Thermo-Elastic Distortion Modelling for Drag-Free Satellite Simulations. Draft

Transcription

1 IN SUPREMÆ DIGNITATIS 1343 UNIVERSITÀ DI PISA Facoltà di Ingegneria Corso di Laurea in Ingegneria Aerospaziale Thermo-Elastic Distortion Modelling for Drag-Free Satellite Simulations Draft Tesi di laurea Anno Accademico Allievo: Montemurro Fabio Relatori: Prof. G. Mengali Dr. W. Fichter Ing. N. Brandt

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3 Contents 1 Introduction The Laser Interferometer Space Antenna Project LISA Pathfinder mission goals The disturbance reduction mechanism requirement Laser metrology requirement Contribution of this work Outline of the thesis Thermo-elastic distortion The thermo-elastic distortion analysis approach Thermo-elastic distortion modelling for LISA PF The choice of the Thermal Areas Accuracy of the modelling approach I Self-Gravity 15 3 Mathematical modelling Nomenclature and definitions Equations of motion Satellite and one Proof Mass The LISA Pathfinder example Specification of forces and torques iii

4 iv CONTENTS Self-gravity Self-gravity requirements LISA and LISA Pathfinder top requirements Other science requirements Spacecraft and Test Mass coupling Spacecraft position control DC force/torque requirements Flow down of top-science requirements Apportioning of force noise requirement to self-gravity field Apportioning of stiffness to self-gravity field gradient Apportioning of dc-force and torque to self-gravity field 32 5 The equations of self-gravity Introduction Reference frames and geometry of the problem Force of a point-like mass on the TM Torque of a point-like mass on the TM The gravity-gradient matrices Linear gravity-gradient matrix for the force Linear gravity-gradient matrix for the torque Angular gravity-gradient matrix for the force Angular gravity-gradient matrix for the torque Conclusions Self-gravity modelling Self-gravity generated by the LTP Force Torque Results

5 Contents v 6.2 Self-gravity generated by the other TM Force Torque Results Self-gravity generated by the compensation masses Results Thermo-elastic distortion Introduction Self gravity due to thermo-elastic distortion Analysis approach The Sensitivity Matrix Accuracy of results and errors Thermal modelling errors Gravity field linearizing error II Optics 65 8 The laser metrology unit Laser metrology unit layout Basic requirements Temperature requirements Thermo-elastic distortion The Sensitivity Matrix for optics elements distortion Optics sensitivity matrix using BeamWarrior The optics law matrix Noise in TM position measurement A Analytical formulation 79 A.1 Force

6 vi CONTENTS A.1.1 Exact formulation A.1.2 Approximate formulation A.2 Gradient A.2.1 Exact formulation A.2.2 Approximate formulation

7 List of Figures 1.1 Artist s view of LISA Scheme of LISA-Pathfinder End-to-end top simulator level architecture Thermal areas of the LTP Satellite and one Test Mass Schematic of axes and layout. Separation between the test masses is along x Schematic for a cube and a point-like source Schematic for torque calculation Displacements for linear stiffness calculation LTP FE Model nodes Schematic the gravity of one TM on the other Preliminary compensation masses system layout Scheme for linearizing error Laser metrology unit interferometers: BeamWarrior 3D-view of the LTP OB with the two TMs LTP frame and Test Masses The required maximum noise level of the interferometer measurement vii

8 viii LIST OF FIGURES

9 List of Tables 3.1 Symbols and definitions Apportioning of force noise due to self-gravity to S/S Apportioning of stiffness due to self-gravity to S/S Apportioning of DC forces due to self-gravity to S/S Apportioning of DC torques due to self-gravity to S/S OB temperature requirements ix

10 x LIST OF TABLES

11 Chapter 1 Introduction 1.1 The Laser Interferometer Space Antenna Project The Laser Interferometer Space Antenna (LISA mission is a ESA joint venture with NASA. Its prime objective is the detection of gravitational waves in the 1mHz to 100mHz band predicted to be emitted by distant galactic sources. It will consist of three spacecraft flying in a quasi-equilateral triangular formation, separated by 5 million km, in a trailing Earth orbit at some 20 behind the Earth. Each spacecraft will carry a measurement system consisting of two proof masses, associated laser interferometer hardware and electronics. Provided that the proof masses are maintained in a disturbance free environment, gravity waves will cause small motions in the test masses relative to one another. Low frequency gravity waves are predicted to produce strain of order 10 21, allowing them to be measured by precision interferometry as path length changes up to 50 pm. 1.2 LISA Pathfinder mission goals Very early, during the various study for LISA, the need for a technology demonstration mission was recognized. Europe will establish a capability through the LISA Pathfinder (formerly 1

12 2 CHAPTER 1. INTRODUCTION Figure 1.1: Artist s view of LISA. known as SMART-2 programme to demonstrate key technology for LISA that cannot be tested on the ground, thus removing risk from the future science programme. The LISA Pathfinder demonstration is to be completed before the start of the LISA implementation phase. The primary mission goal for LISA Pathfinder is to test the key technology critical to the LISA mission. This involves demonstrating the basic principle of the Drag-Free Control System, including the precision acceleration sensor system, the error measurement technique, the control laws and calibration of the µnewton thruster performance. The basic idea behind the LTP is that of squeezing one LISA arm of km to a few centimeters and in placing it on board of a single S/C. Thus the key elements are two nominally flying test masses and a laser interferometer whose purpose is to read the distance between the proof masses. The two proof masses are surrounded by their position sensing electrodes. This position sensing provides the information to a drag-free control loop that operates via a set of micro-newton thruster to center the S/C with respect to one test mass. Accelerations will be derived from measurements

