar irradiance under clear sky conditions is the distance that sunlight has to travel through the atmosphere. This distance is shortest when the sun is at the zenith. The ratio of the actual path length of sunlight to this minimal distance is known as the optical air mass (AM). The value retrieved from the graph is for an optical air mass of 1.5, which corresponds to an angle of 48.2 degrees between the Sun position and the zenith, and it is currently considered the standard spectral distribution. 2.3. Retroreflectors Design 18 Figure 2.10: Plot of Direct Normal Spectral Irradiance, and Hemispherical Spectral Irradiance. Once the solar spectral irradiance is retrieved, it needs to be translated to ground-level where the receiver is placed. Therefore, the atmospheric transmission, Tλ , of the selected wavelength needs to be considered. The atmospheric transmittance at different wavelengths can be retrieved from Figure 2.11. Figure 2.11: Atmospheric transmittance per wavelength. Consequently, the ground-level spectral irradiance can be converted into background daylight spectral radiance by considering a half-sphere portion of the sky, from which the detector sees only a portion. The transformation from spectral solar irradiance to background spectral radiance is summarized in Equation 2.28. I λ · Tλ Nl = (2.28) 2·π During nighttime, the major sources of background noise are the scattering of sunlight reflected by the Moon through the atmosphere and light pollution from nearby city centers or rural areas with a high number of greenhouses. In the Netherlands, multiple sensors have been placed throughout the country to collect data on sky brightness, B, at night. Specifically, in Delft, a sensor was placed on top of the Aerospace faculty building next to the newly built laser communication station. It is important to acknowledge that Delft is surrounded by numerous greenhouses, which makes it one of the most light-polluted regions in the Netherlands. Figure 2.12 shows the data of sky brightness collected by Delft sensor during its first night of operation. 2.3. Retroreflectors Design 19 Figure 2.12: Sky brightness measurements during the night of the 17th of May 2023. Furthermore, Table 2.3 displays the average, 10% percentile, 50% percentile, and the maximum values of sky brightness collected in Delft since the first day of operation. Table 2.3: Average, 10% percentile, 50% percentile, and the maximum value of sky brightness collected in Delft since May 2023. Average 10% Percentile 50% Percentile Maximum 18.0626 mag/arcsec2 18.028 mag/arcsec2 15.480 mag/arcsec2 18.116 mag/arcsec2 Taking the average as a reference value, the sky brightness scale needs to be converted into back- ground spectral irradiance. The first step is to transform mag/arcsec2 into cd/m2 , and then trans- form it into spectral irradiance as shown in Equation 2.29. Bcd/m2 = 10.8 · 104 · 10−0.4·B (2.29) Nl = Bcd/m2 · 1.464 · 10−7 at 555 nm With the background spectral irradiance, Nl , it is possible to compute the background photo-electron rate, Λ, as described before in Equation 2.27. Consequently, the number of background photons en- tering the detector needs to be determined. The detector has two different time settings to collect the returning signal. The range gate time window, τrg , represents the period during which the detector is actively collecting photons. To define a proper range gate, the initial and ending times need to be set based on the accuracy of the orbit prediction model. This ensures that the detector is sensitive only when the returning signal is expected. As reference, 1 µs is considered as range gate, which requires an orbit prediction model accuracy of ±150 m. Using Equation 2.30, the photon count entering the detector within the range gate window can be computed. Nbrg = Λ · τrg (2.30) The second time setting is the integration time, τi , which represents the period during which the detector sums up the received photons. To collect the entire returning signal, the integration time should be set equal to the length of the returning pulse. Using Equation 2.31, the photon count entering the detector within the integration window can be computed. N b i = Λ · τi (2.31) With these two photon counts, the probability of false alarm, Pf a , can be computed using Equation 2.32. Defining a limit value for the probability of false alarm allows the computation of the minimum number of photo-electrons threshold, nt.  nt −1  −Nbrg ·Nb i  (n −1)!  