MASTER THESIS Mission and Thermal Analysis of the UPC Cubesat

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MASTER THESIS

Mission and Thermal Analysis of the UPC Cubesat

Pol Sintes Arroyo

SUPERVISED BY

Josep J. Masdemont Soler

Universitat Politecnica` de Catalunya Master in Aerospace Science and Technology 14 de desembre de 2009

Mission and Thermal Analysis of the UPC Cubesat

BY Pol Sintes Arroyo

DIPLOMA THESIS FOR DEGREE Master in Aerospace Science and Technology

AT Universitat Politecnica` de Catalunya

SUPERVISED BY: Josep J. Masdemont Soler Matematica` Aplicada I

ABSTRACT

The Project is envisioned as a mission control analysis and a thermal control analysis of the Cubesat that is planning to launch the UPC (Universitat Politecnica de Catalunya). A Cubesat is a 10x10x10 cm cubic satellite that weights no more than 1 kg and that is currently used in many countries and educational institutions as an easy access to space.

The primary work to do would be the analysis of the orbit the satellite would perform. In this way, the STK (Satellite Tool Kit) software would provide the mechanism to perform this analysis. Starting with the known values of the project, simulations have been carried out in order to obtain the trajectory the satellite will would perform. Additionally, perturbations in the orbit, comparatives with other models and simulations with different parameters would be studied. All this practical work is completed with a theoretical explanation of what are the models used by the software, what are the parameters used and why we use this ones and no other ones.

The second part of the project presents the thermal analysis of the satellite. In this section, draft calculations of the thermal balance of the satellite are presented along with simulations with the SEET (Space Environment and Effects Tool) module of the STK. In addition, the necessary theoretical explanations are offered in order to conduct a complete thermal analysis in future projects.

Essentially, these two sections plus a theoretical background or state of the art of Cube- Sats or the different types of launches form the Master Thesis. Furthermore, extensive appendixes can be found at the end of the project which act as an ideal complement to the topics presented in the Master Thesis.

Results obtained from the simulations show similarities with other related-type projects and with the theory explained. The inclusion of the UPC Cubesat in the existing network of Cubesat developers is advised. HPOP (High Precision Orbit Propagator) is recommended in future simulations. This propagator shows little differences with SGP4 (Special General Perturbations no 4) but although little, these differences can change signiﬁcantly the orbital elements during a year. Atmospheric drag and the non-spherical shape of the Earth are the ones that affect more the satellite with both secular and periodic changes no matter which orbit is being used. Moreover, among other differences, using the two different shapes of orbit can produce discrepancies in lifetime of 6 or 7 years.

On the other hand results from the thermal analysis show variations of temperature from -85oC to 50 oC for the standard case. Important variations are observed with different values of internal dissipation. Finally, the emissivity/absorptivity ratio is the main parameter which we can play in order to change these temperature variations.

Table of Contents

INTRODUCTION ...... 1

Chapter 1. Introduction to CubeSats ...... 3

1.1. Deﬁnition of a Cubesat ...... 3

1.2. History of CubeSats ...... 5

1.3. The CubeSat of the UPC ...... 7

Chapter 2. Mission Analysis ...... 11

2.1. Introduction to the STK ...... 11

2.2. Theoretical background of the orbital dynamics ...... 13 2.2.1. OrbitDetermination ...... 16

2.3. Perturbation Techniques ...... 18

2.4. Orbit Prediction ...... 20 2.4.1. SGP4Propagator ...... 22 2.4.2. HPOPPropagator ...... 23

2.5. Ground Track and Accessibility ...... 24

2.6. Lighting ...... 24

2.7. Simulation results ...... 25 2.7.1. Totalandindividualaccess ...... 26 2.7.2. Lighting ...... 29 2.7.3. Orbitshapecomparative ...... 30 2.7.4. Elevationanglecomparative...... 32 2.7.5. Propagatorcomparative ...... 33 2.7.6. Simulationtimecomparative ...... 37 2.7.7. Satellitelifetime ...... 37 2.7.8. Orbital elements variation due to perturbations ...... 40 2.7.9. Dispersionanalysis ...... 43

Chapter 3. Thermal Analysis ...... 47

3.1. Theoretical Background ...... 47 3.2. Simpliﬁed results ...... 50 3.2.1. STKsimulation ...... 52

CONCLUSIONS ...... 55

BIBLIOGRAPHY ...... 57

Appendix A. Cubesat launches and participants ...... 61

A.1. Past launches ...... 61

A.2. Upcoming launches ...... 63

A.3. Cubesat participants ...... 63

Appendix B. Cubesat launch vehicles ...... 67

Appendix C. STK modules ...... 71

Appendix D. Additional theory for mission analysis ...... 73

D.1. The n-body problem ...... 73

D.2. The trajectory equation ...... 76

D.3. Type of conics ...... 79

D.4. Types of orbit ...... 80

D.5. Perturbation study ...... 83

D.6. HPOP force models ...... 88

Appendix E. STK user’s guide ...... 91

Appendix F. Simulation tables of the mission analysis ...... 99

F.1. Individual access analysis ...... 99

F.2. Orbit shape comparative ...... 101

F.3. Propagator comparative ...... 102

F.4. Elevation angle comparative ...... 104

F.5. HPOP vs. SGP4 comparative ...... 104 Appendix G. Thermal analysis information ...... 107

G.1. UPCSat Information ...... 107

G.2. Thermal analysis methodology ...... 108

G.3. Software available ...... 109

List of Figures

1.1 TypicalCubesat.Source[2]...... 4 1.2 StandardP-POD.Source[2]...... 4

2.1 Orbitalelements.Source[12]...... 17 2.2 Forcestakenintoaccountbyeachpropagator ...... 21 2.3 Description of the umbra and penumbra effects. Source [17]...... 25 2.4 STK map illustrating all the ground stations of the Cubesat network. Source [STK]...... 26 2.5 Number of accesses for all the ground stations during a year(PartI)...... 28 2.6 Number of accesses for all the ground stations during a year(PartII). . . . . 28 2.7 Totaldurationlightingpercentages ...... 30 2.8 Number of accesses vs. orbit shape comparative for UPC, NTNU and Malaysia 31 2.9 Lighting properties comparative for a 1200x350 and a 350x350kmorbit . . . 32 2.10 Accessschemefordifferentelevationangles ...... 33 2.11 Access comparative for different elevation angles for UPC, NTNU and Malaysia groundstations...... 34 2.12 Mean access duration/day comparative for each propagator for UPC, NTNU andMalaysiagroundstations ...... 35 2.13 Lighting properties comparative for each propagator ...... 36 2.14 SGP4 and HPOP mean access duration/day comparative for UPC, NTNU and Malaysiagroundstationsfora1200x350kmorbit ...... 37 2.15 SGP4 and HPOP lighting properties comparative for a 1200x350kmorbit . . 37 2.16 Apogee, perigee and eccentricity variation during the HPOP 1200x250 km satellitelifetime...... 39 2.17 i comparative between a spherical (blue) and an elliptical (green) model of Earth 41 2.18 ω comparative between a spherical (blue) and an elliptical (green) model of Earth...... 41 2.19 evariationforanellipticalmodelofEarth ...... 41 2.20 evariationforacomplexmodelofEarth ...... 41 2.21 evariationproducedbythird-bodieseffects ...... 42 2.22 Ω variationproducedbythird-bodieseffects ...... 42 2.23 avariationproducedbyatmosphericdrag ...... 42 2.24 ω variationproducedbyatmosphericdrag ...... 42 2.25 avariationproducedbySRP...... 42 2.26 Ω variationproducedbySRP ...... 42 2.27 avariationproducedbyallperturbforces ...... 43 2.28 evariationproducedbyallperturbforces ...... 43 2.29 ivariationproducedbyallperturbforces ...... 43 2.30 Initial conditions of the dispersion analysis ...... 44 2.31 Positionofthesatellitessevendayslater ...... 44 2.32 Finalpositionsofthesatellites ...... 44 2.33 Range variation between UPCSAT and UPC1 for 50m separation ...... 45 2.34 Range within the satellite will be for a 50m margin ...... 46

3.1 Incidentlightcharacteristics ...... 48 3.2 Heat ﬂuxes of a typical satellite orbiting Earth. Source [20]...... 49 3.3 Temperature evolution for the Al. with Kapton with 0.5 W of internal dissipation 54

B.1 Vegalaunchvehicle.Source[6]...... 69

D.1 Orbitalplaneangles ...... 78 D.2 Typesofconicsections...... 80 D.3 Orbital element perturbation changes. Source [19]...... 83 D.4 Nodalregression.Source[19]...... 86 D.5 Apsidalrotation.Source[19]...... 86 D.6 Typical dacay of a satellite for a 300 km circular orbit ...... 87

E.1 STKwindowwithmaincommands...... 91 E.2 Period ...... 92 E.3 Satellitegeographiccoordinates ...... 92 E.4 Facilityconstraints ...... 93 E.5 Propagatorpage ...... 94 E.6 Forcemodelspage...... 94 E.7 Accesspage ...... 95 E.8 Accessreportexample...... 96 E.9 Lightingreportexample ...... 96 E.10 Lifetimepage...... 97 E.11 Lifetimegraphic ...... 98

G.1 Typical operational limits of components of a satellite.Source[23]...... 109 List of Tables

1.1 CurrentCubeSatstatushistory ...... 6

2.1 Datausedinthesimulations ...... 26 2.2 Access time comparative between UPC and global network ...... 27 2.3 Access time comparative between the more an less accessed ground stations 27 2.4 Lightingproperties ...... 29 2.5 Number of accesses vs. orbit shape for UPC, NTNU and Malaysia...... 31 2.6 Lightingpropertiesfora1200x350kmorbit ...... 31 2.7 SGP4data ...... 33 2.8 Mean access duration/day vs. propagator for UPC, NTNU and Malaysia ground stations...... 34 2.9 HPOPdata...... 36 2.10 Simulation time comparative for different gaps of one year for the accessibility propertiesoftheUPCgroundstation ...... 38 2.11 SGP4 and HPOP lifetime comparative for different shapes of orbit. Launch date:1Jul2010 ...... 38 2.12 Accessibility evolution of the ﬁrst four years for the UPCgroundstation . . . . 39 2.13 Theoretical orbital elements changes due to each perturbation force. Source [19]...... 40 2.14 Range variations between UPCSAT and UPC1 for different initial separations 45

3.1 Datausedinthehotandcoldcasecalculations ...... 50 3.2 Hot and cold case temperatures for different materials ...... 51 3.3 Hot and cold case temperatures for different materials with no internal dissipation...... 53 3.4 Hot and cold case temperatures for the Al. with Kapton ...... 53

A.1 CubeSatspastlaunches ...... 61 A.2 CubeSatsupcominglaunches ...... 63 A.3 ListofCubesatparticipants.Source[2]...... 64

B.1 Launchvehiclescomparative. Source[21]...... 69

D.1 Relative accelerations form other bodies to a LEO satellite ...... 75 D.2 Realationship between the type of conic and e, a and ε ...... 80 D.3 Most important accelerations form other bodies to a LEO satellite ...... 84 D.4 ForcemodelsavailableintheHPOP...... 88

F.1 Complete list of the individual access analysis for all groundstations . . . . . 99 F.2 Accessibility analysis vs. orbit shape for UPC ground station ...... 101 F.3 Accessibility analysis vs. orbit shape for NTNU ground station ...... 101 F.4 Accessibility analysis vs. orbit shape for Malaysia groundstation ...... 101 F.5 Lightinganalysisvs.orbitshape ...... 102 F.6 Accessibility analysis vs. propagator for UPC ground station ...... 102 F.7 Accessibility analysis vs. propagator for NTNU ground station ...... 102 F.8 Accessibility analysis vs. propagator for Malaysia groundstation ...... 103 F.9 Lightinganalysisvs.propagator ...... 103 F.10 Elevation angle accessibility analysis for UPC ground station ...... 104 F.11 Elevation angle accessibility analysis for NTNU groundstation ...... 104 F.12 Elevation angle accessibility analysis for Malaysia groundstation ...... 104 F.13 HPOP and SGP4 accessibility analysis for UPC ground station...... 105 F.14 HPOP and SGP4 accessibility analysis for NTNU ground station ...... 105 F.15 HPOP and SGP4 accessibility analysis for Malaysia groundstation...... 105 F.16 HPOPandSGP4Lightingcomparative ...... 105

G.1 Emissivity and absorptivity of different parts of the Cubesat...... 107 G.2 Materialproperties ...... 108 1

INTRODUCTION

The necessity to achieve new goals has kept mankind occupied during his whole history. From immemorial times, men have tried to reach the most extremes points of the planet until deciding to reach the air and later Space. What at the beginning you spent months to arrive to, you now only need a few hours. What in the ﬁrst times only rich people could afford, it is now open for all the population.

Nowadays, access to Space is limited to a few and launching an object into orbit is very expensive. Minimizing this cost and making Space accessible to everyone is the objective of the CubeSats.

CubeSats are small satellites weighting no more than 1 kg which have the purpose of making Space available to small projects, mostly from educational institutions. Precisely, UPC is currently developing a project to launch a Cubesat through the launches of opportunity that offer ESA (European Space Agency). In spite of the dimensions of the satellite, CubeSats need the standard subsystems to work. Depending on the type of the mission one or other subsystems will be necessary. Moreover, in every space mission the contact with the satellite is desired in order to have success. This contact is established by means of the position of the satellite in space so this position must be known and studied. The mission analysis is the one in charge of the study of this position, the calculation of the orbit and the possible perturbations that will affect the satellite.

This project deals with some of the aspects of the mission analysis of the future Cubesat of the UPC. In order to study the orbit, the theory necessary to understand the simulations performed is explained. These simulations provide an extensive knowledge of the performance of the satellite and have been carried out using a commercial software called STK. Moreover, from these simulations it is explained the inﬂuence of the different parameters of the satellite in its orbit apart from explaining how it is affected by the external perturbations. Furthermore, an introduction to the thermal analysis is also present at the end of the project. The thermal analysis pretends to introduce the ﬁrst steps in order to have a complete knowledge of the variation of temperature during the lifetime of a satellite along with its importance to the mission.

The project is structured in three chapters. The first chapter introduces the concept of CubeSats giving information about their state-of-the-art and of the project of the UPC as well. The second chapter is centered in the mission analysis, explaining the necessary theory and the results obtained and finally the third chapter introduces the thermal analysis of the satellite and first simplified calculations of the variation of temperature that will experience the satellite in orbit. It is also important to recall that there also exist some appendixes that try to complete the information of the main body of the project. 2 Mission and Thermal Analysis of the UPC Cubesat Introduction to CubeSats 3

Chapter 1

INTRODUCTION TO CUBESATS

1.1. Deﬁnition of a Cubesat

The CubeSat concept was developed by the California Polytechnic State University (CalPoly) and the Space Systems Development Laboratory of Stanford University [1]. It had the purpose of creating space research opportunities for universities previously unable to access space by deﬁning a standard mechanical interface and deployment system.

A CubeSat (Fig 1.1) is a 10 cm cube with a mass of up to 1 kg and functions fully au- tonomously. Worldwide over 60 universities, high schools and industries are involved in the development of CubeSats [2], with payloads covering a wide range of scientiﬁc, engineering and industrial objectives. The low prize of the satellite and the launch opportunities offered by major satellites launch providers and manufacturers make them an ideal method for educational purposes. Space is fully open to everyone with this concept.

The idea of the launch opportunity reduces the cost of the launching to zero in most of the cases. This concept is based in the utilization of a small part of the launcher that has as the primary mission the launching of a satellite (the normal size concept of satellites). The launch cost is just a little bit higher because you only add 1 kg for each CubeSat but the advantages are really enormous. So the idea is that you launch the CubeSats using the current launches of ’standard’ satellites.

Moreover, the device chosen to deploy the CubeSats will be standard ﬂight-proven Cube- Sat deployment systems. The idea of a standard system has huge advantages. The system would add integrity so the people involved in the project would only deal with only one type of system and they would be able to use the maximum number of launch vehicles as possible. The standardization would beneﬁt both the manufacturers and the launcher companies. The most common standard system is the P-POD (Fig 1.2) (Poly Pico-satellite Orbital Deployer) from CalPoly. Each P-POD is designed to carry three standard CubeSats and serves as the interface between the CubeSats and the launch vehicle. It is a rectangu- lar aluminum tubular frame with an electrically activated spring-loaded door mechanism. After the door is opened the CubeSats are pushed out by a spring along guidance rails, ejecting them into orbit with a separation speed of a few m/s. There are also other versions of the system such as the X-POD from UTIS-SFR (Toronto) that has an independent release mechanism for the spring deployer and feedback to indicate deployment has taken place.

The concept of standardization is often related with ﬂexibility. Pre-qualiﬁed P-POD and launch vehicles interfaces maximize the number of compatible missions, reduce the integration time and minimizes costs. It also minimizes design, analysis and testing for subsequent mission with the idea of repetition. There are other advantages such as the possibility to transfer the CubeSats to a different launch vehicle if the launch is delayed or cancelled. 4 Mission and Thermal Analysis of the UPC Cubesat

The standard 10x10x10 cm basic CubeSat is often called a ’1U’ CubeSat meaning one unit. CubeSats are scalable in 1U increments and larger. CubeSats such as a ’2U’ Cube- Sat (20x10x10 cm) and a ’3U’ CubeSat (30x10x10 cm) have been both built and launched.

Since CubeSats are all 10x10 cm (regardless of length) they can all be launched and deployed using the common deployment system. P-PODs are mounted to a launch vehicle and carry CubeSats into orbit and deploy them once the proper signal is received from the launch vehicle. P-PODs have deployed over 90% of all CubeSats launched to date since the ﬁrst one launched in June of 2003(including un-successful launches), and 100% of all CubeSats launched since 2006. The P-POD has capacity for three 1U CubeSats, however since three 1U CubeSats are exactly the same size as one 3U CubeSat, and conversely two 1U CubeSats are the same size as one 2U CubeSat; the P-POD can deploy 1U, 2U, or 3U CubeSats in any combination up to a maximum volume of 3U.

All educational payload elements shall have their own power supplies and communication facilities. There is no electrical interface to the CubeSats either from the POD or the main payload, and once the CubeSat is installed in the POD prior to launch there will be no opportunity for further battery charging. The CubeSats are expected to be electrically inert once inserted into the POD, with no power dissipation or RF emissions. They are expected to utilize the sensors in the feet of the CubeSat structure to determine launcher separation and begin operations.

Although launch prices have risen quite substantially across the board of launch providers, a CubeSat still forms the most cost-effective independent means of getting a payload into orbit. Several companies and research institutes offer regular launch opportunities in clus- ters of several cubes. The concept of the distribution of costs is commonly used. This concept says that in order to reduce the costs, you need to distribute them over many cos- tumers is needed, deploy multiple spacecrafts per mechanism and use identical, standard systems and not mission speciﬁc devices.

In conclusion, the objectives of the P-POD are the protection of the launch vehicle and the primary payload, the safe and reliable deployment of the CubeSats and the compatibility with many launch vehicles.

Figure 1.1: Typical Cubesat. Source [2]. Figure 1.2: Standard P-POD. Source [2]. Introduction to CubeSats 5

1.2. History of CubeSats

CubeSats were first conceptualized in 1999 by Stanford and Cal Poly as said in the previous section driven by need for student opportunities. The idea was born in the minds of Jordi Puig-Suari and Bob Twiggs. The first, a Catalan aerospace engineer, says in an interview [3] that he moved to the California Polytechnic State University from the Arizona State University with lots of problems and solutions for the satellite engineering. The im- plication of students in the development of satellites was not new but the problems in cost and time were really big. They were building huge satellites of 20 and 30 kilograms, which make the students impossible to finish their thesis before launching the satellites. Then, with the help of Professor Bob Twiggs of the Aeronautic and Astronautic Department of Stanford University, he was able to solve that major problems of the aerospace engineering bringing to life the first version of the CubeSat. Since then, more than 100 projects of CubeSats worldwide have been developed.

Moreover, Cal Poly’s current role is to provide standard interface and systems for deploying CubeSats (P-POD), maintain the CubeSats standard, coordinate launch opportunities and networking ground stations around the world dedicated to CubeSat operations [2]. From the 51 launches of CubeSats, 35 are or were in orbit. The others were lost in two launch failures in Russia and in the United States.

The applications of the CubeSats payloads range from astrobiology experiments to Earth science research. In this ﬁeld it should be remarked the GeneSat mission as an astrobiology experiment, the ionospheric research of the QuakeSat mission or the Pico-inspector testing mission of Aerospace Corporation. A short list of CubeSats applications is shown below giving examples of CubeSat missions with the corresponding application [4].

• Development of CubeSat technology (Testing)

– AAU CubeSat – CanX-1 – AeroCube – CubeSat TestBed 1

• Earth remote sensing

– Libertad-1 – AAUsat-2 – Quakesat

• Tether experiments

– MAST

• Biology

– GeneSat 1 – Pharmasat-1 6 Mission and Thermal Analysis of the UPC Cubesat

All these experiments validate the CubeSat concept to be used in all kind of applications and not only with educational purposes but in industry and research also. A brief description of the missions already launched is shown in Table 1.1 but the complete list of past, current and future missions can be explored in Appendix A.

Table 1.1: Current CubeSat status history

Batch # Date LV No Cont Semi Cont Full Cont Total 1 30 Jun 2003 Rokot 2 1 3 (1 3U) 6 (1 3U) 2 27 Oct 2005 Kosmos-3M 1 0 2 3 3 22 Feb 2006 M-V 0 0 1 (2U) 1 (2U) 4 26 Jul 2006 DNEPR 0 0 0 14 (1 2U) (failure) 5 16 Dec 2006 Minotaur I 0 0 1 (3U) 1 (3U) 6 17 Apr 2007 DNEPR 1 4 (1 3U) 2 7 (1 3U) 7 27 Apr 2008 PSLV 0 0 6 (1 2U + 6 (1 2U + 2 3U) 2 3U 8 03 Aug 2008 Falcon-1 0 0 0 2 (failure) 9 23 Jan 2009 H-IIA 2 0 1 3 10 19 May 2009 Minotaur I 0 0 4 (1 3U) 4 (1 3U) 11 23 Sep 2009 PSLV 0 0 4 4 TOTAL Up to 10/09 16 (1 2U) 6 5 (1 3U) 24 (2 2U 51 (3 2U + 5 3U) + 6 3U) Batch #: Number of the CubeSat group launch (CubeSats are typically launched in group) LV: Launch Vehicle No Cont: Number of CubeSats of the corresponding batch that had no contact with the GS (ground station) Semi Cont: Number of CubeSats of the corresponding batch that had contact for a short time with the GS Full Cont: Number of CubeSats of the corresponding batch that had full contact with the GS Total: Total number of CubeSats of the corresponding batch that didn’t had a launch failure

It can be observed from Table 1.1 that if the failures due to launch vehicles are not taken into consideration the success rate of CubeSats missions is really high. From the 35 that are in orbit now, in 29 there was contact between the satellite and the ground station and form that 29, in 24 there was full contact and most of them are still in operation. It can be deduced from this data that indeed these satellites can be useful for many applications and that cannot be seen just as ’student toys’.

