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Deep Space CubeSat Constellation System 1 2

CubeSats provide a low-cost alternative to static landers and mobile 3 platforms in deep space, which allows for greater launch accessibility and 4 scientific return. Their use in constellation systems allows for complete 5 coverage over a planet or moon, which helps to reduce costs while 6 performing large-scale research for supporting missions. This report shows 7 the design process of a sample deep space CubeSat constellation system, 8 starting with developing subsystem requirements and constraints using 9 existing data from other constellations. Results of design optimization and 10 simulations are shown using set design parameters. Calculations performed 11 in this report can be replicated for other deep space CubeSat constellation 12 designs. 13 14 Keywords: cubesat, constellation, deep-space, spacecraft, 15 nanosatellite 16 17 18

Introduction 19 20 Until recently, two types of spacecraft have been considered for deep 21

space missions: static landers and mobile platforms. Now nanosatellite 22 technology, especially CubeSats, provide a low-cost alternative through greater 23 launch accessibility and scientific return. The ability to carry small instruments 24 through multiple small, disposable spacecraft, as well as access to a variety of 25 terrains, make CubeSats ideal for such missions (Klesh & Castillo-Rogez, 26 2012). 27

The same applies for CubeSat constellation systems, as well as the need 28 for zonal or global coverage of a body such as the Earth. Traditional 29 constellation design methods incorporate continuous or discontinuous coverage 30 of a specific geographic region (Ulybyshev, 2008). Usually, the objectives 31 behind using these systems include providing connectivity across a specific 32 region or performing large-scale scientific research in weather patterns. 33

Though Earth-orbiting constellation systems are becoming more common, 34 there are currently no existing deep space constellation systems. However, 35 deep space nanosatellites are currently in development and are capable of 36 performing such high-risk and high-science return missions (Klesh & Castillo-37 Rogez, 2012). These missions can include communicating with rovers or other 38 larger spacecraft on other planets or moons. However, these types of missions 39 still do not cover as much area over a planet or moon due to the various 40 spacecraft constraints such as size or speed. 41

Using CubeSat constellation systems in deep space missions can help 42 cover larger regions of planets or moons. One CubeSat would act as a mother 43 ship, while the others would act as daughter ships and cover different areas 44 around the planet or moon (Wong, Kegege, Schaire, Bussey, & Altunc, 2016). 45 The more satellites used in the system, the more the propellent mass in each 46 satellite can be reduced, thus reducing the cost-to-orbit (Singh et al., 2020). 47

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The objective of this project is to develop a constellation of CubeSats to 1 augment a deep space mission, such as a mission to the outer solar system. 2 3 4 Literature Review 5

6 This section will cover information on small satellites, including CubeSats, 7

and their deep space missions. Additionally, it will cover various types of 8 constellation systems and their respective missions. 9 10 Small Satellites Overview 11

12 Small satellites (smallsats) in general are classified as satellites with mass 13

less than 180 kg, or the size of a large kitchen fridge. Table 1 shows the five 14 main classes of smallsats: 15

16 Table 1. Classes of small satellites 17 Class Mass (kg) Minisatellite 100 – 180 Microsatellite 10 – 100 Nanosatellite 1 – 10 Picosatellite 0.01 – 1 Femtosatellite 0.001 – 0.01

Source: “What are SmallSats and CubeSats?” 2015. 18 19 One example of a smallsat mission is Janus SIMPLEx, a Lockheed Martin 20

mission which will use a dual small deep space spacecraft system to visit near-21 earth binary asteroids (“Small Spacecraft, Big Universe: Lockheed Martin 22 Selected For The Next Phase Of A Small Spacecraft Mission,” 2019). The 23 system consists of two small satellites weighing around 40 kg each and 24 carrying visible and infrared cameras. One key mission requirement is to meet 25 the planetary launch window, since it is a deep space mission. Some subsystem 26 requirements are that their power systems must handle a range of Sun distances 27 and that their telecommunications systems need to be able to transmit over long 28 distances and be compatible with the Deep Space Network (“Small Spacecraft, 29 Big Universe: Lockheed Martin Selected For The Next Phase Of A Small 30 Spacecraft Mission,” 2019). 31

Nanosatellites are used for extremely low-cost missions and can be used to 32 support larger missions to increase science return. Rather than using static 33 landers or mobile platforms, they allow for an alternative approach for in-situ 34 and close-proximity observations in a variety of terrains through a distribution 35 of multiple small, disposable spacecraft across the surface. Until recently, they 36 were only used in low-Earth orbit missions but can be used as instruments part 37 of larger missions, particularly as interplanetary CubeSats (Klesh & Castillo-38 Rogez, 2012). 39

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CubeSats Overview 1 2 CubeSats are especially popular because their fixed design makes 3

launching easier and allows for a viable market for common components. They 4 also have a low cost of entry into development and a low-cost of failure. One 5 standard form factor is 1U, or a 10x10x10 cm cube weighing 1 or 2 kg. Table 2 6 shows the most common factors along with their mass, volume (before 7 deployment), and power parameters: 8 9 Table 2. Common form factors for CubeSats. 10 Standard Form Factor

Mass (kg)

Volume (cm3)

Power, fixed (W)

Power, deployed (W)

1U 1.5 10x10x10 ~3 - 3U 4.5 10x10x30 ~8 ~25 6U 10 10x20x30 ~14 ~35

Sources: Hunter & Korsmeyer 2015, Klesh & Castillo-Rogez 2012. 11 12 Typical CubeSat characteristics include low mass or inertia and low risk to 13

the primary spacecraft. CubeSats can support science experiments, such as 14 imaging and remote sensing and direct surface sampling, and they are often 15 driven by science applications. Because of these characteristics, they can 16 enable both alternatives and augmentations to larger missions to low Earth 17 orbit, geostationary orbit, and deep space. Other parameters for CubeSat design 18 include the following (Klesh & Castillo-Rogez, 2012): 19

20 • Solar Arrays (fixed and/or deployable) 21 • Battery 22 • Antenna (monopole/dipole, turnstile, patch, 1U dish) 23 • Communications (UHF/VHF, S-Band) 24 • Data Rates 25 • Attitude Control (passive, torquer, drag control, etc.) 26 • Command & Data Handling (software) 27 • Propulsion 28 • Deployable Instruments (antenna, panels, tethers, boom, solar sail) 29 • Demonstrated Lifetime 30 • Payload Instruments (camera, telescopes, etc.) 31 32 The parameters listed above depend on the size of the CubeSat. For 33

example, a 1U CubeSat does not have a propulsion system, but a 3U or 6U 34 CubeSat has a cold gas thruster, electric propulsion, and solar sail (Klesh & 35 Castillo-Rogez, 2012). If the CubeSat is larger, then it has greater performance 36 and can hold more instruments in its payload. A larger CubeSat can also carry 37 a higher-powered battery and antenna. 38

However, CubeSats still have low technological readiness level (TRL) in 39 deep space (Klesh & Castillo-Rogez, 2012; Nieto-Peroy & Emami, 2019). Key 40

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technological developments are being made to support interplanetary missions, 1 such as radiation-tolerant operation strategies, deep-space deployment, and 2 nanosat surface mobility. 3 4 Deep Space CubeSat Mission Example 5

The NASA probe InSight traveled with two CubeSats, together called 6 Mars Cube One (MarCO), in 2018. These CubeSats, each with a mass of 13.5 7 kg, represented the first mission to test CubeSats in deep space. The purpose 8 was to serve as communications relays during InSight’s landing and to send 9 data from the probe to the Mars Reconnaissance orbiter (Scharf, 2018). 10

The CubeSats carried two main scientific instruments: UHF Radio 11 Receiver and X Band Radio Transmitter. Additionally, the CubeSats used a 12 “reflectarray,” a flat antenna patterned to mimic a parabolic dish to help focus 13 transmissions towards Earth (“MarCO (Mars Cube One),” 2019; Scharf, 2018). 14 Critical spare parts, such as experimental radios, antennas, and propulsion 15 systems, will be used in future deep space CubeSat missions (“MarCO (Mars 16 Cube One),” 2019). 17 18 Constellation Systems Overview 19 20

Satellite constellation systems are used for communication, navigation, 21 and remote sensing applications (Ulybyshev, 2008). Communication and 22 navigation applications include Internet connectivity services and Global 23 Positioning Systems (GPS), respectively. Remote sensing applications include 24 research-quality science in systems such as the following: 25

