Investigation of a Concept for a Scout Satellite for a LEO Drag Free Mission

Transcription

1 EXAMENSARBETE 2005:124 CIV Investigation of a Concept for a Scout Satellite for a LEO Drag Free Mission MARTIN FROM MASTER OF SCIENCE PROGRAMME in Space Engineering Luleå University of Technology Department of Space Science, Kiruna 2005:124 CIV ISSN: ISRN: LTU - EX / SE

2 Investigation of a concept for a scout satellite for a LEO Drag F ree mission Master s thesis submitted to the department of Space Science Luleå University of Technology Carried out by Martin F rom Master of Science Programme at the Center of Applied Space Technology and Microgravity (ZARM) University of Bremen under the supervision of Dr. Stephan Theil Silvia Scheithauer

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4 Abstract Two major problems in controlling a Drag F ree Satellite in low Earth orbit during the experimental phase of the mission have been identified and are addressed in this thesis. The first problem is the air drag induced by the residual atmosphere on the satellites orbit. The air drag is the largest disturbance and the most problematic are the unknown density variations constituting the random part of this disturbance. The second problem is charging of the test masses. As the satellite progresses on its way around the Earth a charge is accumulated by the test masses which will reduce measurement accuracy. This charge has to be measured and dumped from time to time by the charge management system. Under nominal conditions this charge can be calculated fairly accurately but if the satellite passes through the South Atlantic Anomaly the estimation of the accumulated charge is difficult. In order to reduce the problems for the drag free control and charge management system a companion satellite is proposed that will fly in a predefined distance in front of the drag free satellite on the same orbit. Using a single drag free sensor this scout could be used to measure non-gravitational disturbances as well as the charging rate. The measurements of the disturbances could be used in a feed forward control by the drag free control system whereas the charge measurements could be used in a continuous charge control. Problems associated with the measurement of air drag are addressed and an error budget for the relative drag to be used in feed forward is derived. Realistic atmospheric density variations are derived using measurement-based models and through simulations using a simplified one degree of freedom drag free controller an increase in performance of the science measurements on a reference mission using a scout satellite is demonstrated. A method for estimating the charge rate for use in a feed forward charge control system is briefly discussed, but due to lack of information no conclusions can be drawn about the performance of such a system. A very simplified baseline configuration of a scout satellite is finally presented. I

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6 Forew ord This report is submitted to the Department of Space Science in partial fulfilment of the requirements for the degree of Master of Science in Space Engineering at Luleå University of Technology, Sweden. The work was carried out in the Space Technology group at the Center of applied Space Technology and Microgravity (ZARM) at the University of Bremen, Germany. First of all I would like to thank my supervisors Stephan Theil and Silvia Scheithauer for the exciting opportunity to come to ZARM and do my thesis, and for giving me support and believing in my work. Especially I would like to thank Silvia for helping me through administrative matters associated with working in Germany. I would also like to thank Alexander Schleicher, who was originally supposed to be my supervisor, but unfortunately changed to a new job a few weeks after I arrived. Thanks also go out to all other members of the Space Technology group, as well as other students doing work there, for making my stay at ZARM a pleasant time. Priya Fernando, my examiner, also deserves a thank you for keeping my interest in space technology alive through my final year at Luleå University and for supporting my choice of thesis subject. Lastly I would like to thank my wonderful family, for supporting me during my student years and my constant moving around the world. III

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8 Contents 1 Introduction Challenge and objective Drag Free Control The STEP Mission A Companion for STEP Analysis The EP experiment Atmospheric Density Atmospheric drag measurement Acceleration Sensors Accuracy of sensor Drag sources Error from other drag sources Uncertainties in spacecraft parameters Drag Coefficient The theoretical drag coefficient Drag Coefficients for control purposes Other uncertainties Parameter variations on STEP Error Analysis Radiation and Charge Measurement Test mass charging Charge control Radiation measurement Radiation sources Trapped radiation and The South Atlantic anomaly Requirements Orbit and formation dynamics Orbit requirements Orbit dynamics spacecraft in formation Calculations and Simulations Orbit simulator Baseline configuration used for simulations Trade-off: Drag Free Control of Companion Drag Free satellite with non-drag Free companion Simulations Two Drag Free satellites V

9 Contents Simulations Discussion Selection Trade-off: Distance between spacecraft Selection Trade-off: Ground Track Atmospheric Drag Radiation Selection Spacecraft Payload Accelerometer Spacecraft requirements Proton monitor Subsystems Propulsion Baseline Attitude and Orbit Determination and Control Baseline Communication Baseline Command and Data Handling Baseline Thermal control Baseline Power Baseline Structure Summary of baseline configuration P erformance Feed Forward of Atmospheric Drag Simulations Performance increase Comments Feed Forward of proton radiation Charge control Feed Forward Comments Conclusions and outlook Summary Suggestions for future work VI

10 List of Acronyms AM Accelerometer Mode AP-8 Trapped proton flux map, version 8 CDH CHAMP DF DFC DM DVM EP FEEP GPS Command and Data Handling CHAllenging Minisatellite Payload Drag Free Drag Free Control Displacement Mode Density Variation Model Equivalence Principle Field Emission Electric Propulsion Global Positioning System HWM93 Horizontal Wind Model 1993 IGRF IR LEO MSIS NRLMSISE-00 RAAN SAA SMAD SREM STEP UT UV International Geomagnetic Reference Field Infra Red Low Earth Orbit See NRLMSISE-00 Naval Research Laboratory, Mass Spectrometer, Incoherent Scatter radar Extended model 2000 Right Ascension of Ascending Node South Atlantic Anomaly Space Mission Analysis and Design Standard Radiation Environment Monitor Satellite Test of the Equivalence Principle Universal Time Ultra Violet VII

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12 1 Introduction 1.1 Challenge and objective A Drag Free satellite is a satellite designed to follow a purely gravitational orbit, by using a control system that cancels out all non-gravitational disturbance forces. Out of these disturbance forces the largest one for low Earth orbits is atmospheric drag. Another problem is the charge that will accumulate on the test mass used for drag free control, which will lead to disturbance forces. In order to reduce the problems for the drag free control and charge management system a companion satellite has been proposed that will fly in a predefined distance in front of the drag free satellite on the same orbit. Using a single drag free sensor this scout could be used to measure non-gravitational disturbances as well as the charge accumulated. The measurements of the disturbances could be used in a feed forward control by the drag free control system whereas the charge measurements could be used in a continuous charge control Drag Free Control The main idea in drag free control is to have a free floating test mass inside a cavity close to the center of gravity of the spacecraft and then control the spacecraft to center this mass. Since the test mass, floating free, will experience no external forces it will follow a purely gravitational orbit. Since the satellite is forced to follow the test mass, it too follows the same gravitational orbit, canceling all non-gravitational forces. In figure 1.1 one can see the concept of the drag-free control. The test mass is centered and in some way coupled to the satellite, here visualized by spring couplings, and by measuring the movement of the test mass relative to the satellite housing the controller can cancel out disturbances by moving the satellite housing to center the mass. The first generation drag free satellites were designed to extend ephemeris prediction for the U.S. Navy s Transit Navigation Satellites but in the last years the drag-free satellites will be used for scientific missions. Examples of current and future drag free satellites are Gravity Probe B, GOCE, MICROSCOPE, STEP and LISA, all of which carry experiments on fundamental and gravitational physics. An extended overview of the history of drag-free satellites is given by Lange [19]. 1.2 The STEP Mission As an example mission for this study the The Satellite Test of the Equivalence Principle (STEP) is used. STEP is a joint European-US space program to investigate one of the most fundamental principle in physics, the equivalence of inertial and passive gravitational mass. Isaac Newton first recognized the identity between these two distinct properties, which represent the quantity of matter in an object, and its weight. A direct consequence of this principle is the universality of the free fall, such that all objects fall with exactly the same acceleration in the same gravity field. The equivalence principle was reinterpreted by Albert 1

13 1.2. THE STEP MISSION Figure 1.1: The Drag Free concept. Source: ZARM [43] Einstein as a consequence of an even broader equivalence between the laws of physics in different accelerated reference frames, a principle which Einstein made the basis for his general theory of relativity. STEP is meant to advance the sensitivity of the equivalence principle tests by six orders of magnitude, into regions where the principle may break down. A violation of equivalence at any level would have significant consequences for modern gravitational theory. The STEP experiment is conceptually a modern version of Galileo Galilei s free-fall experiment, in which he is said to have dropped two weights from the Leaning Tower of Pisa to demonstrate that they fall at the same rate. Any difference in the ratio of inertial to passive gravitational mass of the weights results in a difference in the fall of rate. In STEP, the masses are in free fall in an orbit around the Earth and if there is a violation of the equivalence principle they tend to follow slightly different orbits. The orbiting masses fall all the way around the Earth and never strike the ground, so that any small difference in the rate of fall can produce a measurable displacement. The principle is illustrated by figure 1.2. On STEP the two test masses of the pair are held by frictionless superconducting magnetic bearings, so the motion of the test mass is limited to one degree of freedom. Then a violation of the equivalence principle will produce small differential motions of the masses which will cause an oscillating movement with a slowly rotating satellite. The STEP satellite will carry four of these differential accelerometers in order to test a range of different materials and of course for redundancy. The spacecraft will have a nearly circular orbit at an altitude of 550 km. For thermal stability eclipses have to be avoided. Therefore a sun-synchronous dusk-dawn orbit has been chosen which will prevent the spacecraft from passing through the Earth s shadow during its six month lifetime. An illustration of the satellite can be seen in figure

14 CHAPTER 1. INTRODUCTION Figure 1.2: The concept of the equivalence experiment on STEP. Source: ZARM [43] Figure 1.3: The STEP satellite. Source: ZARM [43] 3

15 1.3. A COMPANION FOR STEP 1.3 A Companion for STEP A companion could be used for two different things. Firstly it could provide measurements of disturbances to STEP for use in a feed forward control of its orbit and the charging of the test masses as mentioned above. This could hopefully improve the quality of measurements made by STEP by reducing the effect of external disturbances from atmospheric drag. Secondly, the Companion would also have a large scientific value. The data from the companion, combined with the data from STEP, could be very valuable for various research work. STEP will already measure accelerations for its Drag Free control, test mass charging and radiation, and combined with the same data from the Companion this could be used for numerous investigations, such as mapping of atmospheric spatial density and radiation, better understanding of test mass charging etc, with higher resolution than only using one satellite. The fact that there are two measurements from different satellites for each point of the orbit can also be very useful. It could for example be used, in addition to drag data from the GRACE mission, for investigating the high frequency changes in atmospheric density which are not well implemented in current models or for investigations of radiation in the South Atlantic Anomaly. The aim is to present a satellite which will measure atmospheric drag and density as accurately as possible. 4