There are going to be upcoming launches in the next months and every year this technology would be accepted more and more as part of the normal Space world. In the second quarter of 2009 there was planned a new launch of the Falcon-1 of SpaceX which carried the RazakSat as the primary payload and was supposed to carry also two triple CubeSats as secondary payload. Due to a last minute problem with vibrations, a mitigation device was installed in the place where the CubeSats were supposed to be [5]. This problem has delayed the two CubeSats to a later launch during 2010 on-board of a Falcon-1 or a still not demonstrated Falcon-9. Introduction to CubeSats 7

Also, late in 2009, the maiden ﬂight of the European Vega LV was planned. This ﬂight has been delayed different times, being October 2010 its current launch date. This LV will carry 11 CubeSats missions as part of the ESA launch opportunity contest [6]. Other missions planned for 2010 are one in the new Minotaur IV with 2 triple CubeSats and another one in the Taurus-XL that will carry 3 CubeSats more. There is a clear increase in the launches of CubeSats and in a few years there would be even a higher increase which means that Space will be open to everyone with a good team, good ideas and willing to work hard.

Although the obvious thing when examining if the CubeSats technology is having success or not is the amount of satellites now in operation in orbit there are others factors to take into consideration. The CubeSat concept was born as an idea to facilitate the access to space to universities or organizations without the resources to do it. As part of this idea, the concepts of networking and standardization were needed. What if all the universities and private and public research institutions building CubeSats share their resources to obtain better results? This is the primary objective of the CubeSat Community [2]. There are more than 80 universities, private companies, government organizations building Cube- Sats and most of them have antennas and tracking devices to follow their satellites in the sky. With this community, each of these institutions shares their capabilities to follow all the CubeSats in orbit. The advantage is incredible because instead of following your satellite some minutes during the day, you can now watch it 10 times more. A complete list of he institutions involved in the CubeSat community can be found in Appendix A.3..

This program is also designed so that students can participate in entire life cycle of a space mission, from the ﬁrst idea of the concept to the launch of the satellite and the study of the data obtained.

In conclusion, the accomplishments made within just 8 years have been spectacular. There are 20 completely operational CubeSats in orbit and just 6 total failures of the CubeSats (not taken into account the failures due to the launch vehicle). There have been successful coordination and launch of 31 satellites on worldwide launch vehicles and each year there is an increase in the number of launch opportunities for CubeSats. Finally and very remarkable, there has been established an international Earth station networking.

1.3. The CubeSat of the UPC

In this context, the Polytechnic University of Catalonia is planning to build a CubeSat and launch it through the ESA launch opportunities. ESA’s launch opportunities are a call for European universities to launch their projects with the new Vega launch vehicle [7]. The opportunity to fly CubeSats on the Vega maiden flight is proposed by the ESA Education Office with the purpose of giving students from European universities and other educational institutions wishing to pursue a career in the space domain a valuable hands-on experience. They can take part in this end-to-end space project including educational activities from design, integration, verification, launch and operations. The educational payloads will be entirely developed by educational institutions, with advice from ESTEC experts if requested and deemed appropriate by the Education Office. All payloads shall fully comply with the Vega general specification for qualification and acceptance test of equipment. 8 Mission and Thermal Analysis of the UPC Cubesat

A ﬁrst Announcement of Opportunity (AO) was published on the education pages of the ESA web portal on 9 October 2007 and the ﬁrst nine CubeSats selected (with two back-up satellites) were announced in 2008 [6]. Although the launch date was planned to be during 2009, the new estimate date is in the end of 2010.

Vega is ESA’s new small launch vehicle which is being developed under the management of the Vega Programme (LAU-PV) in the framework of an ESA optional programme funded by Belgium, France, Italy, Spain, Sweden, Switzerland and The Netherlands. Vega is designed to launch a wide range of missions and payload conﬁgurations in order to re- spond to different market opportunities and therefore provide the ﬂexibility needed by the customers. In particular, it can launch payloads ranging from a single satellite up to one main satellite plus six microsatellites. It is compatible with payload masses ranging from 300 to 2500 kg and can provide launch services for a variety of orbits, from equatorial to Sun-synchronous.

The primary scientific payload on the maiden flight is the LAser RElativity Satellite (LARES) for testing a prediction following from Einstein’s theory of General Relativity, the so-called ’frame-dragging or Lense-Thirring effect’. An educational payload of nine European Cube- Sats is foreseen to be accommodated in three P-PODs, each containing three CubeSats, attached to and deployed from the qualification payload. The AVUM multi-burn facility will be utilized to put the CubeSats into either a 1200x350 km elliptical orbit or a 350 km circular orbit (still to be confirmed). The qualification payload will be separated from the AVUM after the orbital maneuvers.

Regarding the orbital parameters, the LARES satellite is supposed to be placed by Vega into a circular orbit with an altitude of 1200 km and an inclination of 71o. Thereafter, the orbit will be changed by a de-orbit boost of the AVUM. The new orbit will have a perigee of 350 km and an apogee of 1200 km, the inclination is 71o as before. This orbit is more compatible with the capabilities of the planned CubeSats ground stations. The braking effect of the residual atmosphere will lower the apogee by about 40 km per month (this is just a rough estimate, more detailed calculations are ongoing). In this new orbit, the lifetime will be much less than 25 years, therefore compliant with international requirements related to space debris. Currently under investigation is the possibility to change the 350x1200 km orbit to a 350x350 km orbit by an additional ﬁring of the AVUM liquid propellant engine.

At this time it is not known if the deployment will take place in sunlight or eclipse. This may indeed not be known until after the lift off, as a last-minute launch delay could affect the deployment position relative to the sun.

In order to select the CubeSats of the ESA launch opportunities, the criteria include, amongst others:

• The educational content, technical maturity and project objectives of the proposals.

• Letters of commitment by the funding bodies (institutions and/or industry).

• Compliance of the development schedules with the Vega Flight schedule.

• Signing of relevant agreements between the educational institutions (universities) and the Education Ofﬁce. Introduction to CubeSats 9

The idea to launch the UPC CubeSat is to do it through another launch opportunity of the ESA. The process will be very similar to the ﬁrst one so analyzing the steps the other teams have made in this ﬁrst opportunity will be extremely positive to have success in this upcoming mission.

The project is known to be called UPCSat and is currently being developed by the CRAE (Aeronautics and Space Research Center) of the UPC. The project is directed by Adri- ano J. Camps Carmona of the Signal Theory and Communications Department (TSC) but encloses other people and departments of the UPC [8].

CRAE is a specific research centre that encloses 90 researchers from 28 different research groups inside the UPC. UPC has extensively knowledge in the aerospace field but is currently behind other Universities in the development of this kind of technology. This project would benefit the entire University because there is a wide range of payloads and subsystems that can be developed by the research groups of UPC, using the CubeSats as test platforms for their technologies. Moreover, the strategy would be simple. The first missions will be focused in just flying the satellite. No attitude control will be installed in the satellite and the required subsystems will be bought as many as it can be in order to have a quick start-up. Future missions will replace these subsystems with others developed entirely in the University including also, new developments that could be used simultaneously for other scientific experiments. The ground station will also be bought form ISIS Company and installed in one of the faculties of the UPC. Finally testing of the technology will be carried out in some labs of the University and in some external labs of private companies.

The main goal of this project, though, is the collaboration between groups within UPC in space related activities, retain and attract students for ﬁnal project and potentially for Ph D and provide visibility of UPC space-related activities to the society. 10 Mission and Thermal Analysis of the UPC Cubesat Mission Analysis 11

Chapter 2

MISSION ANALYSIS

2.1. Introduction to the STK

In order to understand the different explanations in this section, some basic deﬁnitions should be deﬁned.

Mission analysis is the term used to describe the mathematical analysis of satellite orbits, performed to determine how best to achieve the objectives of a space mission.

Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. The ﬁeld applies principles of physics, historically Newtonian mechanics, to astronomical objects such as stars and planets. It is distinguished from astrodynamics, which is the study of the creation of artiﬁcial satellite orbits.

Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newton’s laws of motion and his law of universal gravitation. It is a speciﬁc and distinct branch of celestial mechanics, which focuses more broadly on Newtonian gravitation and includes the orbital motions of artiﬁcial and natural astronomical bodies such as planets, moons, and comets. Astrodynamics is principally concerned with spacecraft trajectories, from launch to atmospheric re-entry, including all orbital maneuvers, orbit plane changes, and interplanetary transfers.

Astronautics is the branch of engineering that deals with machines designed to work outside of Earth’s atmosphere, whether manned or unmanned. In other words, it is the science and technology of space flight. To perform the mission analysis, some kind of software is needed. The amount of data and calculations needed make it difficult and in some way impossible to achieve without specific software.

There are ways to create your own software by means of programming languages such as C++, Fortran, FreeMat, Numerit Pro or the most common of them Matlab but the mission analysis performed in this project has been done by a commercial software named Satellite Tool Kit.

Satellite Tool Kit, often referred to by its initials STK, is a software package from Ana- lytical Graphics, Inc. (AIG) that allows engineers and scientists to design and develop complex dynamic simulations of real-world problems [9]. Originally created to solve problems involving Earth-orbiting satellites, it is now used in both the aerospace and defense communities.

STK has more than 32,000 installations worldwide, with organizations such as NASA, ESA, CNES, Boeing, JAXA, Lockheed Martin, Northrop Grumman, EADS, DOD, and Civil Air Patrol.

The product is currently used in such areas as: 12 Mission and Thermal Analysis of the UPC Cubesat

• Battlespace Management

• Communications Analysis

• Space Exploration

• Electronic Warfare

• Geospacial Intelligence

• Spacecraft Mission Design

• Missile Defense

• Unmanned Systems (UAVs)

• Spacecraft Operations

In 1989, the three founders of Analytical Graphics, Inc - Paul Graziani, Scott Reynolds and Jim Poland, left GE Aerospace to create Satellite Tool Kit (STK) as an alternative to bespoke project-speciﬁc aerospace software.

The original version of STK ran only on Silicon Graphics computers, but as PCs became more powerful, the code was converted to run on Windows. STK was ﬁrst adopted by the aerospace community for orbit analysis and access calculations but as software was expanded, more modules were added that included the ability to perform calculations for communications systems, radar, interplanetary missions and orbit collision avoidance.

The addition of 3D viewing capabilities lead to the adoption of the tool by military users for real-time visualization of air, land and sea forces as well as the space component. STK has also been used by various news organizations to graphically depict complex events to a wider audience. The interface to STK is a standard GUI display with customizable toolbars and dockable maps and 3D viewports. All analysis can be done through mouse and keyboard interaction.

Each analysis or design space within STK is called a scenario. Within each scenario any number of satellites, aircraft, targets, ships, communications systems or other objects can be created. Each scenario deﬁnes the default temporal limits to the child objects, as well as the base unit selection and properties. All of these properties can be overridden for each child object individually, if necessary. Only one scenario may exist at any one time, although data can be exported and reused in subsequent analysis.

For each object within a scenario, various reports and graphics (both static and dynamic) may be created. Relative parameters, between one object and another can also be re- ported and the effect of real-world restrictions (constraints) enabled so that more accurate reporting is obtained. Through the use of the constellation and chains objects, multiple child objects may be grouped together and the multipath interactions between them inves- tigated.

STK is a modular product, in much the same way as Matlab allows you to add modules to the baseline package to enhance speciﬁc functions. Mission Analysis 13

Since the release of STK 8.0, the STK product range has been reorganized into Editions with additional add-on modules. A brief summary of the editions and modules is listed in Appendix C.

• STK Basic Edition

• STK Professional Edition

• STK Expert Edition

• STK Specialized Analysis Modules

STK Basic Edition is run by a free license obtained from AIG. This license allows the utilization of the STK Basic in an unlimited way and it also permits the evaluation of the Professional Edition and Analysis Modules for 30 days.

Moreover, the ’low cost’ of this license is the reason that in this project the main calculations have been computed with the Basic Edition of STK.

2.2. Theoretical background of the orbital dynamics

It can seem that mission analysis and space engineering is a new development of this past century but history shows us that this race started long ago, in the Greek times. It was Aristarchus who put the ﬁrst stone in the study of the movement of the celestial bodies in space. He was a Greek astronomer who lived in the 300 BC and was the ﬁrst person to present an explicit argument for a heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe. Despite his ideas, the geocentric model (defended by Ptolemy and Aristotle) was the one used until Nicolaus Copernicus (1473-1543) in the XVI century formulated a comprehensive heliocentric model, which displaced the Earth from the center of the universe. Moreover, the heliocentric model was not accepted until Galileo Galilei (1564-1642) in the XVII century discovered the four moons in Jupiter that proved the model.

Meanwhile, two different scientists, Johannes Kepler (1571-1630) and Tycho Brahe (1546- 1601), studied the movement of the planets and other bodies in very different ways. These two scientists marked the history of orbital mechanics with two different personalities. While Tycho was a noble who dedicated part of his life to an accurate observation of the motion of the planets, Kepler was a poor and sickly mathematician who was gifted with the patience and the innate mathematical perception needed to unlock the secrets hidden in Tycho’s data. Thanks to the observational data of Tycho, Kepler could ﬁnally demonstrate his three laws of the movement of celestial bodies in 1609.

• 1st law: The orbit of each planet is an ellipse, with the sun at one focus.

• 2nd law: The line joining the planet to the sun sweeps out equal area in equal times. 14 Mission and Thermal Analysis of the UPC Cubesat

• 3rd law: The square of the period of a planet is proportional to the cube of its mean distance from the sun (1619).

The three Kepler’s laws were just a description, not an explanation of planetary motion. It remained still someone to unveil the mystery.

In 1642, the same year Galileo died, a child born in England was about to change the history of physics and to alter the thought and habit of the world. Isaac Newton (1642-1727) published his ’Principia Mathematica’ in 1687. In this work, Newton described universal gravitation and the three laws of motion which dominated the scientiﬁc view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws by demonstrating the consistency between Kepler’s laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the scientiﬁc revolution.

In Book 1 of the principia Newton introduces his three laws of motion:

• 1st law: Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.

• 2nd law: The rate of change of momentum is proportional to the force impressed and is in the same direction as that force

• 3rd law: To every action there is always opposed an equal reaction.

Newton also formulated his Law of Universal Gravitation by standing that any two bodies attract one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. This law is expressed in Eq. 2.1.

GMm r Fg = − · (2.1) r2 r

In Eq. 2.1 Fg is the force on mass m due to mass M and r is the vector from M to m. The universal gravitational constant G, has the value of 6.67428 · 10−11 m3/kg · s2 .

The application of Newton’s Law of Universal Gravitation to his second law of motion orig- inated the equation of motion for planets and satellites which the N-body problem is a ﬁrst approximation.

Basically, the N-body problem studies the gravitational force that these N bodies exert to one of them [10]. Moreover, it takes the positions, masses, and velocities of some set of n bodies, for some particular point in time, and determines the motions of the n bodies, and ﬁnd their positions in time. Note that in Appendix D there is the explanation of all the theory commented in this section.

The expression obtained is a second order, nonlinear, vector, differential equation of motion that with some simplifying assumptions can be reduced to Eq. 2.2. Mission Analysis 15

n m j r¨ = −G ∑ 3 r ji (2.2) j=1 r ji j6=i

However, in this project it is of interest the study of the motion of a near Earth satellite thus the CubeSat will orbit around the Earth at a low altitude. Therefore, let us rewrite Eq. 2.2 in a different form. If m1 is the Earth m2 is the satellite and all the other masses may be different celestial bodies as the Moon, the Sun, or other planets in the Solar System, then the expression is resduced to Eq. 2.3.

G(m + m ) n r r ¨ 1 2 j2 j1 r12 = − 3 r12 − ∑ Gm j( 3 − 3 ) (2.3) r12 j=3 r j2 r j1

The ﬁrst term of Eq. 2.3 is the force acting between the two main bodies (the Earth and the satellite). The other term is the perturbation of the other bodies affecting the force of the two main ones. However, the acceleration coming from other bodies different from the Earth is very little. Just the effect of the Sun and the Earth oblateness have a considerable value. It is for this reason that some assumptions can be made. Simpliﬁcation hypothesis are made in order to obtain a simpler solution. If just the satellite and the Earth are the only ones in study and they are spherical and homogenous, we obtain:

G(M + m) r¨ = − r (2.4) r3

Moreover, the mass of the satellite is very little compared to the mass of the Earth (M ≫ m). If a parameter µ = G(M + m) ≈ GM is deﬁned, we obtain:

µ r¨ + r = 0 (2.5) r3

This equation is the two-body equation of motion. This equation can be solved analytically, in contrast to the three body problem or higher that can only be solved numerically. This equation must be derived to obtain the trajectory equation.

Taking into account the conservation of the mechanical energy and the angular momentum and operating them with the two-body equation of motion, the trajectory equation is obtained:

h2/µ r = B (2.6) 1 + µ cosv

If e = B/µ, we obtain the polar equation of a conic section (Eq. 2.7), that it is mathematical equal in form to the trajectory equation and widely used in the study of orbits.

p r = (2.7) 1 + ecosv 16 Mission and Thermal Analysis of the UPC Cubesat

In Eq. 2.7 p is the parameter of the orbit and e is the eccentricity. The eccentricity determines the type of conic section that represents the orbit. A review of conic sections can be found in Appendix D.3..

2.2.1. Orbit Determination

Since the ﬁrst method to determine the orbit, the Newton’s one that Halley used to determine when would a comet pass near the Earth, many scientists such as Euler, Lambert, Lagrange, Laplace or Gauss among others have studied the celestial bodies and proved many theories and methods.

In order to determine an orbit, the orbital elements are defined. Orbital elements are the parameters required to uniquely identify a specific orbit [11]. Given an inertial frame of reference and an arbitrary epoch (a specified point in time), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit. This is because the problem contains six degrees of freedom. These correspond to the three spatial dimensions which define position, plus the velocity in each of these dimensions. These can be described as orbital state vectors, but this is often an inconvenient way to represent an orbit, which is why Keplerian elements are commonly used instead.

The Keplerian orbital elements are (Fig. 2.1):

• Semi major axis (a) - Distance between the geometric center of the orbital ellipse with the perigee, passing through the focal point where the center of mass resides.

• Eccentricity (e) - Shape of the ellipse, describing how ﬂattened it is compared with a circle. (not marked in diagram)

• Inclination (i) - Vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node (where the orbit passes upward through the reference plane).

• Right ascencion of the ascending node (Ω) - Represents the angle between the vernal equinox and the point where the orbit crosses the equatorial plane (going north)

• Argument of perigee (ω) - Deﬁnes the orientation of the ellipse (in which direction it is ﬂattened compared to a circle) in the orbital plane, as an angle measured from the ascending node to the perigee.

• True anomaly (ν) - deﬁnes the angle in the plane of the ellipse, between perigee and the position of the orbiting object at any given time.

There are several ways to calculate r and v from the classical orbital elements or in the other way, to calculate the orbital elements form r and v. The following equations are used to pass from r and v to the orbital elements and vice versa.

1. Orbital elements from r and v Mission Analysis 17

Figure 2.1: Orbital elements. Source [12].

h = r × v (2.8)

n = k × h (2.9)

k = (0,0,1) (2.10)

1 v e = v2 − r − (r · v)v (2.11) v h r i

h2 p = (2.12) µ

e = |e| (2.13)

hk cosi = (2.14) h

Ω ni Ω o cos = n n j>0 → <180 (2.15)

ω n·e ω o cos = n·e ek>0 → <180 (2.16) 18 Mission and Thermal Analysis of the UPC Cubesat

ν r·e ν o cos = r·e r · v>0 → <180 (2.17)

2. r and v from the orbital elements

r = rcosνP + rsinνQ (2.18)

p r = (2.19) 1 + ecosν

µ v = [−sinνP +(e + cosν)Q] (2.20) r p

These equations are in the Perifocal coordinate system where P and Q are unit vectors in the direction of xω (perigee) and yω (90o rotated axis in the direction of orbital motion and lies in the orbital plane). Rotations are needed in order to obtain r and v in different coordinates systems.

2.3. Perturbation Techniques

The two-body problem is a good approximation to reality in order to study the motion of a satellite in space, but reality is more complex than that. There exist some forces different from the gravitation of the main body such as the atmospheric drag that perturb the orbit in different ways and that in order to have a complete knowledge of the motion of a satellite in space, must be studied and taken into account. These forces can be summarized in four main ones:

• Non-spherical Earth

• Third-body perturbation

• Atmospheric drag

• Solar radiation pressure

A descrpition of the four types of perturbating forces can be found in Appendix D.5.. More- over, unperturbed orbits have constant orbital elements. When perturbing forces are considered, the classical orbital elements vary with time. To predict the orbit this time variation must be determined using techniques of either special or general perturbation.

General perturbation techniques are an approach to analytically solving some aspects of the motion of a satellite subjected to perturb forces [11] [13]. The original equations of motion are replaced with an analytical approximation, which permits analytical integration. Mission Analysis 19

Because the orbital elements are nearly constant, general perturbation techniques usually solve directly for the orbital elements rather than the inertial position and velocity.They are more difﬁcult and approximate, but they allow a better understanding of the perturbation sources and their effects because the output provides a qualitative analysis of the orbital path. Much faster solutions can also be obtained compared with the special perturbation techniques. The underlying concept of the analytical or general approach is the Variation of Parameters (VOP). Based on this method, the solution of the unperturbed system can represent the solution of the perturbed system, provided that the constants in the solution can be generalized as time-varying parameters.