26 • Space Technology 5 (ST5) Project 27 • the NASA Time-Resolved Observations of Precipitation Structure and 28

Storm Intensity with a Constellation of Smallsats (TROPICS) mission 29 • NASA Hyperspectral CubeSat Constellation 30 31 These systems are ideal for objectives that require coverage for a large 32

geographic region, since they consist of more than one satellites working 33 together as a system. Of the three listed above, ST5 and TROPICS are actual 34 missions; as of 2015, the hyperspectral CubeSat constellation is still a concept 35 presented by NASA scientists at a conference. 36 37 ST5 Project Overview 38

The ST5 Project, launched in March of 2006 and decommissioned after 90 39 days, was a constellation of three microsatellites in a “string of pearls” 40 orientation. Each spacecraft was around 25 kg in mass and 53 cm in diameter 41 by 48 cm in height, and included the subsystems and components shown in 42 Figure 1 below: 43 44 45

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Figure 1. System decomposition of ST5 Project 1

2 Source: Carlisle & Webb 2007 3

4 Several components, such as the nutation damper from the guidance, 5

navigation, and control (GN&C) subsystem and cold gas micro-thruster from 6 the propulsion subsystem, are connected to those in other subsystems through 7 interdisciplinary coupling. For example, both the nutation damper and cold gas 8 micro-thruster are used for transportation purposes. The power subsystem, 9 which operates under low voltage, is responsible for powering all of the 10 components and other subsystems in each of the satellites; therefore, it is the 11 most important subsystem in this constellation system (Carlisle & Webb, 12 2007). 13

The objective of the ST5 mission was to show that a constellation of 14 microsatellites could perform research-quality science. In this case, the 15 spacecraft observed changes in the Earth’s magnetic field, and in doing so, it 16 met its objective. Instruments such as the battery, magnetometer, and ground 17 system components were used in future missions. 18 19 NASA TROPICS Overview 20

The NASA TROPICS mission is a constellation system of six 3U 21 CubeSats in three different low-Earth orbital planes (Braun et al., 2018). Each 22 CubeSat included the subsystems and components shown in Figure 2 below: 23 24 25

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Figure 2. System decomposition of NASA TROPICS mission. 1

2 Source: “Time-Resolved Observations of Precipitation Structure and Storm Intensity with a 3 Constellation of Smallsats” 2015 4

5 The CubeSat bus houses all the other components within the unit, save for 6

the solar panels. The motor controller and the instruments used for the GN&C 7 subsystem help guide each CubeSat to gather data over a specific area using 8 the radiometer payload cube (Braun et al., 2018). Therefore, in this case, the 9 most important subsystem in this constellation system is the GN&C subsystem. 10

The main goal of the TROPICS mission is to provide weather observations 11 of 3D atmospheric temperature and humidity, cloud ice and precipitation 12 horizontal structure, and storm intensity (Braun et al., 2018). These 13 observations will help analyze relationships between the different types of data 14 and various weather patterns to provide more insight into storm processes such 15 as hurricanes (“Time-Resolved Observations of Precipitation Structure and 16 Storm Intensity with a Constellation of Smallsats,” 2015). Currently, the 17 mission is under development. 18 19 NASA Hyperspectral CubeSat Constellation Overview 20

The main goal for a hyperspectral CubeSat constellation is to provide daily 21 or diurnal coverage at any point on the Earth cost-effectively (Mandl et al., 22 2015). This concept was developed by team members of the Earth Observing 1 23 (EO-1) mission, which carried an imaging spectrometer to capture images of 24 natural hazards for more than 15 years. Eventually, they needed a way to take 25 the images daily, which proved impossible using EO-1. 26

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The constellation system has two possible configurations. The first one 1 would contain three CubeSats at one time, angled at 30 degrees each. The other 2 would have three sets of 15 CubeSats observing select spots on any part of the 3 Earth every 46 minutes. Each CubeSat would include the subsystems and off-4 the-shelf components shown in Figure 3 below: 5 6 Figure 3. System decomposition of NASA hyperspectral mission 7

8 Source: Mandl et al. 2015 9

10 The CubeSat system presented in Figure 3 is similar to the two shown in 11

Figures 1 and 2. All three contain a battery and EPS as part of its power 12 subsystem, and all three have similar structural components (bus/box and solar 13 panels). The performance of the nano-hyperspectrometer depends on the 14 location of the CubeSats and data relay. Therefore, the two most important 15 subsystems in this concept are communications and GN&C. 16

Components for the propulsion subsystem were not included in the 17 conceptual design. However, each CubeSat could have a cold gas thruster due 18 to its 6U size. 19

20 Constellation System Design and Coverage 21

There are two main types of constellation design methods: continuous and 22 discontinuous global coverage. To design a constellation system, the following 23 assumptions are made (Ulybyshev, 2008): 24

25 1. The Earth is considered a round body. 26 2. All satellites in a constellation will be at the same altitude with the 27

same number of satellites in each orbit plane. 28 3. All orbit planes in a constellation will have the same orbit inclination. 29

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In simple continuous global coverage, at least one satellite is visible above 1 a minimum elevation angle at every point on the Earth’s surface. Complex 2 continuous coverage can either be full or partial region coverage, useful for 3 telecommunications and navigation purposes. Theoretical designs such as the 4 Draim or Walker constellations can achieve ideal continuous global coverage, 5 which use four satellites and circular orbits, respectively (Singh et al., 2020; 6 Ulybyshev, 2008). The Draim constellation design, especially, can incorporate 7 perturbations only if there is a common period and similar inclination between 8 the satellites. 9

On the other hand, simple discontinuous coverage implies that every point 10 on the surface is viewed with a predefined revisit time, which ranges from a 11 few minutes to a few hours. Similarly, complex discontinuous coverage can 12 either be full or partial region coverage with revisit times. These types of 13 constellation systems have become more prevalent only within the last decade, 14 but they have proved to be especially useful for remote sensing applications or 15 weather observations (Sarno, Graziano, & D’Errico, 2016; Ulybyshev, 2008). 16 17 18 Methodology 19 20

The methodology is as follows: 21 22

• Determine mission and subsystem requirements based on literature 23 review presented, analysis, and interdisciplinary coupling between 24 subsystems 25

• Develop high level designs of the top five subsystems based on the 26 requirements already determined 27

• Perform design of experiments to identify key parameters and 28 performance drivers 29

• Develop simulations of the constellation system 30 31 Design of CubeSat Subsystems 32 33

Based on system decompositions presented earlier, the common 34 subsystems are as follows: 35

36 • Power 37 • Communications 38 • GN&C 39 • Structures 40 • Payload 41 42 Therefore, a deep space CubeSat constellation should include them. 43

Coincidentally, those subsystems are all the ones listed for the two CubeSat 44 constellation systems; the ST5 system also contained the propulsion and 45

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thermal subsystems. Table 3 shows the different components for each 1 subsystem in each CubeSat constellation presented earlier: 2 3 Table 3. Comparison of subsystems in different CubeSat constellations. 4

Constellation System Components Subsystem ST5 System NASA Tropics NASA Hyperspectral

Power

Battery Battery Battery

Solar array CULPRiT chip EPS (not including

battery) EPS (not including

battery) Electronics card

Communications

X-band antenna UHF monopole antenna S-band radio

X-band transponder

RF reflector antenna system CHREC space

processor Amplifer Radio

GN&C

Nutation damper Avionics interface board GPS receiver

Mini sun spinning sensor

ADCS reaction wheel system Attitude determination

system ADCS interface board

Structures Satellite bus 2U bus 6U box Solar array Solar panels Solar panels

Payload

Mini magnetometer Radiometer payload

cube Nano-

hyperspectrometer Deployment boom Other instruments

5 The subsystem components, especially the payload instruments, depend on 6

the scope of the mission. Since the ST5 system comprises microsatellites, each 7 satellite contains far more components in each subsystem than a CubeSat. 8 However, there are some similarities between the components used in the three 9 different systems. For example, all three contain the following: 10

11 • Battery 12 • Satellite bus/box 13 • Solar array 14 15 Additionally, the communications and GN&C subsystems had similar 16

functions, even though the components themselves were different. The NASA 17 Tropics and Hyperspectral systems especially have very similar components in 18 their GN&C subsystem, such as an attitude determination system. 19

Nevertheless, these are not part of deep space missions. Since deep space 20 CubeSats require more instruments and additional subsystems, their size would 21 be at least 6U. Therefore, based on the information presented in the literature 22