16 2 Analysis As mentioned in the introduction, the main idea for having a companion satellite for STEP is to use measurements of atmospheric drag and test mass charging for feed forward control of the drag free and charge control system on STEP. Using feed forward could improve the performance of STEP by measuring the unknown disturbances from high frequency drag variations and for very accurate prediction of entry into the South Atlantic Anomaly, which currently is a major problem for the charge control system. In the following sections methods for measuring these properties are presented, and the expected accuracy of these methods are derived. 2.1 The EP experiment In figure 2.1 the relative movement of a test mass pair in the experiment on STEP if there is an equivalence principle violation can be seen. The experiment has two test masses, one inside the other, that can move freely in one direction thanks to superconducting bearings. A violation will, if STEP is not rotating at all, produce a small relative motion of the two test masses at orbit frequency. STEP will rotate in its orbit at rates varying between minus three and plus three times the orbit rate, which means the violation will occur at the difference between the rotation frequency and the orbit frequency, ω ep = ω r ω o. (2.1) Since atmospheric drag always acts in velocity direction it will also, with the rotating spacecraft, varies mainly at exactly the same frequency. This means that a cancelation of this disturbance by using feed forward could potentially improve performance significantly. In the worst case, with a non-rotating spacecraft, the disturbances from charging will also come at almost exactly the orbit frequency, since charging will mainly come in bursts during passes through the South Atlantic Anomaly once every orbit. More about this further down. The violation will be visible in the frequency contents of the displacement measurement for the two masses, as shown in figure Atmospheric Density For state estimation in a Drag Free control system disturbance models can be used to estimate the disturbances [39]. Atmospheric Drag is the major disturbance in the case of a LEO Drag Free mission, and the drag is related to the atmospheric density as will be shown later. Atmospheric density is varying not only with height above Earth surface, but also with local solar time, time of year, solar activity and other factors. Models for calculating the density at a certain point in space and time exist. One of the more sophisticated such models is the NRLMSISE-00 [26], and density in the 550 km orbit plane calculated using this model can be seen in figure 2.3. The problem with all existing models is that they do not model higher 5

17 2.2. ATMOSPHERIC DENSITY Figure 2.1: Relative motion of the test masses at an equivalence principle violation with no rotation of spacecraft. Figure 2.2: An Equivalence Principle violation will be visible in the frequency contents of the differential mode measurement as a peak at the EP frequency. The common mode measurements are used by the Drag-free control system to minimize disturbances at the EP frequency. 6

18 CHAPTER 2. ANALYSIS Figure 2.3: NRLMSISE-00 Atmospheric density map at 550 km for average atmospheric conditions at noon (UT) close to spring equinox. frequency variations of the density well at all [39, 44], meaning it is very difficult to make an estimation model that estimates the actual disturbance correctly. This is where a Companion satellite would be useful to give a measurement of the density right in front of the Drag Free satellite, giving it a much more accurate estimate of the disturbances than can be given by an internal model Atmospheric drag measurement Atmospheric drag is calculated from density data by the well known formula a D = 1 2 C A D m ρv 2 (2.2) where a D is the drag acceleration experienced by the spacecraft, ρ is the atmospheric density, C D is the drag coefficient, m is the mass of the spacecraft, A is the surface perpendicular to the velocity V. V is here the velocity of the satellite relative to the atmosphere, and not the inertial velocity. Drag coefficient, mass and area is often written as the ballistic coefficient B = m C D A (2.3) The atmospheric density in a 550 km orbit varies between approximately and k g /m 3 (SMAD [41]), which gives a maximum drag of approximately B 1. The easiest way to measure drag/acceleration in orbit is of course by using an on board accelerometer to measure the acceleration experienced. Several types of high accuracy acceleration sensors exist, and Scheithauer [33] gives a good overview of the different acceleration sensor options for a Drag Free mission as well as an overview of how to derive the requirements. 7

19 2.2. ATMOSPHERIC DENSITY Acceleration Sensors In general there are two types of sensor modes, Accelerometer Mode (AM) and Displacement Mode (DM). The DM Concept uses a free-floating test mass and the displacement of this mass relative to the housing (spacecraft) is measured by the sensor. The signal is then used to control the spacecraft to center the mass and the force needed for the control is a measurement of the acceleration. In the AM concept, the test mass is instead forced to follow the satellite. The test mass is centered by an internal control system. Here the force needed to control the test mass is a measure for the acceleration acting on the satellite, usually simpler to measure than for a DM sensor. With an AM sensor the acceleration can also be measured without using Drag Free Control for the spacecraft. The main advantage of using a Displacement Mode sensor is its very high accuracy for drag free control purposes, since no or very little force is needed to suspend the test mass, which lead to very low noise levels. Its drawbacks includes that it is not inherently stable, that it requires very complex control algorithms since the coupled satellite-test mass dynamics have to be taken into account and that a complex discharging mechanism is required, a simple gold wire can not be used. A Drag Free sensor in displacement mode is usually not used as a pure sensor but in most cases is a part of the scientific experiment itself, as is the case for STEP [37]. Having a sensor in DM mode would require that the companion spacecraft uses a Drag Free control system, since this is how the test mass is centered. The only real drawback of an Accelerometer Mode sensor is that its accuracy is lower than a DM sensor for control purposes, but it still has a few advantages over DM sensors. It is inherently stable, it allows for relatively simple control algorithms when used for Drag Free control since the acceleration is directly accessible and it allows for a simple gold wire discharging mechanism. It also allows for acceleration measurement without using Drag Free control of the spacecraft. For the Companion the sole purpose of the sensor is to measure acceleration on the center of gravity of the satellite to use for feed forward control for STEP, and possibly for Drag Free control of the Companion itself. Except for the higher accuracy, a DM sensor has no real advantage over an AM sensor for these purposes. By using a simple gold wire discharging mechanism it might also be possible to measure charge gradients for feed forward to STEP by simply measuring the current in the wire. For more about charge measurement and control, see section 2.3. The best choice for a Companion satellite would probably be an accelerometer in the Acceleration Mode since this allows for a simpler spacecraft design. An accelerator mode sensor developed for another mission could probably be used directly as long as the accuracy is good enough Accuracy of sensor The acceleration measured by the sensor, Γ, is related to the actual acceleration a by Γ = c 0 + c 1 a + noise (2.4) where c 0 is the measurement bias of the sensor, c 1 the scale factor and the noise term is a function of frequency. The error budget for the acceleration is then Γ = ( i c 0i ) + ( i c 1i ) a + noise (2.5) 8

20 CHAPTER 2. ANALYSIS The measurement bias term c 0 and the scale factor c 1 have both constant and variable parts. The variable part of c 0 has two parts, one temperature dependant and one depending of the magnetic field (from magnetic torquers etc). c 1 has, in addition to the constant part, which is usually set to be 1, a variable part which is time and temperature dependant. Looking at the feasibility study for HYPER [34], one can see that a total error of as low as 10 9 m/s 2 can be achieved with already existing sensors (GRADIO from ONERA for example), and this is on the scale of one part in a thousand of the acceleration from air drag. It is concluded that the error from accelerometer bias can be made negligible compared to other errors for density measurement (see section 2.2.4) Drag sources The atmospheric density is not the only force acting on the satellite in the velocity direction, and these other forces need to be considered. Radiation pressure If an assumption is made that the spacecraft will be spherical, as this minimizes the errors in the atmospheric drag measurements, the magnitude of this force can be computed from the so-called cannonball model [18]. The acceleration due to solar radiation pressure is then given by A W a total = C R m c s (2.6) where W is the power density, s is the unit vector in radiation source direction and C R is the radiation pressure coefficient. If the spacecraft reflects light specularly or diffusely, this is expressed as C R = δ (2.7) where δ is the reflection coefficient of the surface material. Assuming the sphere is covered with solar panels the reflection coefficient will be approximately 0.25, depending on type of cells. This would give a total acceleration from solar radiation of about A/m in the direction away from the sun. If a drag coefficient of 3 is assumed this is almost 20 percent of the maximum atmospheric drag. In velocity direction this would only be at most 4 percent of the atmospheric drag, since the orbit of the satellite will be sun synchronous with the orbit plane almost perpendicular to the sun. This is however still a relatively large error. Added to this is also the Earth albedo and IR radiation from Earth. These are more difficult to model since the source cannot be treated as a point for a satellite flying close to Earth. The albedo is sun-light reflected by the surface of Earth, and the coefficient of reflection can vary between 0.2 for a clear sky to 0.6 for a cloudy sky with a mean of about Albedo and IR forces will however mainly act in the direction along the satellite position vector r, pointing from Earth center to satellite center, and it s assumed they will be significantly smaller than solar radiation pressure in velocity direction. See further down for an estimate of modeling errors. Solar wind pressure The solar wind streams off from the sun in all directions at speeds of about 400 km/s. It mainly consists out of protons and has a typical density of about 3 protons per cm 2, 9

21 2.2. ATMOSPHERIC DENSITY resulting in a very small acceleration on a spacecraft. For a worst case scenario, not counting coronal mass ejection, a speed of 1000 km/s and a density of 100 protons per cm 2 are maximum values. This leads to an acceleration of only about A/m, and considering the angle of attack is always at the very least 67 degrees (since the orbit is always sun lit) this will lead to a maximum acceleration in velocity direction of A/m which is far less than 1% of the total acceleration. The force from the solar wind is therefor neglected. Atmospheric velocity Even if the spacecraft inertial velocity can be decided with high accuracy the relative velocity to the atmosphere is not as easy to estimate. A very simple approach is to assume winds are always zero and that the atmosphere just moves with the rotating Earth, but this is not a very good approximation. Wind speeds can reach at least a few hundred meters per second, and adds up to a large error in the measured density when using formula 2.2, especially since velocity is squared. If one uses a wind model such as HWM93 [25] to estimate the winds at orbit height this error can probably be reduced. Zijlstra [44] assumes a 5% error from winds by modelling them with the HWM93 model. Internal disturbance sources There will also be disturbances from sources internal to the spacecraft, such as propellent sloshing, gravity gradients etc. By placing the acceleration sensor in the center of gravity and reduce sloshing by using high I sp propulsion system or propellant stored in solid form this will be far smaller than all other disturbances. It should be noted that for the feed forward control of STEP the drag difference from variations in the Atmospheric velocity should not pose a major problem even if they result in an absolute density value with large errors. For the control system the interesting parameter is really the atmospheric drag pressure and not the density. This is discussed further in section Error from other drag sources For estimating the error in the deduced atmospheric drag from modeling out other drag sources the acceleration from drag is written as a = a D + a E + a I (2.8) where a D is the atmospheric drag, a E is other external drag sources and a I is internal disturbance forces. The atmospheric drag is then estimated with a D = a a E a I (2.9) a I is so small it can be neglected without modeling. The largest constituents of a E are solar, IR and albedo radiation pressure, making up over 97% of the total disturbance [15] when air drag is removed. It can be expressed as a E = a S + a A + a IR + a O (2.10) where a S, a A and a IR are accelerations from solar, albedo and IR radiation respectively and a O is other drag sources. The relative error in the estimate of a E is then given by a E + a E a E = a S a S a S as + a A a A aa + a IR a IR air + a O a O ao (2.11) a E a A a E a IR a E a O a E 10