Special perturbations, on the other hand, employ direct numerical integration of the equations of motion, in which the accelerations are integrated directly to obtain velocity and position [11] [13]. The special techniques are deﬁned in terms of speciﬁc force models and initial conditions. These methods have lately become more popular because of the enhancement in the computational power available to scientists and engineers which reduces the computation time. These types of perturbations use different techniques to calculate the orbit such as the Cowell and the Encke method.

Cowell is the simplest and most straight-forward of all the perturbation techniques [11]. It was developed in the early 20th century but it has been rediscovered in many forms many times. Cowell method simply writes the equations of motion of the object in study including all the perturbations and then integrates them step-by-step numerically. The main advantage of this method relays in the simplicity of formulation and implementation. Any number of perturbations can be handled at the same time but the cost of computation is bigger than that of Encke method especially when it gets close to large attracting bodies. More precisely, this method is in general 10 times slower than Encke method, which is an aspect to take into consideration while designing the mission analysis.

Encke method is in general far more complex than Cowell, although it appeared half a century earlier [11]. In contrast to Cowell, where the sum of accelerations is integrated together, in the Encke method the difference between the primary acceleration and all perturbing accelerations is integrated. Encke method is much faster than Cowell due to the ability to take larger integration step sizes when it is near a large attracting body, which at the end reduces the number of integration steps. Although this method is normally 10 times faster than Cowell, this is only applied to interplanetary trajectories. When talking about Earth satellites, the difference is reduced considerably since Encke is only 3 times faster than Cowell.

Cowell and Encke methods are ways to calculate the same parameters but they both need the integration of their equations in order to get results. Therefore a discussion of numerical integrators is appropriate. This integration can be performed in different ways depending on the problem you are considering. The decision of choosing the best integrator to your problem is extremely important. There are two main types of numerical integrators, single- step and multi-step.

Runge-Kutta is an example of single-step method [11]. It is actually a family of varying order methods where higher ones have higher accuracy. These methods are stable and do not require a starting procedure. They are relatively simple, easy to implement, have a small truncation error and the step size is easily changed. 20 Mission and Thermal Analysis of the UPC Cubesat

The multi-step integrators are predictor-corrector type, where the predictor gives an initial estimate, and the corrector further reﬁnes the result [11]. The predictor-corrector types are known to give better accuracy but at the expense of more complexity. Gauus-Jackson is an example of this type of integration methods. It is also known as the sum squared method and it is one of the best and most used numerical integration methods for trajectory problems of the Cowell and Encke type. Its predictor alone is more accurate than most of the other predictor-corrector methods.

For both methods, the ultimate accuracy is obtained by varying the integrator’s step-size.

2.4. Orbit Prediction

Predicting the orbits of satellites is an essential part of the mission analysis and has impacts on the power system, attitude control and thermal design of the satellite.

Furthermore, it is the starting point in planning whether a proposed mission is feasible and how the satellite needs to be designed. The computation of the orbits of artificial satellites around planets such as the Earth has been studied in detail in the last century. The problem of computing the orbits of satellites, however, is not straight forward. The main factors affecting the orbit of a satellite are the perturbations studied in the previous section. These effects have important considerations for different types of satellite orbits. For example satellites in LEO are strongly affected by the non-spherical nature of the Earth and even atmospheric drag. Satellites out in geostationary orbit, however, are sufficiently far from the Earth and these effects are much more little. The gravitational pull of the Sun and Moon, however, does play a significant role in the evolution of their orbits. Effects such as atmospheric drag and radiation pressure are also very dependent upon the shape, size and mass of the satellite.

Orbits and their models are not perfect, and so are calculated analytically or simulated numerically for future time steps to predict where a satellite will be in the future as we have seen with the perturbations techniques. The purpose of these orbit propagators is to provide high accuracy in predicting the position of a satellite. There exist a wide range of propagators, each one with different levels of accuracy, depending on how many perturb forces are taking into consideration. A list of the propagators used by the simulation software STK is presented below.

• Two-body

• J2

• J4

• HPOP (High Precision Orbit Propagator)

• SGP4 (Simpliﬁed General Perturbations)

• LOP (Long Term Propagator) Mission Analysis 21

• Astrogator

Fig. 2.2 illustrates which perturbating forces each propagator is considering.

Figure 2.2: Forces taken into account by each propagator

Depending on the STK version available for the user, some of the propagators cannot be used. This affect the outputs obtained thus not all the propagators have the same accuracy. Let’s explain ﬁrst the general differences between them further than analyzing the forces each one takes into account.

Extracting the conclusions from Vallado [14] regarding the accuracy of the propagation methods, the propagation techniques can be classiﬁed in low, medium or high accuracy. A breakout like this distinguishes the state-of-the-art (high), from the routine numerical operations (med), and from the analytical ones (low).

• Low >500 m

• Medium 500-10 m

• High <10 m

Low routines are designed for general propagation techniques. The accuracy can be quite limited (as in the two-body case), or have some approximations for drag, resonance, etc, as in the SGP4 examples. The mean element theory in particular can be tailored to give more or less accuracy, depending on the mission needs. Examples of these classes of propagators are the analytical ones of the two-body, J2, J4 or SGP4.

Medium accuracy force models are the ones using the 4th order Runge-Kutta and high accuracy methods are those using the numerical integration of the 8th or 12th order Gauss- Jackson or Adams-Bashforth among others.

An example of ’high’ accuracy method in STK is the HPOP propagator. HPOP uses a numerical integration method to propagate the satellite state in the J2000 reference frame. Available integration techniques in STK are the Runge-Kutta method of 7th or 8th order, the Burlirsch-Stoer method and the Gauss-Jackson method of 12th order. 22 Mission and Thermal Analysis of the UPC Cubesat

The two-body model introduced and analyzed by Kepler is perhaps the ﬁrst mathematical model for any orbit determination. This model can be described as the basic propagator. It does not consider any perturbation so secular and periodic changes are absent.

The J2 and the J4 propagators take in consideration the non-spherical Earth. J2 Pertur- bation includes the point mass effect as well as the dominant effect of the asymmetry in the gravitational ﬁeld. J4 additionally considers the next most important oblateness effects (the second order of J2 term and the J4 in addition to J2). None of these propagators model atmospheric drag, solar radiation pressure or third body gravity but are an advance to the basic two-body propagator.

These propagators are often used in early studies to perform trending analysis. J2 is often used for short analyses (weeks) and J4 for long analyses (months, years). Moreover, the solutions produced by these two propagators are approximate, based upon Keplerian mean elements. In general, forces on a satellite cause these elements to drift over time (secular changes) and oscillate (periodic changes). In particular, the J2 and J4 terms cause only periodic oscillations to semi-major axis, eccentricity and inclination, while producing drift in argument of perigee, right ascension and mean anomaly. In contrast, STK’s J2 and J4 propagators model only the secular drift in the elements.

Because the main propagators used in the simulations are the SGP4 and the HPOP due to they have been analyzed more precisely.

2.4.1. SGP4 Propagator

SGP4 is an acronym for Simpliﬁed General Perturbations No. 4. The SGP4 is a semi- analytical propagator that uses the two-line mean element (TLE) sets to propagate a satellite’s orbit over time.

The original analytic theory, Simpliﬁed General Perturbations (SGP), was developed by Aeronutronic-Ford. In 1965, Max Lane began developing a slightly different analytic theory. His work, along with contributions by Ken Cranford, resulted in the Simpliﬁed General Perturbations Theory No. 4 (SGP4). In the early 1970s, the original SGP analytic theory was replaced by a version of the Air Force General Perturbations Theory No. 4 [15].

In 1980, the equations for a form of analytical propagation (SGP4) were presented in the Spacetrack Report Number 3 [16]. The report contained both equations, and FORTRAN source code. The form of the two-line element sets describing the satellite orbital parameters remains the same today as in 1980. It was not until 1998 that a follow-on paper was finally published that summarized the current mathematical theory of SGP4. The paper (Hoots, 1998) was presented at the US Russian workshop, and it is significant for several reasons, not the least of which is the availability of the mathematical technique for the first time in almost 20 years. It will take time to modify any one of the multitudinous versions of SGP4 that exist on the Internet today because no intermediate formulations were available. This is a quite strange example of a propagator because it has become a ’standard’ even though very little information is generally available.

Regarding Vallado, the accuracy of SGP4 is typically about 1 km in position [14]. At a Mission Analysis 23 distance of for example 300 km this could cause a pointing error of up to 0.2 deg. TLE data older than a few days are likely to be considerably less accurate than this.

While sensor sites and most military users have changed to SGP4, the practice of providing element set data that can be used in either SGP or SGP4 has been preserved. The primary difference between the two element sets is the formulation of mean motion and the atmospheric drag representation. While the SGP is a Kozai-based theory, SGP4 is a Brouwer-based theory.

The Brouwer-based theory is based on the Mean Element Theory. This theory began with the work of Lagrange himself, and has been developed further by many people over many years. It is a formal mathematical theory for approximating motion by separating the effects of fast motions and slow motion.

Moreover, the most widely used form of mean element theory is based upon averaging the differential equations of motion over a fast-moving angular variable, and then using these averaged equations to predict the motion of the slowly varying elements. The term ”mean element” does not refer to a numerical average of a sampling of the element and is not related to statistics at all.

STK currently implements two mean element theories, the Kozai-based theory and the Brouwer-based theory. The Brouwer-based theory has been implemented in two versions. The Brouwer Short Mean Element Refers to the mean elements considering only the short period terms while the Brouwer Long Mean Element type refers to the mean elements considering both the short period and long-period terms.

2.4.2. HPOP Propagator

The HPOP (High Precision Orbit Propagator) is a high accuracy special perturbation propagator. It is not available in the basic edition of the STK. For this reason, the Professional version must be used.

The HPOP uses numerical integration of the differential equations of motions to generate ephemeris [9] [13]. Several different force modeling effects can be included in the analysis, including a full gravitational field model, third-body gravity, atmospheric drag and solar radiation pressure. Several different numerical integration techniques and formulations of the equations of motion are available. The different formulations aid in computational efficiency while preserving accuracy. Because of the many parameter settings available for the user, a precise model of the force model environment for almost any satellite can be specified, and thus a highly precise orbit ephemeris can be generated.

Note that a high precision is not without costs because of two reasons. First, the user is responsible for choosing force model settings appropriate to the situation being modeled and second, ephemeris generation takes more computational time and effort than analytical propagation, which simply evaluates a formula.

Force models can be used to deﬁne a precise representation of a satellite’s force environment for use in HPOP analysis. A complete list of the available force models is presented 24 Mission and Thermal Analysis of the UPC Cubesat

Appendix D.6..

2.5. Ground Track and Accessibility

When designing a satellite, you do not only expect to launch it into orbit but to also contact it while it is in space and obtain data, give him orders or whatever the type of mission requires. It is in this context where the ground track of the satellite becomes important.

A ground track is the projection of the satellite’s orbit onto the surface of the Earth. A satellite ground track is an imaginary path along the Earth’s surface which traces the movement of an imaginary line between the satellite and the center of the Earth. In other words, the ground track is the set of points at which the satellite will pass directly overhead, or cross the zenith, in the frame of reference of a ground observer.

So the importance of it, relay in the fact that the satellite can only be contacted from Earth from parts of the planet that can ’see’ the satellite, which it is in fact, close to the ground track of the satellite. This ability to contact the satellite can be deﬁned as accessibility and the bases where the satellite can be ’seen’ from Earth as ground station. STK allows introducing some constraints to the accessibility. Examples of constraints are the azimuth and elevation angle or the range.

The number of ground stations distributed all over the world would deﬁne the probability that a certain satellite can be contacted in a period of time. This probability would also depend on the type of mission (orbit) of the satellite and on the geographical conditions of the ground stations. By simulating these parameters, the accessibility of a satellite can be obtained.

2.6. Lighting

When the satellite is orbiting the Earth, its position to the Sun is continuously changing. This position affects the satellite in a very important way. If the satellite is directly exposed to the sunlight or it is under the shadow of the Earth will affect in the heat absorbed on the satellite and the solar radiation pressure perturbation force. This amount and variation of heat affects in the thermal design of the satellite and its lifetime.

STK allows the users to perform calculations on the time a satellite would be exposed to direct sunlight or lighting, umbra and penumbra (which is partial light and partial shadow) (Fig. 2.3). Mission Analysis 25

Figure 2.3: Description of the umbra and penumbra effects. Source [17].

2.7. Simulation results

Finally, in this chapter, the simulations performed in the STK are presented in a clear and clean way. The tables and graphics are accompanied by the explanations of the results and the conclusion of them.

Different types of simulations have been carried out. It is recalled that a very huge amount of data has been analyzed during the simulations. The impossibility to include all of it in the ﬁnal report has caused to only prsenet resuming tables and graphics containing the mean results. In order to understand what are the simulations performed and how this simulations have been conducted, a STK user’s guide is presented in Appendix E.

The data that is provided in the website of ESA is the one that has been taken into account in the simulations since the UPC Cubesat is expected to be launched in one of the future launch opportunities of the Vega launch vehicle [6]. Moreover, there has been some data that has been maintained constant in all the simulations while other has been changed to produce comparatives in the simulations. The base data used in the simulation is presented in Table 2.1.

The shape and inclination of the orbit has been extracted from one paper published by ESA where they said they were still deciding between the 350 km circular orbit and another one of 350x1200 km [7]. The comparative of these two orbits can be observed below in this section.

In addition, some propagators have been used during the simulations. The SGP4 has been the standard one due to its reliability when compared to the other analytical ones and because other ones more interesting are not allowed when using the basic version of STK. Moreover, by means of the evaluation license of the STK, another propagator has been used, the HPOP, which is presented and compared with the SGP4 at the end of this section. 26 Mission and Thermal Analysis of the UPC Cubesat

Table 2.1: Data used in the simulations Standard Values Orbit shape 350x350 km Orbit inclination 71◦ (ﬁxed) Mean motion 15.7312 revs/day Propagator SGP4 Elevation angle 15◦ Time of Simulation Beginning 1 Jul 2010 End 1 Jul 2011

Finally, simulations have been carried out during a year in order to obtain consistent results independently of the lifetime of the satellite. The beginning of the simulation has been randomly chosen in a near future.

Note that additional tables and graphics concerning the simulations of this section can be found in Appendix F.

2.7.1. Total and individual access

In this analysis, the accessibility to the satellite is computed. The Cubesat operator can choose either to detect the satellite by its own ground station or to cooperate with the existing worldwide network of Cubesat ground stations. Nowadays, more than eighty universities, colleges and research institutions are involved in it.

Figure 2.4: STK map illustrating all the ground stations of the Cubesat network. Source [STK].

The ﬁrst step in order to obtain results was to insert in STK the UPC ground station to Mission Analysis 27 later introduce the entire ground stations of the worldwide network (see Fig 2.4). Access simulations were carried out in all the ground stations introduced and then classiﬁed in an excel table sorted by time. In other words, the output reports from each ground station were a set of start and stop times of each occasion the satellite was accessed. A ground station can access the satellite an average of 700 times a year, which multiplied by each ground station, gives a total of 56.000 times a year. The problem resides in the comprehension of the data because in instance, a ground station maybe is accessing the satellite at the same moment that another one or maybe more than one. So the total time the satellite was being accessed was not the sum of all the seconds the ground stations were seeing it since some of these seconds were overlapped with others.

The solution was to sort by start time all the accesses independently of the ground station the access was. Then, the last thing to do was to develop a simple algorithm to sum all the seconds comparing if the time to sum was overlapped with the previous accesses. This algorithm gave the total clean time the satellite was accessed during a year. In Table 2.2, a comparative between using the UPC ground station or the entire network is presented along with the duration per day, which gives a better understanding of the time the satellite is being accessed.

Table 2.2: Access time comparative between UPC and global network Total Duration Mean Duration/Day Network 359.3305 sec 9845 sec UPC 140.095 sec 384 sec

It is beyond any doubt that cooperation is of extremely beneﬁt for everyone. Getting involved in the worldwide network can give the UPC operators more than 25 times more access time than using one unique ground station. This time would allow the operators and users to better control and use the satellite. Speaking in terms of a day, the UPC satellite could be operated 2.75 hours instead of 6 minutes a day, which is a considerable increase in time.

Moreover, simulations regarding the differences of access time between ground stations have been performed. The importance of the geographical situation of a ground station along with the type of orbit of a satellite are the major elements of differences when comparing ground stations. A table containing the differences on access between ground stations is presented Appendix F. A table (see Table 2.3) showing the difference between the more and the less accessed ground stations and a graphic (see Fig. 2.5) comparing all of them is presented in the main body of the work.

Table 2.3: Access time comparative between the more an less accessed ground stations Ground Station Number of accesses Max # Accesses NTNU 2130 Min # Accesses Malaysia 528 28 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.5: Number of accesses for all the ground stations during a year (Part I).

Figure 2.6: Number of accesses for all the ground stations during a year (Part II).

The graphics in Fig. 2.5 and Fig. 2.8 allows us to indicate some conclusions about accessibility. It is important to recall that the most accessed ground station, NTNU, is situated at very high latitudes, about latitude 63o ,and the less accessed one, Malaysia, is located, in contrast, in very low latitudes, about 3o. We can extract from these results that for our Mission Analysis 29 case the latitude is someway proportional to the amount of time you can access a satellite at that point. In fact, this is only validated to our case because a satellite orbiting in an equatorial orbit would only be accessed from ground stations close to the equator. In other words, the closer the ground station is in latitude to the inclination of the orbit you are trying to access, the more time you would be able to ’see it’. In our case, the satellite is orbiting with an inclination of 71o, so the ground stations closer to this latitude can access the satellite more time.

If we simulate the accessibility to a ground station situated in some random point of the Earth but at latitude 71o, we obtain that you would access the UPC satellite 17.5 minutes a day. That is an increase of 0.3 minutes to the duration per day to NTNU (the most accessed one) and of 13 minutes to Malaysia (the less accessed one). This is due to the fact that the radius of the equator is the biggest one in terms of latitude and that the radius of the latitudes becomes smaller when approaching both poles. So, a satellite with 71o of inclination would pass through each latitude the same times, but because higher latitudes have smaller radius than smaller latitudes, the satellite would pass through the same point more times in higher ones than in smaller ones. For example, a satellite with a polar orbit would pass through both poles in each revolution but would have to wait a very long time to pass through a determinate point in the equator more than one time.

2.7.2. Lighting

Space is a very harsh environment. Temperatures can change in very little time depending if you are in direct sunlight or in the shadow of some body. In our case, the UPC Cubesat would orbit in a low orbit so at some points it would face directly the Sun but in other cases, it would be at the shadow of the Earth. Depending on which of these situations the satellite is, it would have one temperature or another. The results obtained would be important for the thermal design of the satellite because this subsystem is the one in charge of maintaining a suitable temperature for all the components of the satellite at any time. Tables and graphics about the time the satellite is at lighting (direct sunlight), penumbra (partial sunlight) and umbra (total shadow of the Earth) are presented in Table 2.4 and in Fig. 2.7. Remember that these results are valid for simulations of one year.

Table 2.4: Lighting properties Max Dur. Min Dur. Mean Dur. Total Dur. Mean Dur./Day Lighting 9.7 days 6.4 min 64.5 min 20949334 sec 15.94 h. Penumbra 40.8 min 0.13 min 0.24 min 152889 sec 0.12 h. Umbra 36.4 min 0.4 min 32.2 min 10434988 sec 7.94 h.

Results show us that more than a 66% of the time the satellite is orbiting the Earth, it is directly facing the Sun. It is also of notice that the maximum continuous time the satellite would be facing the Sun is of 9.7 days. This is due to the fact that the orbit at some point would have an orientation in the sense that it would all face to the Sun in some subsequent revolutions.

So this value, 9.7 days, would be the reference value that would be used during the ther- 30 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.7: Total duration lighting percentages mal analysis of the Cubesat. If the components of the satellite can handle the amount of upcoming heat ﬂuxes for 9.7 days without pause, the satellite would resist all other situations.

2.7.3. Orbit shape comparative

The inconsistence of the data given by ESA about the characteristics of the orbit is a problem when designing the mission analysis of the UPC Cubesat. Information given show that it has not been decided yet which orbit the Cubesat would have [7]. At ﬁrst, a circular orbit of 350 km of altitude seemed to be the ﬁnal decision but they later stated that maybe they would switch to an elliptical one with a perigee of 350 km and an apogee of 1200 km. In this section of the results, a comparative between these two orbits is made along with the advantages and drawbacks of each one. Simulations are done with respect to the changes that would occur in accessibility and lighting due to the fact that these two parameters are of great interest in the context of the project but it is important to state that more comparatives regarding other parameters can be performed.

1) Accessibility

Depending on the time you can access the satellite, the operators would be able to perform give more orders to the satellite, download more data or just have more time to control it. The shape of the orbit affects this time independently of its inclination. Simulation results with the most and less accessed ground stations and the UPC ground station are presented in Table 2.5 and Fig. 2.8.

Results prove that indeed, a higher altitude would beneﬁt the control of the satellite. The UPC Cubesat would be able to be seen more time during the day, which is translated to more security to the Cubesat. The main drawback of increasing the orbit apogee to 1200 km would be the more difﬁcult communication with it. A major distance means a bigger Mission Analysis 31

Table 2.5: Number of accesses vs. orbit shape for UPC, NTNU and Malaysia UPC NTNU Malaysia 350x350 km 732 2130 528 1200x350 km 1250 2397 901

Figure 2.8: Number of accesses vs. orbit shape comparative for UPC, NTNU and Malaysia power to transmit and which nearly all the times is translated to a bigger antenna which also means more weight to the satellite. It is true that this increase in distance is not a very big problem to deal with but it is important to take it into account.

2) Lighting

Table 2.6: Lighting properties for a 1200x350 km orbit Max Dur. Min Dur. Mean Dur. Total Dur. Mean Dur./Day Lighting 17.4 days 2.3 min 88.2 min 23437346 sec 17.84 h. Penumbra 40.5 min 0.12 min 0.27 min 142178 sec 0.11 h. Umbra 36.6 min 1.6 min 30.2 min 7957779 sec 6.05 h.