13 1.2. LISA Pathfinder mission goals 3 Figure 1.2: Scheme of LISA-Pathfinder. of distance between two proof test mass within the LISA test package. The key technologies requiring demonstration are: the disturbance reduction mechanism the laser metrology Of these two items, only the first requires space demonstration, but LTP will incorporate both The disturbance reduction mechanism requirement The disturbance reduction mechanism must be able to shield the proof mass from the outside environment in such a way that only gravity waves will cause measurable displacements. This requirements is to allow measurement of gravity waves, given their amplitude and frequency expectations. This disturbance reduction system can only be space demonstrated and results

14 4 CHAPTER 1. INTRODUCTION in a requirement for spacecraft and proof masses acceleration control. LISA Pathfinder mission fundamental technical goal is to demonstrate the nearperfect fall of a Test Mass located inside the body of the spacecraft by limiting the spectral density of acceleration at the test mass to ( ] 2 f Sa 1/ [1 14 m 1 + (1.1 3mHz s 2 Hz for 1mHz f 30mHz This is one order of magnitude bigger than the requirement for LISA, and three orders lower than demonstrated to date. The sources contributing to the acceleration environment of the proof mass arise from both direct effect on the proof mass and effects on the spacecraft that are coupled to the proof mass through the electrostatic suspension system. These are: External forces on the spacecraft, among them: - Thruster force and thruster noise - Difference in gravitational acceleration due to celestial bodies between test mass and spacecraft center of mass - Solar radiation pressure - Interaction with atmosfere, planetary magnetic fields Internal forces acting on the proof mass and the spacecraft, including - Thermal noise - Pressure fluctuation - Electrostatic - Spacecraft self gravity Force that arise from sensor noise feeding into thruster commands.

15 1.3. Contribution of this work Laser metrology requirement An interferometer for precise measurement of variation in distance between the test masses is needed: LISA is expected to detect path length changes of a few picometer within the measurement bandwidth. The interferometric sensing must be able to monitor the test mass position along the measurement axis with a noise level of for relaxing as 1/f 2 towards 1mHz. S 1/2 n 10pm/ Hz (1.2 3mHz f 30mHz The source contributing to the interferometric noise level is exclusively due to the thermal noise affecting the laser metrology system. 1.3 Contribution of this work A full compliant self-gravity tool has been developed. Sources of numerical errors due to the awkward cube shape of the Test Mass have been eliminated. An extensive error estimation has been carried on in order to check the accuracy of the tool. The linearized equation of motion of a Test Mass subjected to the gravity cause by both another Test Mass and the Spacecraft are derived. For the first time in literature, an analytical formulation of self-gravity affected by thermo-elastic distortion is presented. A renewed methodology for deriving sensitivity factors of self-gravity with respect to S/C deformations is proposed. For the first time, a model of the effects of thermo-elastic distortion on the laser metrology unit is developed. A hands-on visualization of the the effect of self-gravity and thermo-elastic distortion on Test Mass movement and optics readout is realized.

16 6 CHAPTER 1. INTRODUCTION Both self-gravity and laser metrology modelling are fully implemented in the end-to-end simulator under development at EADS Astrium GmbH. Figure 1.3 shows the top-level architecture of thermo-elastic simulator for self-gravity and optics. Figure 1.3: End-to-end top simulator level architecture. 1.4 Outline of the thesis First, in Chapter 2 present the thermo-elastic distortion approach for the end-to-end simulator of the LISA Pathfinder Mission. Then, the work is divided into two main parts: the first features the problem of the self gravity environment of the satellite on the Test Masses and how this problem is affected by thermo-elastic deformation; the second part deal with the effect of thermo-elastic deformation on the laser metrology system.

17 1.4. Outline of the thesis 7 In Chapter 3 the equations of motion for a generic drag-free satellite are derived. In Chapter 4 the self-gravity requirements for LISA Pathfinder are recollected from various references. In Chapter 5 and 6 the self-gravity tool is presented and applied to LISA Pathfinder. Chapter 7 describes the methodology to estimate the influence of thermoelastic distortion on self-gravity. In the second Part, Chapter 8 describes the laser measurement system and states its requirement. Then, Chapter 9 proposes an opto-dynamical model which account for thermo-elastic distortion.

18 8 CHAPTER 1. INTRODUCTION

19 Chapter 2 Thermo-elastic distortion model The thermo-elastic distortion analysis approach is based on the experience gained in EADS Astrium GmbH in the GRACE, XMM, and GOCE projects. By applying this approach, a thermo-elastic distortion model is derived and subsequently integrated within the E2E simulator. In fact, this model provides the input data for thermo-elastic distortion analysis for the spacecraft self-gravity and the laser metrology unit. 2.1 The thermo-elastic distortion analysis approach Thermo-elastic distortion calculations of large structures, consisting of a mix of widely ranging CTE and sub-scale details, tend still to be a problem for direct analysis approach and for the implementation within end-to-end simulations. The reason is the huge amount of data coming from FE models and Thermal Mathematical Model (TMM with respect to the reasonable velocity required for real-life performance simulations. In fact many of the thermoelastic analysis done on previous missions relied directly on processing of transient fields of temperature which had to be transferred time-stepwise, one by one, into the respective FEMs, either semi-automated or manually. The problem with this approach is that each change in temperature profile 9