22 CHAPTER 2. ANALYSIS Following Tolson [40] and Jafry [15] realistic values of these become a E + a E a E = 6% % % % % (2.12) meaning the external disturbance forces can be modeled to an accuracy of 13%. The estimation accuracy of the other forces is conservatively set to 100%. Also the direction of the force is not taken into account, meaning in velocity direction the error will be much smaller than 13%. The orbit is an always sunlit sun synhronous orbit at 550 km. For the orbit to have no eclipse the angle of the orbit plane to the sun will be at least 67 degrees. Also IR and albedo radiation will be acting mainly in direction from Earth center. A conservative estimate that all forces act at least with a 67 degree angle gives an error of the estimation in velocity direction of 5%. In velocity direction the total external force minus air drag will for solar maximum be about 5% of the atmospheric drag force. This means that if the mission was flown at solar maximum the relative error from external force modeling errors would be about 0.25%. For solar minimum however, the non-air-drag related forces will be larger than the atmospheric drag, 125%, translating to an error of up to 6.25% in the estimate of the air drag. It is assumed that the mission will probably fly closer to solar maximum than minimum, since a maximum of the 11 year cycle will occur around Using the average solar conditions that are used for simulations later in this study of F 10.7 = 150 sfu and a p = 4 the error will be a little less than 1%. These indexes are explained later in section Uncertainties in spacecraft parameters The biggest uncertainty in the measurements of density is the drag coefficient. For a satellite of complex shape both drag coefficient C D and area A are not well defined. C D is highly related to the shape and the way atmospheric molecules interacts with the spacecraft surface, and is also dependant of orbit height and the atmospheric composition which varies with solar activity [3, 27, 29]. A is not well defined since the angle of attack is not always well known, but this is much less of a problem than the drag coefficient if one only keeps track of attitude and/or uses a symmetrical shape. The attitude needs to be kept track of anyway, to be able to know in what direction the acceleration measured is Drag Coefficient The drag coefficient, C D in equation 2.2, is often determined experimentally by fitting measurements of acceleration with an atmospheric model [44]. For example, using density values ρ m from the NRLMSISE-00 [26] model a least square fit can be made: N min ( 2ma V 2 i=1 i Ax ρ m) 2 (2.13) where the drag coefficient then is C D = 1/x. This is often expected to give an accuracy of 5-10% for the drag coefficient [29, 44]. Since what is needed for the control system of STEP is the density in real-time this fit can not be made for the control measurement, at least not without spending a lot of valuable science time on calibration and it would be desirable to know the coefficient already before launch. 11

23 2.2. ATMOSPHERIC DENSITY The theoretical drag coefficient A way to theoretically find the drag coefficient is by using kinetic gas theory if the interaction between air molecules with the satellite surface can be found. Although much research seems to have gone into this all models found seem to be based on coarse assumptions. One of the most well known and used models is Schamberg s model [32]. In this model the drag force is divided into two parts. The force from incident molecules is F i = A ρ V 2 i sinθ i da (2.14) and a force for reflected molecules F r = ρ V i sinθ i V r Φ(φ 0 ) cos(θ i + θ r )da (2.15) A where Φ(φ 0 ) is a function describing the angular distribution of the reflected molecules. The velocity of the reflected molecules is defined as V r = V i 1 + α( T w T i 1) (2.16) where α is the energy accommodation coefficient, which is defined as α = E i E r E i E w = T i T r T i T w (2.17) Here E i is the kinetic energy transported to a unit area of the satellite surface by incident molecules, E r is the energy carried away by reflected molecules and E w is the energy that would be carried away if the molecule had left the surface at surface temperature. T i, T r and T w are the corresponding absolute temperatures. By inserting the total forces from 2.14 and 2.15 in equation 2.2 the value of the drag coefficient can be found. It is evident from this model that attitude of a non-spherical spacecraft will affect the drag coefficient, meaning the satellite would need to keep the same attitude with regard to the velocity vector throughout the mission to get good accuracy. Using a completely spherical spacecraft would also solve this problem. The biggest problem for getting good accuracy from this model is knowing the value of the energy accommodation coefficient and the angular distribution of reflected molecules. Work by Langmuir [20] and Roberts [31] has lead to the conclusion that α is much smaller on clean surfaces than on contaminated ones. A satellite in orbit heights where oxygen is the main atmospheric constituent (such as STEPs orbit) will have its surface contaminated, since oxygen is highly reactive and will stick to almost all surface materials. This also means that a low orbit will have a higher accommodation coefficient than a higher orbit, since the concentration of atomic oxygen is higher, meaning the surface is more contaminated, for low orbits than for higher ones. Moe et al [29] concludes that for orbits below 200 km the accommodation coefficient is close to 1, but it is lower for higher orbits. No exact measurements seem to exist for orbits above 200 km. Only one recorded measurement of the angular distribution Φ(φ 0 ) was found [42], showing that for a 225 km orbit 98% of the molecules were reemitted in a diffuse manner, and only 2% quasi-specular (see figure 2.5). Laboratory experiments [8] have shown that as the surface coverage of adsorbed molecules decreases, the fraction of molecules that are diffusely reflected decrease monotonically, and this is believed to be true also in orbit, meaning that for a higher orbit there will be a higher percentage of molecules reflected in a quasi-specular manner. 12

24 CHAPTER 2. ANALYSIS Figure 2.4: Drag coefficient uncertainty for cylindrical and spherical spacecraft. Source: Moe et al [29] Figure 2.5: Diffuse and quasi-specular particle reflection patterns. Source: Moe et al [29] 13

25 2.2. ATMOSPHERIC DENSITY Figure 2.6: Variations in the yearly average of the ballistic coefficient of several satellites due to solar activity. Source: Bowman [3] Moe et al expects that Sentman s model, which assumes a complete diffuse reemission, will give an accuracy of C D of 5% or better at low altitude orbits. As seen in figure 2.4 the accuracy will be worse as the altitude goes up because of the unknown surface interactions at higher altitudes. It is also evident from these two plots that a spherical spacecraft gives a much smaller uncertainty than a cylindrical one, meaning the drag coefficient of a spherical spacecraft would be less affected by environment changes than a cylindrical. As also pointed out by Moe in an earlier report [27] the uncertainty is actually minimized by a spherical geometry. If one would use a spherical satellite and find a way to measure the energy accommodation coefficient and the angular distribution in-orbit the accuracy of the drag coefficient in the companion orbit could probably be brought down to these 5%, hopefully even better depending on how accurate measurements of α and Φ(φ 0 ) can be made. This might lead to the possibility of having more accurate atmospheric density measurement than possible today [29], as the model-fitting method described in the previous section only gives an estimated accuracy of about 10 percent, and this is assuming that the existing atmospheric model is correct. The possibility of measuring these properties in orbit should be examined further for the possibility of higher accuracy aeronomy experiments Drag Coefficients for control purposes As shown by Bowman [3] the drag coefficient in an orbit at 550 km seems to be heavily affected by solar activity, as very low solar activity will cause the dominant atmospheric species at altitudes as low as 500 km to change from atomic oxygen to helium. As the theoretic drag coefficient of a sphere with respect to atomic oxygen is approximately 2.2, while with respect to helium it is 2.8, this will lead to large uncertainties in the drag coefficient. For the case where the values are to be used in control for STEP, the variations in C D due to changes in environment should at least alter the coefficients in the same direction for both spacecraft, improving the precision in the C Dstep /C Dscout ratio (an explanation why this 14

26 CHAPTER 2. ANALYSIS ratio is important can be found in section 2.2.4). It has been found that for the GRACE mission, consisting of two identical spacecraft in formation separated by about 220 km, the drag measured differed by at most about 1% between the two spacecraft [10], suggesting that uncertainties in the drag coefficients would at least not cause errors in the C D2 /C D1 ratio to be larger than 1%. In the case of STEP and its companion however, the two spacecraft will not be identical. However, Bowman s work shows that for 5 differently shaped satellites at orbits of approximately 400 km the changes in Ballistic Coefficient (C D A/m) due to solar activity measured in percent are strikingly similar (see figure 2.6), almost identical. Taking this into account, the variations in the C Dstep /C Dscout ratio is definitely lower than 5%, and maybe even lower than the 1% found in GRACE, because the difference measured there is definitely not all due to drag coefficient variations. Backing this up are findings by Moe et al [28] that satellites of very different shape, compact satellites versus arrow-like satellites, give variations in drag measurements of no more than ±2% when the drag coefficients have been fitted to an atmospheric model, which seems probable considering the 1% achieved for two identical spacecraft. Since the shapes of both STEP and the companion are probably closer to the compact shape than the arrow-like, the difference in STEP and the companion would probably be less than these 2%. But because of all the uncertainties a conservative maximum error of 2% will be assumed for the C Dstep /C Dscout ratio due to the variations mentioned earlier. This assumes that the ratio can be calibrated in-orbit during the initial phase of the mission, since the drag coefficients themselves can not be determined with high enough accuracy. This is true even if α and Φ(φ 0 ) can be measured, as discussed earlier, since these measurements can not be processed in real-time but probably need time to be analyzed on ground after the mission Other uncertainties Another uncertainty is normally the cross sectional area, A. The attitude of the satellite must be very well known and controlled to get a precise value for the area. This problem is totally solved by using a spherical spacecraft as earlier mentioned, making sure the cross sectional area is constant and well defined, not depending on the attitude of the spacecraft at all and almost totally eliminating this uncertainty. Even if a non-spherical spacecraft is used, the attitude of a spacecraft can, with a star sensor on board, be determined down to an accuracy of 0.1 degree, translating to a very small error in a calculated area for a known shape. However, a symmetric shape that will at least have a constant cross sectional area in velocity-direction is recommended, and optimal would be a spherical shape. Yet another minor uncertainty is the mass of the satellites. Even if it can be decided very accurately before launch the mass is not constant. Outgassing and fuel consumption will change the mass of the spacecraft over time, but by keeping track of fuel consumption this might not pose that big of a problem. It is estimated that this error is much smaller than the error from drag coefficient variations and it is therefore seen as part of the 2% error in the ballistic coefficient ratio Parameter variations on STEP As will be shown in the next section in equation 2.27, the area and drag coefficient of STEP are also used to calculate the expected drag. While the dewar containing the experiment has a symmetric almost cylindrical shape that will always be constant in velocity direction the Service Module has an octagon shape, whose area will vary depending on the pitch angle. Since STEP will be spinning its cross sectional area will not be constant in velocity direction. 15