The comparative (see Table 2.6 and Fig. 2.9) shows that with a higher orbit, the amount of lighting is increased considerably. The effect of this increase in direct upcoming solar radiation is translated in a harsher environment to the Cubesat. The maximum duration of direct sunlight is increased from 9.7 to 17.4 days, which is almost the double. The thermal control of the satellite should be more effective and important in this situation because the components inside the satellite would be operating in a wider range of temperatures.

Moreover, umbra and penumbra times are smaller now due to the fact that the satellite is more far away from the Earth and the shadow of it would be smaller. Anyway, values are nearly the same as in the 350 km circular orbit so little changes to the thermal design of the UPC Cubesat should be considered because of umbra and penumbra effects. 32 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.9: Lighting properties comparative for a 1200x350 and a 350x350 km orbit

2.7.4. Elevation angle comparative

The elevation angle is the constraint we insert in the simulations in order to obtain a given degree of reliability of the results. A picture describing this effect is shown in Fig. 2.10.

This elevation angle can have any value you want to insert from 0o to 90o. In a lot of simulations and reports this elevation angle is set to 15o. This angle is thought to be the one where objects orbiting around the Earth can be ’clearly seen’. In other simulations this angle is reduced to 10o, 5o or even 0o, which is the line of sight, because the equipment and the simulations they are performing allow doing that. In the UPC Cubesat simulations the default value is 15o but we wanted to compare what differences arise when changing the elevation angle to 5o and 0o (See Fig. 2.11).

The lighting of the satellite would not be affected due to this angle so accessibility is the only parameter this comparative is considering. Note, that a comparative with respect to the shape of the orbit or other parameters can be also performed.

Clearly, a smaller value of elevation angle is translated to more accessibility. It is enormous the change with only 15o but certainly, the mean values show that with an elevation angle of 5o, we get almost 3 times more access time than with the default value of 15o. Moreover, if the constraint applied is the one of the line of sight, 0o, which means indeed no constraint but the one of the Earth, we get about 5 times more access time than with 15o. So the question is why the constraint is not applied to the lower one, and the answer is clear, Mission Analysis 33

Figure 2.10: Access scheme for different elevation angles because the data obtained during the time the satellite is between 0o and 15o it is not reliable due to reception and transmission errors.

2.7.5. Propagator comparative

STK basic edition allows the user to simulate using four different propagators. These are the two-body, the J2, the J4 and the SGP4 propagators. Each one has different levels of reliability depending on which perturbations effects take into account as seen in the previous section. The values used with the SGP4 propagator are resumed in Table 2.7.

Table 2.7: SGP4 data B*term 8e-5 (fixed) Atmospheric density 6.98e-12 (fixed) Cd 2.2 (fixed)

SGP4 computes the atmospheric drag with a simple model that uses parameters such as the B* term (only used in the SGP4 code) [15] [18]. The B* term is related to the ballistic coefficient of the satellite and needs the value of the drag coefficient of the satellite and the atmospheric density at the altitude of the orbit to be computed. These three values, the B* term, the drag coefficient and the atmospheric density have been used only when performing simulations with the SGP4 propagator and have been set to the fixed values shown in Table 2.7.

A comparative to show the differences between the propagators of the basic edition is performed along with a comparative between the SGP4 the HPOP propagator. As in the previous comparative, simulations are performed with respect to accessibility and lighting for one year.

1) Accessibility 34 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.11: Access comparative for different elevation angles for UPC, NTNU and Malaysia ground stations

Results (see Table 2.8 and Fig. 2.12) show different aspects of the simulations done. First, it can be concluded that the J2 and the J4 propagator give nearly the same results. The J4 term is really little compared to the J2 term so when simulating, very little differences are obtained from them. Moreover, the two-body propagator also gives very similar results to the ﬁrst two ones. This propagator does not take into account any perturb force other than the gravitational force between the satellite and the Earth. Results are similar to the J2 and the J4 propagator but it is true that are signiﬁcantly different due to this fact. The two-body propagator simulation gives more access time to the satellite in all ground stations.

Finally, the SGP4 propagator does give considerably different results than the other three propagators. This propagator is the most accurate of the four because it takes into ac-

Table 2.8: Mean access duration/day vs. propagator for UPC, NTNU and Malaysia ground stations UPC NTNU Malaysia SGP4 6.397 min 17.213 min 4.469 min J4 8.216 min 22.568 min 5.612 min J2 8.215 min 22.566 min 5.612 min Two-Body 8.225 min 22.596 min 5.614 min Mission Analysis 35

Figure 2.12: Mean access duration/day comparative for each propagator for UPC, NTNU and Malaysia ground stations count not just the gravitational force from the Earth, the J2 and the J4 terms but also the atmospheric drag and the gravity coming from the Sun and the Moon. Because of all these effects, the duration a ground station is accessing the satellite is smaller than that of the other propagators. In each revolution, the satellite is closer to the Earth because of the atmospheric drag so the time passing through the ground stations would be smaller each time.

2) Lighting

When simulating the effects of propagators in lighting (see Fig. 2.13), similarities arise from those regarding accessibility. J2 and J4 propagators have nearly exact results while the other ones, the two-body and the SGP4 propagators are signiﬁcantly different. We can extract from Fig. 2.13 that when using the SGP4 propagator less lighting is obtained due to the constant approach to the Earth because of the effect of the atmospheric drag. Using the two-body propagator, we obtain also less lighting probably because of the inconsis- tency of the data thus just the gravitational force of the Earth is taken into account as a perturb acceleration.

3) SGP4 vs. HPOP

Basic simulations have been performed using the SGP4 propagator but because this is not the most accurate one, STK professional has been used in order to be able to use a more accurate one, in this case the HPOP, described in section 2.4.2.. A comparative between these two propagators in accessibility and lighting is performed. Values used 36 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.13: Lighting properties comparative for each propagator in this comparative are the previous mentioned for the SGP4 propagator. For the HPOP propagator the values used are resumed in Table 2.9.

Table 2.9: HPOP data Force model WGS84 EGM96 Radiation coefﬁcient 1.5 Shadow model Dual cone Third bodies Sun and Moon Integrator Gauss-Jackson 12th Interpolation Lagrange Cd 2.2 (ﬁxed) Atmospheric model Jacchia-Roberts

Note that in contrast to the previous propagator comparative, this one has been carried out using the 1200x350 km orbit because in order to simulate the circular 350 km orbit using HPOP it is necessary to reach at least one year of lifespan, which it is not obtained, as it will be seen in the lifespan simulation.

Signiﬁcant differences (see Fig. 2.14) are observed between the two propagators in terms of accessibility which are due to the accuracy of each of them. HPOP in contrast to the SGP4 propagator takes into account the solar radiation pressure. Moreover, the atmospheric drag simulation is more accurate in the HPOP thus the SGP4 one uses a very simple one. Also, the HPOP propagator uses a numerical simulation in contrast to the analytical one used in the SGP4. This numerical simulation gives us a better accuracy but in the other hand needs more computational time to obtain the results.

Lighting (see Fig. 2.15) is observed to be the same no matter which propagator is being Mission Analysis 37 used so when referring to the thermal analysis of the satellite it is not a big problem which propagator is being used.

Figure 2.14: SGP4 and HPOP mean access duration/day comparative for UPC, NTNU and Malaysia ground stations for a 1200x350 Figure 2.15: SGP4 and HPOP lighting prop- km orbit erties comparative for a 1200x350 km orbit

2.7.6. Simulation time comparative

Up to this point, all the simulations have been carried out for the same time duration, one year, as it has been said at the beginning of this section. In this subsection, different start and stop times, all of them with duration of one year, will be simulated in order to understand the importance of the launch date. Table 2.10 shows the comparative between ﬁve different simulations times.

Little differences are observed from the simulations performed for the UPCSat. It is important to know which is the exact launch date but this parameter is not vital in order to conduct the simulations for the accessibility of a satellie. Moreover, the launch date is a ﬂexible parameter due to the fact that the launch depends of lots of factors such as the weather conditions for instance. By all these reasons, a perfect simulation is impossoble to perform until the launch is ﬁnally performed.

2.7.7. Satellite lifetime

The lifetime or lifespan of a satellite is one of the most important issues while designing the mission analysis. This is the time the satellite will be orbiting the Earth without falling to the atmosphere and burn up. The satellite decay is inﬂuenced by the atmospheric drag of the 38 Mission and Thermal Analysis of the UPC Cubesat

Table 2.10: Simulation time comparative for different gaps of one year for the accessibility properties of the UPC ground station Start time - Stop time # AY # AD TDY (sec) MDA (sec) DD (min) 1/07/2010-1/07/2011 1352 3.704 618909.291 457.773 28.260 1/07/2011-1/07/2012 1353 3.706 619918.150 458.180 28.306 1/07/2012-1/07/2013 1352 3.704 619269.088 458.039 28.277 1/07/2013-1/07/2014 1352 3.704 619521.971 458.226 28.288 1/07/2014-1/07/2015 1356 3.715 619681.348 456.992 28.295 AY: Access/year AD: Access/day TDY: Total duration/year MDA: Mean duration/access DD: Duration/day

Earth. This atmospheric drag is greater the more closer you are to the Earth, so satellites orbiting in LEO will be more prone to this effect. In other words, if you want your satellite in LEO to have a lifetime of many years you need a propulsion system in order to power up your satellite when approaching to dangerous altitudes. Moreover, this propulsion system is most of the times so heavy that Cubesat cannot handle it so no propulsion can be added to them. In conclusion, CubeSats would have a little lifespan that has to be calculated in order to deﬁne the mission duration.

To simulate this parameter the Lifetime module form the STK professional edition is needed so it has been simulated using the evaluation license provided by STK. Propagators used in this simulation have been the SGP4 and the HPOP, previously commented. Moreover, simulations have been performed in the two cases of the orbit shape. The atmospheric density model and the radiation coefﬁcient used is the same as for the HPOP simulations.

Table 2.11: SGP4 and HPOP lifetime comparative for different shapes of orbit. Launch date: 1 Jul 2010 Propagator Reentry Date # orbits Lifetime 350x350 SGP4 23 Nov 2010 2313 145 days HPOP 04 Oct 2010 1522 95 days 1200x350 SGP4 05 Jan 2018 40962 7.5 years HPOP 02 Sep 2016 33745 6.2 years

The difference in lifespan (see Table 2.11) is considerable different when orbiting in the circular 350 km or in the 1200x350 km orbit independently of the propagator used. A difference of years it is observed which it is due to the effects of the atmospheric drag. So if the satellite operators want a longer lifespan they must use the 1200x350 km because with the other one they would only be able to operate the satellite less than half a year.

The propagator comparative shows that using the HPOP it is obtained a lower lifetime. The difference between them relays mainly in the different atmospheric drag model used Mission Analysis 39 by each one. These models are commented previously and this gives the difference it is observed in lifespan. In case of doubt, it is advised to take for real the value of the HPOP, which it has a more reliable drag model. A graphic of the apogee, perigee and eccentricity variation of the 1200x350 km orbit using the HPOP propagator is presented in Fig. 2.16.

Figure 2.16: Apogee, perigee and eccentricity variation during the HPOP 1200x250 km satellite lifetime

Finally, an analysis of the evolution of the accessibility during the lifetime of the satellite adviced. The satellite will be colser to the Earth in each revolution, and this variation in altitude will cause a change in the accessibility to the satellite. Table 2.12 shows the accessibility variation in the UPC ground station for its ﬁsrt four years of lifetime. The simulations have been carried out for the 1200x350 km orbit with the HPOP. Note that every year that passes, the access time to the ground station is lower due to the fact that the apogee is lower. The explanation is similar to the one of the orbit shape comparative subsection.

Table 2.12: Accessibility evolution of the ﬁrst four years for the UPC ground station Year # AY # AD TDY (sec) MDA (sec) DD (min) 1st year 1352 3.704 618909.291 457.773 28.260 2nd year 1197 3.279 472752.965 394.948 21.586 3rd year 1109 3.038 382026.196 344.478 17.444 4th year 985 2.698 283595.182 287.914 12.949 AY: Access/year AD: Access/day TDY: Total duration/year MDA: Mean duration/access DD: Duration/day 40 Mission and Thermal Analysis of the UPC Cubesat

2.7.8. Orbital elements variation due to perturbations

The variation of the classical orbital elements due to each of the perturb forces is studied now. The HPOP allows to choose which perturb forces are considered during the simulations. Simulations comparing the changes that the perturb forces produce in each orbital element are performed along with the conclusions that can be extracted from it.

Changes can be either secular or periodic. Existent literature has proved that these changes can be summarized in Table 2.13 [19].

Table 2.13: Theoretical orbital elements changes due to each perturbation force. Source [19]. Non-spehrical Third-bodies Atm. drag SRP a P P P S P e P P P S P i P P P S P Ω P S P S P P S ω P S P S P P S ν P S P S P P S S: Secular P: Periodic

Some of these variations are very little compared to other ones but at the end, little or big, these should be the variations that would appear in the simulations.

Furthermore, several simulations have been performed with the HPOP in the 1200x350 km orbit in order to understand the effects produced by the non-spherical Earth, the third- bodies, the atmospheric drag and the solar radiation pressure.

Simulations have started with no perturbations and then the perturb forces have been added gradually in order to understand which effects are producing to the orbital elements. The effects of perturbations in the orbital elements have been compared when they are applied and when the Earth does not have any perturbations.

So the ﬁrst thing that has been done, has been to deactivate the effects produced by third- bodies, atmospheric drag and solar radiation pressure. Then the only thing to change is the shape of the Earth. To simulate the effects of a spherical Earth, the central body gravity of the Earth has been changed. In the potential gravity model (see Eq. 2.21), n is the degree of the harmonics and m is the number of the harmonics. By giving values to these two terms, you are telling STK where to truncate the series.

∞ n µ R⊕ 2 U = [1 − ∑ ∑ Jnm( ) ρnm(senφ)cos(m(λ − λnm))] (2.21) r n=2 m=0 r

Simulations have been performed for the spherical model of the Earth (n=1), the ellipsoidal (n=2) and the default value of STK (n=21). Mission Analysis 41

Results prove that a spherical Earth model without perturbations do not produce any changes to the orbital elements as expected. When simulating for an elliptical model and a more complex (n=21) model of Earth, periodic changes appear in the semi-major axis, the eccentricity and the inclination. Secular variations are observed in the argument of perigee and the right ascension of the ascending node (RAAN). Graphics comparing the inclination and the argument of perigee in a spherical and an elliptical model of Earth are presented in Fig. 2.17 and Fig. 2.18 respectively.

Figure 2.17: i comparative between a spher- Figure 2.18: ω comparative between a ical (blue) and an elliptical (green) model of spherical (blue) and an elliptical (green) Earth model of Earth

Very little differences arise between the elliptical model of Earth and the complex model of Earth. The secular changes of the argument of perigee and the RAAN are the same. The same occur with the periodic changes of the semi-major axis and the inclination. The only difference appears in the eccentricity where a new periodic pattern is added as it can be seen in Fig. 2.19 and Fig. 2.20.

Figure 2.19: e variation for an elliptical model Figure 2.20: e variation for a complex model of Earth of Earth

Third-bodies effects produce periodic changes to all orbital elements and also secular effects to all of them in contrast to what it is said in the theory, where only secular changes are appreciated in the argument of perigee and in the RAAN. It is true, though, that all this changes are very little and do not produce important consequences. Examples in the eccentricity and the RAAN are presented in Fig. 2.21 and Fig. 2.22 respectively.

When talking about the atmospheric drag, the theoretical assumptions are correct. Peri- odic changes are produced in all orbital elements, being the ones in the semi-major axis, eccentricity and inclination little compared to their secular variations. So, secular variations are produced in those three orbital elements. Graphics presenting the variations in 42 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.21: e variation produced by third- Figure 2.22: Ω variation produced by third- bodies effects bodies effects the semi-major axis and the argument of perigee are presented in Fig. 2.23 and Fig. 2.24 respectively.

Figure 2.23: a variation produced by atmo- Figure 2.24: ω variation produced by atmospheric drag spheric drag

Finally, the solar radiation pressure effect is studied. Effectively, periodic variations occur mainly in the semi-major axis, the eccentricity and the inclination. Curiously, these changes have a period of exactly one year. Moreover, the argument of perigee and the RAAN have secular variations and little periodic changes. Examples of it can be seen in Fig. 2.25 and Fig. 2.26 where the changes produced by the solar radiation pressure in the semi-major axis and the RAAN respectively are presented.

Figure 2.25: a variation produced by SRP Figure 2.26: Ω variation produced by SRP

To conclude this section, the effects produced by all the perturb forces together is simulated. Results show that the main perturbations that affect our satellite are those of the Mission Analysis 43 atmospheric drag and the non-spherical Earth. Although as we have seen, all the perturb forces affect our satellite, the ones of the third-bodies and the solar radiation pressure are so little that are not appreciated when simulated with the other ones. Graphics with the semi-major axis, eccentricity and inclination are presented in Fig. 2.27, Fig. 2.28 and 2.29 respectively. The argument of perigee and the RAAN are mostly the same as in the non-spherical Earth simulation.

Figure 2.27: a variation produced by all per- Figure 2.28: e variation produced by all perturb forces turb forces

Figure 2.29: i variation produced by all perturb forces

2.7.9. Dispersion analysis

Finally, a dispersion analysis is performed for our satellite. We deﬁne as dispersion the differences observed between two or more bodies relatively close to each other in a determinate period of time. In other words, we have simulated the orbits of three satellites and then we have observed the variations in their position relative to each other during one week. The main satellite is the UPCSat and the two other ones (UPC1 and UPC2) are possible variations of the CubeSat separated 50 m up and down respectively of the main one (see Fig. 2.30).

In order to simulate the dispersion, the inputs introduced in STK are in the spherical coordinate type, only changing the value of the range for the two additional satellites. Simula- tions, as have been said previously, have been performed for seven days and the results obtained can be seen in Fig. 2.31. The propagator used is the HPOP for a 1200x350 km orbit.

In just seven days, a difference in range of approximately 90 km is observed. It is important to mention that this difference is not in altitude. If we compute the altitude difference in one 44 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.30: Initial conditions of the dispersion analysis

Figure 2.31: Position of the satellites seven days later week, the results have a lower value as can be seen in Fig. 2.32. The evolution of the range in time can be seen in Fig. 2.33 for the difference between the UPCSat and UPC1. It is important to underline that there is an increasing periodic variation along with the linear secular variation of about 12 km/day. Changing the perturbations taken into account does not make visible changes to the results so only the distance travelled by the satellite is the one affecting these differences in range. It is important to mention though, that satellites placed at 50 m. in the longitudinal plane of the main satellite will only have small periodic variations and no secular variations. It can be said, that the distance will keep constant at the initial range.

Figure 2.32: Final positions of the satellites

Changing the initial values of separation gives different results (see Table 2.14). It is evi- dent that for less initial range, fewer differences in time will be obtained. Mission Analysis 45

Figure 2.33: Range variation between UPCSAT and UPC1 for 50m separation

Table 2.14: Range variations between UPCSAT and UPC1 for different initial separations Initial Range Final Range 10 m 13.5 km 50 m 85 km 100 m 176 km 500 m 900 km 1 km 1790 km

Moreover, the dispersion analysis can be seen as a ’safety’ simulation for the UPCSat because it tells you the possible difference you will obtain with a certain margin, in this case, a margin of 50 meters. In other words, the simulations are not perfect and so are simulated with some safe margins in order to obtain a range of values where the results can be. This margin can be more or less constraint that would depend on the reliability of the data and the simulator.

For the UPCSat, a margin of 50 meters has been studied for a simulation of seven days. It is highly probable that the real position of the satellite in these seven days will be within the range obtained. It is necessary to distinguish between the vertical range and the horizontal range, which is deﬁned in Fig. 2.34.

The ellipsoid presented is the range within the values of the real position of the Cubesat will be in seven days in the future. 46 Mission and Thermal Analysis of the UPC Cubesat

Figure 2.34: Range within the satellite will be for a 50m margin Thermal Analysis 47

Chapter 3

THERMAL ANALYSIS

3.1. Theoretical Background

Thermal analysis is a branch of materials science that studies the properties of materials as they change with temperature. Moreover, in a satellite deﬁnition, the thermal subsystem helps protect electronic equipment from extreme temperatures due to intense sunlight or the lack of sun exposure on different sides of the satellite’s body.

In general, there are three main heat transfer modes [20]: conduction, radiation and convection. In space, due to the extremely low residual pressure, only conduction and radiation modes are present.

Conduction is the process by which heat is transferred through a solid and it is governed by Fourier’s Law (see Eq. 3.1).

q¯ = −k∇¯ T (3.1)

Where q¯ is the heat ﬂow rate vector, k the thermal conductivity and T the temperature.

On the other hand, radiation heat transfer is governed by Stefan-Boltzmann’s Law stating that the black-body irradiance is proportional to the fourth power of its temperature (see Eq. 3.2).

E = σ · T 4 (3.2)

Where σ is the Stefan-Boltzmann constant, 5.67 · 10−8W/m2K4.

All these formulas concern the black-body which is an idealized object absorbing all radiant energy from any direction or wavelength and emitting in any direction isotropically. The radiated energy of the black-body only depends on its temperature, but a real body can absorb, reflect and transmit radiation energy so that absorptivity a, transmittivity t and reflectivity r quantities are defined, all wavelength and angular dependent (see Fig. 3.1) [20].

As there is no perfect black-body in practice, the emissivity ε(l) is deﬁned as the ratio between the energy emitted by a surface to that of a black body at the same temperature.

Absorptivity and emissivity can either be hemispherical or directional and either total or spectral. The second Kirchhoff’s law states that for a given direction q, directional spectral absorptivity and emissivity are equal (see Eq. 3.3). 48 Mission and Thermal Analysis of the UPC Cubesat

Figure 3.1: Incident light characteristics

ε(q,l) = α(q,l) (3.3)

But, in general, this is not true with total hemispherical values mainly because of their strong wavelength dependence. Both α and ε varies with the angle of incidence but they are assumed to follow the Lambert’s law stating that directional absorptivity/emissivity is proportional to cos(q) (maximum for normal incident angles and null for tangential ones).