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review, one example of a deep space CubeSat system decomposition can be as 1 shown in Figure 4. 2 3 Figure 4. System decomposition for deep space CubeSats 4

5 6 The system decomposition in Figure 4 is similar to those shown earlier, 7

save for their payload subsystems. Therefore, it shall be used as the system 8 decomposition for this mission. The N2 diagram shown in Figure 5 shows the 9 relationship between the subsystems. 10 11 Figure 5. N2 diagram of CubeSat constellation system 12

13

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Figure 5 above shows the inputs and outputs for the subsystems. For 1 example, one input for the structures subsystem is the power in daylight, which 2 is an output from the power subsystem. 3 4 Power Subsystem 5

The power subsystem contains the battery and solar arrays. The solar 6 arrays provide power generation, and the battery provides power storage. 7 Together, along with wires for power conversion and distribution, they form an 8 EPS. 9 10 Power Generation from Solar Arrays 11

12 There are two main types of solar arrays: deployed and fixed. The power 13

generation from the solar panels depends on the amount of sunlight received. 14 Equation 1 is used to determine the amount of power needed from the solar 15 array during daylight to power the spacecraft in orbit. Assume that battery 16 charging losses are excluded. 17

18

𝑃𝑃𝑠𝑠𝑠𝑠 =𝑃𝑃𝑒𝑒𝑇𝑇𝑒𝑒𝑋𝑋𝑒𝑒

+𝑃𝑃𝑑𝑑𝑇𝑇𝑑𝑑𝑋𝑋𝑑𝑑

𝑇𝑇𝑑𝑑 (1) 19

20 Inputs: Pe = Spacecraft’s power requirement during eclipse, 21 Pd = Spacecraft’s power requirement during daylight, 22 Te = Period length of eclipse orbit, 23 Td = Period length of daylight orbit, 24 Xe = Transmission efficiency during eclipse, 25 Xd = Transmission efficiency during daylight 26 Output: Psa = Power needed from solar array during daylight 27 28

The efficiencies Xe and Xd depend on the type of power regulation, shown 29 in Table 4. 30 31 Table 4. Transmission Efficiencies for Different Types of Power Regulation 32

Type of Power Regulation Transmission Efficiencies Direct Energy Transfer Peak-Power Tracking Xe 0.65 0.6 Xd 0.85 0.8

Source: Space Mission Analysis and Design 1999. 33 34 Efficiencies corresponding to direct energy transfer are larger because 35

peak-power tracking requires a power converter between the solar arrays and 36 the loads. 37

The beginning-of-life power per unit area PBOL is calculated using 38 Equation 2. 39

40

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𝑃𝑃𝐵𝐵𝐵𝐵𝐵𝐵 = 𝑃𝑃𝐵𝐵𝐼𝐼𝑑𝑑𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃) (2) 1 2 Inputs: PO = Ideal solar cell performance, 3 Id = Inherent degradation, 0.77 (nominal), 4 θ = Sun incidence angle, 23.5° for worst case, 90° for maximum power 5 Output: PBOL = Beginning-of-life power per unit area 6

The ideal solar cell performance PO depends on the type of solar cell used. 7 8 By multiplying the solar flux with the efficiency value corresponding to 9

the desired cell type in Table 5, the value for PO can be found. 10 11 Table 5. Efficiencies for different types of solar cells 12 Cell Type Efficiencies Silicon 0.148 Gallium Arsenide (GaAs) 0.185 Indium Phosphide 0.18 Multijunction GaInP/GaAs 0.22

Source: Space Mission Analysis and Design 1999. 13 14

The life degradation of the solar array Ld can be found using Equation 3. 15 16

𝐿𝐿𝑑𝑑 = (1 − 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑐𝑐𝑑𝑑)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑙𝑙𝑠𝑠 (3) 17 18 Inputs: Degradation = solar cell degradation/year, 0.375 for silicon, 0.275 for 19 GaAs, 20 Satellite life = usually length of the mission 21 Output: Ld = life degradation of the solar array 22 23

The end-of-life power per unit area PEOL is calculated with Equation 4. 24 𝑃𝑃𝐸𝐸𝐵𝐵𝐵𝐵 = 𝑃𝑃𝐵𝐵𝐵𝐵𝐵𝐵𝐿𝐿𝑑𝑑 (4) 25

Inputs: PBOL = Beginning-of-life power per unit area, 26 Ld = life degradation of the solar array 27 Output: PEOL = End-of-life power per unit area 28 29 Power Storage from Batteries 30

31 There are two types of battery cells: primary and secondary. Primary 32

batteries are useful for short missions and tasks that use very little power. 33 Secondary batteries are useful for long-term missions and can provide power 34 during eclipse periods. Therefore, the CubeSats in this project’s constellation 35 system will use secondary batteries. The required battery capacity Cr for each 36 spacecraft can be estimated using Equation 5. 37

38 𝐶𝐶𝑟𝑟 = 𝑃𝑃𝑒𝑒𝜏𝜏𝑒𝑒

(𝐷𝐷𝐵𝐵𝐷𝐷)𝑁𝑁𝑁𝑁 (5) 39

40

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Inputs: Pe = Spacecraft’s power requirement during eclipse, 1 τe = Eclipse duration 2 DOD = depth of discharge, ranges from .20 to .80 3 N = number of batteries, between 2 and 5 4 n = transmission efficiency of the battery 5 Output: Cr = required battery capacity 6

7 The mass of the batteries can be calculated by dividing the battery capacity 8

Cr by the specific energy density. Table 6 shows the specific energy densities 9 for the different types of secondary battery couples. 10 11 Table 6. Specific Energy Densities for Different Types of Secondary Batteries 12

Secondary Battery Type Specific Energy Density (W-hr/kg)

Nickel-Cadmium (NiCd) 25 - 30 Nickel-Hydrogen, individual pressure vessel design (NiH) 35 - 43 Nickel-Hydrogen, common pressure vessel design (NiH) 40 - 56 Nickel-Hydrogen, common pressure vessel design (NiH) 43 - 57 Lithium-Ion (Li-Ion) 70 - 110 Sodium-Sulfur (NaS) 140 - 210

Source: Space Mission Analysis and Design 1999. 13 14 Communications Subsystem 15

The communications subsystem primarily includes an antenna system 16 (antenna, transmitter, and receiver), as well as software to transmit data 17 between the CubeSats and to and from Earth. The transmitter sends downlink 18 signals to the other satellites and the ground station. The receiver gets uplink 19 signals from other satellites and the ground station. 20

The amount of data can be calculated using Equation 6. 21 22

𝐷𝐷𝐷𝐷 = 𝐷𝐷𝐷𝐷×(𝐹𝐹𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚−𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑚𝑚𝑖𝑖𝑒𝑒)𝑀𝑀

(6) 23 Inputs: DR = data rate, 24 F = fractional reduction in viewing time, greater than 0.5 25 Tmax = maximum time in view, 26 Tinitiate = time required to initiate a communications pass, 2 minutes 27 M = margin needed to account for missed passes, estimated at 2 to 3 28 Output: DQ = quantity of data 29 Equation 7 can be used to size a data link. 30 31

𝐸𝐸𝑏𝑏𝑁𝑁𝑜𝑜

= 10 𝑙𝑙𝑐𝑐𝑑𝑑 𝑃𝑃𝐵𝐵𝑙𝑙𝐺𝐺𝑖𝑖𝐵𝐵𝑠𝑠𝐵𝐵𝑚𝑚𝐺𝐺𝑟𝑟𝑘𝑘𝑇𝑇𝑠𝑠(𝐷𝐷𝐷𝐷) (7) 32

33 Inputs: P = transmitter power, 34

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Ll = transmitter-to-antenna line loss, 1 Gt = transmit antenna gain, 2 Ls = space loss, 3 La = transmission path loss, 4 Gr = receive antenna gain, 5 k = Boltzmann’s constant, 1.380 x 10-23 J/K, 6 Ts = system noise temperature, 7 DR = data rate 8 9 Output: Eb/No = ratio of received energy-per-bit to noise density, usually 50-10 100 11

12 The expression PLlGt can also be expressed as the effective isotropic 13

radiated power EIRP, which is related to the power flux density Wf. For an S-14 band antenna system, Wf is -137/4 kHz with the following uplink and downlink 15 frequency ranges (Larson & Wertz, 1999): 16