27 2.2. ATMOSPHERIC DENSITY Figure 2.7: The geometry of the STEP Service Module and its projected area for a pitch angle of 0 and 22.5 degrees. For 22.5 degrees the area will be about 8 percent larger than for 0 degrees A/A pitch angle Figure 2.8: Change in Area of STEP Service module due to pitch angle. A 0 is area at angle 0, defined as when two of the sides of the module are parallel to the velocity vector. It is however very simple to model the variations for a simple shape such as an octagon, as done in figure 2.8. The change in attitude will also affect the drag coefficient as pointed out in the previous section. Equations 2.14 and 2.15 are used to estimate how the attitude change on STEP would affect the drag force on the Service Module. Both diffuse and specular reflection will be considered. Completely diffuse reflection can be expressed as having a reflection angle of 90 degrees from the surface plane and an angular distribution function that is constant at 2/3. Reflected particles are assumed to have the same velocity as incoming. Equations 2.14 and 2.15 for a single surface element of the Service Module will then become F i = ρv 2 i sinθ i A (2.18) F r = ρv i V r sinθ i 2 3 cos(θ i + 90)A (2.19) 16

28 CHAPTER 2. ANALYSIS Drag Force variations F/F Best case Worst case pitch angle Figure 2.9: Relative variations in Drag Force on Service Module of STEP due to pitch angle. meaning the total force on one element would be F element = ρv i sinθ i (V i 2 3 V rcos(θ i + 90))A (2.20) In figure 2.9 the total velocity direction force dependance on pitch angle can be seen. It is here assumed that reflection velocity equals incoming velocity for the worst case. Also the best case, explained further down, is plotted in the same figure. An assumption that all reflection is diffuse is good for very low orbits, as pointed out earlier, but for higher orbits a larger part of the particles will be reflected in a specular manner. For specular reflection the force on one element reduces to F element = ρv i (V i + V r )Asinθ i. (2.21) The angular distribution function is in this case 1. For the Service Module geometry this always totals to F tot = 2ρV i (V i + V r )A (2.22) over all exposed surface elements in velocity direction. This will make the variations of the force on the Service Module smaller than found above, but how large the fraction of specularly reflected particles is, as mentioned in previous sections, unknown at these altitudes. For a best case scenario the findings of Harrison and Swinerd [29] that for satellites at km the speed of reemission was about 2 km/s and the angular distribution was half diffuse and half specular will be used. Drag coefficient variations for the service module in the worst case, defined as only diffuse reflection with reemission velocity equal to incoming velocity, and this best case scenario can be seen in figure Average incoming velocity is assumed to be equal to the spacecraft velocity. Combined with the variations in area this will however not lead to variations in the experienced drag larger than 0.4% for the worst case scenario, as seen in the force variations in figure 2.9. The Service Module is also only about a third of the total spacecraft, and with the rest having attitude-independent drag force due to symmetry the total force will vary much less than this. Considering this variation is very small compared to the 2% variation expected in the drag coefficient ratio, it could be assumed that the area and drag coefficient is constant with pitch angle, since the calculated force using this assumption will have a worst-case error of less than 0.2%. 17

29 2.2. ATMOSPHERIC DENSITY 1 Drag Coefficient variations C D /C D Best case Worst case pitch angle Figure 2.10: Relative variations in Drag Coefficient on Service Module of STEP due to pitch angle Error Analysis Assuming the aforementioned errors in the parameters, summed up in table 2.2, a maximum error in the measured density of approximately 12% is expected. If we consider the case when the measurements will be used for feed forward control the winds can be neglected and the error reduces to 4% (see table 2.1). The reason for not considering the wind is that for control purposes the density is not really the interesting parameter, but atmospheric drag pressure is. Wind speed at these altitudes is expected to be almost constant over the time between the two satellites. The calculation of expected drag for STEP is done by assuming it will experience the same atmospheric density and relative atmospheric velocity as the Companion. That is, V s = V c (2.23) ρ s = ρ c (2.24) where subscript s corresponds to STEP and c to the companion satellite. These assumptions together with equation 2.2 give a Ds = 1 2 C A s Ds ρ c Vc 2. (2.25) m s V c is assumed to be measurable and known, and ρ c is derived from equation 2.2, All together this gives ρ c = 2 Which, with equation 2.3 can be written as a Dcm A c C Dc V 2 c. (2.26) a Ds = a c C Ds C Dc A s m c m s A c. (2.27) a Ds = a c B c B s. (2.28) All of the parameters needed to calculate the expected drag on STEP except for the ratio of the drag coefficients are known. The ballistic coefficient ratio can however be estimated to 18

30 CHAPTER 2. ANALYSIS Error source Error Ballistic Coefficient ratio 2% Unmodeled Drag Forces 1% Other errors 1% T otal error 4% Table 2.1: Expected error for drag estimation for STEP. Other errors include accelerometer measurement errors etc. Does not include ratio calibration errors. Error source Error Ballistic Coefficient 5% Unmodeled Drag Forces 1% Winds 5% Other errors 1% T otal error 12% Table 2.2: Expected error for absolute density measurements from the Companion satellite an accuracy of about 2% during the initial phase of the mission as discussed earlier in section The relative error budget for the measured acceleration from atmospheric drag, with ballistic coefficient quota as a single variable B, can be written as a Ds a Ds = a c a c + B B + err (2.29) where unmodelled drag sources and other errors are grouped together in the err term. a c is the accelerometer error on the companion. See table 2.1 for the estimated error this leads to in the expected drag experienced for STEP. The error budget for absolute density variations are calculated in the same way and summed up in table Radiation and Charge Measurement Another large problem for STEP is the electrical charging of the test mass. Free masses may acquire charge from particle radiation and through interactions with the environment this charge will lead to movements of the test mass that needs to be kept at a minimum. In addition to being used for charge control on STEP measurements can be valuable for mapping of the trapped proton radiation environment in the 550 km orbit plane, studies of test mass charge behavior etc. Having two spacecraft flying in the same orbit with a short distance in-between will also allow higher resolution radiation measurements and the ability to compare test mass charging for two spacecraft in the same environment and to investigate temporal variations in the flux rates. A radiation monitor will also fly on STEP Test mass charging Cosmic rays, solar protons and geomagnetically trapped protons with energies in excess of about 100M ev can penetrate spacecraft structures and deposit heat, momentum and electrical charge on test masses used in experiments or for Drag Free control [14]. Charging is 19

31 2.3. RADIATION AND CHARGE MEASUREMENT the most significant of these disturbances as it leads to Coulomb forces between the test mass and surrounding conducting materials as well as Lorentz forces due to the motion through a planets magnetic field if the masses are not electromagnetically shielded. Higher energy charged particles will result in higher net charges, as e+/e- pair production and cascaded nuclear reactions take over [1]. For STEP this is a major problem as disturbance levels need to be extremely low for its high accuracy measurements. The momentum transferred to the test mass from incoming particles gives disturbances small enough to be negligible [37] Charge control The charge accumulated has to be measured and dumped before it gets large enough to cause large disturbances in the experiment. Charge control of STEP is done by UV light which releases photoelectrons from the surface. These electrons are then redistributed in order to neutralize the charge on the test mass. Charge levels are measured by applying an altering dither voltage to each test mass. If the mass carries a charge, this produces a force which alternates in time with the voltage. The masses accelerate in response, and this acceleration can be measured. It is not necessary to actually calculate the charge to control it, but a proportional control system is supposed to reduce the measured charge down to the noise level [37]. The period of the alternating voltage on STEP is 100 seconds. Problems have been identified in this control approach with large charge gradients during passage through the SAA (see section ). As one EP measurement on STEP requires 20 orbits [37], this means 8 to 10 unavoidable passes through the SAA for each measurement. If a companion spacecraft could measure the radiation environment in front of STEP and use this for feed forward, the charge control system on STEP would know exactly when to expect higher charging rates, be able to calculate expected charge rate and act accordingly. Hopefully this can lead to improved performance of the charge control system. The possibilities and accuracy of predicting charging needs to be examined further in future studies Radiation measurement Test mass charging occurs when the spacecraft is subjected to proton radiation of energies larger than 100MeV. This can be measured by a particle monitor designed to measure proton flux of about 100MeV and above. As the amount of charge deposited on the test masses in a given event is dependent on the energy of the incident particle, it would also be beneficial to have spectral information [1]. The number of energy ranges and accuracy requirements of a sensor needs to be decided based on an investigation on how accurately test mass charging can be calculated from proton flux data. One candidate could be the ESA Standard Radiation Environment Monitor (SREM), which has an accuracy of better than 1% and comparators with fifteen detection ranges[2, 7] Radiation sources The only two cases where significant amounts of proton radiation with energies above 100M ev is received is during a pass through the South Atlantic Anomaly and during solar flares. There is also a region around each pole (the auroral regions) where some radiation will be encountered. Radiation from Galactic Cosmic rays is negligible in comparison [14]. 20

32 CHAPTER 2. ANALYSIS Figure 2.11: Charged particle motion in the Van Allen Belts. Source: ESTEC [6] Trapped radiation and The South Atlantic anomaly Charged particles, especially protons and electrons, can be trapped by the Earth s magnetic field. Their motions are a blend of three periodic motions which take place simultaneously. A gyration around magnetic field lines, typically thousands of times each second, a slower back-and-forth bounce along the field line, typically lasting 1/10 second and a slow drift around the magnetic axis of the Earth, from the current field line to its neighbor. Typical time to circle the Earth is a few minutes. The area where these particles are contained is known as the Van Allen belt. An illustration of the particle motions can be seen in figure The South Atlantic Anomaly, illustrated in figure 2.12, is the region where Earth s inner van Allen radiation belt makes its closest approach to the planet s surface. Or, for a given altitude, the radiation intensity is higher over this region than elsewhere. It is produced by a dip in the Earth s magnetic field at that location, caused by the fact that the center of Earth s magnetic field is offset from its geographic center by 450 km. Other radiation sources (not counting solar flares) are negligible in comparison to the trapped proton flux for the STEP orbit [14]. It is also the only source of radiation that would be useful to measure by a scout satellite since other sources (solar, cosmic rays etc) will reach both satellites at the same time, while the flux from trapped protons varies with Earth coordinates. In figure 2.13 the trapped proton flux for energies above 100 MeV at orbit height (550km) during solar minimum is shown. The South Atlantic Anomaly is clearly visible as the only place where high fluxes of trapped proton radiation is received. For protons, a larger flux occurs at solar minimum, but for electrons the situation is reversed [12]. The radiation belt model used to generate the figure is AP-8 [24]. The input to this model is the L-value, or dimensionless distance, defined by L = r 0 /R E (2.30) where r 0 is the radial distance to the field-line in the equatorial plane and R E is the Earth equatorial radius, and the ratio between local magnetic field strength B and the field strength at the equator B 0. So the proton flux is a function of these two, F p = f(l,b/b 0 ) (2.31) 21