The source temperature of incident radiation is different to that of the satellite, so it is important to distinguish the spectra that it is being used in every moment. As the temperature of a spacecraft lies in the 70K-700K range, the emitted radiation is infrared (IR wavelengths range from 5 to 50 µm). But the source of the main incident radiation is the Sun which can be considered as a blackbody emitting at 5776K. This temperature lies in the visible spectrum (UV wavelengths range from 0.2 to 2.8 µm). Actually, thermal engineers adopted a standard convention. These call ε the hemispherical emissivity in infrared wavelengths and α the solar absorptivity in visible wavelengths. A speciﬁc material has speciﬁc values of α and ε, which determine its thermal behavior.

Furthermore, in order to perform the thermal analysis of a satellite, the heat sources of the satellite must be studied. A satellite orbiting Earth has several heat sources [20]. A scheme of the heat ﬂuxes of a satellit is presented in Fig. 3.2.

• Direct solar ﬂux depending on the distance to the Sun, with a mean value around 1367 [W/m2] at 1AU (1414 [W/m2] at winter solstice and 1322 [W/m2] at summer solstice).

• Albedo planetary reﬂected radiation. For Earth, the mean reﬂectivity is assumed to be near 30% .

• Earth infrared radiation. Earth can be modeled as an equivalent black-body emitting at a mean temperature of 255 K with a ﬂux of 237 [W/m2].

• Internal dissipated power in electronic components.

During the eclipse, only two heat sources are still present. These are Earth’s infrared and internal dissipation and the spacecraft will be cooler. The temperatures of the satellite tend Thermal Analysis 49

Figure 3.2: Heat ﬂuxes of a typical satellite orbiting Earth. Source [20]. thus to vary in a cyclic way along the orbit, rising in sunshine and dropping during eclipse. The sky, called Deep Space, is the main source of cold and can be seen as a black body emitting at 3K. This temperature represents the radiation of the stars, the galaxies and the Cosmic Microwave Background.

This change in temperature along the orbit of a satellite is the main reason to require a thermal control for a satellite. But why we need it? Mainly because the electronics and equipments of the satellite can only operate in certain temperatures ranges. Typical limits are:

• Electronics: -15oC - +50oC

• Mechanisms: 0oC - +50oC

• Rechargeable batteries: 0oC - +20oC

So the design goal is to maintain these temperatures ranges. In other words, is to have a thermal balance in the satellite (Total of absorbed plus generated heat equal to heat lost to space - see Eq. 3.4) [20].

QSun + QAlbedo + QEarth + Qinternal = Qradiated (3.4)

Q is the heat generated by all these elements. In practice, in a grey body with a determinate emissivity and absorptivity, the heat ﬂux absorbed and radiated (J) would be the ones of the expressions of Eq. 3.5 and Eq. 3.6 respectively.

Jabsorbed = α · Jincident (3.5)

4 Jradiated = εσT (3.6)

So in order to achieve thermal balance we can rewrite Eq. 3.4 as Eq. 3.8 and Eq. 3.8.

4 Js · Asolar · α + Ja · Aalbedo · α + Jp · Aplanet · α + Qinternal = Ar · εσT (3.7) 50 Mission and Thermal Analysis of the UPC Cubesat

Qinternal + α(Js · Asolar + Ja · Aalbedo + Jp · Aplanet) T 4 = (3.8) Ar · ε · σ

Note that the areas of the Eq. 3.7 and Eq. 3.8 refer to the areas of the satellite facing the upcoming heat ﬂux.

Moreover, by taking a look to Eq. 3.8, it can be assumed that balance temperature can only be controlled by varying the ratio of α to ε. By calculating this temperature for all the possible cases the satellite can be involved in, you will get the range of temperatures for each part of the satellite.

3.2. Simpliﬁed results

The thermal balance of a satellite can be computed by several ways by means of the theory explained in the previous section. In this section, the thermal balance of the Cubesat of UPC is computed with some simpliﬁcations in its most critical cases. In Appendix G the steps to follow are shown in order to have a more realistic analysis.

First, the cases for which the satellite will be simulated must be presented. The thermal analysis is performed to obtain among others the range of temperatures the satellite and its components will be operating in. So, in order to obtain this range, the maximum and minimum temperatures must be calculated, these are the critical temperatures. These temperatures will be obtained when simulating the critical cases.

The critical cases known as cold and hot cases are the ones where there is a minimum and maximum incoming heat ﬂux respectively. In Table 3.1, there is a sample of the values used in the two cases.

Table 3.1: Data used in the hot and cold case calculations Hot Case Cold Case Orbit Direct sunlight Total eclipse Solar heat flux 1414 W/m2 0 Albedo coefficient 0.35 0.25 Earth heat flux 260 W/m2 220 W/m2

The hot case will be that in which the Sun is constantly radiating the satellite with its maximum heat flux, the greatest albedo coefficient and the Earth is also radiating with its maximum heat flux. By contrast, in the cold case no incoming heat flux of the Sun would appear because the satellite will be in eclipse, there will not be albedo too and the Earth temperature would emit its lower heat flux.

Note that internal dissipation is not taken into account in the calculations. This is the first simplification of the problem. The second simplification of the problem is to assume the satellite as a sphere of radius r (see Eq. 3.10). With this second simplification we will obtain the same areas for the incident radiation coming from the Sun, the Earth’s albedo Thermal Analysis 51 and the Earth radiation (see Eq. 3.9). So, the only parameters considered are the heat fluxes of the Sun, albedo and Earth and the values of emissivity and absorptivity of the materials.

2 A = Asolar = Aalbedo = Aplanet = πr (3.9)

2 Ar = 4πr (3.10)

The heat ﬂux coming from the albedo can be expressed by Eq. 3.11 where F is the view factor (fraction of radiation arrives at one surface after leaving another surface).

Ja = Albedocoe f f · F · Js (3.11)

We solve the equations for a black-body (emissivity and absorptivity equal to one) considering low thermal inertia (see Eq. 3.12).

α · A · (Js + Ja + Jp) Js + Ja + Jp α T 4 = =( ) · (3.12) Ar · ε · σ 4 · σ ε

Js + Ja + Jp 1414 +(0.35 · 0.15 · 1414) + 260 HotCase ⇒ T 4 = = ⇒ T = 23.16oC 4 · σ 4 · 5.67x10−8 (3.13)

Js + Ja + Jp 0 +(0.25 · 0.15 · 0) + 220 ColdCase ⇒ T 4 = = ⇒ T = −90.94oC (3.14) 4 · σ 4 · 5.67x10−8

So, a satellite considered as a black-body would have a temperature ranges from -91 to 23 oC (see Eq. 3.13 and Eq. 3.14) which is quite a lot for the design requirements of the components of the satellite. However, the UPCSat is not a black body so new calculations will be performed with the values of emissivity and absorptivity the materials of our Cube- sat. Results are exposed in Table 3.2. Note that the values of emissivity, absorptivity and other characteristics of the materials of the UPCSat can be found in Appendix G.1..

Table 3.2: Hot and cold case temperatures for different materials Surface Al. frame Al. panel (with Kapton) Solar panels α 0.08 0.87 0.91 ε 0.15 0.81 0.81 Hot Case -20oC 28.5oC 32oC Cold Case -117.4oC -87.7oC -85.6oC

Temperatures ranges are not that different from the black-body example. The Kapton layer used in the aluminum panel has improved the cold case temperature and with the 52 Mission and Thermal Analysis of the UPC Cubesat solar panels can form an interesting multilayer face. However, the temperatures shown are for simulations with no thermal inertia. Thermal inertia is the key property controlling the diurnal and seasonal surface temperature variations and is typically dependent on the physical properties of the materials. The temperature of a material with low thermal inertia changes signiﬁcantly during the day, while the temperature of a material with high thermal inertia does not change as drastically.

When thermal inertia is taken into account, cold case temperatures are higher due to the fact that the satellite is most of the time in sunlight and the reduction of temperature during eclipse is slower. Transient analysis is necessary to determine the effects of thermal inertia. However, there is still one parameter to take into account. Internal dissipation causes the temperatures to vary signiﬁcantly and a most realistic analysis should be performed. This analysis has to be performed after the simpliﬁed one and the steps to follow it are presented in the appendix.

3.2.1. STK simulation

By means of the Space Environment and Effects Tool (SEET) module of STK released to commercial uses in the ﬁnal version of STK at the end of 2009 a more complete thermal analysis has been performed. SEET module offers users to perform analysis in the following ﬁelds [9].

• Radiation environment

• Vehicle temperature

• South Atlantic anomaly (SAA)

• Magnetic ﬁled

• Particle impacts

Among a wide range of simulations, SEET module can compute the expected radiation dose rate and total dose due to energetic particle fluxes for a range of shielding thicknesses and materials, the spacecraft’s entrance and exit times through the SAA, the total mass distribution of meteor and orbital debris particles that impact a spacecraft along its orbit during a specified time period or the local magnetic field at the current location. But the most important feature considered in the thermal analysis is the vehicle temperature.

Using the known thermal balancing equations, the vehicle temperature simulation determines the mean temperature of a space vehicle due to direct solar flux, reflected and infrared Earth radiation, and the dissipation of internally-generated heat energy. The user may specify spherical objects, or planar objects with particular orientation, for the computation of temperature. SEET treats the spacecraft as a single isothermal node, where the user specifies bulk thermal characteristics. On the other hand there are several simplifi- cations in the STK calculations. First of all, the satellite is considered as one isothermal node so no multi-nodal analysis can be performed. Heat transfer analysis can’t also be Thermal Analysis 53 made because they depend on the shape of the satellite and STK is also considering the Cubesat as a sphere.

So once considered all the issues presented above, simulations can be run in STK. Several simulations have been performed with different values of emissivity and absorptivity of the materials presented in this section (see Table 3.3) and with simulations with different values of internal dissipation (see Eq. 3.4).

Table 3.3: Hot and cold case temperatures for different materials with no internal dissipation Surface Al. frame Al. panel (with Kapton) Solar panels α 0.08 0.87 0.91 ε 0.15 0.81 0.81 Hot Case -7oC 50oC 50oC Cold Case -100oC -100oC -100oC

Table 3.4: Hot and cold case temperatures for the Al. with Kapton Q 0W 0.5 W 1W Hot Case 50oC 50oC 55oC Cold Case -100oC -85oC -78oC

Table 3.3 refers to the maximum and minimum temperature for each material with no internal dissipation. We can deduce from the graphics that the cold case temperature does not vary with the type of material but that the hot case temperature has important variations.

When analyzing the effects produced by the internal dissipation of the satellite for a determinate material we can conclude that it majorly affects the cold case temperature obtaining lower values for higher internal dissipations. Fig. 3.3 shows the evolution of temperature during a year for the Kapton case with 0.5 W of internal dissipation. It can be extracted from the results that the maximum temperature is obtained for the maximum incoming solar heat ﬂux and the minimum temperature is obtained during the times the satellite is at the shadow of Earth.

Although the results obtained with STK are more accurate than those done previously with de direct application of the equations, these results are still not accurate enough. A multi-nodal analysis with the speciﬁc shape of the satellite must be carried out with other available software. Speciﬁc information about the multi-nodal analysis and the available software can be found in Appendix G. 54 Mission and Thermal Analysis of the UPC Cubesat

Figure 3.3: Temperature evolution for the Al. with Kapton with 0.5 W of internal dissipation Thermal Analysis 55

CONCLUSIONS

The objective of this work has been to analize some aspects the mission analysis of the Cubesat the UPC is willing to launch through the launch opportunities of ESA. In order to perform this analysis, a complete theory reminder of orbital dynamics has been explained along with the issues concerning propagators and perturbations. Simulations have been performed using the commercial software STK which provides a great variety of parameters to simulate with really big reliability. As a complement of the mission analysis, the ﬁrst steps to perform a thermal analysis have also been presented in this work. Basic theory of thermal design of the satellite has been explained along with basic calculations of thermal balance by means of the equations available. Additionally, the state-of-the-art of the Cubesat program has also been explained.

In addition, it is important to remark that this project is one of the ﬁrst ones regarding the project of the Cubesat of the UPC. Information contained is not the ultimate one and changes may arise during the project in the next months or years. In this manner, this Master Thesis is one of the ﬁrst steps of the project, which can be seen as a guideline for future studies concerning the UPC Cubesat.

Furthermore, results have shown similarities with other related-type projects and with the theory explained. The inclusion of the UPC Cubesat in the existing network of Cubesat developers is advised in order to have more time to access the satellite while orbiting the Earth. It is also recommended to use the HPOP in future simulations. This propagator shows little differences with SGP4 but although little, these differences can change signif- icantly the orbital elements during a year. Perturbations affecting the satellite have been studied separately and all together concluding that the atmospheric drag and the non- spherical shape of the Earth are the ones that affect more the satellite with both secular and periodic changes no matter which orbit is being used. Actually, the shape of the orbit is really important in the mission analysis. Among other differences, using the two different shapes of orbit can produce discrepancies in lifetime of 6 or 7 years.

Alternatively, results from the thermal analysis show variations of temperature from -85oC to 50oC for the standard case. It is important to recall that the results obtained are sim- pliﬁed ones, and that important characteristics of the simulation such as the shape of the satellite and the internal dissipations should be studied deeply. However, important variations are observed for different values of internal dissipation and the emissivity/absorptivity ratio is the main parameter which we can play in order to change these temperature variations.

Finally, this project and its results could be a great help and a useful information tool for other studies concerning the Cubesat of the UPC. Further work should be done when knowing the speciﬁc requirements of the launch vehicle in matter of orbit in order to obtain more accurate results. It is also highly recommended to continue the thermal analysis of the satellite form the end point of this project using available simulation tools as ESATAN, ESARAD or ANSYS. 56 Mission and Thermal Analysis of the UPC Cubesat BIBLIOGRAPHY 57

BIBLIOGRAPHY

[1] http://directory.eoportal.org/presentations/7053/8502.html, CubeSat Concept and De- ployer Services.

[2] CubeSat community website, http://www.cubesat.org/

[3] http://www.publico.es/ciencias/181834/cubesat/satelite/elevado/cubo, Interview with Jordi Puig-Suari (Spanish).

[4] List of CubeSat satellite missions, http://mtech.dk/thomsen/space/cubesat.php

[5] Space Exploration Technologies Corporation - SpaceX, http://www.spacex.com/

[6] ESA education website, Call for CubeSats on the Vega maiden ﬂight, http://www.esa.int/SPECIALS/Education/SEMSJ8QR4CF 0.html

[7] ESA Education and Vega Programme Ofﬁce, Call For CubeSat Proposals, ESA, 2008

[8] Camps, A., UPCSat: A CubeSat Project , UPC CubeSat Project Presentation, 2008

[9] Analytical Graphics, Inc. website, http://www.stk.com/

[10] Chobotov, V. A., Orbital Mechanics, Washington AIAA, 1996.

[11] Bate, R. R., Mueller, D. D. & White, J. E., Fundamentals of astrodynamics, New York Dover, 1971.

[12] NASA website, http://spaceﬂight.nasa.gov/realdata/elements/graphs.html

[13] Farahmand, M., Orbital propagators for horizon simulation framework, California Poly- technic State University, 2009.

[14] Vallado, D. A., A summary of astrodynamics standards, AAS/AIAA Astrodynamics Specialists Conference, Quebec city, 2001.

[15] Vallado, D. A., SGP4 Orbit Determination, American Institute of Aeronautics and As- tronautics, 2007.

[16] Hoots, F. R. & Roehrich, R. L., Spacetrack report NO. 3, Defense Documentation Center, 1980.

[17] Umbra, penumbra and antumbra, Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Umbra

[18] Canadian Astronomy, Satellite Tracking and Optical Research, http://www.castor2.ca/03 Astronomy/03 TLE/B Star.html

[19] Vallado, D. A., Fundamentals of astrodynamics and applications, 3rd ed., Microcosm Press, Hawthorne, Ca, 2007.

[20] Fortescue, P. W., Stark, J. & Swinerd, G., Spacecraft systems engineering, 2nd edition, John Wiley and Sons, 2003. 58 Mission and Thermal Analysis of the UPC Cubesat

[21] Astronautix, Encyclopedia Astronautica, http://www.astronautix.com/

[22] The CubeSat Kit of Pumpkin, Inc., http://www.cubesatkit.com/

[23] Jacques, L., Thermal Design of the Oufti-1 nanosatellite, University of Liege,` Master Thesis, 2009.

[24] ESA Thermal Control website, http://www.esa.int/TEC/Thermal control/SEMTT0CE8YE 0.html

[25] ANSYS, Inc. webpage, http://www.ansys.com/

[26] ANSYS, Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/ANSYS APPENDIXES

Cubesat launches and participants 61

Appendix A

CUBESAT LAUNCHES AND PARTICIPANTS

A.1. Past launches

Table A.1: CubeSats past launches

Batch # Date LV Launch Site Prim. Payload 1 30 Jun 2003 Rokot Plesetsk, Russia MIMOSA, MOST Name Institution Type Deployer Status CUTE-1 TITECH Single T-POD Operational XI-IV U.Tokyo Single T-POD Operational CanX-1 U.Toronto Single P-POD No Contact DTUsat DTU Single P-POD No Contact AAU CubeSat AAU Single P-POD Some Contact QuakeSat Stanford Triple P-POD Operational 2 27 Oct 2005 Komos-3M Plesetsk, Russia SSETI Express Name Institution Type Deployer Status NCube2 NTNU Single T-POD No Contact UWE-1 U.Wurzburg¨ Single T-POD Operational XI-V U.Tokyo Single T-POD Operational 3 22 Feb 2006 M-V Uchinoura, Japan ASTRO-F (AKARI) Name Institution Type Deployer Status CUTE-1.7+APD TITECH Double T-POD Operational 4 26 Jul 2006 DNEPR Baikonur, Kazaskstan Belka Name Institution Type Deployer Status ION U.Illinois Double P-POD Launch failure Sacred U.Arizona Single P-POD Launch failure KUTEsat U.Kansas Single P-POD Launch failure ICE Cube 1 Cornell Single P-POD Launch failure RINCON 1 U.Arizona Single P-POD Launch failure SEEDS Nihon Single P-POD Launch failure HAUSAT 1 Hankuk Single P-POD Launch failure Ncube 1 NTNU Single P-POD Launch failure MEROPE Montana Single P-POD Launch failure AeroCube-1 Aerospace Single P-POD Launch failure CP2 CalPoly Single P-POD Launch failure CP1 CalPoly Single P-POD Launch failure ICE Cube 2 Cornell Single P-POD Launch failure Mea Huaka U.Hawaii Single P-POD Launch failure 5 16 Dec 2006 Minotaur MARS,USA TacSat-2 Continued on next page 62 Mission and Thermal Analysis of the UPC Cubesat

Table A.1 – continued from previous page Batch # Date LV Launch Site Prim. Payload Name Institution Type Deployer Status GeneSat-1 Ames Triple P-POD Operational 6 17 April 2007 DNEPR Baikonur, Kazakstan EgyptSat Name Institution Type Deployer Status CP4 CalPoly Single P-POD Partial operations AeroCube-2 Aerospace Single P-POD Short contact CSTB-1 Boeing Single P-POD Operational MAST Tethers Single P-POD Partial operation CP3 CalPoly Single P-POD No contact CAPE-1 U.Lousiana Single P-POD Partial operation Libertad-1 Sergio Arbol. Single P-POD Operational 7 28 Apr 2008 PLSV SDSC, India CartoSat-2A, IMS-1 Name Institution Type Deployer Status CanX-2 U.Toronto Triple X-POD Operational CUTE-1.7+APD2 TITECH Double X-POD Operational Delﬁ-C3 TUDELFT Triple X-POD Operational AAUsat-2 AAU Single X-POD Operational Compass One FH Aachen Single X-POD Operational SEEDS 2 Nihon Single X-POD Operational 8 3 Aug 2008 Falcon-1 Omelek, Marshall Is. Trailblazer Name Institution Type Deployer Status NanoSail-d Ames Triple P-POD Launch failure PREsat Ames Triple P-POD Launch failure 9 23 Jan 2009 H-IIA TNSC, Japan GOSAT, SDS-1 Name Institution Type Deployer Status KKS-1 JAXA Double T-POD No contact STARS JAXA Single T-POD No contact PRISM JAXA Single T-POD Operational 10 19 Jan 2009 Minotaur MARS, USA TacSat-3 Name Institution Type Deployer Status AeroCube-3 Aerospace Single P-POD Contacted HawkSat Hawk Inst. Single P-POD Contacted CP6 CalPoly Single P-POD Contacted PharmaSat Ames Triple P-POD Contacted 11 23 Sep 2009 PLSV SDSC, India OceanSat-2 Name Institution Type Deployer Status Uwe-2 U.Wurzburg¨ Single P-POD Contacted ITUpSAT1 ITU Single P-POD Contacted SwissCube EPFL Single P-POD Contacted BEESAT TU Berlin Single P-POD Contacted Cubesat launches and participants 63

A.2. Upcoming launches

Table A.2: CubeSats upcoming launches

Batch # Date (approx) LV Launch Site Prim. Payload 1 TBC-delayed TBC (Space X) TBC TBC Name Institution Type Deployer Status InnoSat Malaysia Triple P-POD TBL CubeSat ATSB Triple P-POD TBL 2 23 Jan 2010 Taurus-XL Vandenberg Glory Name Institution Type Deployer Status Explorer-1 PRIME Montana Single P-POD TBL KySat-1 U.Kentucky Single P-POD TBL HERMES U.Colorado Single P-POD TBL ASTREC 1 (Backup) U.Florida Single P-POD TBL 4 Feb 2010 Minotaur IV Kodiak FASTSAT Name Institution Type Deployer Status OREO Ames Triple P-POD TBL RAX U.Michigan, SRI Triple P-POD TBL 3 Oct 2010 VEGA (maiden) Kourou LARES Name Institution Type Deployer Status SeissCube-2 EPFL Single P-POD TBL Xatcobeo U.Vigo Single P-POD TBL UniCubesat U.Roma Single P-POD TBL Robusta UM2 Single P-POD TBL atmoCube U.Trieste Single P-POD TBL e-st@r U.Torino Single P-POD TBL OUFTI-1 U.Liege Single P-POD TBL Goliat U.Bucharest Single P-POD TBL PW-Sat Pol.Warsaw Single P-POD TBL UWE-3 (Backup) U.Wurzburg¨ Single P-POD TBL HiNCube (backup) NUC Single P-POD TBL TBC: To be conﬁrmed TBL: To be launched

A.3. Cubesat participants 64 Mission and Thermal Analysis of the UPC Cubesat

Table A.3: List of Cubesat participants. Source [2].