17 • Uplink: 2.65 – 2.69 GHz 18 • Downlink: 2.5 – 2.54 GHz 19 20 Likewise, for an X-band antenna system, Wf is -142/4 kHz with the 21

following uplink and downlink frequency ranges (Larson & Wertz, 1999): 22 23 • Uplink: 7.9 – 8.4 GHz 24 • Downlink: 7.25 – 7.75 GHz 25 26 For all of the uplink and downlink frequencies shown above, the system 27

noise temperatures Ts are as follows (Larson & Wertz, 1999): 28 29 • Uplink: 614 K 30 • Downlink: 135 K 31 Also, based on the frequencies, the transmission path loss La is -0.3 dB 32

(Larson & Wertz, 1999). 33 The space loss Ls, antenna pointing loss Lθ, and antenna half-power 34

beamwidth θ can be found using Equations 8, 9, and 10. 35 36

𝐿𝐿𝑠𝑠 = 20 𝑙𝑙𝑐𝑐𝑑𝑑 𝑐𝑐4𝜋𝜋𝜋𝜋𝑙𝑙

(8) 37

𝐿𝐿𝜃𝜃 = −12 𝑠𝑠𝜃𝜃2 (9) 38

𝜃𝜃 = 21𝑙𝑙𝐺𝐺𝐺𝐺𝐺𝐺𝐷𝐷

(10) 39 40 Inputs: c = speed of light, 3 x 108 m/s 41 S = path length, 42

f = frequency in GHz, 43 e = pointing error, 44

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D = antenna diameter 1 Outputs: Ls = space loss, 2 Lθ = antenna pointing loss, 3

θ = antenna half-power beamwidth 4 5 GN&C Subsystem 6

The GN&C subsystem mainly contains an ADCS system and a navigation 7 system. The ADCS system contains sensors and actuators such as reaction 8 wheels or torquers. The navigation system, similar to the communications 9 subsystem, includes a receiver and transmitter. This section will focus more on 10 the ADCS system design. 11

Each spacecraft will use a spin stabilization control type; the entire 12 spacecraft rotates while the angular momentum vector remains fixed in space. 13 Reaction wheels in each ADCS system will act as the actuators to adjust 14 orientation. They spin in either direction, while providing one axis of control 15 for each wheel. The following shows the typical specifications for reaction 16 wheels (Larson & Wertz, 1999). 17

18 • Weight: 2 – 20 kg 19 • Power: 10 – 110 W 20 • Performance Range: 0.01 to 1 N-m (max torque) 21 22 The sizing of the reaction wheels can be determined in two different ways. 23

One way is to account for disturbance rejection in Equation 11. 24 𝑇𝑇𝐷𝐷𝑅𝑅 = 𝑇𝑇𝐷𝐷(𝑀𝑀𝑀𝑀) (11) 25

Inputs: TD = worst-case disturbance torque, 26 MF = margin factor 27 Output: TRW = reaction-wheel torque due to disturbance rejection 28

29 The disturbance torque TD is caused by gravity gradients, solar radiation, 30

magnetic fields, and aerodynamic disturbance. This is shown in Equation 12 31 (Larson & Wertz, 1999). 32

𝑇𝑇𝐷𝐷 = 𝑇𝑇𝐺𝐺 + 𝑇𝑇𝑠𝑠𝑠𝑠 + 𝑇𝑇𝑚𝑚 + 𝑇𝑇𝑠𝑠 (12) 33 Inputs: TG = maximum gravity torque, 34 Tsp = solar radiation torque, 35 Tm = magnetic torque, 36 Ta = aerodynamic torque 37 Output: TD = worst-case disturbance torque 38 Another way to size the reaction wheels is to find the slew torque Tslew using 39 Equation 13. 40

𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 4𝜃𝜃𝜃𝜃𝑠𝑠2

(13) 41 Inputs: θ = slewing angle, 30 degrees, 42 I = moment of inertia of CubeSat, 43 t = minimum maneuver time 44 Output: Tslew = slew torque 45

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The wheel momentum h can be estimated using Equation 14. 1 ℎ = 𝑇𝑇𝐷𝐷𝜏𝜏

4√2 (14) 2

Inputs: TD = disturbance torque, 3 τ = orbit period 4 Output: h = wheel momentum 5

6 Both Sun sensors and gyroscopes (inertial sensors) will also be used to 7

supplement the CubeSats’ navigation system. Sun sensors use visible light to 8 measure angles between their mounting base and incident sunlight. Gyroscopes 9 are used to measure the speed or angle of rotation from an initial reference. The 10 following shows characteristics for both (Larson & Wertz, 1999). 11

12 • Weight Range: 1 to 15 kg (gyro), 0.1 to 2 kg (Sun sensor) 13 • Power: 10 to 200 W (gyro), 0 to 3 (Sun sensor) 14 • Performance: Gyro drift rate 0.003 to 1 deg/hr, sun sensor accuracy 15

0.005 to 3 deg 16 17 Propulsion Subsystem 18

The main component in the propulsion subsystem is the cold gas thruster 19 due to its low cost and simplicity. The propellant (N2, NH3, Freon, He) 20 performance characteristics are as follows (Larson & Wertz, 1999): 21

22 • Vacuum specific impulse (Isp): 50-75 seconds 23 • Thrust range (F): 0.05-200 N 24 • Average bulk density: 0.28, 0.60, and 0.96 g/cm3 25 26 This subsystem would only maintain attitude control and orbit maintenance 27

and maneuvering. Therefore, staging will not be considered for this project. 28 Since a single thruster has such a low specific impulse, multiple will need to be 29 used. Figure 6 shows a schematic of a single cold gas thruster propulsion 30 system. 31 32 Figure 6. Schematic of Cold Gas Propulsion System 33

34 Source: Anis 2012. 35

36

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The propulsion system consists of a gas tank, gas feed line, valve, and 1 thruster. The gas tank is either cylindrical or spherical in shape. The thruster 2 itself is a convergent-divergent nozzle. Key assumptions for the design include: 3

4 • The flow is isentropic, quasi, and one-dimensional 5 • The propellant is an ideally, thermally, calorically perfect gas 6

7 8 Equation 15 is used to find the thrust generated by the system. 9

𝑀𝑀 = 𝑚𝑉𝑉𝑠𝑠 + 𝐴𝐴𝑠𝑠𝑃𝑃𝑠𝑠 (15) 10 Inputs: ṁ = propellant mass flow rate, 11

Ve = propellant exhaust velocity, 12 Ae = nozzle exit area, 13 Pe = gas pressure at nozzle exit 14

Output: F = thrust 15 The thrust is used to find the specific impulse, shown in Equation 16. 16

𝐼𝐼𝑠𝑠𝑠𝑠 = 𝐹𝐹𝑚𝑔𝑔

(16) 17 Inputs: F = thrust, 18

ṁ = propellant mass flow rate, 19 g = acceleration due to gravity, 9.81 m/s2 20

Output: Isp = specific impulse 21 The propellant exhaust velocity Ve can be found using Equation 17. 22

𝑉𝑉𝑠𝑠 = 2𝛾𝛾𝛾𝛾−1

𝐷𝐷𝐴𝐴𝑀𝑀𝑊𝑊

𝑇𝑇𝑐𝑐 1 − 𝑃𝑃𝑒𝑒𝑃𝑃𝑐𝑐𝛾𝛾−1𝛾𝛾 (17) 23

Inputs: γ = specific heat ratio for gas, 24 RA = universal gas constant, 8.314 J/K-mol, 25 MW = molecular weight of gas, 26 Tc = chamber temperature, 27 Pe = gas pressure at nozzle exit 28 Pc = chamber pressure 29

Output: Ve = propellant exhaust velocity 30 The nozzle’s area expansion ratio ε can be found using Equation 18. 31

𝜀𝜀 = 𝐴𝐴𝑒𝑒𝐴𝐴𝑖𝑖

(18) 32 Inputs: Ae = nozzle exit area, 33 At = nozzle throat area 34 Output: ε = area expansion ratio 35

36 The characteristic velocity c* and thrust coefficient cf can be found using 37

Equations 19 and 20. They are used to evaluate the propulsion system 38 performance. 39

𝑐𝑐∗ = 𝑃𝑃𝑐𝑐𝐴𝐴𝑖𝑖𝑚

(19) 40

𝑐𝑐𝑙𝑙 = 𝐹𝐹𝑃𝑃𝑐𝑐𝐴𝐴𝑖𝑖

(20) 41 Inputs: PC = chamber pressure, 42

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At = nozzle throat area, 1 ṁ = propellant mass flow rate 2 F = thrust, 3 Pc = chamber pressure, 4 At = nozzle throat area 5