20 10 CHAPTER 2. THERMO-ELASTIC DISTORTION requires a time-consuming complete new FEM analysis run. Previous projects at EADS Astrium GmbH as GRACE, XXM, and GOCE purse the strategy to calculate the primary satellite distortion shape with reasonable accuracy with the help of sensitivity factors and to cover the potential small scale influences by an adequate uncertainty factor. The novel approach calculates sensitivity factors on individual sets of structural nodes. This is done by applying unit heat load case on defined nodal areas (thermal areas of the FEM and determining the static displacement and rotation of selected nodes of interest due to the heat loads. approach are: The advantages of this as many temperature profiles as needed can be calculated and recalculated from the TMM without a new FEM analysis run as the sensitivity factors are calculated only once the amount of data can be overseen and still adequately judged by the designers each effect can be easily traced bach to a small number of causes, giving the possibility to easily identify the major mechanical/thermal design drivers the needed uncertainty factor can be assessed from previous projects. The sensitivity factors can be arranged in a linear, static transfer matrix as follow: r 1 α 1 T T A1. (t = [D]. (t (2.1 r n T T Ak α n For k thermal areas T A and n selected nodes of interest, the thermoelastic sensitivity matrix is given by [D]. This matrix simply relates linearly the temperature changes of defined thermal areas to the distortion of the selected nodes, thus superimposing the unit load case results. Note that the sensitivity matrix dimension is defined solely by the number of thermal areas

21 2.2. Thermo-elastic distortion modelling for LISA PF 11 k and the number of nodes n whose deformation, linear and angular, are of interest. That is, the size of the FEM has no influence on the size of the sensitivity matrix. This leads to manageable matrix sizes w.r.t. numerical evaluation. 2.2 Thermo-elastic distortion modelling for LISA Pathfinder In order to model thermo-elastic distortions within the LISA Pathfinder E2E simulator the analysis approach, as detailed above, has been applied. In order to calculate the sensitivity matrices, the following steps are followed: 1. the FEM of the whole LTP (ca nodes is divided in 80 thermal areas and the FEM s congruent set of structural nodes is allocated to each thermal area, see Figure 2.1. (Note that an FEM of the spacecraft will be included as soon as it is available Besides, a reference temperature of 20 C is chosen. 2. Starting from the reference temperature, the temperature of all FE Model nodes within an individual thermal area is increased by 1 C whilst all the other nodes are kept at the reference temperature. 3. According to this unitary temperature variation, the translational and rotational displacements of the selected nodes of interest are calculated These steps are repeated for each TA. The sensitivity matrix of thermo-elastic distortion is used for two purpose: 1. Self-gravity analysis. The nodes of interest for this case are all the FEM nodes; being this number still considerably high, a further processing of the sensitivity matrix made by a self-gravity tool is required to obtain a remarkable model reduction. 2. Opto-dynamical model. The nodes of interest are the one defining the position and the orientation of all the elements belonging to the optical path. In this case, the sensitivity matrix can be used straightforwardly.

22 12 CHAPTER 2. THERMO-ELASTIC DISTORTION Figure 2.1: Thermal areas of the LTP. Using a temperature time series from the TMM as an input to the sensitivity matrix, scaled by the reference temperature, thermo-elastic distortion over time can be determined. Up to date, because no appropriate TMM is available, temperature noise models act as inputs to sensitivity matrices. 2.3 The choice of the Thermal Areas TBC For the LTP a preliminary number of 80 Thermal area has been chosen. The choice is mainly suggested by the experienced acquired in previous project; nevertheless some guidelines can be introduced: each stand-alone element must contain at least one thermal area any region surrounding a lumped heat generators (e.g. electrical boxes, photo-diodes must be model as a thermal area the more a region is subject to environmental changes, the higher the number of thermal areas in it must be

23 2.4. Accuracy of the modelling approach Accuracy of the modelling approach The modelling approach makes use of the LTP FEM and TMM. FEMs, primarly designed to calculate the structural dynamics of the major static load path, have shown to be able predicting realistically the fundamental distortion not requiring an additional higher degree of discretization. TMMs have been shown generally precise enough, in terms of nodes and details, to serve as input into the FEM for elastic distortion calculation as well. The whole approach is based on the linearity of the finite elements analysis, which is given by definition, allowing for the superposition of all numeric solutions. The non-linearity within the temperature field calculation is completely taken into account in the TMM. Hence there is no additional loss in accuracy. Past experiences show that the biggest errors are in missing details of the simulation FE and TM Models and in the deviations from actual material parameters as stiffness and CTE data. According to the deviations between numerical and test results of former missions as GRACE, SOHO, and XMM, a safety factor of 2 is recommended.

24 14 CHAPTER 2. THERMO-ELASTIC DISTORTION

25 Part I Self-Gravity 15

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27 Chapter 3 Mathematical modelling of a Drag-Free satellite In this chapter the equations of motion (EoM of a drag-free satellite are derived analytically. The approach reported in [5] is followed. In general, a drag-free controlled satellite consists of the following rigid bodies: - the rigid satellite body (6 DoF - one or more rigid test masses (6 DoF each - fixed or moving rigid test mass housing (3 DoF each if moving The LISA Pathfinder satellite is a particular case of a drag-free satellite: it features two test masses and two fixed rigid test mass housings, which, along with the S/C, constitute a 18 DoF system. From now and then, any test mass housing will be always considered fixed. 3.1 Nomenclature and definitions For the derivation of the equations of motion the scheme depicted in Figure 3.1 is followed. The reference frames used are hereby listed: The inertial reference frame Σ J 17

28 18 CHAPTER 3. MATHEMATICAL MODELLING Figure 3.1: Satellite and one Test Mass. The spacecraft (body fixed reference frame Σ B ; it is attached to the CoM of the S/C The housing reference frame Σ H ; it is attached to a generic point of the housing frame and it features a generic orientation w.r.t the spacecraft frame; anyway, being the housing supposed fixed, this orientation is constant. This frame is used for the TM dislocation measurement The Test Mass (body fixed reference frame Σ M ; it is attached to the CoM of the TM. The vectors notation here adopted is defined by the following rules. A vector named r X is the vector for the body X given in its local frame (i.e. the vector origin, as defined in Figure 3.1. A vector named r XY gives the vector position of the body X w.r.t the body Y in the local frame of the latter. Further, a vector named r Z X means that the vector coordinates are given in the reference frame defined by the index Z.