Institution Complete Name Location TITECH Tokyo Institute of Technology Tokyo, Japan U.Tokyo The University of Tokyo Tokyo, Japan U.Toronto University of Toronto Toronto, Canada DTU Technical University of Denmark Lyngby, Denmark AAU Aalborg University Aalborg, Denmark Stanford Stanford University Palo alto, CA, USA NTNU Norwegian University of Science and Tech. Trondheim, Norway U.Wurzburg¨ University of Wurzburg¨ Wurzburg,¨ Germany U.Illinois University of Illinois Illionois, USA U.Arizona The University of Arizona Tucson, AZ, USA U.Kansas The University of Kansas Lawrence, KS, USA Cornell Cornell University Ithaca, NY, USA Nihon Nihon University Tokyo, Japan Hankuk Hankuk Aviation University Goyang, Korea Montana Montana State University Montana, USA Aerospace The Aerospace Corporation USA CalPoly California Polytechnic State Univesity San Luis Obispo, CA, USA U.Hawaii The University of Hawaii Hawaii, USA Boeing The Boeing Company Chicago, II, USA Tethers Tethers Unlimited Bothell, WA, USA U.Lousiana University of Lousiana Lousiana, USA Sergio Arbol. Universidad Sergio Arboleda Bogota,´ Comlobia TUDELFT Delft University of Technology Delft, The Netherlands FH Aachen Fachhochschule Aachen Aachen, Germany Ames NASA Ames Research Center Moffett Field, CA, USA JAXA Japan Aerospace Exploration Agency Japan Hawk Inst. Hawk Institute for Space Sciences Pocomoke City, MD, USA Malaysia-USM Universiti Sains Malaysia Pulua Pinang, Malaysia Malaysia-UTM Universiti Teknologi Malaysia Johor Bahru, Malaysia Malaysia-UniMAP Universiti Malaysia Perlis Arau Perlis, Malaysia ATSB Astronautic Technology (m) Sdn BHD Shah Alam, Malaysia U.Kentucky University of Kentucky Lexington, KN, USA U.Colorado University of Colorado Boulder, CO, USA U.Florida University of Florida Gainesville, FL, USA EPFL Ecole Polytechnique Federale de Lausanne Lausanne, Switzerland U.Vigo Universidad de Vigo Vigo, Spain U.Roma Universita` degli Studi di Roma ”La Sapienza” Roma, Italy UM2 Universite´ de Montpellier 2 Montpellier, France U.Trieste Universita` degli Studi di Trieste Trieste, Italy U.Torino Universita` degli Studi di Torino Torino, Italy U.Liege Universite´ de Liege` Liege,` Belgium U.Bucharest University of Bucharest Bucharest, Romania Continued on next page Cubesat launches and participants 65

Table A.3 – continued from previous page Institution Complete name Location Pol. Warsaw Warsaw University of Technology Warsaw, Poland NUC Narvik University College Narvik, Norway U.Michigan University of Michigan Michigan, USA SRI SRI International California, USA ISU Istanbul Technical University Istanbul, Turkey TU Berlin Technical University of Berlin Berlin, Germany 66 Mission and Thermal Analysis of the UPC Cubesat Cubesat launch vehicles 67

Appendix B

CUBESAT LAUNCH VEHICLES

This appendix presents the current launch vehicles that have launched at least one Cube- sat with a short description of its most important characteristics and a ﬁnal table (Table B) comparing the main characteristics of each one.

Rokot

Rokot is a Russian space launch vehicle that can launch a payload of 1,950 kilograms into a 200 kilometer orbit with 63◦ of inclination. It is a derivative of the UR-100N intercontinental ballistic missile (ICBM), supplied and operated by Eurockot Launch Services. The ﬁrst launches started in the 1990s from Baikonur Cosmodrome out of a silo. Later commercial launches commenced from Plesetsk Cosmodrome using a launch ramp specially rebuilt from one of the Kosmos-3M rocket. The cost of a commercial launch is about $14 million.

Kosmos-3M

The Kosmos-3M is a Russian space launch vehicle. It is a liquid-fueled two-stage rocket, ﬁrst launched in 1967 and with over 410 successful launches to its name. It uses nitrogen tetroxide as an oxidizer to lift roughly 1400 kg of payload into orbit. PO Polyot has manufactured these launch vehicles in the Russian town of Omsk for decades, though the latest digitally controlled rockets are now ofﬁcially referred to as ’Kosmos 3MU’. It is scheduled to be retired from service in 2011. By the meantime, it is launched from Plesetsk and Kapustin Yar Cosmodromes.

Dnepr

Dnepr is a three-stage Ukrainian space launch vehicle named after the Dnieper River. It is a converted ICBM used for launching artiﬁcial satellites into orbit, operated by launch service provider ISC Kosmotras. The ﬁrst launch, on April 21, 1999 and can launch up to 4500 kilograms into LEO. Its current launch sites are Baikonur and Dombarovsky.

Minotaur I and IV

The Minotaur is a family of American solid fuel rockets derived from converted Minuteman and Peacekeeper intercontinental ballistic missiles. They are built by Orbital Sciences Cor- poration. Minotaur I is an American expendable launch system derived from the Minute- man II missile. It is used to launch small satellites for the US Government. Initially Minotaur I launches were conducted from the Vandenberg Air Force Base. Starting with the launch of TacSat-2 in December 2006, launches have also been conducted from the Mid-Atlantic Regional Spaceport on Wallops Island. The Minotaur IV, also known as Peacekeeper is an American expendable launch system derived from the Peacekeeper missile. It is scheduled to make its maiden ﬂight in early 2010, with the SBSS satellite for the United States Air Force and some CubeSats. Minotaur IV launches will be conducted from Vandenberg Air Force Base, the Mid-Atlantic Regional Spaceport, and from the Kodiak Launch Complex. 68 Mission and Thermal Analysis of the UPC Cubesat

M-V

The M-V rocket was a Japanese solid-fuel rocket designed to launch scientiﬁc satellites. The Institute of Space and Astronautical Science (ISAS) began developing the M-V in 1990 at a cost of 15 billion yen. It has three stages and it was capable of launching a satellite weighing 1.8 tons into an orbit as high as 250 km. The ﬁnal launch was that of the Hinode on 22 September 2006. It was launched from the Uchinoura Space Center.

H-IIA

The H-IIA is a family of Japanese liquid-fuelled rockets providing an expendable launch system for the purpose of launching satellites into geostationary orbit. It is manufactured by Mitsubishi Heavy Industries (MHI) for the Japan Aerospace Exploration Agency, or JAXA. Launches occur at the Tanegashima Space Center.

PSLV

The Polar Satellite Launch Vehicle commonly known by its abbreviation PSLV is an Indian expendable launch system developed and operated by the Indian Space Research Orga- nization (ISRO). It was developed to allow India to launch its Indian Remote Sensing (IRS) satellites into sun synchronous orbits. PSLV can also launch small size satellites into geostationary transfer orbit (GTO). The PSLV has launched 41 satellites into a variety of orbits till date. In April 2008, it successfully launched 10 satellites in one go, breaking a world record previously held by Russia Each launch costs 17 million USD. It is launched form the Satish Dhawan Space Centre.

Falcon 1 and 9

The Falcon 1 and 9 are two rockets developed by Space X, a private American space transport company. The Falcon 1 is a partially reusable launch system designed and it is the ﬁrst successful fully liquid-propelled orbital launch vehicle developed with private funding. Falcon 1 achieved orbit on its fourth attempt, on 28 September 2008, with a mass simulator as a payload. Its current launch site is in Omelek Island, in the Republic of the Marshall Islands but it would also use the Vandenberg facility Falcon 9 is a reusable two-stage-to-orbit, liquid oxygen and rocket-grade kerosene powered launch vehicle. It is scheduled to have its maiden launch in late 2009. Multiple variants are planned with payloads of between 10,450 kg and 26,610 kg to low Earth orbit , and between 4,540 kg and 15,010 kg to geostationary transfer orbit. Moreover, the Falcon 9 will be the launch vehicle for the SpaceX Dragon spacecraft. It will be launched from Cape Canaveral and form Omelek.

Vega

Vega (Fig. B.1) is a European expendable launch system being developed for Arianespace jointly by the Italian Space Agency and the European Space Agency. Development began in 1998 and the ﬁrst launch, which will take place from the Guiana Space Centre in the late 2010. Vega is a single-body launcher composed of three solid-propellant stages and a re-startable liquid-propellant fourth stage. It is 30 m high, has a maximum diameter of 3 m and weighs 137 tons at lift-off. Vega has three sections: the Lower Composite, the Upper Module and the Payload Composite. The Upper Module, known as the Altitude and Cubesat launch vehicles 69

Vernier Upper Module (AVUM), is itself composed of a Propulsion Module and an Avionics Module. The AVUM provides attitude control and axial thrust during the final phases of Vega’s flight to allow the correct orientation and orbit injection of multiple payloads, with final de-orbiting of the stage once maneuvers are completed.

Figure B.1: Vega launch vehicle. Source [6].

Table B.1: Launch vehicles comparative. Source [21].

LV. Country Stages Fuel Payload (T.) Launch Site Launch Price* Rokot Russia 3 Liquid 1’95 LEO Plesetsk 14 million USD Kosmos 3-M Russia 2 Liquid 1’5 LEO Plesetsk 10 million USD 0.775 SSO Dnepr Ukraine 3 Liquid 4’5 LEO Baikonur 10-13 million Dom- USD barovsky Minotaur I USA 4/5 Solid 0’58 LEO Vandeberg 12’5 million USD 0’331 SSO MARS Minotaur IV USA 4 Solid 1’735 LEO Vandeberg -*** MARS Kodiak M-V Japan 3 Solid 1’8 LEO Uchinoura 60 million USD H-IIA Japan 2 Sol./Liq. 4’1/6 GTO Tanegashima 190 million USD PSLV India 4 Sol./Liq. 3’25 LEO Satish 30 million USD 1’06 GTO Dhawan Continued on next page 70 Mission and Thermal Analysis of the UPC Cubesat

Table B.1 – continued from previous page LV Country Stages Fuel Payload (T.) Launch Site Launch Price* Falcon 1 Space X 2 Liquid 0’67 LEO Omelek 7 million USD 0’43 SSO Vandeberg Falcon 9** Space X 2 Liquid 10’45 LEO Cape 3.365 USD LEO 4’54 GTO Canaveral 10.000 USD GTO Omelek Vega Europe 4 Sol./Liq. 1.5 LEO Kourou 23.5 million USD * Launch prices are approximate and depends on the configuration of each launch and its payload. ** Falcon 9 has different configurations. Characteristics are considered for the basic Flacon 9 configuration. *** No price has been found. No launches have been carried out for this LV yet. STK modules 71

Appendix C

STK MODULES

STK Basic is free to all users and is also the core module for all other STK modules [9]. It allows access calculations to be performed between satellites and ﬁxed points on the Earth’s surface (or between satellites). There is also the ability to import satellites from the NORAD public satellite database (which can be updated online from within STK).

STK Professional adds to the Basic Edition with the following abilities:

• High ﬁdelity trajectories

• Aircraft performance models

• Multi-point and group inter-visibility

• Constrained inter-visibility

• Complex sensor modeling

• Integrated 3D analysis tools

• Custom reports and graphs

STK Expert adds a host of additional modules to what is provided in the STK Professional Edition:

• STK/Integration Module (ability to integrate STK with external tools through Connect and/or through Matlab as well as embed STK into a custom application)

• STK/Terrain, Imagery and Maps Module (high-resolution imagery and terrain of the whole world)

• STK/Analyzer (perform trade-off studies and what-if analyses)

• STK/Attitude (integrate custom attitude control laws into STK)

• STK/Communications (high-ﬁdelity link budget analysis)

• STK/Coverage (multi-point access calculations and navigation system analysis)

• STK/Radar (ground, space and airborne radar systems simulation)

Each of the above modules can be purchased individually and added to either the STK/Basic Edition or the STK/Professional Edition.

STK Specialized Analysis Modules are modules that can be added to the Basic, Profes- sional or Expert editions: 72 Mission and Thermal Analysis of the UPC Cubesat

• STK/Astrogator (interplanetary orbits, orbit maintenance, LEOP simulation)

• STK/Conjunction Analysis Tools (In-orbit collision prediction)

• STK/Missile Modeling Tools (missile simulation)

• STK/PODS (Precision Orbit Determination System - to reconstruct satellite orbits through observational data)

• STK/Scheduler (use scarce resources in the most efﬁcient way)

• STK/Space Environment (radiation dose, debris ﬂux, thermal loading) Additional theory for mission analysis 73

Appendix D

ADDITIONAL THEORY FOR MISSION ANALYSIS

D.1. The n-body problem

The N-body problem is effectively a class of problem, which can be described as the problem of taking an initial set of data that gives the positions, masses, and velocities of some set of n bodies, for some particular point in time, and then using that set of data to determine the motions of the n bodies, and to ﬁnd their positions at other time [11] [10].

In this case, the bodies in Space exert a gravitational force to all the other bodies around. The magnitude of this force, as the equation of gravity shows, would depend on the masses of the bodies and the distance between them. So if we take into consideration an N-body problem we would study the gravitational force that these N bodies exert to one of them. In addition, the gravitational force is not the only one acting in a celestial body. There are other forces such as the non-spherical shape of the planets, the propulsive forces, the atmospheric drag, etc.., which should be considered in the problem.

So for each pair of bodies i and j, we have a force j acting on i (see Eq. D.1 and Eq. D.2).

¯ Gmim j r¯ji Fg j = − 2 · (D.1) ri j r ji

r¯ji = r¯i − r¯j (D.2)

If all the n-bodies are applied to the particle i, it is obtained Eq. D.3.

n ¯ m j Fg = −Gmi · ∑ 3 · r¯ji (D.3) j=1 r ji j6=i

This F¯g is the gravitational force of the n-bodies acting on the bodies of study but there are still others forces to consider that it will be called F¯others, so the total force acting on the body is Eq. D.4.

F¯total = F¯g + F¯others (D.4)

Newton’s second law of motion (Eq. D.5) is now ready to be applied.

∑F = mr¨ (D.5) 74 Mission and Thermal Analysis of the UPC Cubesat

Thus we obtain Eq. D.6.

d (m v ) = F (D.6) dt i i total

This time derivative can be expanded to Eq. D.7.

dvi dmi m + v = F (D.7) i dt i dt total

So ﬁnally it is obtained Eq. D.8.

Ftotal m˙ i r¨i = − r˙i (D.8) mi mi

Eq. D.8 is a second order, nonlinear, vector, differential equation of motion which has de- ﬁed solution in its present form. It is here therefore that we have to make some simplifying assumptions and depart from the realities of nature.

It would be assumed no perturbation forces (F¯others = 0) and no external forces such as dmi the propellant coming out from the rocket (vi dt = 0). So the only forces that it would be taken into account are the gravitational ones. Eq. D.8 would be reduced to Eq. D.9.

n m ¨ j ri = −G · ∑ 3 · r ji (D.9) j=1 r ji j6=i

To solve Eq. D.9 in an analytical way, 6n 2nd order equations are needed. For example, for the three-body problem, 18 equations will be needed to solve the problem. The problem relays that there are only 10 equations: six of center of masses (position and acceleration in the three axes, one of conservation of energy and three for conservation of momentum. This is the reason because there is no analytical solution for the three (or more) body problem and it must be performed a numerical simulation to obtain an approximate solution.

Continuing now with the previous equation, it is of interest the study of the motion of a near Earth satellite thus the CubeSat will orbit around the Earth at a low altitude. Therefore, let us rewrite the equation in a different form. If m1 is the Earth m2 is the satellite and all the other masses may be different celestial bodies as the Moon, the Sun, or other planets in the Solar System, then we obtain the following expressions.

n m ¨ j r1 = −G · ∑ 3 · r j1 (D.10) j=2 r j1

n m ¨ j r2 = −G · ∑ 3 · r j2 (D.11) j=1 r j2 j6=2 Additional theory for mission analysis 75

r12 = r2 − r1 (D.12)

r¨12 = r¨2 − r¨1 (D.13)

So:

Gm n m Gm n m ¨ 1 j 2 j r12 = −[ 3 · r12 + G ∑ 3 · r j2] − [ 3 · r21 + G ∑ 3 · r j1] (D.14) r12 j=3 r j2 r21 j=3 r j1

G(m + m ) n r r ¨ 1 2 j2 j1 r12 = 3 · r12 − ∑ Gm j( 3 − 3 ) (D.15) r12 j=3 r j2 r j1

The ﬁrst term of Eq. D.15 is the force acting between the two main bodies (the Earth and the satellite). The other term is the perturbation of the other bodies affecting the force of the two main ones.

In order to determine the force between the Earth and the satellite, the accelerations of the other bodies must be calculated. In Table D.1 there is a comparison of the relative acceleration for a 200 NM Earth satellite.

Table D.1: Relative accelerations form other bodies to a LEO satellite Earth 0.89 Sun 6.0x10-4 Mercury 2.6x10-10 Venus 1.9x10-8 Mars 7.1x10-10 Jupiter 3.2x10-8 Saturn 2.3x10-9 Uranus 8.0x10-11 Neptune 3.6x10-11 Moon 3.3x10-6 Earth oblateness 10-3

From Table D.1, some conclusions can be taken. The acceleration coming from other bodies different from the Earth is very little. Just the effect of the Sun and the Earth oblateness have a considerable value. It is for this reason that some assumptions can be made.

The two body restricted problem is a simpliﬁcation of the n-body problem. The simpliﬁca- tions hypothesis made are the following ones:

• The two bodies are the only ones in Space

• The bodies are spherical and homogenous 76 Mission and Thermal Analysis of the UPC Cubesat

• There are not any external or internal force acting to the system

Therefore, using these assumptions and if the mass of the satellite is m and the mass of the Earth is M, the equation will result as Eq. D.16.

G(M + m) r¨ = r (D.16) r3

The mass of the satellite is very little compared to the mass of the Earth (M ≫ m). Ifa parameter µ = G(M +m) ≈ GM is deﬁned, the equation would result as follows Eq. D.17.

µ r¨ + r = 0 (D.17) r3

µ ≡ GM (D.18)

Eq. D.17 is the two-body equation of motion. This equation can be solved analytically, in contrast to the three body problem or higher that can only be solved numerically.

D.2. The trajectory equation

In order to obtain the trajectory equation, the two body equation of motion must be derived. Previously, some useful information regarding the nature of the orbit must be explained. First, a gravitational ﬁeld is conservative, that is, an object moving under the inﬂuence of gravity alone does not lose or gain mechanical energy but only exchanges one form of energy to another one (kinetic to potential energy). Secondly, the gravitational force is always directed radially toward the center of the large mass. If it is recalled that it takes a tangential component of force to change the angular momentum of a system in rotational motion, it is expected that the angular momentum of the satellite about the center of our reference frame, the center of the Earth, does not change. So the conservation of mechanical energy and of angular momentum would be the constants of motion needed to derive the two body equation of motion and obtain the trajectory equation [11] [10].

1) Conservation of mechanical energy

Multiply the two body equation of motion by r˙:

µ r˙ · r¨ + r˙ · r = 0 (D.19) r3

Since in general a · a˙ = aa˙, v = r˙ and v˙ = r¨, then:

µ v · v˙ + r · r˙ = 0 (D.20) r3 Additional theory for mission analysis 77

So:

µ vv˙+ rr˙ = 0 (D.21) r3

2 Noticing that d v and d −µ µ : dt ( 2 ) = vv˙ dt ( r ) = r2 r˙

d v2 d −µ ( ) + ( ) = 0 (D.22) dt 2 dt r

d v2 µ ( − + c) = 0 (D.23) dt 2 r

C is an arbitrary constant referring to the origin of the energies. Normally, this constant is o because the origin of the energies is at the inﬁnity.

This expression must be a constant that is called speciﬁc mechanical energy (ε). The expression means that the sum of the kinetic energy per unit mass and the potential energy per unit mass of a satellite remains constant along its orbit, neither increasing nor decreasing as a result of its motion.

v2 µ ε = − (D.24) 2 r

2) Conservation of angular momentum

If r is cross multiplied to the two body equation of motion:

µ r × r¨ + r × r = 0 (D.25) r3

Since in general a × a = 0:

r × r¨ = 0 (D.26)

d Noticing that dt (r × r¨) = r˙ × r˙ + r × r¨:

d d (r × r¨) = 0 or (r × v) = 0 (D.27) dt dt

The expression r × v which must be a constant of the motion is simply the vector h, called speciﬁc angular momentum. Therefore, we have shown that the speciﬁc angular momentum h of a satellite remains constant along its orbit and that the expression for h is:

h = r × v (D.28) 78 Mission and Thermal Analysis of the UPC Cubesat

These equations also tells that the satellite must be conﬁned to a plane which is ﬁxed in space, or in other words, in the orbital plane. There are some angles in the orbital plane that must be explained before continuing with the explanation of the trajectory equation.

Figure D.1: Orbital plane angles

γ = Zenith angle

φ = Flight-path angle

From the deﬁnition of the speciﬁc angular momentum:

h = rvsinγ (D.29)

If the expression is expressed in terms of the ﬂight-path angle:

h = rvcosφ (D.30)

Finally, the integration of the two body equation of motion can start. If the original equation is multiplied by h:

µ r¨ × h = − (r × h) (D.31) r3

d The left side of the equation is clearly dt (r˙ × h) and the right side of the equation can be operated as follows:

µ µ µ µ µr˙ (r × h) = (r × v) × r = [v(r · r) − r(r · v)] = v − r (D.32) r3 r3 r3 r r2

Taking into account that times the derivative of the unit vector is also d r µ v µr˙r µ µ dt ( r ) = r − r2 and that r · r˙ = rr˙, the equation can be rewrite as the following expression: Additional theory for mission analysis 79

d d r (r˙ × h) = µ ( ) (D.33) dt dt r

Integrating both sides:

r r˙ × h = µ( ) + B (D.34) r

Where B is the vector constant of integration. Multiplying this equation by r:

r r · r˙ × h = r · µ( ) + r · B (D.35) r

Simplifying this equation:

h2 = µr + rBcosν (D.36)

Where ν is the angle between the constant vector B and the radius vector r. The trajectory equation is obtained by solving this equation for r.

h2/µ r = (D.37) B ν 1 +( µ )cos

If e = B/µ, we obtain the polar equation of a conic section, that it is mathematical equal in form to the trajectory equation and widely used in the study of orbits:

p r = (D.38) 1 + ecosν

Where p is the parameter of the orbit and e is the eccentricity. The eccentricity determines the type of conic section that represents the orbit.