Outputs: c* = characteristic velocity, 6 cf = thrust coefficient 7

8 Structures Subsystem 9

Basic analysis for the structures subsystem will mainly be done on the 10 CubeSat box, cold gas thruster tank, and solar arrays. 11 12 CubeSat Box 13

14 The CubeSat box will have a monocoque structure and will be made of 15

aluminum 7075-T73 sheets. This type of material is most often used for 16 spacecraft due to its easy machinability, ductility, low density, and high 17 strength versus weight. The following material properties will be useful in 18 conducting structural analysis for the main CubeSat box (Larson & Wertz, 19 1999). 20

21 • Young’s Modulus: E = 71 x 109 N/m2 22 • Poisson’s Ratio: ν = 0.33 23 • Density: ρ = 2.8 x 103 kg/m3 24 • Ultimate Tensile Strength: Ftu = 524 x 108 N/m2 25 • Yield Tensile Strength: Fty = 448 x 106 N/m2 26 27 Assuming uniform thickness, Equations 21 and 22 can be used to find the 28

axial frequency fnat,a and lateral frequency fnat,l. 29

𝑓𝑓𝑁𝑁𝑠𝑠𝑠𝑠,𝑠𝑠 = 0.250 𝐴𝐴𝐸𝐸𝑚𝑚𝐵𝐵𝐵𝐵

(21) 30

𝑓𝑓𝑁𝑁𝑠𝑠𝑠𝑠,𝑠𝑠 = 0.560 𝐸𝐸𝜃𝜃𝑚𝑚𝐵𝐵𝐵𝐵3

(22) 31

32 Inputs: A = cross-sectional area of CubeSat box, 33 E = Young’s Modulus, 34 mB = estimated distributed mass of CubeSat box, 35 L = length of CubeSat box, 36 I = area moment of inertia 37 Outputs: fnat,a = axial frequency, 38 fnat,l = lateral frequency 39

40 The estimated distributed mass of the CubeSat box mB includes the box 41

itself, as well as the components located inside. 42 The area moment of inertia for the CubeSat box can be found using Equation 43 23. 44

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𝐼𝐼 = 𝑏𝑏ℎ3

12 (23) 1

Inputs: b = base of cross section, 2 h = height of cross section 3 Output: I = area moment of inertia 4 5 The limit load, or the equivalent axial load Peq, can be found using Equation 6 24. 7

𝑃𝑃𝑠𝑠𝑒𝑒 = 𝑚𝑚𝐵𝐵𝑑𝑑(𝐿𝐿𝑀𝑀) + 4𝑀𝑀𝑏𝑏

(24) 8 Inputs: mB = estimated distributed mass of CubeSat box, 9 g = acceleration due to gravity, 9.81 m/s2, 10 LF = loading factor, 11 M = bending moment, 12 b = base of cross section 13 Output: Peq = equivalent axial load 14 15 The bending moment used in Equation 24 can be calculated using Equation 25. 16

𝑀𝑀 = 𝑚𝑚𝐵𝐵𝑑𝑑𝑔𝑔 (25) 17 18 Inputs: mB = estimated distributed mass of CubeSat box, 19 g = acceleration due to gravity, 9.81 m/s2, 20 y = moment arm, or distance from center of mass 21 Output: M = bending moment 22

23 The ultimate and yield equivalent axial loads (Peq,u and Peq,y, respectively) 24

can be found using their respective factors of safety FSu and FSy, shown in 25 Equation 26. 26

𝑃𝑃𝑠𝑠𝑒𝑒,𝑢𝑢 𝑐𝑐𝑑𝑑 𝑃𝑃𝑠𝑠𝑒𝑒,𝑦𝑦 = 𝑃𝑃𝑠𝑠𝑒𝑒 × (𝑀𝑀𝑆𝑆𝑢𝑢 𝑐𝑐𝑑𝑑 𝑀𝑀𝑆𝑆𝑦𝑦) (26) 27 Inputs: Peq = equivalent axial load, 28 FSu or FSy = ultimate or yield factor of safety 29 Output: Peq,u or Peq,y = ultimate or yield equivalent axial load 30

31 The factors of safety depend on the type of structure being designed. Since 32

the CubeSat box is being designed for strength and will be affected by 33 temperature changes during the mission, the factors of safety will be as follows 34 (Larson & Wertz, 1999): 35

36 • Ultimate factor of safety FSu = 1.25 37 • Yield factor of safety FSy = 1.1 38

39 The critical buckling load Pcr for the structure is calculated using Equation 27. 40

𝑃𝑃𝑐𝑐𝑟𝑟 = 𝜋𝜋2𝐸𝐸𝜃𝜃4𝐵𝐵2

(27) 41 Inputs: E = Young’s Modulus, 42 I = area moment of inertia, 43 L = length of CubeSat box 44

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Output: Pcr = critical buckling load 1 2 Both the ultimate equivalent axial load Peq and critical buckling load Pcr 3

are used in Equation 28 to determine the stability of the structure. 4 𝑀𝑀𝑆𝑆 = 𝑃𝑃𝑐𝑐𝑟𝑟

𝑃𝑃𝑒𝑒𝑒𝑒− 1 (28) 5

Inputs: Pcr = critical buckling load, 6 Peq = equivalent axial load 7 Output: MS = margin of safety 8

9 If the margin of safety MS is less than zero, then the structure is not 10

adequate for the mission. The inputs to calculate the applied loads will need to 11 be modified until the margin of safety MS is greater than or equal to zero. 12

The mass of the box’s shell mbox, not including fasteners or attachments, is 13 calculated using Equation 29. 14

𝑚𝑚𝑏𝑏𝑏𝑏𝑏𝑏 = 𝜌𝜌𝑉𝑉 (29) 15 Inputs: ρ = density of material, 16 V = volume of CubeSat box only 17 Output: mbox = mass of CubeSat box shell only 18 19 Gas Tank 20 21

The structural design of the tank depends on its shape. As mentioned 22 earlier, a tank for a cold gas thruster system is either cylindrical or spherical in 23 shape. Equation 30 is used to find the spherical tank stress. 24

𝜎𝜎 = 𝑃𝑃𝑟𝑟2𝑠𝑠

(30) 25 Inputs: P = maximum expected operating pressure, 26

r = radius, 27 t = thickness 28

Output: σ = allowable stress 29 30 Cylindrical pressure vessels depend on both the longitudinal and hoop 31

stresses. Equation 30 can be used to find the longitudinal stress of a cylindrical 32 pressure vessel, and Equation 31 can be used to find the hoop stress. 33

𝜎𝜎 = 𝑃𝑃𝑟𝑟𝑠𝑠

(31) 34 Inputs: P = maximum expected operating pressure, 35

r = radius, 36 t = thickness 37

Output: σ = allowable stress 38 39

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Solar Arrays 1 2 The size of the solar arrays can be found using values calculated for the 3

power subsystem. Equation 32 is used to calculate the solar-array area Asa 4 needed to support each spacecraft’s power requirement. 5

𝐴𝐴𝑠𝑠𝑠𝑠 = 𝑃𝑃𝑠𝑠𝑚𝑚𝑃𝑃𝐸𝐸𝐸𝐸𝐸𝐸

(32) 6 Inputs: Psa = Power needed from solar array during daylight, 7

PEOL = End-of-life power per unit area 8 Output: Asa = required solar-array area 9 The mass of the solar arrays can also be found using Equation 33. 10

𝑚𝑚𝑠𝑠 = 0.04𝑃𝑃𝑠𝑠𝑠𝑠 (33) 11 Input: Psa = Power needed from solar array during daylight 12 Output: ma = mass of solar arrays 13 14 Payload Subsystem 15

The payload subsystem is made up of different instruments and depends 16 on the type of mission and mission objective. Therefore, it drives the mission 17 requirements and specifications. For a deep space, remote sensing mission, the 18 instruments would be required to perform imaging, intensity measurement, and 19 topographic mapping. Therefore, the payload would include an imager/camera, 20 radiometer, and altimeter used for analysis in deep space (Larson & Wertz, 21 1999). 22

The data rate DR used as an input for the communications subsystem can 23 be calculated using Equation 34. 24

𝐷𝐷𝐷𝐷 = 𝑍𝑍𝐶𝐶𝑍𝑍𝐴𝐴𝐵𝐵 (34) 25 Inputs: ZC = Number of cross-track pixels, 26 ZA = Number of swaths in one second 27