Pf a = 1 − exp  ∑ t N m  (2.32) nt −1 bi m=0 m! 2.3. Retroreflectors Design 20 Considering the receiving signal as the sum of the mean number of returning photo-electrons and the background noise photons, as given by Equation 2.33, the probability of detection, Pd , can be computed approximating it with a Poisson distribution, as shown in Equation 2.34. N = npe + nbr (2.33) t −1 n∑ Nm Pd = 1 − e−N · (2.34) m=0 m! By setting a lower limit for Pd , the required mean number of returning photo-electrons, npe , can be determined. 2.3.3. Optical Cross Section The OCS of a retroreflector, denoted as σ, when oriented normal to the incident light and in the far-field limit (rcc /R → 0), can be computed using Equation 2.35. In this case, the Far-Field Diffraction Pattern (FFDP) of the reflected wave is the Airy function. [ ]2 4 · π · A2cc 2 · J1 (k · rcc · sin (θr )) σ =ρ· · (2.35) λ2 k · rcc · sin (θr ) In this equation, θr represents the returning signal orientation error, ρ is the retroreflector reflectivity, 2 k is the propagation number (2π/λ), and Acc = πrcc denotes the area of the retroreflector circular aperture with radius rcc. The function J1 is a Bessel function of the first kind. Bessel functions are part of the set of solutions for Bessel’s differential equation and can be expressed generally as shown in Equation 2.36. ∑∞ (−1)k ( x )n+2k Jn (x) = · (2.36) k! · Γ(n + k + 1) 2 k=0 Where Γ(n) = (n − 1)!, and n represents the order of the Bessel function. For normally incident light, the peak on-axis OCS of an unspoiled, θr = 0, retroreflector is given by Equation 2.37. 4·π 4 · π · A2cc π 3 · ρ · Dcc 4 σθr =0 = ρ · Acc · =ρ· = (2.37) Ω λ 2 4·λ 2 Where, 4π/Ω is the on-axis gain and Ω is the effective solid angle occupied by the FFDP of the retrore- flector. However, when the retroreflector is not oriented normal to the incident light, an incident angle is created, θi , which decreases the actual area of the circular aperture of the retroreflector, Acc. The decreasing factor, η, is given by Equation 2.38. [ ( ) ] 2 l η(θi ) = · sin−1 (µ) − · µ · tan(θref ) · cos(θi ) (2.38) π rcc √ Where θi is the incidence angle and l/rcc is the corner-cube ratio which is set to its limit value of 2. θref is the internal refracted angle as determined in Equation 2.39 by Snell’s Law. ( ) −1 sin(θi ) θref = sin (2.39) n Here n is the cube index of refraction. The quantity µ in the decreasing factor equation can be computed with Equation 2.40. √ ( )2 l µ= 1− · tan2 (θref ) (2.40) rcc Thus, the peak OCS of a single retroreflector not oriented normal to the incident light falls off as de- scribed in Equation 2.41. σef f = η 2 · σ (2.41) 2.3. Retroreflectors Design 21 And, if the single reflector is unspoiled, its effective peak OCS can be calculated with Equation 2.42 2 4 · π · [Acc · η] σef f θr =0 = ρ · (2.42) λ2 While, if velocity aberration is considered, inducing an off-axis reflection, its effective OCS can be calculated with Equation 2.43. [ ]2 4 · π · (Acc · η)2 2 · J1 (k · rcc · η · sin (θr )) σef f =ρ· · (2.43) λ2 k · rcc · η · sin (θr ) 2.3.4. Velocity Aberration The retroreflectors are built following the principle of reflecting light in the same direction as the incoming beam, provided no spoiling is applied. Therefore, the beam reflected by a satellite should ideally reach the ground station with no deviation. However, in reality, this is not the case due to the relative velocity, v, between the satellite and the ground station. As a consequence, the direction of the reflected beam is angularly shifted towards the relative velocity vector by an amount α. This effect is known as velocity aberration and can result in significant losses in the signal. Figure 2.13 illustrates a schematic of the principle of velocity aberration. Figure 2.13: Schematic of velocity