29 3.2. Equations of motion 19 Furthermore, in order to clarify the meaning of the angular velocities, the following definitions are given: ω B : angular velocity of the satellite w.r.t the inertial frame ω H : angular velocity of the frame Σ H w.r.t Σ B ; when the cage is not moving, ω H is identically equal to zero. ω M : angular velocity of the TM w.r.t Σ M. Notice that the angular velocity ω X for the body X is, by definition, always given in its own body frame Σ X. Further symbols are given in Table 3.1 Symbol E i i I X q T XY A t Description Unit diagonal matrix of size i Matrix of inertia around the CoM for body X Generalized coordinate vector Transformation matrix from Y to X reference frame Transpose of matrix A Table 3.1: Symbols and definitions As custom, given a generic vector v, then it is 0 v z v y ṽ = v z 0 v x v y v x 0 and so it can be written ω r = ωr and ω ω = ω 2. The notation r is used to define a differentiation w.r.t. the inertial frame, whereas ṙ is the derivative in the local body frame. 3.2 Equations of motion In order to derive the EoM of the satellite and of the proof masses d Alembert principle is used. According to this principle, and using the Newton-Euler equations of rigid body dynamics, it leads to: [ JT t i (ṗ i f ei JR t i ( L ] i l ei = 0 (3.1 i

30 20 CHAPTER 3. MATHEMATICAL MODELLING In this equation i stands for the generic i th body of the system. Then ṗ i = m i r i represent the impulsive term differentiated and expressed in the inertial frame, using the CoM of the respective body as its reference point. The term L i = I i ω i + ω i I i ω i represents the angular momentum of the CoM of the i th body, expressed in body coordinates. The terms f ei the applied forces acting on the body, expressed in the inertial frame, while l ei are the applied torques acting on the body, [ ] expressed in[ the] respective ṙ body frame. The Jacobian matrices J Ti = i ω and J q t Ri = i resemble q t the gradient w.r.t the generalized coordinates q i. are The derivation process for the equations of motion follows the steps: 1. Define a set of generalized coordinates for the problem 2. Describe the rotational and translational kinematics of each body 3. Differentiate the translational kinematics (2 nd order and the rotational kinematics (1 st order 4. Evaluate the Jacobian matrices 5. Set up the system according to Eq Evaluate each row and write the EoM in the desired form. 3.3 Satellite and one Proof Mass As a first step, only one TM will be considered. Results can be easily extended to two or more TM. The generalized coordinates are chosen as follows: ( q t rb = ω B ṙ M ω M (3.2 The rotational kinematics of the CoM of each body are to be expressed in body coordinates, this is due to definitions of Eq As far as the satellite

31 3.3. Satellite and one Proof Mass 21 body, they are already given by ω B. The absolute rotational kinematics of the test mass are defined by 1 : ω J M J = ω J B + ω J H +ω J M = T }{{} JB ω B + T JM ω M (3.3 =0 ω M M J = T MB ω B + ω M (3.4 and by: ω J M J = T JB ω B + T JM ω MJ ω M + T JM ω M (3.5 ω M M J = T MB ω B + T MB ω B ω M + ω M (3.6 The translational kinematics of the satellite body are already expressed in the inertial frame by the definition of r B. The translational kinematics of the test mass CoM are given by: r MJ = r B + r J H + r J M = r B + T JB r H + T JH r M (3.7 r MJ = r B +T JB ω B r H + T JH ω HJ r M + T JH ṙ M ω HH = T HB ω B, ṙ H = 0 r MJ = r B +T JB ω B (r H + T BH r M + T JH ṙ M r MB = r H + T BH r M = r B +T JB ω B r MB + T JH ṙ M = r B +T JB ω 2 Br H + T JB ω B r H + T JH ω 2 H J r M + T JH ω HJ r M + T JH ω HJ ṙ M + T JH ω HJ ṙ M + T JH r M (3.8 = r B +T JB ( ω 2 B + ω B r MB + 2T JH T HB ω B ṙ M + T JH r M The Jacobian matrices can evaluated as follows: J TM J TB J RB J RM 1 The following relation are used: = r B q t = [ E ] = ω B q t = [ E ] = r MJ q t = [ E 3 3 T JB r MB T JH ] = ω M J q t = [ T MB E 3 3 ] T MJ T JB = T MB T MJ T JM = E 3 3 (3.9 (3.10 (3.11 (3.12

32 22 CHAPTER 3. MATHEMATICAL MODELLING By using the formulations derived above, Eq. 3.1 gives: E (m B r B f eb + r MB T BJ E 3 3 T HJ ( m B r MJ f em E (I B ω B + ω B I B ω B l eb + T BM (I M ω MJ + ω MJ I M ω MJ l em = E 3 3 (3.13 Evaluating the first row of Eq leads to: (m B + m M r B m M T JB r MB ω B + m M T JH r M + The second row is: + m M T JB ω 2 Br MB + m M 2T JH T HB ω B ṙ M = f eb + f em (3.14 m M r MB T BJ r B + [ I B + I M + m M r t M B r MB ] ωb + m M r MB T BH r M + + T BM I M ω M + m M r MB ω 2 Br MB + m M 2 r MB T BH T HB ω B ṙ M + The third row is : + ω B I B ω B + T BM I M T MB ω B ω M + T BM ω MJ I M ω MJ = m M T HJ r B m M T HB r MB ω B + m M r M + The fourth and final row leads to: = r MB T BJ f em + l eb + T BM l em ( m M T HB ω 2 Br MB + m M 2 T HB ω B ṙ M = T HJ f em (3.16 I M T MB ω B + I M ω M + I M T MB ω B ω M + ω MJ I M ω MJ = l em (3.17 The equations written above have already been sorted in a certain way to write the EoM of the satellite-proof mass system in the following standard form for 2 nd order differential equations: M(q q + g(q, q = k(q, q