B = Number of bits per pixel 28 Output: DR = data rate 29

The number of cross-track pixels ZC and number of swaths ZA can be 30 calculated using Equations 35 and 36. 31

𝑍𝑍𝐶𝐶 = 2𝜂𝜂𝜃𝜃𝐹𝐹𝐵𝐵𝐼𝐼

(35) 32

𝑍𝑍𝐴𝐴 = 𝐼𝐼𝑔𝑔×1 𝑠𝑠𝑠𝑠𝑐𝑐

𝜃𝜃𝐹𝐹𝐵𝐵𝐼𝐼×𝐻𝐻 𝜋𝜋180°

(36) 33

Inputs: η = maximum sensor look angle, 34 Vg = spacecraft ground-track velocity, 35 IFOV = Instantaneous field of view, 36 H = orbit altitude 37

Outputs: ZC = Number of cross-track pixels, 38 ZA = Number of swaths in one second 39

The instantaneous field of view IFOV can be calculated using Equation 37. 40 𝐼𝐼𝑀𝑀𝐼𝐼𝑉𝑉 = 𝑌𝑌𝑚𝑚𝑚𝑚𝑚𝑚

𝐷𝐷𝑠𝑠180°

𝜋𝜋 (37) 41

Inputs: Ymax = maximum along-track ground sampling distance, 42 Rs = slant range 43

Output: IFOV = Instantaneous field of view 44

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1 Orbital Mechanics and Constellation Design 2 3

In designing a constellation system, several key design factors need to be 4 considered: 5

6 • Number of satellites 7 • Constellation pattern 8 • Altitude 9 • Number of orbit planes 10 • Minimum elevation angle 11 • Inclination 12 • Eccentricity 13 • Spacing between satellites 14 15 Of those listed above, the three most important are the altitude, inclination, 16

and minimum elevation angle. The altitude and minimum elevation angle can 17 be assumed as design variables, while the inclination depends on either 18 analytical calculations or constellation pattern. 19

In designing the constellation structure, all assumptions listed in the 20 literature review will be considered. Therefore, the following will be true for 21 this project: 22

23 1. Since the planet or moon will be considered a round body, no oblate 24

body perturbations will be incorporated. However, other perturbations 25 such as solar radiation or finite burns may be incorporated depending 26 on the mission. 27

2. All satellites will be at the same altitude, and each orbit plane will 28 contain the same number of satellites. 29

3. All orbit planes will have the same inclination, either determined 30 through analytical calculations or the chosen constellation pattern. 31

32 Forces and Perturbations 33

To better simulate reality, the following outside forces or perturbations 34 must be considered in the simulations. 35

• Finite burns 36 • Atmospheric drag 37 • Solar radiation 38 39 Solar pressure and atmospheric drag can affect the values of the orbits’ 40

Keplerian elements. Finite burns, however, have low impact on orbital changes 41 but should still be considered (Curtis, 2014; Larson & Wertz, 1999). 42 43 Keplerian Elements 44

The satellites’ orbits are generally defined by the following Keplerian 45 elements (Curtis, 2014): 46

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Length of semi-major axis, a 1 • Eccentricity, e 2 • Argument of periapsis, ω 3 • True anomaly, θ 4 • Inclination, i 5 • Right ascension of ascending node, Ω 6 7 The latter four elements are illustrated in Figure 7. Note that the equatorial 8

plane shown below is of the Earth. 9 10

Figure 7. Orientation of Orbit Plane with respect to Equatorial Plane 11

12 Source: Curtis 2014. 13

14 For both types of coverage, continuous or discontinuous, the orbits can be 15

assumed as circular, due to its more widespread use than elliptical ones 16 (Ulybyshev, 2008). Therefore, a is equal to the radius of the orbit, which is the 17 sum of the altitude and the radius of the planet or moon. For circular orbits, e is 18 equal to zero. 19

θ and ω can be combined to obtain the argument of latitude u, which can 20 range from 0 to 360 degrees (“Basic Concepts of Manned Spacecraft Design: 21 Describing Orbits,” 2018). The orbital period τ, shown in Equation 38, can be 22 considered as another Keplerian element (Curtis, 2014). 23

𝜏𝜏 = 2𝜋𝜋𝑠𝑠3

𝐺𝐺𝑀𝑀 (38) 24

Inputs: a = length of semi-major axis, 25 G = gravitational constant, 6.67 x 10-20 km3/kg-s2, 26

M = mass of the planet or moon 27 Output: τ = orbital period 28

29

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The remaining two Keplerian elements depend on the type of constellation 1 structure. The inclination also depends on the angular momentum, which is 2 found using Equation 39. 3

ℎ = 𝑑𝑑 × 𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 (39) 4 Inputs: a = length of semi-major axis, 5

vsatellite = velocity of the satellite 6 Output: h = angular momentum of the satellite 7 8 Constellation Structure 9

The structure of a constellation system is symmetric; therefore, there 10 should be an equal number of satellites in each orbit plane. Usually, 11 constellations with one or two planes are more responsive to changes based on 12 user needs than those with more than two (Larson & Wertz, 1999). Therefore, 13 it is better to place more satellites in a smaller number of planes rather than 14 placing less satellites in a larger number of planes. Nevertheless, the structure 15 and number of constellation planes should allow for maximum coverage with 16 the smallest possible number of satellites. 17

There are two common types of constellation structures that use circular 18 orbit patterns: 19

20 • Streets of Coverage pattern 21 • Walker-Delta pattern 22

23 Figure 8 shows an example of a Streets of Coverage pattern. 24 25 Figure 8. Streets of Coverage Pattern, i = 90° 26

27 Source: Space Mission Analysis and Design 1999. 28

29 Planes in the Streets of Coverage pattern all have the same inclination at 30

90 degrees, since it is a polar constellation (Hu, Maral, & Ferro, 2002). One 31 main advantage is that most of the orbital planes in this pattern are evenly 32 spaced, so they have similar Ω values. The example shown in Figure 8 uses 33 five planes in the pattern, all intersecting at the poles. The image shows the 34 main disadvantage of this pattern; coverage is more concentrated in areas 35 closer to the poles than the equator (Hu et al., 2002; Larson & Wertz, 1999). 36

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Figure 9 shows an example of a Walker-Delta pattern. 1 2 Figure 9. Walker-Delta pattern, i = 65° 3

4 Source: Space Mission Analysis and Design 1999. 5

6 Planes in the Walker-Delta pattern all have the same inclination relative to 7

the equatorial plane, which allows for equally distributed coverage everywhere 8 over the surface. The example shown in Figure 9 uses 15 satellites in total, 9 distributed across five planes. Each plane is at an inclination of 65 degrees with 10 respect to the equator. 11

However, one main disadvantage of the Walker-Delta pattern is its design 12 constraints; both the orbital planes and satellites have to be equally spaced 13 from each other, which adds more difficulty in optimizing the design (Hu et al., 14 2002). Therefore, for maximum coverage and simplicity, the constellation 15 system for this project will use the Streets of Coverage pattern. Figure 10 16 shows a view of the example shown in Figure 8 from the north pole. 17 18 Figure 10. Streets of Coverage Pattern View from North Pole 19

20 Source: Space Mission Analysis and Design 1999. 21

22 Satellites traveling over half of the planet or moon travel northward, while 23

those over the other half travel southward. Equation 40 is used to calculate the 24 maximum perpendicular separation between two adjacent orbital planes going 25 in the same direction. 26

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𝐷𝐷𝑚𝑚𝑠𝑠𝑏𝑏𝑆𝑆 = 𝜆𝜆𝑠𝑠𝑠𝑠𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 + 𝜆𝜆𝑚𝑚𝑠𝑠𝑏𝑏 (40) 1 Inputs: λstreet = street of coverage angle, 2

λmax = maximum central angle 3 Output: DmaxS = maximum perpendicular separation between two adjacent 4

orbital planes going in the same direction 5 6 There is a seam between the two halves in Figure 10, which has much 7

narrower spacing than other areas in the pattern. The Ω values for those two 8 planes will be different from those of the other planes. Equation 41 is used to 9 calculate the maximum perpendicular separation between two adjacent orbital 10 planes going in opposite directions. 11