aberration. To calculate the magnitude of the angular displacement in the FFDP, α, due to velocity aberration, Equation 2.44 can be used. √ α = αmax · cos2 (ω) + Γ2 · sin2 (ω) (2.44) Where Γ is given by Equation 2.45 and ω by Equation 2.46. √ ( )2 Re · sin(θzen ) Γ= 1− (2.45) R e + hs ω = cos−1 [(r̂ × p̂) · v̂] (2.46) Here, r̂ is the unit vector to the satellite from the geocenter, p̂ is the unit vector from the station to the satellite, and v̂ is the unit vector in the direction of satellite velocity. The maximum value of the angular shift, αmax , can be computed with Equation 2.47. √ 2 · vs 2 g · Re2 αmax = = · (2.47) c c R e + hs While, the minimum value of the angular shift, αmin , can be calculated using Equation 2.48, and it depends on the maximum zenith angle for tracking, θzenmax. αmin = αmax · Γ(hs , θzenmax ) (2.48) 2.3. Retroreflectors Design 22 The latter equations ignore the small contribution of station motion due to Earth rotation, ∼ 0.46 km/s, to the relative velocity which typically reduces α by 4 or 5 µrad for LLR but is negligible for LEO to GEO satellites. To compensate for velocity aberration, several techniques are available, with the most common being the spoiling of retroreflectors. This technique involves inducing an angular shift, γ, in the reflected signal, θr , which is equal in magnitude but opposite in direction to the angular shift caused by veloc- ity aberration, α. Achieving this compensation requires aligning the attitude of the reflector with the direction of motion. To achieve spoiling, the faces within the retroreflector are connected at slightly perturbed angles instead of the usual right-angle configuration. The angle perturbation is known as the dihedral angle, δ, and can be applied to one or all faces, nδ = 1 or 3. The induced shift in the reflected signal can be computed with Equation 2.49.  √  4  · 6 · δ f or nδ = 3 γ= 3 √ (2.49)   2 · 6 · δ f or nδ = 1 3 This results in a shifted velocity aberration,αs , as shown in Equation 2.50, which is used in Equation 2.43 as the returning signal orientation error, θr. αs = α − γ (2.50) Nevertheless, spoiling the retroreflector reduces the on-axis peak OCS, σθr =0 , as described by Equa- tion 2.51. σθr =0 σef f = (2.51) (2 · nδ )2 Another technique involves decreasing the diameter, D, hence, the retroreflector aperture. This ac- tion results in an increase in the FFDP angle, θ, leading to a broadening of the diffraction pattern, as illustrated by Equation 2.52. 3.8 · λ θ= (2.52) π·D However, while this adjustment places the OCS above the threshold level within the velocity aberration range if α < θ. It decreases it in other regions, thereby limiting the range of acceptable tilt angles and imposing higher constraints on the attitude control of the satellite. Another way of compensating for velocity aberration consists in broadening the diffraction pattern by tilting the retroreflector away from the station in the direction of the relative velocity vector to achieve a narrower effective aperture in the direction of the velocity aberration. In fact, tilting the retroreflec- tor, will lead to an effective decrease of the retroreflector aperture, D, by η, the factor introduced in Equation 2.38. Consequently broadening the diffraction pattern as described in Equation 2.52, with the drawback of decreasing the overall OCS response. 2.3.5. Retroreflector Types Generally, there are three type of retroreflectors: corner retroreflectors, cat eye reflectors, and non- linear retroreflectors. The retroreflector that is most commonly implemented in space is the corner retroreflector in the form of corner cube prisms, where reflections occur on three mutually orthogonal mirror surfaces. The corner cube retroreflectors can either be solid back-coated, solid uncoated, or hollow. Each configuration has its own advantages and disadvantages which are displayed in Table 2.4. 