33 3.3. Satellite and one Proof Mass 23 where M(q is the system mass matrix, g(q, q contains apparent forces and torques, and k(q, q is the force and torque vector. The EoM written in this form are shown in the next equation: (m B + m M E 3 3 m M T JB r MB m M T JH r B m M r MB T BJ I B + I M + m M r t M r MB m B M r MB T BH T BM I M ω B m M T HJ m M T HB r MB m M E r + M I M T MB I M ω M m M T JB ω 2 B r M B + m M 2T JH T HB ω B ṙ M m + M r MB ω 2 B r M B + m M 2 r MB T BH T HB ω B ṙ M + ω B I B ω B + T BM I M T MB ω B ω M + T BM ω MJ I M ω MJ m M T HB ω 2 B r M B + m M 2T HB ω B ṙ M = I M T MB ω B ω M + ω MJ I M ω MJ f eb + f em = r MB T BJ f em + l eb + T BM l em T HJ f em (3.18 l em Now, a more compact form can be derived; in fact, subtracting the 1 st row by T JH 3 rd row, gives the translational orbit movement equation of the satellite: m B r B = f em (3.19 The latter expression times T HJ and inserted in the 3 rd row results in the equation describing the relative acceleration of an inertial sensor: r M = T HB ( ω 2 B + ω B r MB 2T HB ω B ṙ M T HJf eb m B + T HJf em m M (3.20 Then, subtracting the 2 nd row by T BM 3 rd row and r MB T BH 4 th row results in the angular momentum equation of the satellite body: I B ω B + ω B I B ω B = l eb The fourth row cannot be simplified any further, since it already resemble the angular momentum of the test mass inside the satellite body in its most general form. Combining the above derived simplifications in a matrix-vector form, it results in a much more decoupled differential equation system: m B E r B I B ω B m M T HB r MB m M E r M I M T MB I M ω M f eb ω B I B ω B + m M T HB ω 2 Br MB + m M 2T HB ω B ṙ M = l eb T HJ f em m M mb T HJ f eb (3.21 T MB ω B ω M + ω MJ I M ω MJ l em I M

34 24 CHAPTER 3. MATHEMATICAL MODELLING 3.4 The LISA Pathfinder example In order to extend the EoM written in Eq.3.21 to the LISA Pathfinder satellite, it must simply introduce another test mass. The second-order nonlinear equations of motion for the satellite (B with two test masses (M 1 and M 2 in their respective housing H 1 and H 2 are: m B E r B I B ω B m M1 T H1 B r MB m M1 E r M I M1 T M1 B I M ω + M m M2 T H2 B r MB m M2 E r M I M2 T M2 B I M2 ω M ω B I B ω B f eb m + M1 T H1 B ω 2 Br BM1 + m M1 2T l eb H1 Bω B ṙ M1 I M1 T M1 Bω B ω M1 + ω M1J I M1 ω M1J = T H1 Jf em1 m M 1 m B T H1 Jf eb l em1 m M2 T H2 B ω 2 Br BM2 + m M2 2T H2 Bω B ṙ M2 T H2 Jf em2 m M 2 m B T H2 Jf eb I M2 T M2 Bω B ω M2 + ω M2J I M2 ω l em2 M2J ( Specification of forces and torques The forces and torques acting on the satellite and the two proof masses are broken down according to what reported in [6]. This outline resembles the actual way force and torque are schematized in the simulator. Actions on the proof masses are: gravitational forces and torques due to celestial bodies forces and torques due to satellite and proof mass coupling (stiffness; their origin can be: - gravitational - electrostatic - magnetic

35 3.4. The LISA Pathfinder example 25 actuation forces and torques (suspension control loops mutual gravitational interaction forces and torques between the two proof masses other undefined environmental forces and torques. Actions on the satellite are: gravitational forces and torques due to celestial bodies solar pressure forces and torques forces and torques due to satellite and proof mass coupling (stiffness and TM actuation actuation forces and torques (FEEP other undefined environmental forces and torques Self-gravity A first estimation of the accelerations, and therefore the force and torques, acting on LISA Pathfinder and its test masses is carried out in [6]. As far as regards the acceleration of one test mass w.r.t the satellite (i.e. the housing, the self-gravity (i.e. the gravity between one TM and the rest of the S/C is among the leading sources. Therefore a detailed modelling of self-gravity field is required; this model must include: The self-gravity on one TM due to the S/C itself; the variation of selfgravity due to thermo-elastic distortion must be accounted for, as well. The self-gravity on one TM due to the other TM; the variation of selfgravity due to the movement of both TM must be also considered. This model is presented in the following Chapters.