D.3. Type of conics

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. The three types of conics are the hyperbola, ellipse, and parabola (see Fig. D.2). The circle can be considered as a fourth type (as it was by Apollonius in the 200 BC) or as a kind of ellipse. The circle and the ellipse arise when the intersection of cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the ﬁgure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves, though often one is ignored [11] [10]. 80 Mission and Thermal Analysis of the UPC Cubesat

Figure D.2: Types of conic sections

All conic sections can be deﬁned in terms of eccentricity (e). The type of conic section is also related to its semi-major axis (a) and the energy (ε). Table D.2 shows the relationship between eccentricity, semi-major axis energy and type of conic section.

Table D.2: Realationship between the type of conic and e, a and ε Conic Section Eccentricity (e) Semi-major axis (a) Energy (ε) Circle 0 = radius < 0 Ellipse 00 < 0 Parabola 1 ∞ 0 Hyperbola >1 <0 > 0

The eccentricity of a conic section is thus a measure of how far it deviates from being circular.

D.4. Types of orbit

Orbits can be classiﬁed in different types depending on some of their nature. These clas- siﬁcations would only describe the type of orbits that are of interest to the project. Note that may be other orbits not described below.

A ﬁrst classiﬁcation can be made regarding which body is one of the focuses of the orbit.

• Heliocentric orbit: An orbit around the Sun. In our Solar System, all planets, comets, and asteroids are in such orbits.

• Geocentric orbit: An orbit around the planet Earth, such as the Moon or artiﬁcial satellites. Currently there are approximately 2,465 artiﬁcial satellites orbiting the Earth. Additional theory for mission analysis 81

Geocentric orbits can also be classiﬁed regarding to the distance to Earth, to its altitude with respect to Earth.

• Low Earth orbit (LEO): Ranging in altitude from 0-2,000 km.

• Medium Earth orbit (MEO): Ranging in altitude from 2,000 km to just below geosynchronous orbit at 35,786 km.

• High Earth orbit (HEO): Above the altitude of geosynchronous orbit 35,786 km.

Orbits can also be classiﬁed regarding its inclination. An inclined orbit is an orbit whose inclination in reference to the equatorial plane is not 0. A special type of these orbits is the polar orbit.

• Polar orbit: An orbit that passes above or nearly above both poles of the planet on each revolution. Therefore it has an inclination of approximately 90 degrees.

In contrast, a non-inclined orbit is an orbit whose inclination is equal to zero with respect to some plane of reference. A special type of these orbits is the equatorial orbit.

• Equatorial orbit: A non-inclined orbit with respect to the equator.

There is also a classiﬁcation regarding its eccentricity.

• Circular orbit: An orbit that has an eccentricity of 0.

• Elliptic orbit: An orbit with an eccentricity greater than 0 and less than 1 whose orbit traces the path of an ellipse.

• Parabolic orbit: An orbit with the eccentricity equal to 1. Such an orbit has a velocity equal to the escape velocity and therefore will escape the gravitational pull of the planet and travel until its velocity relative to the planet is 0. If the speed of such an orbit is increased it will become a hyperbolic orbit.

• Hyperbolic orbit: An orbit with the eccentricity greater than 1. Such an orbit also has a velocity greater than the escape velocity and as such, will escape the gravitational pull of the planet and continue to travel inﬁnitely.

A Synchronous orbit is an orbit where the satellite has an orbital period that is a rational multiple of the average rotational period of the body being orbited and in the same direction of rotation as that body. This means the track of the satellite, as seen from the central body, will repeat exactly after a ﬁxed number of orbits. Important types of these orbits are the geosynchronous and the sun-synchronous orbits. 82 Mission and Thermal Analysis of the UPC Cubesat

• Geosynchronous orbit (GEO): An orbit around the Earth with a period equal to one sidereal day, which is Earth’s average rotational period of 23 hours, 56 minutes, 4.091 seconds. For a nearly circular orbit, this implies an altitude of approximately 35,786 km (22,240 miles). If both the inclination and eccentricity are zero, then the satellite will appear stationary from the ground. If not, then each day the satellite traces out an analemma in the sky, as seen from the ground.

• Graveyard orbit: This orbit, also called a super-synchronous orbit, junk orbit or dis- posal orbit, is an orbit signiﬁcantly above synchronous orbit where spacecraft are intentionally placed at the end of their operational life.

• Sun-synchronous orbit (SSO): Geocentric orbit which combines altitude and inclination in such a way that an object on that orbit ascends or descends over any given point of the Earth’s surface at the same local time. The surface illumination angle will be nearly the same every time. This consistent lighting is a useful characteristic for satellites that image the Earth’s surface and for other remote sensing satellites.

Other types of orbits with different characteristics from the above mentioned are the following.

• Molniya orbit: Highly elliptical orbit with an inclination of 63.4 degrees and an orbital period of about 12 hours. These orbits are named after a series of Soviet/Russian Molniya (”Lightning” in Russian) communications satellites which have been using this type of orbit since the mid 1960s. A satellite placed in a Molniya orbit spends most of its time over a designated area of the earth as a result of ’apogee dwell’.

• Geostationary orbit (GSO): A circular geosynchronous orbit with an inclination of zero. To an observer on the ground this satellite appears as a ﬁxed point in the sky.

• Escape orbit (EO): A high-speed parabolic orbit where the object has escape velocity and is moving away from the planet.

• Capture orbit: A high-speed parabolic orbit where the object has escape velocity and is moving toward the planet.

• Retrograde orbit: An orbit with an inclination of more than 90◦. In other words, an orbit counter to the direction of rotation of the planet.

• Hohmann transfer orbit: An orbital maneuver that moves a spacecraft from one circular orbit to another using two engine impulses.

• Halo orbits and Lissajous orbits: These are orbits around a Lagrangian point. Orbits near these points allow a spacecraft to stay in constant relative position with very little use of fuel. Orbits around these points are used by spacecraft that want a constant view of the Sun or by missions that always want both the Earth and Sun behind them. Additional theory for mission analysis 83

D.5. Perturbation study

There are four main factors that perturb the orbit.

• Third-bodies

• Non-spherical Earth

• Atmospheric drag

• Solar radiation pressure

In this section we will study how this factor perturbs the orbit by analyzing the changes they induce to the orbital elements. These changes can be classiﬁed as:

• Secular

• Large-Periodic

• Short-Periodic

Secular are non-periodic changes that monotonously vary in time. The large-periodic and short periodic changes are terms which its variation is repeated with a large period (greater than the orbital period) and short period (same order than the orbital period) respectively [19].

Although it is true that the periodic changes can have interest, in general, the most important ones are the secular changes.

Figure D.3: Orbital element perturbation changes. Source [19].

In Fig. D.3 it can be appreciated the three different types of orbital element changes. The straight line shows the secular effects. The large oscillating line shows the secular plus long-periodic effects and the small oscillatory line, which combines all three, shows the short-periodic effects. 84 Mission and Thermal Analysis of the UPC Cubesat

The four main perturbation forces affecting a satellite orbiting the Earth are presented with a description of the changes each one creates to the orbital elements [11] [10]. a) Third-body perturbation

Gravitational forces of other celestial bodies different from the Earth cause important periodic variations in all of the orbital elements. The most important accelerations from other bodies are those from the Sun and the Moon, thus these two bodies would be the ones that would cause a major change to the satellite orbiting around the Earth. A brief table is presented below to illustrate the fact that the Sun and the Moon are the main bodies that would affect a satellite orbiting around Earth at an altitude of 200 NM (see Table D.3). A more complete table can be found in the appendix.

Table D.3: Most important accelerations form other bodies to a LEO satellite Body Acceleration Earth 0.89 Sun 6.0x10-4 Venus 1.9x10-8 Mars 7.1x10-10 Moon 3.3x10-6

Secular variations are only experienced by the right ascension of the ascending node, the argument of perigee and the mean anomaly. These secular variations in the mean anomaly are much smaller than the mean motion and have little effect on the orbit; however, the secular variations in the right ascension of the ascending node and the argument of perigee are important, especially for high-altitude orbits.

The equations for the secular rates of change of an Earth-centered satellite orbit resulting from the Sun and the Moon are presented below (Eq. D.39, D.40, D.41 and D.42).

cosi Ω˙ = −0.00338 (D.39) Moon n

cosi Ω˙ = −0.00154 (D.40) Sun n

4 − 5sin2i ω˙ = 0.00169 (D.41) Moon n

4 − 5sin2i ω˙ = 0.00077 (D.42) Sun n

Where i is the orbital inclination, n is the number of orbit revolutions per day and Ω˙ and ω˙ are in deg/day. These equations are only approximate, they neglect the variation caused by the orbital plane’s changing orientation with respect to the Moon’s orbital plane and the ecliptic plane. Additional theory for mission analysis 85 b) Non-spherical Earth perturbation

When developing the two-body equation of motion, the Earth is assumed to be a spheri- cally symmetrical, homogenous mass. In fact, the Earth is neither homogenous nor spherical. The most dominant features are a bulge at the equator, a slight pear shape and a ﬂattening at the poles. For a potential function of the Earth, a satellite’s acceleration can be found by taking the gradient of the potential function. A widely used from of the geopotential function is shown in Eq. D.43.

∞ µ RE j φ = [1 − ∑ Jj( ) Pj sin( jL)] (D.43) r j=1 r

Where RE is the Earth equatorial radius in km, Pj the Legendre polinomials, L the geocentric latitude in degrees and Jj the dimensionless geopotential coefﬁcients or harmonics, of which the most important are:

−3 J2 = 1.083 · 10

−6 J3 = −2.534 · 10

−6 J4 = −1.620 · 10

The first term of the geopotential function represents the sphere, while the other ones represent the deviation from the spherical model. This form of the geopotential function depends on latitude, and its coefficients (Jj) are called zonal coefficients. Other, more general expressions for the geopotential include sectoral and tesseral terms in the expansion. The sectoral terms divide the Earth into slices and depend only on longitude. The tesseral terms depend on both longitude and latitude and divide the Earth into a checkboard pattern of regions that alternately add to and subtract from the two-body potential.

The potential generated by the non-spherical Earth causes periodic variations in all orbital elements. However, the secular variations are the dominant ones, precisely in the right ascension of the ascending node and in the argument of perigee because of the Earth oblateness, represented by the J2 term. The rates of change of Ω and ω due to J2 are the following ones in Eq. D.44 and Eq. D.45 respectively.

RE cosi Ω˙ = −1.5nJ ( )2( ) (D.44) J2 2 a (1 − e2)2

2 RE 4 − 5sin i ω˙ = 0.75nJ ( )2( ) (D.45) J2 2 a (1 − e2)2

The ﬁrst phenomena it is known as nodal regression (Fig. D.4) while the second one is called apsidal rotation (Fig. D.5). In the nodal regression, the perturbation manifest itself through a change in the angular momentum vector, and the node regress much like a precessing top. It changes the orbital plane continuously. The apsidal rotation modiﬁes the geographic location of the line of apsides (apogee and perigee). 86 Mission and Thermal Analysis of the UPC Cubesat

Figure D.4: Nodal regression. Source [19]. Figure D.5: Apsidal rotation. Source [19].

The J3 coefﬁcient produces long period periodic effects and the J4 accounts also for secular variations in the orbit elements due to Earth oblateness. Secular variations of J4 are 100 times smaller than those coming from J2. c) Atmospheric drag

When the orbit perigee height is below 1000 km, the atmospheric drag effect becomes increasingly important. Drag, unlike other perturbations forces, is a non conservative force and will continuously take energy away from the orbit. Thus, the orbit semi-major axis and the period are gradually decreasing because of the effect of the drag.

Since drag is greatest at perigee, where the velocity and atmospheric density are greater, the energy drain is also greater at this point. Under this dominant negative impulse at perigee, the orbit would become more circular in each revolution. Moreover, decreasing energy causes the orbit to shrink, leading to further increases in drag. Eventually the orbit’s altitude becomes so small that the satellite reenters the atmosphere. The atmospheric drag acceleration can be modeled with Eq. D.46.

1 CDA a = − ρ( )v2 (D.46) D 2 m

3 Where ρ is the tmospheric density in kg/m , CD is the drag coefﬁcient (2-3), A is the satellite cross sectional area in m2, m is the satellite mass in kg and v is the satellite’s velocity with respect to the atmosphere in m/s.

Density depends on the altitude and it is very difﬁcult to model. For the same altitude, different values ﬂuctuate between a maximum and a minimum. There exist different atmospheric density models, each one taken into account different considerations with different accuracies. A list of existing atmospheric density models is presented in the appendix.

Moreover, this perturbation induces important secular variations to the semi-major axis and the eccentricity. It also produces secular changes to the inclination and periodic changes to all of them. Additional theory for mission analysis 87

The changes in semi-major axis and eccentricity per revolution can be approximate by Eq. D.47 and Eq. D.48 respectively, obtained from the Bessel functions.

CDA ∆a = −2π )a2ρ e−c[I0+2eI1] (D.47) rev m p

CDA c I e I I ∆e = −2π )aρ e− [ 1+ 2 ( 0+ 2)] (D.48) rev m p

3 Where ρp is the atmospheric density at perigee in kg/m , c is ae/H, H the density scale height in meters and Ii the Modiﬁed Bessel Functions of order i and argument c

This effect is typically for low altitudes the one responsible of the lifetime of the satellite. This lifetime can also be roughly estimate by the Eq. D.49.

−H L = (D.49) ∆erev

Fig. D.6 shows the lifetime of a satellite. It can be seen how the apogee and the perigee altitude decay with time. For a circular 300 km orbit, the average lifetime is less than 35 days.

Figure D.6: Typical dacay of a satellite for a 300 km circular orbit d) Solar radiation pressure

Solar radiation pressure effects induce periodic variations in all orbital elements, even exceeding the effects of atmospheric drag at altitudes above 900 km. Let’s see how this effect works.

Solar radiation pressure is the mechanical effect produced by the incidence of solar ﬂux (photons) in a surface. In the Earth, the average solar radiation ﬂux is typically I = 1358W/m2. Then, the mechanical pressure is computed by Eq. D.50, where c is the speed of light in vacuum. 88 Mission and Thermal Analysis of the UPC Cubesat

p = I/c = 4.5 · 10−6N/m2 (D.50)

Moreover, the force in a flat panel will be F = pA(1 + ε)cosϕ where A is the area, ε is the coefficient of reflectivity and ϕ the angle of incidence of the solar flux. Then, the perturb force can be approximate by Eq. D.51 where m is the satellite mass in kg.

−4.5 · 10−6 a ≈ (1 + ε) · cosϕ (D.51) R m

The albedo of the Earth (reﬂection of the solar ﬂux in the Earth) should be also taken into account in the calculation along with the direct radiation coming from the Earth.

The solar radiation pressure is complex to treat analytically but it can be demonstrated that there are only secular variations in Ω and ω while there are periodic variations in all the orbital elements with a period of one year. The induced changes in perigee height can have signiﬁcant effects on the satellite’s lifetime, but the typical radiation pressure effect on satellite orbits is the long-term sinusoidal variations on eccentricity. For a typical satellite at geosynchronous orbits, the eccentricity may vary from 0.001 to 0.004 in six months as a result of this effect. Its effect is strongest in satellites with low ballistic coefﬁcient.

Finally, it is important to mention that the solar radiation pressure must only be taken into account when the satellite is facing the Sun. In other words, when the satellite is in eclipse or umbra (shadow of the Earth) this force must not be taken into account.

D.6. HPOP force models

The available force models that can be used in the HPOP are presented in the table below [9].

Table D.4: Force models available in the HPOP

Force model Types EGM-96 (70x70) GEM-T1 (36x36) JGM-2 (70X70) JGM-3 (70x70) Gravity WGS-84 (70x70) WGS-84/EGM-96 (70x70) WGS-84 old (WGS-84 version 1) (12x12) GGM01C (90x90) GGM02C (90x90) WGS72 ZonalsToJ4 (4x0) Continued on next page Additional theory for mission analysis 89

Table D.4 – continued from previous page Force model Types 1976 Standard: A table look-up model based on the satellite’s altitude, with a valid range of 86km - 1000 km. Harris-Priester: Takes into account a 10.7 cm solar flux level and diurnal bulge. Valid range of 0 - 1000 km. Jacchia 1970: The predecessor to the Jacchia 1971 model. Valid range is 90 km - 2500 km. Atmospheric Jacchia 1971: Computes atmospheric density based on the com- density model position of the atmosphere, which depends on the satellite’s altitude as well as a divisional and seasonal variation. Valid range is 100km - 2500 km. Jacchia-Roberts: Similar to Jacchia 1971 but uses analytical methods to improve performance. CIRA 1972: Empirical model of atmospheric temperature and densities as recommended by the Committee on Space Research (COSPAR). Similar to the Jacchia 1971 model but uses numeric integration rather than interpolating polynomials for some quantities. Jacchia 1960: An earlier model by Jacchia that uses the solar cycle to predict a value for the F10.7 cm flux and accounts for the effects of the dirunal bulge. RK 4: Runge-Kutta integration method of 4th order with no error control for the integration step size. RKF 7(8): Runge-Kutta-Fehlberg integration method of 7th order with 8th order error control for the integration step size. Integration Bulirsch-Stoer: Integration method based on Richardson extrap- method olation with automatic step size control. Gauss-Jackson: 12th order Gauss-Jackson integration method for second order ODEs. There is currently no error control implemented for this method meaning that a fixed step size is used. VOP: Uses a special interpolator that deals well with ephemeris produced at a large step size, which happens frequently when using the VOP formulation. The interpolator itself uses a VOP formulation. The VOP mu value is the gravitational parameter used by the formulation. You can also specify the interpolation order. Interpolation Lagrange: Uses the standard Lagrange interpolation scheme, interpolating position and velocity separately. You can also specify the interpolation order. Hermitian: Uses the standard Hermitian interpolation scheme, which uses the position and velocity ephemeris to interpolate position and velocity together (i.e., using a polynomial and its derivative). You can also specify the interpolation order. 90 Mission and Thermal Analysis of the UPC Cubesat STK user’s guide 91

Appendix E

STK USER’S GUIDE

In order to understand how the simulations have been performed in the STK, a brief user’s guide explaining the steps to complete a basic simulation is presented. At the end of this section, you would be able to make basic simulations in STK and also to understand what has been done in the results section of the main body report.

For this simulation, we have performed the simulations for access, lighting and lifetime with just the UPC ground station using the HPOP propagator and an elliptic 1200x350 km orbit.

Step 1) Create the scenario

In STK, the ﬁles are known as scenarios. The scenario is the highest-level object in STK. It includes the 2D and 3D Graphics windows and contains all other STK objects (e.g., satellites, facilities, etc.).

So we create a new scenario in the STK by clicking the corresponding icon.

Figure E.1: STK window with main commands

Highlight the scenario in the Object Browser (see Fig. E.1), and click the Properties button on the toolbar to display its Properties Browser. In the scenario properties page, we should select the time period of our simulations, this is the interval of time we want the satellite to simulate (see Fig. E.2). We have introduced some future values for it, because the UPC Cubesat would ﬂy at some time in the future. So the values entered are the following.

• Start: 1 July 2010 at 00:00h. 92 Mission and Thermal Analysis of the UPC Cubesat

• Stop: 1 July 2011 at 00:00h.

• Epoch time: 1 July 2010 at 00:00h.

Figure E.2: Period

Step 2) Creating the ground station

The ﬁrst thing to introduce in our simulation should be the objects necessary to perform it, the satellite and the ground station. In order to do it, we need to use the object catalog icon. In that icon, we can specify which object we want to introduce. First, we will introduce the ground station, so we will create a new facility that we can rename for instance UPC. In the facility properties page we will introduce the geographical characteristics of our facility, this is, the latitude and the longitude of the ground station (see Fig. E.3). In our case, for Barcelona we have introduced:

• Latitude: 41.53o

• Longitude: 2.17o

Figure E.3: Satellite geographic coordinates

There exist a different way of deﬁning facilities. You can insert facilities from the STK database with no need of setting the geographical characteristics. You just specify the name of the city and the STK will perform a search in its database. In order to do it, you must select insert from the main toolbar and then city from database.

Moreover, we will specify the constraint we will apply in our simulations. Our simulations will be computed with an elevation constraint of 15o. So we open the basic constraints STK user’s guide 93

Figure E.4: Facility constraints label in the properties page and we specify a minimum elevation constraint of 15o (see Fig. E.4).

Step 3) Create the Satellite

The next object to introduce will be the satellite that we will rename as UPCSat. We use the Object Catalog and select satellite. In the satellite properties page, we will specify which propagator we want to use and the necessary terms to carry out the simulation. As we have said before, we will use the HPOP propagator so in the propagator menu we select HPOP (see Fig. E.5). Then, we should introduce the values needed for the simulation which are resumed below.

• Apogee altitude: 1200 km

• Perigee altitude: 350 km

• Inclination 71o

• Argument of perigee: 0o

• RAAN: 45o

• Mean anomaly: 30o

Leave the other values as default.

Then, the force models should be speciﬁed (see Fig. E.6). If we open the Force models page, it would appear all the perturbations the HPOP can handle. Our satellite will be affected by the atmospheric drag, the solar radiation pressure and the gravities of the Sun and the Moon. So, we will select all these forces in the force models page with the following values for each one. Leave the other default values as they appear.