2.3. Retroreflectors Design 23 Table 2.4: Characteristics of three different type of CCRs Solid Back-Coated Solid Uncoated Hollow Reflectivity 0.78 0.93 0.99 No with metal coat- Polarization No Yes ings. Yes with di- Sensitive electric coatings Far Field Wide Wide Narrow Pattern Weight Heavy Heavy Light Metal coatings absorb sun- Leak of signal at incident an- Thermal heating Issue light creating thermal gradi- gle higher than 17◦. Polariza- and gradient effects ents. Not well shielded, sus- tion effect can decrease optical on joints ceptible to radiation effects. cross section by a factor of 4. For the design of the array to be placed on top of the new TU Delft platforms, the solid back-coated retroreflector type will be utilized. This decision is influenced by the fact that the uncoated type ex- hibits leakage starting from 17◦ of incident angle, thereby limiting the array configuration and imposing stringent requirements on the attitude control of the satellite. Furthermore, back-coated types are the most commonly used within the space industry, offering high accessibility of Commercial Off-The- Shelf (COTS) components. While, the hollow retroreflectors have never been implemented for visible wavelengths. For the array of the new TU Delft platforms, the specular retroreflector prisms with backside gold coat- ing from Thorlabs , and the aluminum coated fused silica corner cubes from Edmund Optics are considered. The Thorlabs retroreflectors are made of N-BK7, and are available in different di- ameters, namely 12.7 mm, 25.4 mm, and 50 mm. While the Edmund retroreflectors are made of fused silica and are available with diameters of 7.16 mm, 12.7 mm, 25.4 mm, and 50.8 mm. The diameter of the retroreflector will be a design consideration during the design of the array. In Ta- ble 2.5 and Table 2.6, the available specifications of the two type of retroreflectors are displayed. Table 2.5: Main specifications of the specular retroreflector Table 2.6: Main specifications of the aluminum coated fused prisms with backside gold coating from Thorlabs. silica corner cubes from Edmund Optics. Parameter Value Parameter Value Prism material N-BK7 Prism material Fused Silica Wavelength range 800 − 2000 nm Wavelength range 400 − 2000 nm Reflective coating Gold with black over-paint Reflective coating Aluminum Diameter tolerance ±0.1 mm Diameter tolerance ±0.1 mm Surface quality 40 − 20 Scratch-dig Surface quality 20 − 10 Scratch-dig Beam deviation < 3 ′′ Beam deviation < 5 ′′ Damage threshold 1.25 J/cm2 Damage threshold 0.3 J/cm2 Index of refraction 1.517 at 587.6 nm Index of refraction 1.45 at 587.6 nm 2.3.6. Radiation Effect When orbiting in space, satellites and all their components are subject to radiation. Particularly note- worthy is the impact of radiation, such as gamma rays, on the optical components of a satellite, which can result in a decrease in performances or, in some cases, in mission failure. For instance, in the con- text of the retroreflector of Thorlabs introduced in the previous section, N-BK7, the material of which the prism is made, is susceptible to radiation, leading to a reduction in its reflectivity and consequently 2.3. Retroreflectors Design 24 diminishing the intensity of the reflected beam. A study conducted by T.L. Griffiths et al. examined how the intensity of a laser beam passing through different lenses made of BK7, as well as coated and uncoated fused silica, changed due to gamma ray radiation. Figure 2.14 and Figure 2.15 illustrate the variation in intensity at various levels of radiation exposure across different wavelengths for BK7 glass and uncoated UV-grade fused silica (UVFS) respectively. Figure 2.14: Spectral output of the deuterium-halogen lamp when transmitted via unirradiated, 5 kGy, 25 kGy and 50 kGy irradiated BK7 glass collection optics. Figure 2.15: Spectral output of the deuterium-halogen lamp when transmitted via unirradiated, 5 kGy, 25 kGy and 50 kGy irradiated UV-grade fused silica (UVFS) glass collection optics.. From the BK7 graph, it can be observed that the visible range of wavelengths is severely affected already at the first level of radiation. However, for higher laser wavelengths, the effect on the laser intensity is negligible. Therefore, implementing a 1064 nm laser is preferable over a classic laser in the visible range when implementing retroreflector prisms made of N-BK7. However, most of the ILRS