36 26 CHAPTER 3. MATHEMATICAL MODELLING

37 Chapter 4 Self-gravity requirements This Chapter describes the disturbance reduction system requirements for the LISA Pathfinder mission. Then, it is showed how there requirements are apportioned to self-gravity and to each S/S of the LISA Pathfinder S/C. 4.1 LISA and LISA Pathfinder top requirements LISA will be the first high sensitivity space-borne gravitational wave detector. LISA sensitivity goal is a strain power spectral density of / Hz at around 3 mhz. Its sensitivity performance is limited at low frequency by stray force perturbing the TM s out of their geodesics. The equation of motion of the two end-mirror masses, of mass m, of one interferometer arm in LISA can be drastically simplified if the following assumptions are made: long wavelength limit for the gravitational signals small signals Then, if x is the separation between the two mass, it is: m d2 x dt 2 = F x + ml d2 h dt 2 (4.1 where h(t is the gravitational wave strain signal and L is the unperturbed value of the TM separation, F x is the differential force either of non gravi- 27

38 28 CHAPTER 4. SELF-GRAVITY REQUIREMENTS tational origin or due to local sources of gravitational field and acting along the measurement axis x. A meaningful explanation of the role played by force noise is obtained converting Eq. 4.1 to the frequency domain: any force noise with spectral density S Fx would mimic a gravitational wave noise density S 1/2 h = S1/2 F mlω = S1/2 a (4.2 2 Lω 2 where ω = 2πf, f is the frequency of the measurement and a is the relative acceleration of the TMs in the inertial reference frame. It is therefore clear that is a top objective to minimize the force noise on the Test Masses. LISA primary goal is achieved only if each TM falls under the effect of the large scale gravitational field only, within an acceleration noise, relative to a free falling frame, whose power spectral density (PSD is less than: ( ] 2 f Sa 1/ [1 15 m 1 + (4.3 3mHz s 2 Hz for 0.1mHz f 0.1Hz along the sensitive axis of each TM of each S/C. LISA Pathfinder primary goal is to verify that a TM can be put in a pure gravitational free-fall within an order of magnitude from the requirement for LISA in Eq.4.3. So the mission is considered satisfactory if the acceleration noise is less than: for S 1/2 a [1 + ( f ] 2 3mHz 1mHz f 30mHz along the sensitive axis of the two TMs. m 1 (4.4 s 2 Hz 4.2 Other science requirements The following Figure depicts the layout of the TMs within the LTP.

39 4.2. Other science requirements 29 Figure 4.1: Schematic of axes and layout. Separation between the test masses is along x Spacecraft and Test Mass coupling The coupling (i.e. the stiffness between the spacecraft and the TM along the sensitivity axis, if no actuation is turned on must be: ( ] 2 f ωp 2 < [1 6 + s 2 3mHz for 1mHz f 30mHz (4.5 Motion of the spacecraft relative to the test mass creates a force on the Test Mass through a parasitic coupling (electrostatic, self gravity gradient. Gradients in the force experienced by the test mass lead to changes in the acceleration of the Test Mass if its position changes Spacecraft position control The part of the residual jitter x n between the TM ans the S/C, which is not correlated with any direct force on the TM, must be: S 1/2 x n < 5nm/ Hz

40 30 CHAPTER 4. SELF-GRAVITY REQUIREMENTS for 1mHz f 30mHz (4.6 As can be easily understood, this requirement is closely connected to the stiffness allocation DC force/torque requirements On LISA Pathfinder, DC forces and torques are compensated by a low frequency suspension based on capacitive actuation. DC force compensation with electric field poses a series of problem. The leading ones are listed in the following. Electric DC force is applied by the capacitive actuation according a control loop with vanishing gain within the MBW. Fluctuation of the voltage supply V within the MBW are not within the control loop and directly convert into a force fluctuation as: δf DC 2F DC δv V where δ stands for a fluctuating quantity. A requirement for F DC is needed. Similar formulas can be obtained also for the rotational DoF. The second effect relates to stiffness. For the capacitive actuation geometry along the sensitivity axis, by applying a force to the TM, it also means to apply a stiffness of order 2F DC /d where d is the sensor gap. If one wants to limit this stiffness to the required values, a requirement follows for F DC too. The requirements for DC forces and torque are given by the following: - maximum dc difference of force between the TMs along x must be: F x m m/s 2 (4.7 - maximum dc difference of force between the TMs along y must be: F y m m/s 2 (4.8

41 4.3. Flow down of top-science requirements 31 - maximum dc difference of force between the TMs along z must be: F z m m/s 2 (4.9 - maximum dc torque of force between the TMs along ϕ must be: T ϕ I ϕ s 2 ( maximum dc torque of force between the TMs along η must be: T η I η s 2 ( maximum dc torque of force between the TMs along ϑ must be: T ϑ I ϑ s 2 ( Flow down of top-science requirements In order to demonstrate the requirements from (1.1 to (4.4 an error budget analysis must be carried out. The approach is: 1. the error budget is apportioned to the various sources of disturbance (gravity field, magnetic field, etc. 2. the noise from each source of disturbance is apportioned to each S/S. For convenience we distinguish three major subsystems: - the inertial sensor (called IS proper - the remaining parts of the LTP, including the second IS (called LTP - the remaining parts of the S/C including the DRS (merely called S/C.

42 32 CHAPTER 4. SELF-GRAVITY REQUIREMENTS Allocated value [ of noise to self-gravity field ( f 2 ] m 1 3mHz s 2 Hz IS LTP S/C Total Table 4.1: Apportioning of force noise due to self-gravity to S/S Apportioning of force noise requirement to selfgravity field Force noise requirement in Eq.1.1 is apportioned to self-gravity field as in Table??. Contributions add according to the quadratic sum Apportioning of stiffness to self-gravity field gradient Stiffness requirement stated in Eq.4.4 is apportioned to self-gravity field gradient as in Table 4.2. Contributions are added up linearly. Allocated value of stiffness [ to gravitational gradient ( f 2 ] s 2 3mHz IS LTP S/C Total Table 4.2: Apportioning of stiffness due to self-gravity to S/S Apportioning of dc-force and torque to self-gravity field DC-forces/torques requirements from Eq are apportioned to selfgravity field as in Tables??, 4.3. Contributions add linearly along each axis.