33 2.3. RADIATION AND CHARGE MEASUREMENT Figure 2.12: Location of the South Atlantic Anomaly in the 550 km orbit plane. SPENVIS [36] Source: The magnetic field strength has both an internal part from the Earth s magnetic field and an external part that is due to extraterrestrial sources such as the ring current and the solar wind. The internal field varies very slowly, on the scale of 100 years. The external field can however vary rapidly on timescales as low as milliseconds, but even during extremely large magnetic storms the external field makes up only about 1 percent of the total field at LEO altitudes. It is assumed that the field will be constant over the short time period between the passes of the Companion and STEP in this study and the AP-8 model will be used. For modeling the magnetic field, the static IGRF [13] model is used. The external field is considered to be negligible. In figure 2.14 flux of protons with energies above 100 MeV is plotted as a function of time for one day in orbit. The spikes in the plot are from passes through the SAA Requirements The requirements for STEP is that no test mass is allowed to accumulate a charge giving variations larger than C at signal frequency and C at DC [37]. Simulations made by Jafry et al [14] for the geometry of STEP shows that for one orbit a charge of at most about C will be accumulated, almost all in the SAA, a small part being from cosmic rays. In the worst case, with a non-rotating spacecraft the science signal frequency will be the same as the orbit frequency. Since there is one pass through the SAA for each orbit for several orbits straight the charge deposits will also come at close to the signal frequency. This means that it might not be possible to keep the requirement of charging at signal frequency unless charge is dissipated during each pass through the SAA. During an average solar flare a charging of about C/kg is expected and a large flare can lead to charging of up to C/kg. A companion would not be as helpful in the case of solar flares since it will not be able to measure the radiation before it reaches STEP. It could however be a little helpful since the slow charge measurement system will discover 22

34 CHAPTER 2. ANALYSIS Latitude AP 8 MIN Flux >100 MeV [cm 1 s 1 ] Longitude Figure 2.13: Proton radiation flux over entire Earth for energies above 100MeV during solar minimum (worst case) using the IGRF and AP-8 models Proton Flux > 100 MeV [cm 2 s 1 ] Figure 2.14: Proton (100 MeV) radiation flux for one day of the STEP/Companion orbit. The South Atlantic Anomaly gives large spikes in the flux. Time [s] 23

35 2.3. RADIATION AND CHARGE MEASUREMENT the charge increase much later than the raise in proton flux is detected. Since flares occur relatively seldom and it will be known when a flare has occurred, data from times coinciding with a flare will just be ignored [14]. The concept of Feed Forward and control by using proton flux measurements is explained a little further in section 5.2. Unfortunately not enough details on the test mass charging and control for STEP was available to do a more detailed analysis of such a system. 24

36 3 Orbit and formation dynamics In this chapter the orbit requirements will be presented as well as trade-offs for deciding the exact orbit and the type of orbit control. A dynamics model is created to simulate formation keeping performance for the trade-offs. 3.1 Orbit requirements The orbit of the companion will be a sun synchronous dusk-dawn orbit nearly identical to the orbit of STEP. Parameters for the orbit of STEP are given in table 3.1. In a sun-synchronous orbit the satellite passes over the same part of the Earth at roughly the same local time each day. In a dusk-dawn orbit, the satellite trails the Earth s shadow such that satellite never moves into the Earth s shadow making sure the satellite is always sunlit. The fact that Earth is not a perfect sphere is taken advantage of to create a sun synchronous orbit. An orbit that is close to being polar will be affected asymmetrically by the bulge at the equator and this asymmetry acts to slowly rotate the plane of the orbit about the axis of the Earth. In a sun-synchronous orbit the orbit plane executes one full rotation about the axis of the Earth in one year. The rate of rotation of the Right Ascension of the Ascending Node (RAAN), Ω, because of the flattening of the Earth is expressed by the equation Ω = a 7/2 (cos i)(1 e 2 ) 2 (3.1) where a is the semimajor axis, i is the inclination and e is the eccentricity of the orbit [41]. For a 550 km circular orbit the inclination where the orbit becomes sun synchronous is 97.6 degrees. There are several considerations to make when deciding the exact orbit for the Companion. For drag free control purposes the optimal would probably be to have the Companion flying as close as possible in front of STEP to get as current measurements as possible. There might however be factors that requires the distance between the spacecraft to be larger, and having a large distance then raises the question wether an identical inertial orbit is desirable or if orbits using the same ground track would be better. This will depend on if the disturbances that need to be measured are following Earth s rotation or not. At the end of this chapter trade-offs are made to decide more exactly the orbit of a Companion satellite. A trade-off is also made to establish wether a Companion should fly Drag Free in front of STEP, or if a simpler design with only periodical orbit corrections would suffice. 3.2 Orbit dynamics The basic equation of motion for a satellite in an Earth orbit is given by r = a G + a D (3.2) where r is the Earth centered inertial (ECI) position, a G is acceleration due to gravity and a D is the acceleration due to other disturbance forces. For more information on how to model 25

37 3.2. ORBIT DYNAMICS Figure 3.1: Definition of the inclination i and the Right Ascension of Ascending Node (RAAN) Ω. The vernal equinox is a constant direction in the Earth Centered Inertial (ECI) coordinate frame. Parameter V alue Orbit height 550 km Inclination 97.6 degrees Eccentricity < 0.02 RAAN 90 degrees from sun vector Table 3.1: Orbit Parameters for the near-circular sun-synchronous STEP orbit given in Phase A study report [37] 26

38 CHAPTER 3. ORBIT AND FORMATION DYNAMICS main spacecraft Circular orbit r R 0 scout Earth Figure 3.2: The coordinate system for Hill s equations with main spacecraft in origo. ξ-axis is along the radial direction from Earth center, η is velocity direction. ζ is pointing orthogonal to the others into the image. gravity and disturbances see section For calculating initial conditions Keplerian orbit elements are used, which give a good enough approximation spacecraft in formation Relative motion of spacecraft in close formations in circular orbits can be described by Hill s equations [39]. These are linearized equations of the difference between the gravitational forces acting on the two spacecraft. The linearization is done about a perfectly circular orbit in a coordinate frame that rotates with the spacecraft. The main spacecraft is placed in origin and the ξ-axis is along the radial direction from Earth, η is along the orbit path and ζ is orbit normal (see figure 3.2). The motion of secondary spacecraft relative to the main one is described by Hill s equations, ξ 2n η 3n 2 ξ = a ξ (3.3) η + 2n ξ = a η (3.4) ζ + n 2 ζ = a ζ (3.5) where a i is a disturbance acceleration in i-direction and n is the orbit rate. n = µ/a 3 (3.6) where a is the radius of the orbit. µ is the product of the gravity constant and the mass of Earth. µ = Gm E (3.7) 27

39 3.3. CALCULATIONS AND SIMULATIONS Hill s equations give fairly good approximations of relative motion for small distances along ζ- and ξ-axis, and any distance in the η-direction (along-track). For evaluating the relative motion of spacecraft formations it is often assumed that the disturbance forces a ξ,η,ζ are zero. This is an even better approximation if the spacecraft in the formation are all flying drag free. The analytical solution to Hill s equations making this assumption is ξ ξ(t) = 0 n sin nt (3ξ 0 + 2η 0 n ) cos nt + (4ξ 0 + 2η 0 n ) (3.8) η 0 η(t) = (6ξ n 2ξ 0 ) sin nt + n (6nξ 0 + 3η 0 )t + (η 0 2 ξ 0 n ) (3.9) ζ ζ(t) = ζ 0 cos nt + 0 sin nt (3.10) n These functions can be used to get an estimation of the relative positions of two perfectly drag free spacecraft at time t with given initial positions, where one is flying in a perfect circular orbit (in origin) and the other has initial conditions ξ 0, η 0, ζ 0, ξ 0, η 0 and ζ 0 relative to the first. Worth noting is that ζ(t) is decoupled from the other equations. For estimating the relative motion due to some bias error in the atmospheric drag measurement on one of the spacecraft one could see this as constant accelerations a ξ, a η and a ζ of the size of the maximum error acceleration. The solutions for Hill s equations then become ξ ξ(t) = 0 n sin nt (3ξ 0 + 2η 0 n ) cos nt + (4ξ 0 + 2η 0 n ) 1 3n 2 a ξ + 2 n a ηt (3.11) η 0 η(t) = (6ξ n 2ξ 0 ) sin nt + n (6nξ 0 + 3η 0 )t + (η 0 2 ξ 0 n ) 3 2 a ηt 2 (3.12) ζ(t) = ζ 0 cos nt + ζ 0 n sin nt + a ζ n 2 (3.13) The only difference from equations 3.8 to 3.10 being the last terms of each. These can be used to estimate effects of disturbances on the companion when assuming STEP flies in a perfect orbit. 3.3 Calculations and Simulations Orbit simulator For studying the behaviour of Drag Free satellite formations a satellite dynamics model was developed in Matlab/Simulink, based on a simple 6 degrees of freedom dynamics model. Motion in x,y and z direction is described in the model by equation 3.2. The original model was modified to include forces of gravitation of a non-spherical Earth. Since a Drag Free satellite is designed to follow a purely gravitational orbit no forces need to be added to the model except for Earth s gravity. Gravity potential is given by the two-body potential plus the J2 term [16], U = µ J2 [1 + r 2 (R E r )2 (1 3 sin 2 θ)]. (3.14) where r is distance from Earth center, R E is Earth s equatorial radius and θ is colatitude, measured from the north pole. 28