stations do not implement a 1064 nm laser therefore limiting the number of stations with which ranging measurements can be gathered. Regarding fused silica lenses, the effect of gamma ray radiation is much less significant as observed in Figure 2.15. Furthermore, to put this into perspective, the ionizing radiation expected on the retroreflector of Delfi-Twin is estimated using Spenvis, ESA’s Space Environment Information System. The orbital parameters are input into the tool, and the ionizing dose for simple geometries is simulated with the SHIELDDOSE-2 model, assuming SiO2 as the target material. With the absorber thickness set to the smallest simulated value, a total mission dose of 1.396·106 rad, equivalent to 13.96 kGy, is obtained. Consequently, implementing N-BK7 as the material 2.4. Ground Station 25 would lead to a significant reduction in performance over the mission duration. This would suggest a preference for implementing retroreflectors with a fused silica prism like the one from Edmund Optics. However, fused silica components are extremely expensive and scarce on the market. Manufacturers typically produce limited batches over the span of years. Nevertheless, for the implementation on the new TU Delft platforms it would be preferable to implemented the retroreflectors from Edmund Optics rather than the one from Thorlabs. 2.4. Ground Station This section presents the main specifications of four possible SLR stations with which the retroreflector array will be designed to operate. The first station is the Mini-SLR optical system developed by DLR in Stuttgart, the second and third one are station 7941 in Matera, and station 14473 in Potsdam, which are part of the ILRS network. Lastly, the new TU Delft laser satellite communication station is included. Although this station is primarily intended for laser communication, it could also be utilized for ranging measurements. Table 2.7, Table 2.8, Table 2.9, and Table 2.10 respectively list the main specifications of the Mini-SLR, the new TU Delft laser communication station, Potsdam station, and Matera station. Table 2.7: Main specifications of the Mini-SLR optical system. Table 2.8: Main specifications of TU Delft lasercom station. Parameter Symbol Value Parameter Symbol Value Transmit aperture Dt 7.5 cm Transmit aperture Dt 40.5 cm Beam diameter Dbeam 5 cm Beam diameter Dbeam 10 cm Receiver aperture Dr 20 cm Receiver aperture Dr 40.5 cm Obscuration γ 25% Obscuration γ 44% Laser pulse energy Ep 85 µJ Laser Power Plaser 0.5 W Laser repetition rate flaser 50 kHz Laser repetition rate flaser N/A Laser pulse width ts 4 ns Laser pulse width ts T BD Operating wavelength λ 1064 nm Operating wavelength λ 1560 nm Half-angle beam div. θd 50 µrad Half-angle beam div. θd 9.93 µrad Beam stability ∆θp 25 µrad Beam stability ∆θp 10 µrad Transmitter efficiency ηt 0.6 Transmitter efficiency ηt 0.7* Receiver efficiency ηr 0.1 Receiver efficiency ηr 0.7* Efficiency of detector ηd 30% Efficiency of detector ηd 80%* *Variables with assumed values. Table 2.9: Main specifications of station 14473 in Potsdam of Table 2.10: Main specifications of station 7941 in Matera of the ILRS network. the ILRS network. Parameter Symbol Value Parameter Symbol Value Transmit aperture Dt 15 cm Transmit aperture Dt 1.5 m Beam diameter Dbeam 15 cm Beam diameter Dbeam 0.01 m Receiver aperture Dr 44 cm Receiver aperture Dr 1.5 m Obscuration γ 0.34 Obscuration γ N/A Laser pulse energy Ep 10 mJ Laser pulse energy Ep 200 mJ Laser repetition rate flaser 10 Hz Laser repetition rate flaser 10 Hz Laser pulse width ts 50 ps Laser pulse width ts 50 ps Operating wavelength λ 532 nm Operating wavelength λ 1064 nm Full-width beam div. θd 25 µrad Full-width beam div. θd 218 µrad Beam stability ∆θp 0.41 µrad Beam stability ∆θp 0.41 µrad Transmitter efficiency ηt 0.8 Transmitter efficiency ηt 0.75 Receiver efficiency ηr 0.4 Receiver efficiency ηr 0.87 Efficiency of detector ηd 8% Efficiency of detector ηd 13% Since the new TU Delft laser communication station is still in development phase, some parameters are currently unknown. Therefore, assumptions are made: the transmitter and receiver