𝐷𝐷𝑚𝑚𝑠𝑠𝑏𝑏𝐸𝐸 = 2𝜆𝜆𝑠𝑠𝑠𝑠𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠 (41) 12 Input: λstreet = street of coverage angle 13 Output: DmaxO = maximum perpendicular separation between two adjacent 14

orbital planes going in opposite directions 15 16 The “street of coverage” is an area or swath with continuous coverage that 17

is centered on the ground track. The street of coverage angle, half of a swath, 18 depends on both the space between satellites in each plane and the maximum 19 central angle. This relationship is shown in Equation 42. 20

𝑐𝑐𝑐𝑐𝑐𝑐(𝜆𝜆𝑠𝑠𝑠𝑠𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠) = 𝑐𝑐𝑏𝑏𝑠𝑠(𝜆𝜆𝑚𝑚𝑚𝑚𝑚𝑚)

𝑐𝑐𝑏𝑏𝑠𝑠𝑆𝑆2 (42) 21

Inputs: λmax = maximum central angle, 22 S = space between satellites in each plane 23

Output: λstreet = street of coverage angle 24 25 The relationships between both the maximum central angle and space 26

between satellites help determine whether coverage is continuous or 27 discontinuous, which are as follows: 28

29 • If S < 2λmax, then coverage in the swath is continuous. 30 • If S > 2λmax, then coverage in the swath is discontinuous. 31 32 Using the number of satellites per plane NP, Equation 43 can be used to 33

find the revisit time trev for discontinuous coverage (Singh et al., 2020). 34 𝑑𝑑𝑟𝑟𝑠𝑠𝑟𝑟 = 𝜏𝜏

𝑁𝑁𝑃𝑃 (43) 35

Inputs: τ = orbital period, 36 NP = number of satellites per plane 37

Output: trev = revisit time 38 39 40 Results 41 42

The equations presented earlier will be used to determine key design 43 parameters, which will help determine the best design scenarios for the system. 44

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Additionally, the parameters of the constellation system will be used to 1 generate simulations in Systems Tool Kit (STK). Results from sample 2 calculations, which use Mars as the central body, will also be discussed. 3 4 Analysis of CubeSat Subsystems 5 6 Power Subsystem Analysis 7

The power subsystem depends on the spacecraft life, solar array 8 characteristics, and orbit characteristics. Each CubeSat can be assumed to have 9 a lifespan of two to four years to function in deep space. If the lifespan is 10 higher, then both Ld and PBOL are lower. Both Pe and Pd can be assumed to be a 11 fixed power of 14 W, taken from literature review. 12

For all solar array calculations, peak-power tracking was used for power 13 regulation. Therefore, Psa is higher than if direct energy transfer was used. Both 14 PBOL and PEOL depend on the type of solar cell used. GaAs solar cells have the 15 best efficiency and the lowest degradation, and they contribute to the best PBOL 16 and PEOL results. 17

The batteries used for the CubeSats in this project are based on those used 18 on MarCO. Li-Ion batteries are predominantly used on CubeSats today because 19 they are rechargeable, light, and have a high specific energy density. Each 20 CubeSat will contain two Li-Ion batteries with a total battery mass of around 21 0.7 kg. 22

The orbit characteristics depend on the central body. If the central body 23 has a smaller angular radius, then the eclipse duration is smaller. Additionally, 24 if the central body is farther from the Sun, then the solar flux is smaller. These 25 characteristics are determined concurrently with simulation results. 26

Table 7 shows results from the sample calculations along with 27 explanations for each. 28 29 Table 7. Power Subsystem Sample Results 30 Results Values Comments Psa 41.027 W Orbit around Mars, peak-power tracking PBOL 77.075 W Mars solar flux, GaAs solar cell type Ld 0.381 year CubeSat life of 3 years PEOL 29.372 W Cr 191.49 W-hr Battery capacity for each battery Mass of batteries 0.723 kg/battery Li-Ion battery type

31 Communications Subsystem Analysis 32

The communications subsystem indirectly depends on orbit characteristics, 33 since one of the design parameters is the payload’s data rate. The size and 34 performance in the communications instruments will be based on similar ones 35 in other CubeSat missions. 36

An X-band antenna system has higher frequency ranges for both uplink 37 and downlink, but those frequencies can contribute to a higher Lθ if there is a 38

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pointing error. However, this type of antenna system was used on MarCO, so it 1 will be used in this mission. 2

Table 8 shows the results using the desired antenna system and other 3 parameters. 4

5 Table 8. Communications Subsystem Sample Results 6 Results Values Comments DQ 27871800 bits DR = 31529 bps (see Section 4.1.6), F =

0.77, maximum time is 1 hour Eb/No (uplink) 225 dB Frequency = 7.9 GHz Eb/No (downlink) 231 dB Frequency = 7.25 GHz Lθ (uplink) -16982 dB D = 1 mm, pointing error of 0.0001 Lθ (downlink) -14303 dB

7 The Eb/No ratio for both uplink and downlink is more than two times 8

higher than the desired value, probably due to the data rate. Still, it is higher for 9 the downlink than the uplink. The downlink Lθ is less than that of the uplink. 10

11 GN&C Subsystem Analysis 12

The orbit characteristics, especially those of the central body, greatly 13 impact the sizing of the reaction wheels of the GN&C subsystem. As 14 mentioned earlier, their sizing is determined by TRW or Tslew. The wheel 15 momentum is found through TD and Tslew. 16

If the central body has little to no impact on the CubeSat’s magnetic or 17 aerodynamic torque, then Tslew should be used as a design parameter. 18 Otherwise, TRW must be compared with the maximum torque (0.01 to 1 N-m) 19 to determine the better design parameter. If TRW is less than the maximum 20 torque, then Tslew must be considered as the main design parameter for the 21 reaction wheel torque. 22

Table 9 shows TRW, Tslew, and the wheel momentum for a CubeSat orbiting 23 Mars. 24

25 Table 9. GN&C Subsystem Sample Results 26 Results Values Comments

TRW 1.098 x 10-6 N-m Mainly influenced by solar radiation and aerodynamic torques

Tslew 1.396 x 10-6 N-m Min maneuver time of 5 seconds Wheel momentum 0.001433 N-m-s

27 As seen in the table above, TRW is less than the maximum torque and Tslew. 28

Therefore, Tslew should be the main design parameter for the reaction wheel 29 torque. Additionally, the wheel momentum is very small; therefore, the mass of 30 the reaction wheels should be less than 2 kg. 31

32

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Propulsion Subsystem Analysis 1 The performance of the propulsion system depends on the propellant used. 2

Therefore, the major design variables are the thrust, specific impulse, 3 molecular weight, and specific heat ratio. Out of the four propellants 4 mentioned, N2 and NH3 produce the best results because of their light weight 5 and lower specific heat ratio. NH3 has the best of the two, since it produces the 6 highest exhaust velocity. Therefore, the propulsion system in each CubeSat 7 will use NH3 as its propellant, which has the following characteristics: 8

9 • γ = 1.32 10 • MW = 0.01703 kg/mol 11

12 The size and other characteristics of the cold gas thruster is based on 13

similar models used on other CubeSats. For this project, the area ratio of the 14 nozzle will be 50. Since a cold gas thruster has lower performance than other 15 propulsion systems, the chamber temperature will be relatively low (between 16 200 and 320 K). The higher the chamber temperature, the higher the exhaust 17 velocity. 18

Table 10 shows results using the selected propellant. 19 20

Table 10. Propulsion Subsystem Sample Results 21 Results Values Comments

Ve 790.5 m/s Assume chamber and exit conditions to be Tc = 300 K, Pc = 800 kPa, Pe = 40 kPa

ṁ -2.53 kg/s F = 0.05 N, Isp = 50 seconds c* -316.213 m/s Assume At = 0.001 m2, Ae = 0.05 m2

cF 0.167 22 Structures Subsystem Analysis 23

Parameters for CubeSat structure depend on mB, including the mass of the 24 propulsion system, batteries, and other instruments. The mass of the solar 25 arrays is not included, since they are not stored inside the box. The mass of the 26 box shell is around 16.8 kg, so the distributed mass should be assumed greater 27 than that. For calculations, mB is assumed to be around 25 kg. 28

Given a maximum operating pressure and gas tank thickness, a spherical 29 tank would have a lower allowable stress than a cylindrical tank’s hoop stress. 30 However, a spherical tank is more structurally stable than a cylindrical one, 31 since there are no edges. Therefore, the gas tank for the cold gas thruster 32 system will be spherical. Using an average bulk density of 0.96 g/cm3 and 33 volume of the tank, the propellant mass is found to be around 0.025 kg. 34