40 CHAPTER 3. ORBIT AND FORMATION DYNAMICS A module for atmospheric density was also added to the model to be able to observe differences in orbits of Drag Free satellites and satellites without Drag Free control. Atmospheric drag is by far the largest disturbance force at altitudes interesting for this study and all other forces are assumed to be negligible in comparison. The module uses the NRLMSISE-00 model [26] for calculating atmospheric density, and the acceleration experienced by the spacecraft depends on the atmospheric density as described by equation 2.2. For the simulations the atmosphere is assumed to rotate with Earth (winds are neglected) so that V = V ECI V atm where V ECI is the velocity in the inertial frame and V atm is the velocity of the atmosphere, V atm = ω e r (3.15) where ω e is Earth s rotation vector (pointing in direction of the rotation axis). The NRLMSISE- 00 model used for density calculation in simulations requires the F 10.7 and a p indexes as input (apart from position and time). F 10.7 is the solar radio flux observed at a wavelength of 10.7 cm and this corresponds to a radio emission line for iron and is normally reported in solar flux units (sfu) where one solar flux unit is W/m 2 H z. It varies from about 50 sfu at solar minimum to 240 sfu at solar maximum with the 11-year solar cycle. The short-term geomagnetic activity dependence is accounted for by the 3-hour, semilogarithmic K p or its linearized form a p. This index represent magnetic-field disturbances induced by changes in the solar wind and correlate with short-term variations of the upper atmosphere. The a p values were selected so that they correspond to the maximum variations in the Earth s surface magnetic field at midlatitudes in a 3-hour period. The a p index range from a minimum value of zero to a maximum of 400. Since a p is ultimately related to solar activity, there is a weak correlation between them and F For normal atmospheric conditions to minor magnetic storms the a p index is below 50. A worst case value quoted as used for the space station design [12] is 140 for a p and 230 sfu for F The model is implemented with these indexes as constant parameters, but it could be updated to have these as input parameters to allow simulation of the past atmosphere in cases where these are known Baseline configuration used for simulations For the dynamics simulations of a Drag Free satellite its mass and shape are not important since the drag free control will make sure the orbit is purely gravitational. For the simulations run with atmospheric drag several different values of the ballistic coefficient B are considered. Average values for input to the atmospheric density model are used. Real disturbances can vary both ways depending on atmospheric conditions during the actual flight. Torques and attitude rates and are not considered for this study (although the simulink model on which the simulator is based includes them) and therefore the inertia matrix and attitude rates are irrelevant. Ballistic coefficients are varied between 25 to 400 and the time difference between the two satellites will be varied between 1 to 200 seconds. The orbit used is the orbit for STEP, as described in section 3.1. For modelling the density in the worst case values of F 10.7 of 230 sfu and a geomagnetic index number of 140 are used in the NRLMSISE-00 model. Simulations are also made with a more average atmosphere with a F 10.7 value of 150 sfu and a p value of 4, corresponding to a quiet magnetosphere at a time in between solar maximum and minimum. 29

41 3.4. TRADE-OFF: DRAG FREE CONTROL OF COMPANION Figure 3.3: The Simulink satellite dynamics model for both Main Drag Free spacecraft and a Companion with optional atmospheric drag. 3.4 Trade-off: Drag Free Control of Companion In this section a trade-off is made to examine if the Companion needs to fly Drag Free, like STEP does, or if the performance can be good enough even if its orbit is subject to disturbances from air drag and other drag sources, which could simplify the design Drag Free satellite with non-drag Free companion A possibility might be to have a companion without drag free control. This could lead to a simpler, and therefore lower-cost, spacecraft design. A non-drag free satellite would be subject to several orbit disturbances, but since air drag is by far the largest one, it is the only one considered for calculations and simulations in this study. Considering an average density at a 550 km orbit of kg/m 3 equation 2.2 gives an acceleration of ab out m/s 2 for a spacecraft of mass 100 kg, cross section area of 1 m 2 and a drag coefficient of 3. Assuming ξ 0 = ξ 0 = η 0 = a ξ = 0 equation 3.12 gives η(t) = η a ηt 2 (3.16) Setting a η = m/s 2 gives an along-track deviation from η 0 of ab out 2700 meters after 1 day. A w orst case density of kg/m 3 [41] w ould give a deviation of almost meters in one day. 30

42 CHAPTER 3. ORBIT AND FORMATION DYNAMICS s 10 s 25 s 50 s 100 s 200 s 1000 Distance variation [m] time [s] x 10 4 Figure 3.4: Absolute separation of Drag Free Satellite and non-drag free companion over one orbit due to J2 perturbations and air drag depending on initial time distance Simulations A numerical simulation with average atmospheric conditions was also run and the result can be seen in figure 3.4. The result from the approximate calculation above is confirmed and a drift in distance between the two spacecraft of about 2700 meters over the first day is observed, following the shape of a square function. Also visible is the effect of J2 perturbations which will cause periodic distance variations over each orbit. More interesting for the application of a scout satellite are the variations in time difference between the two spacecraft. To be able to measure this a definition needs to be made as to how the difference should be measured, since the spacecraft will not pass through exactly the same points in space. The chosen definition is the time difference between the passage over a latitude line on Earth. Figure 3.5 shows how the time difference will vary with time, assuming an average atmosphere and an initial time difference of 100 s for several different values of the ballistic coefficient. For a small, heavy spacecraft average atmospheric drag will result in a lag of less than 0.1 seconds in one day, but for a larger and lighter spacecraft the lag will be as much as half a second after one day. In figure 3.6 the separation in a worst case atmosphere is shown. The variations will be much larger in this case, up to almost three seconds after one day. Depending on the accuracy requirements the orbit would need to be corrected at a regular interval, probably several times per day if very high accuracy is required Tw o Drag F ree satellites Using two drag free satellites in the same orbit would in theory mean that no additional manoeuvres would be needed to correct the orbits. Perfectly identical orbits are not achievable in reality however, and effects on the formation from small differences between the orbits will be examined through calculations and simulations. 31

43 3.4. TRADE-OFF: DRAG FREE CONTROL OF COMPANION separation [s] time [t] x 10 4 Figure 3.5: Time separation of Drag Free Satellite and Companion over one day due to air drag for several ballistic coefficients for average atmospheric conditions separation [s] Figure 3.6: Time separation of Drag Free Satellite and Companion over one day due to air drag for several ballistic coefficients, in a worst case atmosphere with F 10.7 = 230sf u and a p = 140. time [s] x

44 CHAPTER 3. ORBIT AND FORMATION DYNAMICS Along track separation [m] s 10 s 25 s 50 s 100 s 200 s time [s] Figure 3.7: Separation of two Drag Free Satellites over one orbit due to J2 perturbations depending on initial time distance. Again, equations 3.12 and 3.11 can be used to study the behavior of the formation. Assuming both satellites fly perfectly Drag Free (a ξ = a η = 0) and ξ 0 = ξ 0 = η 0 = 0 these reduce to ξ(t) = 0 (3.17) η(t) = η 0 (3.18) That is, the formation will be perfectly stable and static. In reality the acceleration sensors and control systems on STEP and the Companion will not be perfect, and they have small biases in the measurements. For more information about acceleration measurements errors, see section If a maximum bias difference of 1% of the average air drag used for calculations in section is assumed for the two spacecraft, equation 3.16 gives an along track separation deviation of only 27 meters over 1 day. This translates to about 4 ms in time separation difference making the same assumptions as previously. A maximum residual acceleration of m/s 2 should not be a problem to achieve using Drag Free Control Simulations Figure 3.7 shows the variations in initial distance between two drag free satellites in identical orbits depending on initial time difference. As seen in the figure the distance between the two spacecraft will vary with several kilometers even when in perfectly identical orbits if the initial distance between the spacecraft is larger than a few seconds. This is due to the Earth not being a perfect sphere. Non-perfect Drag Free control can also be simulated as one perfect drag free spacecraft and one that always experiences 1% of the average air drag (see section ). In this case STEP is assumed to fly in a perfect orbit and the companion will have an error resulting from a non-perfect drag free control. This is illustrated in figure 3.8. Again the previous 33

45 3.4. TRADE-OFF: DRAG FREE CONTROL OF COMPANION separation [s] time [s] x 10 4 Figure 3.8: Separation of two Drag Free Satellites over one day due to 1% bias errors in acceleration sensor for several ballistic coefficients. calculations and simulations show consistency. As seen this results in time errors on the scale of 1% of the case where no drag free control was used for the companion. One measurement for STEP requires 20 orbits [37], and during these 20 orbits the total maximum time error would be less than 10 ms. Since the sampling frequency is 10 Hz, this difference is small enough to not require other orbit control than the Drag Free control during a measurement Discussion Mainly because of the fact that a Companion flying Drag Free would mean no orbit corrections would need to be made during a measurement phase this would be a better choice from a formation keeping point of view. A high accuracy accelerometer needs to be included in the payload in any case, and this can also be used for Drag Free control, which means it does not add much extra complexity over a non-drag free alternative, basically only software. Even if no Drag-free control is used automatic orbit corrections between each measurement phase would be needed, which would still require a somewhat complex control system. According to Leitner et al [21] there is also a lower total V for formations that use Drag Free control compared to a formation where the orbit would be corrected at regular intervals, though for heights above 450 km, this improvement is very small, less than 1%. A problem with having a Drag Free control system on a satellite designed to measure the drag, as opposed to actually counteracting it, is that since the thrusters will tend to cancel the air drag very accurate thrust measurements must be available. According to Jafry [15] it would be possible to determine this thrust with an accuracy as good as 1% in the best case for the helium thruster type used on STEP and another Drag-free mission, GP-B. This however puts strict design demands on the satellite, and thrusters in particular. Using electric propulsion thrusters this estimation error can probably be brought down even lower. The helium thrusters used in Jafry s study give errors related to nozzle calibration issues. Since many electric propulsion systems, such as Field Emission Electric Propulsion (FEEP) thrusters, has no valve and no moving parts, it generates no mechanical noise [4] meaning 34

46 CHAPTER 3. ORBIT AND FORMATION DYNAMICS Con trol choice Adv an tages/disadv an tages Non-Drag Free companion + Simpler control system design - Bad formation keeping - Frequent orbit updates required Drag Free companion + Excellent formation keeping - Possibly lower accuracy for density measurement Choice Drag Free control Table 3.2: Trade-off for orbit control the calibration will not pose as large a problem as for helium thrusters. It is assumed the error induced by this will not be larger than 1%. It is also possible that attitude corrections would need to be made during science phases, meaning the thrusters may need to be used anyway. The alternative could be to make a very dense spacecraft and thereby minimize the orbit drift by minimizing the atmospheric acceleration. A spacecraft with a ballistic coefficient of about 100 would still lead to acceptable drift levels so that the orbit would not need to be corrected during a 20 orbit measurement. The decay in height during a 20 orbit measurement phase for a spacecraft with a ballistic coefficient of 100 is less than 20 meters under average conditions, which is too small to give any significant difference in atmospheric density encountered, and likely less than the formation insertion accuracy. Also there is no experiment onboard that requires a drag free environment. In simulations that will be presented further in section and figure 5.4, the performance increase of a drag free companion with 1% larger measurement error is almost exactly identical to the performance increase when using a decaying orbit Selection Drag-free control provides superior formation keeping performance since the main spacecraft is also flying drag free. Using a non drag free companion would also still require automatic orbit updates using GPS measurements or other means, meaning the only difference between the two cases would basically be the software of the control system. Based on this the choice is to have a companion flying Drag-free in front of the main Drag-free satellite. 3.5 Trade-off: Distance b etween spacecraft Quality of measurements for control purposes is of course better with smaller distances, but what distance to choose is also dependant of how the control systems works and how soon before the control they would need the measurement. Since this is currently unknown for the charge control system no definitive choice can be made at this time. One might also look at what would be interesting for scientific measurements of density and radiation to take that into account. In section and figure 5.5 a comparison is made for performance of the Feed Forward control of drag for different time differences, and it is, as expected, obvious that a shorter distance is better than a larger Selection The baseline distance chosen for evaluation in this study is 10 seconds, however this is subject to change depending on requirement for the charge control system and scientific interest. A 35