efficiencies are taken to be 0.7 , and the implemented detector is assumed to be an InGaAs photoreceiver with 2.5. TU Delft Satellite Platforms 26 80% quantum efficiency. The laser power of the available Exail laser is 0.5 W. Furthermore, since this ground station is primarily built for laser communication, the laser is continuous. However, if On- Off Keying (OOK) modulation is assumed to be implemented a duration of each pulse in the OOK modulation can be retrieved leading to a repetition rate. Knowing the repetition rate makes it possible to compute the energy per pulse dividing the laser power by the laser length. 2.5. TU Delft Satellite Platforms Under the Delfi program started in 2004, the Technical University of Delft entered the field of Cube- Sats and became one of the first to demonstrate their benefits within the professional space sector with the launch of Delfi-C3 in 2008. In the following years, the number of universities and other insti- tutes developing CubeSats increased rapidly, and TU Delft launched a second CubeSat, Delfi-n3XT, in 2013. With these satellites, TU Delft aims to provide a real-life platform for students to understand the dynamics behind a space project, as well as to perform demonstrations of small innovative space technology, thus boosting small satellite bus development. In fact, in recent years, TU Delft has been refocusing on miniaturization of satellites, launching Delfi-PQ, a triple-unit PocketQube, in 2022. The mini-satellite was fully produced by the Delfi team, from circuit boards to micro-propulsion systems, as COTS components were too large to fit in the small bus of the satellite. Figure 2.16, Figure 2.17, Fig- ure 2.18 display the entire TU Delft satellite legacy starting with Delfi-C3, then Delfi-n3xt, and Delfi-PQ. Figure 2.16: Delfi-C3. Figure 2.17: Delfi-n3Xt. Figure 2.18: Delfi-PQ. The next satellite platform of TU Delft, Delfi-Twin, will be built on top of the Delfi-PQ design, maintain- ing its same size 5 x 5 x 18 cm and its orbit altitude of 525 km implementing new state-of-the-art Pock- etQube technologies. TU Delft aims to utilize this new platform to perform laser ranging measurements by implementing an array of retroreflectors on top of the satellite, coupled with a laser communication station under construction on top of the Aerospace faculty building. In order to design the retroreflector array, it is important to understand how the attitude determination and control system of the satellite will be designed. The plan is to implement a completely passive control system consisting of two ex- tendable flaps positioned at one end of the satellite, which will provide stability utilizing the principle implemented on a shuttlecock. Nevertheless, this system will constrain the yawing and pitching rota- tion axes, it will not stop the satellite from rolling. This insight is of primary concern for the design of the retroreflector array, as a smart design will need to be implemented to achieve full coverage even if the satellite is rolling. Another platform is currently being developed at TU Delft by the Da Vinci Satellite project, an initiative led by a non-profit student team from Delft University of Technology. This team is participating in the ESA program ”Fly Your Satellite.” In this case, the satellite follows a CubeSat format, with its attitude designed to remain fixed, ensuring that the bottom face consistently points towards Nadir. Another important limitation factor for the design of the retroreflector array is the launch case which will be utilized during the launch of the satellite. The casing for Delfi-Twin is a standardized case suggested by ESA, which consists of a 3P casing where the satellite is locked inside through rails present at all four corners of the cube. Thus, this case limits the amount of space the array could take on the outside of the main body of the satellite. Figure 2.19 shows the 3P casing configuration and its dimensions in 2.5. TU Delft Satellite Platforms 27 mm, while Figure 2.20 displays the protrusion allowances on each side of the PocketQube within the case. From the drawings, the following ESA