Both the area and the mass of the solar arrays strongly depend on the 35 design parameters of the power subsystem. The area depends on the type of 36 solar cells used; GaAs solar cells contribute to a smaller area, which still makes 37 them the best choice for the solar arrays. The mass, on the other hand, depends 38 on the power required and orbit characteristics. Using the given power 39 requirement, the area and mass are less than 2 m2 and 2 kg, respectively. 40

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Payload Subsystem Analysis 1 Payload instruments are based on similar ones in other CubeSat missions 2

due to the size, weight, and power requirements. Therefore, the most important 3 calculation was the data rate, which depends on the orbit altitude and 4 spacecraft speed. This also influences other parameters in the constellation 5 system. A higher spacecraft speed gives a higher number of swaths in one 6 second, which then leads to a higher date rate. Assuming a Ymax of 50 m and a 7 slant range of 1200 m, the data rate was found to be around 32327 bps with 8 8 bits per pixel. 9

10 Analysis of Constellation System 11 12 The velocity and orbital period of each spacecraft affect subsystems such as 13 communications, GN&C, and the payload. The scalar angular momentum can 14 be found using the velocity. The following orbit parameters were determined 15 earlier: 16 17

• Eccentricity, e = 0 18 • Inclination, i = 90 degrees 19 20 Other orbit parameters, u and Ω, are determined based on the size of the 21

swaths and the space between the satellites. 22 Using Mars as the central body and an orbit altitude of 500 km, the 23

following parameters are found: 24 25 • τ = 7386 seconds 26 • vsatellite = 3.314 km/s 27 • h = 12914 km2/s 28

29 The constellation system will have between three and five planes for 30

maximum possible coverage and simplicity. Constellation design parameters 31 such as the number of satellites and space between satellites depend on the 32 desired amount of coverage and the radius of the central body. Each plane 33 should have a minimum of nine satellites per plane, so that coverage within a 34 swath can be continuous. A higher number of satellites per plane leads to lower 35 revisit time. 36

The sample calculations assume 3 planes with 18 satellites in each. Table 37 11 shows the resulting parameters for the simulation. 38

39 40 41

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Table 11. Sample Constellation Parameters, NP = 18, 3 Planes 1 Parameters Values Comments S 20 degrees For satellites within each plane trev 410.3 seconds For discontinuous coverage

λmax 20.9 degrees Assume 10° elevation angle, Coverage in swath is continuous

λstreet 18.4 degrees Dmax,S 39.26 degrees Size of swaths away from seam Dmax,O 36.8 degrees Size of swaths near seam

2 The parameters above will be used to generate a simulation. Other 3

simulations will also be done to show constellations with different number of 4 satellites per plane. 5

6 Simulations 7 8

This section presents three different simulations, each with different Ω and 9 u values for each orbit. The first simulation uses data presented in Table 11, 10 while the other two are shown with their respective constellation data. All other 11 inputs (i.e. CubeSat communications parameters) are the same for all three 12 simulations. 13

14 Sample Constellation System 15

Using the parameters shown in Table 11, Figure 11 shows a top view of 16 the constellation system at the start of the simulation. 17 18 Figure 11. Constellation system over Mars, NP = 18, 3 planes 19

20

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The image above shows that this constellation is only useful if the mission 1 requires coverage over a specific area during specific times of the day. The 2 area labeled as Place1 is seen by at most six CubeSats for various durations 3 over 24 hours. Place1 is not seen by any CubeSats for around one hour and 40 4 minutes, when it passes between plane 3 and plane 1. This is because the space 5 between the two orbits is larger than Dmax,S and Dmax,O. 6

The three-plane constellation system is especially useful for follow-up 7 missions to places such as Mars or the Moon, where certain areas are already 8 known. Keeping all other parameters the same, five planes are more useful for 9 complete coverage over the central body. 10 11 Second Constellation System 12

Parameters for another three-plane constellation model are shown in Table 13 12. 14 15 Table 12. Second Constellation Parameters, NP = 18, 3 Planes 16 Parameters Values Comments S 20 degrees For satellites within each plane trev 410.3 seconds For discontinuous coverage

λmax 24.73 degrees Assume 5° elevation angle, Coverage in swath is continuous

λstreet 22.73 degrees Dmax,S 47.47 degrees Size of swaths away from seam Dmax,O 45.5 degrees Size of swaths near seam

17 Additionally, the corresponding constellation system is shown in Figure 12 18

below. 19 20

Figure 12. Second Constellation System over Mars, NP = 18, 3 Planes 21

22 23

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For this constellation, the area labeled as Place1 is seen by at most seven 1 satellites for various durations over a 24-hour period. Place1 is not seen by any 2 CubeSats for about 25 minutes, when it passes between plane 3 and plane 1. 3 This period of time is far less than that of the previous constellation system due 4 to larger swath size, which is in turn due to the lower elevation angle. 5

Lower elevation angles helps with using a lower number of planes to cover 6 a lot of area. Still, keeping all other parameters the same for this system, one 7 additional plane is needed for complete coverage over the central body. 8 9 Third Constellation System 10 Parameters for the third three-plane constellation are shown in Table 13. 11 12 Table 13. Third Constellation Parameters, NP = 18, 3 Planes 13 Parameters Values Comments S 20 degrees For satellites within each plane trev 410.3 seconds For discontinuous coverage

λmax 28.36 degrees Assume 1° elevation angle, Coverage in swath is continuous

λstreet 26.68 degrees Dmax,S 55.04 degrees Size of swaths away from seam Dmax,O 53.36 degrees Size of swaths near seam

The corresponding constellation system is shown in Figure 13 below. 14 15 Figure 13. Third Constellation System over Mars, NP = 18, 3 Planes 16

17 18

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For this constellation, the area labeled as Place 1 is seen by at least one 1 CubeSat throughout the entire 24-hour duration. Therefore, this constellation 2 system can be considered as fully continuous. 3 4 5 Discussion 6 7

Constellations with only one or two planes are better responsive to user 8 needs. However, for this project, three to five planes were considered to 9 achieve better coverage of the central body. Of the three constellation systems 10 presented in this chapter, the third one requires the fewest number of planes to 11 cover the most possible area. Therefore, it is the most ideal system, provided 12 that there are no restrictions on the elevation angle. 13

For communications satellites, the elevation angle should be at least 5 14 degrees (Larson & Wertz, 1999). In this case, a version of the second 15 constellation system would be the most ideal, which would have four planes 16 instead of three to achieve maximum possible coverage. A version of the 17 sample constellation system, with five planes instead of three, would also 18 work; however, it would not be as ideal because it is less responsive to user 19 needs. 20

Note that the results only correspond to an altitude of 500 km over Mars. If 21 the central body was a larger planet such as Venus, for example, the 22 constellation system would require more planes to cover the entire area. A 23 smaller body such as the Moon would require no more than two planes 24 regardless of the elevation angle. 25

If the altitude was decreased over Mars, more planes would be needed 26 since the CubeSats would cover a smaller portion than before. The opposite is 27 true if the altitude was increased, since the CubeSats would cover more area. 28 29 30 Conclusion 31 32

Constellation systems depend on user needs and specific mission 33 requirements, as well as the type of CubeSats used. Given the calculations and 34 simulations presented in this paper, the number of planes and CubeSats in the 35 constellation system depend on the central body and orbit characteristics. 36 Larger bodies generally need more planes for complete coverage, as do 37 constellation systems with smaller orbits. 38

The CubeSats themselves also depend on the user needs and mission 39 requirements, which determine the type of instruments carried. Deep space 40 CubeSats need to communicate with ground stations on Earth, so their 41 communication instruments should allow them to transmit and receive data 42 over long distances. They have different power requirements than those 43 surrounding the Earth, and the size of the CubeSat box should be able to 44 support the total mass of all the instruments. The payload depends on the 45

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mission. For this project, the payload includes an imager/camera, altimeter, and 1 radiometer. 2

A deep space CubeSat constellation system, such as those simulated in this 3 project, provides coverage over large areas of a central body. Therefore, it 4 would help gather data for future deep space missions. Using Mars as the 5 central body for the simulations presented, the potential constellation systems 6 could help determine where future missions could focus on exploring. Perhaps, 7 these constellation systems can help determine the best areas for humans to 8 physically explore next. 9 10 11 References 12 13 Anis, A. (2012). Cold Gas Propulsion System - An Ideal Choice for Remote Sensing 14

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