47 3.6. TRADE-OFF: GROUND TRACK main spacecraft companion Figure 3.9: Two orbits with the same ground track. 10 second separation would also allow for some orbit drift without risking the companion ending up behind the main spacecraft. 3.6 Trade-off: Ground Track There are two different possibilities for the formation of the two spacecraft. On a first glance one might think it s clear that the companion should fly in front of STEP in the same inertial orbit, as one wants to measure the variations at the exact same point that STEP will soon pass through. When considering how the variations will behave however, one realizes that it might also make sense to use the same Earth-fixed ground track instead, since some disturbances, especially the South Atlantic Anomaly, will move with the Earth. In an orbit where the separation is more than a few seconds this might lead to unacceptably large differences in environment for good feed forward control between the times when the spacecraft pass through the same point. On the other hand, atmospheric density is heavily dependant on local solar time, which will almost not change at all when using the same inertial orbit, but will change by the separation time when using the same ground track. A trade off needs to be made to see how big of an impact the two orbits will have to the performance and to decide which of these orbits (or something in between) would minimize environment changes during the time between the two passes. To get the same ground track one needs only change the right ascension of the ascending node of the companion by the same angle that the Earth rotates during the time between the two spacecraft. That is 36

48 CHAPTER 3. ORBIT AND FORMATION DYNAMICS Ω c = Ω s + ω e t d (3.19) where ω e is Earth s angular velocity and t d is the time separation of the two spacecraft. For inertial (ECI) coordinates the position and velocity vectors of the companion compared to STEP will then be rotated around the z-axis by Ω c Ω s = ω e t d. (3.20) New coordinates for the companion can then be calculated from the coordinates for identical inertial orbits x = x cos (ω e t d ) + y sin (ω e t d ) (3.21) y = y cos (ω e t d ) x sin (ω e t d ) (3.22) z = z (3.23) ẋ = ẋ cos (ω e t d ) + ẏ sin (ω e t d ) (3.24) y = ẏ cos (ω e t d ) ẋ sin (ω e t d ) (3.25) z = ż (3.26) In the (ξ,η,ζ) coordinate frame this would mean that the companion would move in an 8 around the point where it would be if the two orbits were identical during 1 orbit. It will not lead to any variations in the time difference for two drag free satellites since the two orbits are still J2-invariant. This means only the quality of measurements for the two alternatives needs to be considered when deciding the orbit A tmospheric Drag According to preliminary calculations, done with the assumption that the NRLMSISE-00 model gives a good approximation of density variations, a 200 second separation will give maximum differences in density experienced of about one part in a thousand for the case where the satellites follow the same inertial orbit, and one part in a hundred when following the same ground track. For a 10 second separation there will be maximum differences of only 0.05 percent when using the same ground track. The same inertial orbit would then give differences of less than 0.01 percent. Even for the case where the difference is up to 1 percent, this is small compared to the total error in the measurement (see section 2.2.4). Simulations in section with results shown in figure 5.4 show almost no difference in the performance for the two cases for a 10 second formation distance Radiation As discussed in section 2.3 proton flux is almost static with geographic coordinates, which means a same ground track orbit would be optimal, leading to very small differences in the flux that would be encountered between the two spacecraft. Looking at figures 2.14 and 2.13 from the previous chapter one can see that at the edges of the South Atlantic Anomaly radiation will vary very quickly. Using the same inertial orbits for the two satellites and a 10 second formation distance this would lead to rather small differences in the flux encountered at the same point, less than 2% of the maximum flux according 37

49 3.6. TRADE-OFF: GROUND TRACK Control choice Same inertial orbit Same Ground Track Choice Advantages/disadvantages + Accurate density measurments - Relatively large difference in radiation encountered at SAA edge + Accurate particle radiation measurments - Small differences in density encountered Same ground track Table 3.3: Trade-off for orbit to simulations, but with larger distance the difference will become more significant. If the formation distance is 100 seconds simulations show that the differences will be up to about 6%, which is significantly larger than the error that would result in drag measurements from using a same ground track orbit. When using the same ground track simulations show very small differences in the flux encountered since a static magnetic field is assumed (see section ). The temporal variations should however be investigated further, as the error estimate for the AP-8 model is quoted as a factor of 2 [23], so it is not a very accurate and for this application trustworthy model Selection Given that the proton flux varies very fast with small distances at the edges of the South Atlantic Anomaly it is important for accurate feed forward that the satellites follow the same ground track, meaning they will encounter as similar radiation environments as possible. A time distance of 10 seconds would give an maximum inertial separation of about 650 meters between the point where the main spacecraft passes through and the point where the companion made its measurement 10 seconds earlier. Atmospheric density variations are very small over such distances and this should not pose a problem. A same ground track orbit is chosen. 38

50 4 Spacecraft In this chapter a simple baseline configuration of a scout satellite for a Drag Free mission will be proposed. For most parts it will be based on coarse assumptions and each subsystem needs of course to be studied in greater detail before actually designing and selecting components for a mission using a scout. It should however give an idea of how such a satellite could be designed and serve as the baseline for a coarse cost estimate. An illustration of the suggested configuration can be seen in figure P ayload The main payload of a companion will be the accelerometer used for drag measurement and a proton monitor to monitor flux of protons with energies of over 100 MeV for estimation of charging rates Accelerometer To reach the objective of measuring air drag, it is really only necessary to measure acceleration in one direction, namely the direction of movement. It would however be good to have measurements in all 3 degrees of freedom for redundancy and for the Drag-free control system. The most sensitive axis of the accelerometer should however be pointed in velocity direction. To reach the accuracy levels derived in section the total error due to accelerometer inaccuracies must be less than one percent of the average air drag. The total error budget is (from equation 2.5) Γ = ( c 0i ) + ( c 1i ) a + noise (4.1) i i Average drag will, in accordance with assumptions made earlier in this report, be about m/s 2 and if it is assumed that each of the three parts are one third of the total allowable error this means the errors in velocity direction should be at most m/s 2 for each of the three parts. Most important are the time- and temperature variational parts, as constant scale factor and bias errors could be calibrated out. The accelerometer should optimally be able to sample at 10 Hz, the same frequency as is used onboard STEP, though simulations show that a sample frequency of 1 Hz would not mean any degradation of the performance demonstrated in the next chapter. Error source Allow able error [m/s 2 ] Bias, variable part Scale factor, variable part Noise T otal error Table 4.1: Error budget for accelerometer for use on companion satellite. 39

51 4.1. PAYLOAD Figure 4.1: A near-spherical spacecraft - The PowerSphere nano-satellite from The Aerospace Corporation Spacecraft requirements For acceleration measurements there are a few requirements this puts on the overall spacecraft design. As pointed out in section an ideal spacecraft would be a perfect sphere, or at the very least have a constant attitude relative to the velocity vector, to avoid variations in cross section area and drag coefficient. A realistic shape would be using many flat pentagonand hexagon-shaped plates forming a near-spherical shape, such as the PowerSphere nanosatellite from The Aerospace Corporation depicted in figure 4.1. For higher accuracy more and smaller surface plates can be used. The accelerometer would also need to be placed in the center of gravity of the satellite, since this is where acceleration needs to be measured for calculating drag, and it minimizes disturbances from gravity gradients. For the disturbance levels to be as low as possible no moving parts should be allowed on the spacecraft, meaning reaction and momentum wheels can not be used for attitude control. The sensor should also be magnetically shielded to avoid disturbances from Lorentz forces Proton monitor For estimation of charge rates a proton monitor designed for energies of about 100 MeV and above would be required. As the amount of charge deposited on the test masses in a given event is dependent on the energy of the incident particle, it would also be very beneficial to have spectral information. Other considerations in designing a particle monitor include the accuracy of determination of charging rates, which will depend on the type and volume of the detector implemented. It could also be that anisotropies in the flux could result in disturbances and an ideal monitor would be designed to be sensitive to this [1]. To be able to make a more detailed specification of requirements for a proton monitor detailed studies should be made to investigate how accurately charge rate can be estimated from proton flux data and how accurately energies, direction etc needs to be measured to reach this accuracy. For the power and mass budgets ESA s Standard Radiation Environment Monitor (SREM) is used in this study. The SREM can detect high energy protons above 30 MeV (and electrons) in 15 energy ranges and has an accuracy of 1% [2, 7]. This proton monitor might or might not be a good candidate for use on the companion, but is assumed to give a representative estimate of mass and power consumption of a particle monitor. 40

52 CHAPTER 4. SPACECRAFT Figure 4.2: Suggested companion spacecraft as seen from the sun. Grey line marks velocity direction and orange spheres mark position of the four thruster clusters. 4.2 Subsystems Propulsion For precision formation insertion and Drag-free control it is crucial that the thrusters can produce low (micro Newton) and very precise thrust levels. The thrusters would of course be required to give a maximum thrust larger than the largest disturbance, and a lowest thrust lower than the smallest. The atmospheric drag in a 550 km orbit would vary between about minimum 2µN and maximum 50µN. Larger thrusts on the millinewton scale would also be desirable for larger orbit corrections, such as during orbit injection or for recovering from errors. A good overview of different available micro Newton thruster systems is given by Kilter [17]. Field Emission Electric Propulsion (FEEP) is an advanced electrostatic thruster capable of delivering very low thrust with very high accuracy and controllability. FEEP thrusters are suggested for many future Drag Free satellites [4, 9] since they are considered ideal for several reasons. The advantages over other technologies include very high specific impulse (I sp ) leading to very low propellant weight, on the scale of grams for the entire mission, and low mass and dimensions. The low propellant requirement is ideal for the companion since this will prevent disturbances from propellent sloshing and minimizes problems associated with estimating mass. The technology is also fairly mature and will fly on several missions in the near future. It might however be desirable to choose an even more proven and mature system if costs are a major concern. This should be investigated in further detail. One problem with FEEP thrusters is its relatively high power consumption of about 60W for a system producing a maximum of 1 mn [4, 17], though as seen in the power budget further down, this can be handled. 1 mn is enough to be able to correct for the drift of a measurement period of 20 orbits in less than a half orbit. During orbit correction phases instruments can also be shut down giving more power to the thrusters for larger thrust levels even if the size of the spherical satellite needs to be smaller than here suggested. During drag free control less power is needed as maximum forces then are on the micronewton level. 41