requirements are set : SYS-PHY-06: Components and parts may be installed on the +X, -X, +Y, +Z, -Z surfaces of the baseline PocketQube configuration, provided that they do not protrude more than 7 mm from the main body baseline envelope, and are separated more than 3.4 mm from the +Y surface of the sliding plate. SYS-PHY-07: Components and parts may be installed on the -Y surface of the baseline Pock- etQube configuration, provided that they do not protrude more than 7 mm from the baseline envelope, and are separated more than 3 mm from the +X, -X, +Z, -Z edges of the sliding plate. The case to be implemented by the Da Vinci satellite is also recommended by ESA. From the docu- mentation of the ”Fly Your Satellite” program, the following requirement is retrieved : CDS-2.2.5: Rails shall have a minimum width of 8.5mm measured from the edge of the rail to the first protrusion on each face. Figure 2.19: 3P PocketQube case baseline configuration. Figure 2.20: PocketQube protrusions envelope allowance.  3 Methodology This chapter outlines the methodology used to conduct the simulations and tests throughout this thesis. It provides the framework that integrates the rationale behind the simulations with the underlying re- quirements in relation to the research questions and objectives. The high-level overview of the thesis, its building blocks, key steps, and methods employed are discussed. Firstly, in section 3.1 the top-level requirements set for the design of the system are listed and a rationale behind each requirement is provided. Secondly, in section 3.2, the methodical approach implemented in the thesis to answer the research objectives is discussed describing the main steps. 3.1. Requirements In this section, the top-level requirements of the analyzed system are listed. These serve as a baseline for the simulations performed during the thesis. REQ-SYS-1: The retroreflector array shall fit on the new TU Delft platforms. Rationale: The thesis aims to design a retroreflector array to be integrated on the new TU Delft platforms, as defined by RO-1 and RO-2. Verification: Inspection, analysis. – REQ-SYS-1.1: The retroreflector array shall fit on Delfi-Twin. Rationale: The new satellite being developed by the Department of Space Engineering (DSE) requires the implementation of an array of retroreflectors to achieve ranging mea- surements. Verification: Inspection, analysis. * REQ-SYS-1.1.1: The retroreflector array shall consist of a maximum of 8 retroreflectors. Rationale: Due to budget and size constraints, the DSE set a limit on the maximum number of retroreflectors within the array. Verification: Inspection, analysis. * REQ-SYS-1.1.2: Components and parts may be installed on the +X, -X, +Y, +Z, -Z surfaces of the baseline PocketQube configuration, provided that they do not protrude more than 7 mm from the main body baseline envelope, and are separated more than 3.4 mm from the +Y surface of the sliding plate. Rationale: This requirement comes from the ESA PocketCube recommendations for the use of the PocketCube launch case, as shown in Figure 2.20. Verification: Inspection, analysis. * REQ-SYS-1.1.3: Components and parts may be installed on the -Y surface of the base- line PocketQube configuration, provided that they do not protrude more than 7 mm from the baseline envelope, and are separated more than 3 mm from the +X, -X, +Z, -Z edges of the sliding plate. Rationale: This requirement comes from the ESA PocketCube recommendations for the 28 3.1. Requirements 29 use of the PocketCube launch case, as shown in Figure 2.20. Verification: Inspection, analysis. * REQ-SYS-1.1.4: The retroreflector array shall accommodate a satellite rolling rate not higher than 10 deg/s. Rationale: Delfi-PQ had an attitude control system that allowed the satellite to freely roll. The maximum recorded rolling rate of Delfi-PQ was 10 deg/s, and the same is expected for Delfi-Twin. Therefore, the retroreflector array is sized using an upper boundary for the rolling rate of 10 deg/s. Verification: Test, analysis. * REQ-SYS-1.1.5: The retroreflector array shall accommodate a satellite rolling rate not lower than 5 deg/s.