• Cd: 2.2 94 Mission and Thermal Analysis of the UPC Cubesat

• Area/Mass ratio: 0.01 m2/kg

• Atm. Density model: Jachia-Roberts

• Cr: 1.5

• Shadow model: Dual cone

Figure E.5: Propagator page

Figure E.6: Force models page

In the integrator page inside the one of the propagator, we will select the Gauss-Jackson method with the default values. STK user’s guide 95

Note that by changing all these values, different results will be obtained. These values have been selected taking into account all the characteristics of the satellite and all the theory explained in this project.

Step 4) Obtaining results

Returning to the main propagator properties page and selecting apply, the simulation would be computed. Note that it may take several seconds, so do not attempt to obtain results until the computation has ﬁnished.

Then, you may run the simulation with the play button in the 2-D or in the 3-D graphics window. There, you would be able to see how the satellite is orbiting around the Earth with the speciﬁed orbital elements.

Results can be obtained either with a report or a graphic. These two options can be obtained dynamically (dynamic display and strip chart) or in steady-state (report and graph).

So highlight the ground station and display the facility menu form the main toolbar. There, you can select which option you want to compute. First, we will calculate the access to our satellite from the UPC ground station. You must select access in the facility menu and a new window will appear (see Fig. E.7).

Figure E.7: Access page

In order to obtain the access, you should select the element to which you want to calculate the access, in our case, the UPCSat and then select compute. Then we will select Access in the reports menu to obtain the list of accesses to our satellite during a year. A new window will appear with the list of accesses and at the bottom of it, global statistics of the simulation such as the maximum time of one single access, the minimum time or the mean time in seconds (see Fig. E.8). 96 Mission and Thermal Analysis of the UPC Cubesat

Figure E.8: Access report example

We can see that there are 1352 accesses during a year with a total duration of 618909 seconds. These values are exactly the same as the ones obtained during the simulations of this project.

There are other results that are independent of the ground station and that we are going to compute now. First of all, the lighting of the satellite will be computed. This is the time the satellite spends facing directly to the Sun, in umbra or in penumbra. To perform this calculation we must highlight the satellite in the Objects Browser and then display the satellite menu in the main toolbar. There, you must select reports.

Different types of reports can be obtained depending on the property you want to study. For our case, we will study the lighting of the satellite, so you must select lighting times in the report type and then create. A list similar to the one of access will appear with all the intervals the satellite is in lighting ﬁrst, penumbra, and umbra at the end (see Fig. E.9).

Figure E.9: Lighting report example

The global statistics of each property appear at the end of the intervals of each one. For lighting, penumbra and umbra we obtain a total duration of 23.587.878, 123.049 and 7.827.523 seconds respectively. STK user’s guide 97

Finally, we are going to compute the lifetime of our satellite. In satellite menu, there is an option that is called lifetime. We must go there. Several terms must be introduced before computing the lifetime (see Fig. E.10).

Figure E.10: Lifetime page

The drag coefficient and the reflection coefficients will be left as in the propagator options. The drag area and the area exposed to the Sun are set to 0.01 m2 because we assume that there is only one face facing the Sun.

A = 10cm · 10cm = 0.01m2 (E.1)

Finally, the atmospheric density model is set to the Jacchia-Roberts, the same as in the propagator computation. Then, by clicking compute, a new window appears telling you the lifetime of your satellite, in our case, this is 6.2 years. If you want a more visual description of the lifetime, you can select the graph option in the same page and a graphic with the evolution of the apogee and perigee altitudes and the eccentricity over time will appear (see Fig. E.11). 98 Mission and Thermal Analysis of the UPC Cubesat

Figure E.11: Lifetime graphic Simulation tables of the mission analysis 99

Appendix F

SIMULATION TABLES OF THE MISSION ANALYSIS

F.1. Individual access analysis

Table F.1: Complete list of the individual access analysis for all ground stations

GS # AY # AD TDY (sec) MDA (sec) DD (sec) Aachen 934 2.558 177562 190.109 486.470 Aalborg 1181 3.235 225548 190.981 617.939 Arizona 618 1.693 118728 192.116 325.281 Auburn 645 1.767 120922 187.475 331.292 Beirut 645 1.767 122788 190.369 336.404 Berlin Tech 986 2.701 187795 190.462 514.507 Boston 760 2.082 141995 186.835 389.026 Bucharest 786 2.153 149340 190.000 409.151 Buenos Aires 667 1.827 128929 193.297 353.231 Cal Poly 647 1.772 123609 191.05 338.654 Carleton 804 2.202 152679 189.899 418.000 Chen Kung 559 1.531 107318 191.983 294.022 Chicago 729 1.997 139575 191.461 382.397 Colorado 717 1.964 136116 189.841 372.921 Cornell 753 2.063 142225 188.878 389.658 Delft 968 2.652 184628 190.732 505.830 Dnipropetrovsk 870 2.383 165187 189.871 452.568 DTU 1116 3.057 212995 190.856 583.549 Embry-Riddle 612 1.676 115288 188.38 315.858 George Washington 703 1.926 133060 189.274 364.547 Goyang 689 1.887 129584 188.075 355.024 Hawaii 564 1.545 105891 187.749 290.111 Illionois 721 1.975 136724 189.631 374.586 Imperial College 950 2.602 181844 191.415 498.202 Iowa State 761 2.084 141285 185.657 387.082 Istambul 741 2.030 138593 187.035 379.706 Kansas 698 1.912 132507 189.838 363.033 Laussane 816 2.235 156573 191.879 428.967 Lousiana 619 1.695 116298 187.88 318.624 Malaysia 528 1.446 97863 185.346 268.117 Michigan Tech 837 2.293 159674 190.769 437.462 Continued on next page 100 Mission and Thermal Analysis of the UPC Cubesat

Table F.1 – continued from previous page GS # AY # AD TDY (sec) MDA (sec) DD (sec) Montana 800 2.191 153944 192.431 421.765 NC State 671 1.838 126810 188.987 347.425 New Delhi 601 1.646 114185 189.991 312.834 NM State 631 1.728 119339 189.126 326.954 North Dakota 871 2.386 163898 188.172 449.035 NTNU 2130 5.835 376957 176.975 1032.759 Oklahoma 643 1.761 123045 191.361 337.109 Porto 743 2.035 138447 186.335 379.306 Purdue 729 1.997 137084 188.044 375.572 Roma 742 2.032 140784 189.736 385.709 Sergio Arboleda 530 1.452 98823 186.459 270.749 Sherbrooke 807 2.210 152836 189.388 418.729 Siegen 934 2.558 177821 190.386 487.180 Stanford 682 1.868 129636 190.083 355.168 Stellenbosch 668 1.830 128045 191.684 350.808 SUPSI 813 2.227 155165 190.854 425.108 Sydney 678 1.857 129150 190.487 353.836 Taylor 729 1.997 136965 187.881 375.246 Texas 626 1.715 116568 186.211 319.363 Texas AaM 617 1.690 116712 189.161 319.759 Texas Christian 635 1.739 119829 188.707 328.299 Tokyo 661 1.810 125361 189.653 343.454 Toronto 775 2.123 146835 189.465 402.288 Trieste 806 2.208 153308 190.208 420.020 Tsinghua 724 1.983 136750 188.881 374.657 UNOPAR 602 1.649 112229 186.428 307.477 UPC 732 2.005 140095 191.387 383.823 Utah State 737 2.019 140647 190.837 385.333 Warsaw 969 2.654 185708 191.65 508.790 Was-StLouis 694 1.901 132179 190.459 362.133 Washington 849 2.326 161246 189.924 441.768 Wurzburg 907 2.484 172110 189.758 471.534 Mean Values 767.6 2.103 145169 189.369 397.723 GS: Ground Station AY: Access/year AD: Access/day TDY: Total duration/year MDA: Mean duration/access DD: Duration/day Simulation tables of the mission analysis 101

F.2. Orbit shape comparative

1) Accessibility

Table F.2: Accessibility analysis vs. orbit shape for UPC ground station UPC Analysis Orbit Shape # AY # AD TDY (sec) MDA (sec) DD (sec) 350x350 732 2.005 140095 191.387 383.823 1200x350 1250 3.424 544962 435.969 1493.045 AY: Access/year AD: Access/day TDY: Total duration/year MDA: Mean duration/access DD: Duration/day

Table F.3: Accessibility analysis vs. orbit shape for NTNU ground station NTNU Analysis Orbit Shape # AY # AD TDY (sec) MDA (sec) DD (sec) 350x350 2130 5.835 376957 176.975 1032.759 1200x350 2397 6.567 1074035 448.075 2942.561

Table F.4: Accessibility analysis vs. orbit shape for Malaysia ground station Malaysia Analysis Orbit Shape # AY # AD TDY (sec) MDA (sec) DD (sec) 350x350 528 1.446 97863 185.346 268.117 1200x350 901 2.468 401865 446.022 1101.001

2) Lighting 102 Mission and Thermal Analysis of the UPC Cubesat

Table F.5: Lighting analysis vs. orbit shape Orbit Shape MaxD (sec) MinD (sec) MeanD (sec) TD (sec) MDA (sec) 350x350 Lighting 9.730 6.390 64.491 20949334 15.943 Penumbra 40.837 0.130 0.237 153750 0.117 Umbra 36.394 0.442 32.230 10434988 7.941 1200x350 Lighting 17.359 2.264 88.196 23437346 17.836 Penumbra 40.499 0.122 0.268 142178 0.108 Umbra 36.593 1.612 30.156 7957779 6.056 MaxD: Maximum Duration MinD: Minimum Duration MeanD: Mean Duration TD: Total duration MDA: Mean duration/day

F.3. Propagator comparative

1) Accessibility

Table F.6: Accessibility analysis vs. propagator for UPC ground station UPC Analysis Propagator # AY # AD TDY (sec) MDA (sec) DD (min) SGP4 732 2.005 140095 191.387 6.397 J4 828 2.268 179922 217.297 8.215 J2 826 2.263 179919 217.820 8.215 TwoBody 829 2.271 180117 217.270 8.224 AY: Access/year AD: Access/day TDY: Total duration/year MDA: Mean duration/access DD: Duration/day

Table F.7: Accessibility analysis vs. propagator for NTNU ground station NTNU Analysis Propagator # AY # AD TDY (sec) MDA (sec) DD (min) SGP4 2130 5.835 376957 176.975 17.212 J4 2233 6.117 494232 221.331 22.567 J2 2231 6.112 494205 221.517 22.566 TwoBody 2230 6.109 494846 221.904 22.595 Simulation tables of the mission analysis 103

Table F.8: Accessibility analysis vs. propagator for Malaysia ground station Malaysia Analysis Propagator # AY # AD TDY (sec) MDA (sec) DD (min) SGP4 528 1.446 97863 185.346 4.468 J4 574 1.572 122894 214.101 5.611 J2 576 1.578 122892 213.354 5.611 TwoBody 578 1.583 122730 212.335 5.604

2) Lighting

Table F.9: Lighting analysis vs. propagator Propagator MaxD (sec) MinD (sec) MeanD (sec) TD (sec) MDA (sec) SGP4 Lighting 840701.59 383.45 3869.47 20949334 57395.43 Penumbra 2450.22 7.84 14.22 153750 421.23 Umbra 2183.67 26.57 1933.83 10434987 28589.00 J4 Lighting 893306.21 2017.94 4237.05 21740323 59562.53 Penumbra 1546.78 7.97 14.85 152144 416.83 Umbra 2161.15 43.13 1888.1 9646607 26429.06 J2 Lighting 898726.98 2017.94 4239.65 21745174 59575.82 Penumbra 1556.65 7.97 14.88 152402 417.54 Umbra 2161.14 33.53 1887.17 9641552 26415.21 Two-Body Lighting 3787.92 62.29 3497.44 20089343 55039.29 Penumbra 1315.66 7.99 11.83 135906 372.34 Umbra 2158.41 1672.41 1970.63 11313401 30995.62 MaxD: Maximum Duration MinD: Minimum Duration MeanD: Mean Duration TD: Total duration MDA: Mean duration/day 104 Mission and Thermal Analysis of the UPC Cubesat

F.4. Elevation angle comparative

Table F.10: Elevation angle accessibility analysis for UPC ground station UPC Analysis Angle # AY # AD TDY (sec) MDA (sec) DD (min) % Dif 15o 732 2.005 140095 191.387 6.397 5o 1251 3.427 396008 316.554 18.082 282.6 0o 1687 4.621 709455 420.543 32.395 506.4 AY: Access/year AD: Access/day TDY: Total duration/year MDA: Mean duration/access DD: Duration/day Dif: Difference

Table F.11: Elevation angle accessibility analysis for NTNU ground station NTNU Analysis Angle # AY # AD TDY (sec) MDA (sec) DD (min) % Dif 15o 2130 5.835 376957 176.975 17.212 5o 2577 7.060 907789 352.266 41.451 240.8 0o 2886 7.906 1374340 476.209 62.755 364.5

Table F.12: Elevation angle accessibility analysis for Malaysia ground station Malaysia Analysis Angle # AY # AD TDY (sec) MDA (sec) DD (min) % Dif 15o 528 1.446 97862 185.346 4.468 5o 859 2.353 273680 318.603 12.496 279.6 0o 1162 3.183 489444 421.209 22.349 500.1

F.5. HPOP vs. SGP4 comparative

1) Accessibility Simulation tables of the mission analysis 105

Table F.13: HPOP and SGP4 accessibility analysis for UPC ground station UPC Analysis Propagator # AY # AD TDY (sec) MDA (sec) DD (min) SGP4 1250 3.424 544961 435.969 24.884 HPOP 1336 3.660 608790 455.682 27.798 AY: Access/year AD: Access/day TDY: Total duration/year MDA: Mean duration/access DD: Duration/day

Table F.14: HPOP and SGP4 accessibility analysis for NTNU ground station NTNU Analysis Propagator # AY # AD TDY (sec) MDA (sec) DD (min) SGP4 2397 6.567 1074035 448.075 49.042 HPOP 2491 6.824 1223882 491.322 55.885

Table F.15: HPOP and SGP4 accessibility analysis for Malaysia ground station Malaysia Analysis Propagator # AY # AD TDY (sec) MDA (sec) DD (min) SGP4 901 2.468 401865 446.022 18.350 HPOP 901 2.468 391115 434.090 17.859

2) Lighting

Table F.16: HPOP and SGP4 Lighting comparative Propagator MaxD (sec) MinD (sec) MeanD (sec) TD (sec) MDA (sec) SGP4 Lighting 17.359 2.264 88.196 23437346 17.836 Penumbra 40.499 0.122 0.268 142178 0.108 Umbra 36.593 1.612 30.156 7957779 6.056 HPOP Lighting 16.095 2.695 89.327 23480637 17.869 Penumbra 20.732 0.122 0.232 121845 0.092 Umbra 36.497 2.015 30.348 7935623 6.039 MaxD: Maximum Duration MinD: Minimum Duration MeanD: Mean Duration TD: Total duration MDA: Mean duration/day 106 Mission and Thermal Analysis of the UPC Cubesat Thermal analysis information 107

Appendix G

THERMAL ANALYSIS INFORMATION

G.1. UPCSat Information

In order to determine the thermal balance of the UPCSat, the characteristics of the Cube- sat must be known and studied. Moreover, it is important to know the values of emissivity and absorptivity for each material related to the Cubesat.

If the Cubesat is launched through a P-POD, the use of Aluminum 7075 is suggested for the main structure [22]. If other materials are used, the thermal expansion must be similar to that of the P-POD (Aluminum 7075-T73). However, the Cubesat kit is composed by different types of materials. The base plate, chassis and cover plate are made from 5052- H32 aluminum. The two plates are made from 0.060” material, the chassis from 0.050”. The chassis is hard anodized and alodyned in a manner that leaves the rail surfaces hard anodized, and the rest of the structure alodyned, so that it remains conductive and therefore gives you a Faraday cage. If it were completely hard anodized, it would be an electrical insulator, which is undesirable in this application. Panels use the 7175 aluminum alloy, and are alodyned as said before The CubeSat Kit’s feet are machined from 6061-T6 aluminum, and are also hard anodized. This alloy has roughly the same coefﬁcient of thermal expansion as the 5000-series material used in the structure. Finally, all fasteners (screws and captive fasteners) are stainless steel.

Although these are the initial inputs, the structure can be completed with other materials in order to change the values of emissivity/absorptivity or to enhance the power subsystem with for example solar cells, which is the case of the UPCSat. Moreover, the use of Kapton foil is typical for insulation purposes between the solar cells and the panels.

Typical values of emissivity and absorptivity for the different parts of the structure and the thermal characteristics of the two types of aluminum alloys are presented in Table G.1 and Table G.2 respecitvely [23].

Table G.1: Emissivity and absorptivity of different parts of the Cubesat Part Material Thermal ﬁnish α ε Aluminum frame 5052 Al alloy Alodine 0.08 0.15 Aluminum frame 5052 Al alloy Hard anodized 0.88 0.88 rails Aluminum panels 7075 Al alloy Alodine 1200 0.1 0.1 Aluminum panels 7075 Al alloy Kapton foil on Alodine 0.87 0.81 (outside) 1200 Solar panels GaAs cells Anti reﬂective coating 0.91 0.81 108 Mission and Thermal Analysis of the UPC Cubesat

Table G.2: Material properties Alloy ρ(kg/m3) c(J/KgK) k(W/mK) 5052 2672-2698 963-1002 140-152 7075 2770-2830 913-979 131-137

G.2. Thermal analysis methodology

Several simpliﬁcations have been made in chapter 3 to obtain a ﬁrst approximation of the solution. These are:

• Cubesat is a sphere with radius r

• No internal dissipation is observed

• There is no thermal inertia

Treating the Cubesat as a sphere reduces the computations for the angle of incidence of the heat ﬂux by making the angle constants because of the same area of incidence. Values of emissivity and absorptivity will change due to this factor which must be taken into account.

Moreover, thermal inertia as explained in chapter 3 must be also studied because the difference in temperature evolution. To do it, transient analysis must be performed.

Finally, an accurate calculation of temperatures of all internal parts is needed in order to obtain the internal dissipation of the satellite. However, this is extremely complex so the problem is again simpliﬁed to a thermal mathematical model (TMM) which assumes that the spacecraft is a set of blocks represented by nodes, and each block has no thermal gradient (isothermal). Results obtained from STK were through node analysis and took into account the internal dissipation of the satellite. However, it was just a single node analysis which it is not the most realistic one.

Radiation and conduction in a complex structure of various materials should be studied [20]. The heat ﬂow between any pair of nodes i and j is presented in Eq. G.1 where h is the thermal conductance.

Qci j = hi j(Ti − Tj) (G.1)

So, a TMM is a mathematical model that forms this network of nodes and calculates all the heat fluxes numerically to reach a steady state. Radiation heat flow has a similar situation as the environmental thermal balance taking into account that emission between two surfaces reflects each other [20].

ε σ 4 4 Qri j = AiFi j i j (Ti − Tj ) (G.2) Thermal analysis information 109

εiε j εi j = (G.3) εi + ε j − εiε j

So the goal of the thermal design is to keep the critical systems within its operating limits and not to achieve a speciﬁc temperature. To do it, the worst-case conditions should be used and a network of TMM nodes should be deﬁned. Then, the thermal balance temperature limits are obtained and they must be compared to the operating limits (see Fig. G.1) in order to apply or not a control method to bring temperatures back inside the limits.

Figure G.1: Typical operational limits of components of a satellite. Source [23].

TMM calculation is a quite complex calculation so available software is used to perform this part of the thermal analysis. Equations can be introduced by the user in software languages as Matlab which will give accurate results depending on the inputs introduced. In addition, there exists commercial software that performs this type of analysis also. Exam- ples of software are ESATAN, ESARAD and ANSYS that are presented in the next section of the appendix.

G.3. Software available

ESATAN [24] is a software package for the prediction of temperature and heat ﬂows using a thermal lumped parameter network. It is the standard European thermal analysis tool used to support the design and veriﬁcation of space thermal control systems. It provides the following capabilities.

• Steady state and transient analysis. 110 Mission and Thermal Analysis of the UPC Cubesat

• One, two and three dimensional models.

• Conduction, convection and radiation heat transfer.

• Condensation and boiling heat transfer.

• Facilities to allow the user to model phase change phenomena.

ESARAD [24] is a thermal-radiative analysis and pre/post-processing tool that provides the following functionality:

• Deﬁne and visualize a 3D external surface model of a space-craft with thermo-optical and thermo-physical properties.

• Compute view factors, radiative exchange factors and couplings.

• Deﬁne space-craft trajectories (orbits), attitude and pointing.

• Optionally add articulation (rigid body kinematics).

• Compute solar and planetary heat ﬂuxes for a selected set of positions along a space-craft trajectory.

Moreover, it is a pre and post process for ESATAN. It produces the thermal-radiative part of an ESATAN lumped parameter thermal network model, and can be used to visualize the results of an ESATAN run on the geometric model.

ANSYS is an engineering simulation software provider [25] [26]. It develops general- purpose ﬁnite element analysis and computational ﬂuid dynamics software. While ANSYS has developed a range of computer-aided engineering (CAE) products, it is perhaps best known for its ANSYS Mechanical and ANSYS Multiphysics products.

ANSYS Mechanical and ANSYS Multiphysics software are non exportable analysis tools incorporating pre-processing (geometry creation, meshing), solver and post-processing modules in a graphical user interface. These are general-purpose ﬁnite element modeling packages for numerically solving mechanical problems, including static/dynamic structural analysis (both linear and non-linear), heat transfer and ﬂuid problems, as well as acoustic and electro-magnetic problems. ANSYS software can also be used in civil engineering, electrical engineering, physics and chemistry.

There exist other software but this three are the most common used in thermal analysis. However, there are some differences between them. While ESATAN and ESARAD are softwares that can be downloaded for evaluation, ANSYS is a software that must be bought and then paid for being able to use it. Functionality of each of them is quite different also. ESATAN and ESARAD are software developed for the thermal analysis purposes uniquely while ANSYS is an engineering software that provides thermal analysis among a wide range of other uses. This capability of ANSYS makes it more complete than the other ones because you can perform other analysis and coupled analysis to your satellite but on the other hand it is more complex to use because there are a lot of parameters to take into account. Thermal analysis information 111

In conclusion, there are many programs that can be used for thermal analysis, each of them with its own characteristics but that at the end they must give similar results.