53 4.2. SUBSYSTEMS Using a football shaped satellite, thrusters can be placed symmetrically in four clusters of four as seen in figure 4.2. A recommended configuration is using 12 or 16 thrusters, for a system that will be able to produce thrust and momentum in all degrees of freedom as required by a 3-axis stabilized satellite plus redundancy. According to Marcuccio [22] a system like this would have a total weight of 33.2 kg including thruster assembly, neutralizer and electronics unit. To see why high specific impulse, defined as the ratio of thrust to weight flow rate, F/ṁg, leads to low fuel mass one can look at a form of the rocket equation [41] m p = m 0 (1 e ( V /Ispg) ) (4.2) where m p is propellant mass, m 0 is mass of the satellite and V is the required velocity change. The higher I sp is, the lower the propellant mass will be. For FEEP thrusters typical values for the specific impulse is s 1. An estimate from simulations is that even if no drag free control was used, which would lead to larger total V [21], the orbit would need to be raised by about 20 m each day. This can be done with a total V of about 0.01 m/s. Assuming this is done every day for the 6 month mission and even with a safety margin of 100 times, this still leads to only 0.5 kg of propellent for the entire mission for an I sp value of Baseline The baseline for the companion satellite is to use a FEEP thruster system. For power and mass budgets information was gathered from Centrospazio [4] that is assumed to give representative values Attitude and Orbit Determination and Control Accurate attitude determination and control will be required for the measurements of drag. Ultimately the most sensitive axis of the accelerometer needs to always point in velocity direction. This requires the spacecraft to rotate at orbit frequency. A star sensor pointed in a direction normal to the orbit plane can be used to determine attitude. This would mean it should point in the direction away from the sun. The pointing accuracy requirement is set to 0.25 degrees, which would translate to less than a 0.5% error in the acceleration measurement. A good star sensor is able to determine attitude with far higher accuracy than this. The solar panels on the surface of the sphere shaped satellite could be used as a coarse sun-sensor for attitude determination during orbit insertion phase. The main control mode of the satellite is Drag-free control which will mean the formation will be kept very accurately without intervention as the main spacecraft is also flying Drag-free. This requires however that the satellite is inserted into a very accurate orbit before starting its Drag-free control. Since high accuracy positioning would be required for formation keeping a GPS receiver could serve as a good position sensor. This would allow the companion to know its position in the formation within a few decimeters, which would be more than accurate enough for formation keeping and orbit insertion. Autonomous orbit control using GPS has been successfully demonstrated in the past on the UoSat-12 satellite [11], keeping its position to better than 1 km using only 53 very small engine burns in one month for a comparable orbit. A requirement for good acceleration measurements is that there are no moving parts on board, meaning momentum wheels cannot be used to keep attitude. This is also not needed, 42

54 CHAPTER 4. SPACECRAFT as the attitude and rotation could be kept stable by the drag free control system only by using thrusters. To minimize the need to use thrusters for attitude adjustment moments of inertia should be chosen to get an Earth oriented attitude as stable as possible Baseline For the baseline configuration the use of a star sensor for fine attitude determination, pointed away from the sun. A GPS receiver is used for high accuracy position determination and attitude control by a drag-free control system using FEEP thrusters is suggested for pointing and maneuvers. For the power and mass budgets in this study the Oersted Advanced Stellar Compass [30] and SSTL SGR-10 Space GPS receiver [38] system is used as a baseline, the same sensor used for the demonstration of GPS based orbit control mentioned earlier Communication A communication link would be needed to the main spacecraft for forwarding of measurement data to be used by its control system. Down-link to Earth could be made either through this same link via STEP or directly to ground. Probably the same antenna, receiver and transmitter could be used both for ground communication and communication with the main spacecraft. Assuming accelerometer data is sampled at 10 Hz with 128 bits including time stamps etc., and proton flux data is sampled at 1 Hz for 10 channels at 128 bit precision this adds up to a data transfer rate of 2560 bits per second. Adding a safety factor of about 100 percent for control bits, position information, other telemetry information etc the link should be able to handle 5 kbit/s to the main spacecraft. The system should be designed to be able to communicate with the existing communication system on the main satellite Baseline A data transfer rate of 3 kbit/s is required and the system needs to be designed to communicate with the main spacecraft. The details of such a system needs to be studied further Command and Data Handling The primary task of the CDH system will be to provide control inputs to the thrusters based on attitude and position data, to route power between subsystems, and handle all data processing that would be needed for the forwarding of data to the main spacecraft and downlink to Earth. The exact requirements on the system depend on the complexity of the control system and other things Baseline A baseline requirement for the data handling system is not available at this time Thermal control Except for the usual need for thermal control for maintaining equipment within operating limits, the accelerometer accuracy is dependant on thermal stability [33]. Because of the always sun lit sun-synchronous orbit thermal stability will not be a big problem. Sunshine will be constant, and the IR earthshine is more or less constant too. Albedo would vary, 43

55 4.2. SUBSYSTEMS Radiation source Flux [W/m 2 ] Solar radiation 1367 ± 5 Earth albedo 410 ± 68 Earth IR 237 ± 21 Equilibrium Temp erature 32 ± 3.5C Table 4.2: Radiation sources and the corresponding flux and equilibrium temperatures this gives for a spherical spacecraft covered by solar panels. Values from reference [41]. although probably not much since the orbit would follow close to the line separating the dark half of Earth from the lit half. To estimate the equilibrium temperature without any internal heat sources the following equation is used [41] T = ( A pg r α A r ɛσ )1/4 (4.3) where A p is the projected area in radiation source direction, A r total radiating area of satellite, G r is radiation flux, α absorptivity of the surface and η is the infrared emissivity. σ is Stefan-Boltzmann s constant. Assuming the entire satellite is covered in solar cells with emissivity of and absorptivity of and using flux values as summed up in table 4.2 this yields a temperature of about 30 degrees Celsius. This indicates that passive thermal control could easily keep the payload at operating temperatures only by choice of surface material with appropriate emissivity and absorptivity for parts that do not need to be covered with solar panels. Temperatures will in reality be slightly higher than this because of internal heat sources such as the power system and thrusters. Typical operating temperatures for most accelerometers is around room temperature [34] and the spacecraft needs to be kept at optimally about 20 degrees Celsius. This could be achieved by using a paint with high emissivity on parts of the shadow side of the satellite. All instruments aboard should be able to operate at this temperature Baseline The baseline is to use simple passive control, with thermal isolation of the accelerometer due to its sensitivity to thermal gradients. Paint could be used on the half of the satellite that is not sun-lit to control emissivity and thereby the equilibrium temperature. If temperature variations at the accelerometer are found to be to large an active electrical heater could be used to keep the temperature stable Power Power can be generated by mounting solar panels on at least the half of the spherical satellite that will be facing the sun. A power budget is presented in table 4.3 and from this the required area of solar panels needed can be calculated. Using high quality multi-junction cells an efficiency of up to 22 percent can be reached [41]. There is also an inherent degradation that is taken into account when sizing a solar array, due to temperature dependency, aging etc. With an average solar flux of 1367 W/m 2, 20 percent efficiency and an inherent degradation of 23 percent [41], a cross sectional area of 0.76 m 2 is needed to produce the 159 W required by the satellite. This area includes a large margin, 50 percent, because of the many assumptions made. The margin also includes conversion losses etc in the power subsystem. This sets the 44

56 CHAPTER 4. SPACECRAFT Consumer Usage [W ] Source FEEP Thrusters 60 Centrospazio [4] Communication 10 Computer 10 Star tracker 7.6 Oersted ASC [30] GPS receiver 6.3 SSTL SGR-10 [38] Accelerometer 10 Estimate from [34] Proton Monitur 2 SREM[7] Sum Consumers Safety Margin, 50% 53 Solar panels Table 4.3: Power Budget for the companion spacecraft. When not noted otherwise numbers are estimates from typical small satellites [41] smallest possible size of the spherical satellite, meaning it must have a radius of at least 0.49 m for surface covering solar panels to produce enough power Baseline Power will be generated through solar panels mounted on the satellite surface. A battery is not required since there is no eclipse in the orbit, but one should be included to buffer large currents occuring when switching on and off devices, for backup etc Structure As the spacecraft ideally needs to be spherical to minimize drag coefficient uncertainty a practical shape would be that of a football, with five- and six-sided plates making up the surface of the satellite. Inside all instruments and electronics boxes must be placed so that the accelerometer is placed in the center of gravity, and preferably also in the geometrical center. An illustration of a suggested satellite structure and placement of instruments and subsystems can be seen in figure 4.3. The estimated mass of the structure that could hold up solar panels and all instruments and boxes is 30 kg. As seen in the mass budget in table 4.4 the total mass of a scout satellite is kg (excluding launch adapter). Propellent mass for the FEEP thrusters is conservatively set to 1 kg, though it will be less than this in reality thanks to the very high specific impulse of the thruster system. A complete V analysis needs to be made to get a more precise value for propellent mass. This configuration, with a cross section area of 0.76m 2 and a mass of 169.5kg gives a ballistic coefficient of 87, assuming a drag coefficient of 2.5, which is a guess based on work by Moe et al [27] in estimating drag coefficients of spherical spacecraft. It is put slightly higher than can be seen in figure 2.4 since the shape is not a perfect sphere and the altitude is higher than that figure reaches. These values will be used for simulations done to estimate the performance of an added scout satellite with feed forward in the next chapter. 4.3 Summary of baseline configuration The suggested configuration is a football-shaped satellite covered in solar panels. The payload is an accelerometer that requires its most sensitive axis to be pointed in velocity direction, 45

57 4.3. SUMMARY OF BASELINE CONFIGURATION System/part Mass [kg] Source Structure 30 Feep thruster system 33.2 Centrospazio [4] Computer 10 Communication 6 Star tracker 1.5 Oersted ASC [30] GPS system 1.2 SSTL SGR-10 [38] Accelerometer 20 Estimate from [34] Proton Monitur 2.5 SREM[7] Propellent 1 Conservative estimate Power and cabling 25 Sum Margin of error, 30% 39.1 Total Mass Table 4.4: Mass Budget for the companion spacecraft. When not noted otherwise numbers are estimates from typical small satellites [41]. Star Tracker Proton Monitor Orbit path Thrusters Electronics/computer/communication Accelerometer Figure 4.3: Illustration of companion satellite with possible payload and systems placement. Star tracker will always be pointed in the direction away from the Sun. 46