43 4.3. Flow down of top-science requirements 33 Allocated absolute value of dc-force per unit mass ( 10 9 m/s 2 IS LTP S/C Total x y z Table 4.3: Apportioning of DC forces due to self-gravity to S/S Allocated absolute value of dc-torque per unit moment of inertia ( 10 9 s 2 IS LTP S/C Total ϕ η ϑ Table 4.4: Apportioning of DC torques due to self-gravity to S/S

44 34 CHAPTER 4. SELF-GRAVITY REQUIREMENTS

45 Chapter 5 The equations of self-gravity In the two following chapters the self-gravity calculation tool for the LISA Pathfinder mission is introduced. This tool is used to calculate linear acceleration, angular acceleration and accelerations gradients on each TM caused by the surrounding elements of the spacecraft. 5.1 Introduction The force due to gravity between a test mass of density ρ T M, located at position (x,y,z and a spacecraft element of density ρ source at (X,Y,Z is calculated from Gρ T M ρ source F = (x X2 + (y Y 2 + (z Z dv 2 T M dv source (5.1 V source V T M The inner integral (i.e the one over the TM can be performed both analytically and numerically. As it will be explained further, a hybrid approach is chosen, featuring both analytic and numerical methods. The outer integral (on the source is performed by summing over a discrete nodal mass distribution. The nodal mass distribution is provided from the FE model used for structural analysis. The same applies to torque and force gradient calculation. Because ρ T M is supposed to be constant, it can be written: m i F Gρ T M (x X2 + (y Y 2 + (z Z dv 2 T M (5.2 i V T M 35

46 36 CHAPTER 5. THE EQUATIONS OF SELF-GRAVITY Figure 5.1: Schematic for a cube and a point-like source. where m i is the i th mass of the nodal mass distribution. Therefore, the steps are: 1. calculation of gravity effects (linear acceleration, angular acceleration and accelerations gradients of a point-like mass on the TM (inner integral 2. integration over the mass distribution (outer sum. The first step is illustrated in this Chapter; the integration over the actual mass distribution is presented in the following one. 5.2 Reference frames and geometry of the problem For a more general approach, a parallelepiped TM is used. The TM features a mass M and dimensions L x L y L z ; then define a reference frame Σ M located at the geometric center of the TM and whose axis are parallel to the edges of the TM. This frame is body-fixed and follows the TM while it moves. The nominal position of the TM is supposed to be in the geometric center of its own housing. The reference frame Σ M0 corresponds to this position. In the simulator for LISA Pathfinder, the housing frame Σ H and the

47 5.3. Force of a point-like mass on the TM 37 frame Σ M0 coincide. The tool calculates linear and angular accelerations (i.e. force and torque on the TM in its nominal position due to the surrounding mass distribution. These accelerations are always expressed in the frame Σ M0. Then, stiffness matrices due to the movement of the TM and the mass distribution w.r.t. their nominal position are calculated. 5.3 Force of a point-like mass on the TM The first step of the gravitational tool is the calculation of the force of a point-like mass on the TM. Take a point-like source (simply named source of mass m whose position in the frame Σ M0 that the source exerts on the TM is: F(X, Y, Z = GMm L x L y L z L x/2 L y /2 L z/2 L x/2 L y/2 L z/2 Consider now, for instance, only the x-component: is given by {X, Y, Z}. The force dx dy dz (x X2 + (y Y 2 + (z Z 2 (5.3 F x = = GMm L x L y L z GMm L x L y L z L x /2 L y /2 L z /2 L x/2 L y/2 L z/2 L y/2 L z /2 L y /2 L z /2 [ dx dy dz x (x X2 + (y Y 2 + (z Z = 2 ] Lx 2 1 (x X2 + (y Y 2 + (z Z 2 L x 2 dy dz The integral in y can be solved explicitly as, except for the multiplicative factor, it holds: Ly 2 = ln 1 ( L y Lx 2 2 X ( L y 2 Y + Lx 2 X ( L y 2 Y + Lx 2 X 2+(y Y 1 ( 2 +(z Z 2 2+ ( L y 2 Y 2+(z Z 2 2+ ( L y 2 Y 2+(z Z 2 dy = 2+(y Y Lx 2 +X 2 +(z Z 2 ( ln L y 2 Y + Lx 2 +X ( L y 2 Y + Lx 2 +X 2+ ( L y 2 Y 2+(z Z 2 2+ ( L y 2 Y 2+(z Z 2 (5.4

48 38 CHAPTER 5. THE EQUATIONS OF SELF-GRAVITY Now, the integration along z can be solved as it is: ln [a + ] a 2 + b 2 + z 2 = d { [ z ln ( a + a dz 2 + b 2 + z 2 ] 1 + a ln ( z + ( } a 2 + b 2 + z 2 bz + b arctan a 2 + b 2 + a a 2 + b 2 + z 2 (5.5 By applying Eq. 5.5 to the two logarithms in Eq. 5.4, the analytic expression for the x component is found. In order to simplify the formulation, some auxiliary variables are introduced: a + = L x 2 X a = L x 2 X (5.6 b + = L y 2 Y b = L y 2 Y (5.7 c + = L z 2 Z c = L z 2 Z (5.8 So the component of the force along the x axis is given by: F ( x =c + [ln(b + + a 2 + +b2 + +c2 + 1]+b + ln c + + a 2 + +b2 + +c2 + +a + arctan c [ln(b + + a 2 + +b2 + +c2 1] b + ln(c + a 2 + +b2 + +c2 a + arctan c + [ln(b + a 2 + +b2 +c2 + 1] b ln(c + + a 2 + +b2 +c2 + a + arctan ( +c [ln(b + a 2 + +b2 +c2 1]+b ln c + a 2 + +b2 +c2 +a + arctan c + [ln(b + + a 2 +b2 + +c2 + 1] b + ln(c + + a 2 +b2 + +c2 + a arctan ( +c [ln(b + + a 2 +b2 + +c2 1]+b + ln c + a 2 +b2 + +c2 +a arctan ( +c + [ln(b + a 2 +b2 +c2 + 1]+b ln c + + a 2 +b2 +c2 + +a arctan c [ln(b + a 2 +b2 +c2 1] b ln(c + a 2 +b2 +c2 a arctan expect for the common factor GMm L x L y L z a + c + + a 2 + +b2 + +b + a 2 + +b2 + +c2 + a + c + a 2 + +b2 + +b + a 2 + +b2 + +c2 a + c + + a 2 + +b2 +b a 2 + +b2 +c2 + a + c + a 2 + +b2 +b a 2 + +b2 +c2 a c + + a 2 +b2 + +b + a 2 +b2 + +c2 + a c + a 2 +b2 + +b + a 2 +b2 + +c2 a c + + a 2 +b2 +b a 2 +b2 +c2 + a c a 2 +b2 +b a 2 +b2 +c2 (5.9

49 5.4. Torque of a point-like mass on the TM 39 The analytical formula for the y and z component can be obtained by just swapping (x, X and (y, Y and (x, X and (z, Z respectively (see Eq. A.2,A.3 in the Appendix. 5.4 Torque of a point-like mass on the TM The resultant gravitational force on the TM due to a point-like source acts at the center of gravity 1 of the TM itself (See Figure 5.2. In general, the center of mass and the center of gravity are distinct points. The resultant force F applied the CG is equivalent to the same force applied to the CoM plus the moment r cg F acting on the CoM. Figure 5.2: Schematic for torque calculation. The CG of a body of mass m in presence of a source mass m s can be evaluated as follow: r cg = r s v Define than u = F F, and R = ( Gmms F 1/2 1 By definition the center of gravity (CG of a body is the point, non necessarily inside the body itself, at which the gravitational potential energy of the body is equal to that of a single particle of the same mass located at that point and through which the resultant of the gravitational forces on the component particles of the body acts.

50 40 CHAPTER 5. THE EQUATIONS OF SELF-GRAVITY so that it gives r cg = r s Ru The torque is given by T = r cg F = (r s Ru F = r s F where it is Ru F = 0, being u F by definition. Torque components are given by: T(X, Y, Z = Y F z (X, Y, Z Z F y (X, Y, Z Z F x (X, Y, Z X F z (X, Y, Z X F y (X, Y, Z Y F x (X, Y, Z ( The gravity-gradient matrices When the TM is located in its nominal position, as well as the source, the actions on the TM are given by DC force and torque. Perturbation to DC self-gravity may be caused by: TM motion (w.r.t. its nominal position: translation dr M rotation dα M source motion (w.r.t. its nominal position; being point-like, only translation dr s is considered In order to evaluate the perturbation to the DC action on the TM, a linearized approach is used; this means that for each source, force and torque can be written as: F s F DC s T s T DC s + F s r M dr M + F s α M dα M + F s r s dr s ( T s r M dr M + T s α M dα M + T s r s dr s (5.12 The linearized approach holds because the small variation of TM and source position. On principle, six gravity-gradient matrices for each TM should

51 5.6. Linear gravity-gradient matrix for the force 41 be evaluated; nevertheless any source translation can be traced back to an equivalent TM translation, as it is shown in the following Section. Therefore only four gravity-gradient matrices for each TM are considered: the linear gravity-gradient matrix for the force Γ lin the linear gravity-gradient matrix for the torque Ω lin the angular gravity-gradient matrix for the force Γ ang the angular gravity-gradient matrix for the torque Ω ang By definition, the gravity-gradient matrices are computed considering the TM and the source in their own nominal positions. 5.6 Linear gravity-gradient matrix for the force The force on the TM due to a single source is expressed as: F s = F s (X, Y, Z where (X, Y, Z are the coordinate of the source in the frame Σ M0. Notice that, according to Eq. 5.3, the force is always given in the body-fixed frame Σ M. Assume the TM moves of dr M and the source by dr s (see Figure 5.3. These displacements are always given in the frame Σ M0 The initial position of the source w.r.t the TM is given by r s while the final one is r s. As it deals only with translation, the force expressed in Σ M is the same as expressed in Σ M0. This means that TM and source displacement are equivalent to a sole source displacement as can be easily seen in Figure 5.3. dr s = dr s dr M The linear gravity-gradient matrix for the force on the TM can be obtained straightforwardly by deriving analytically the force equation, that is: X Γ lin,s = [ ] F x F y F z Y Z

52 42 CHAPTER 5. THE EQUATIONS OF SELF-GRAVITY Figure 5.3: Displacements for linear stiffness calculation. where the minus sign is necessary as the derivatives are made with respect to the coordinates of the source, while we are calculating the gradient on the TM. The results are shown in Eq. A.7, A.8 in the Appendix. It is worthy to say that the linear gravity-gradient matrix for the force is symmetric. The variation of force on the TM due to linear stiffness is then given, for each source, by: F s = Γ lin,s (dr M dr S This variation is expressed in the nominal TM frame. 5.7 Linear gravity-gradient matrix for the torque The same scheme used for the force is applied to the derivation of the linear stiffness for the torque. In fact it is Ω lin,s = T x X T y X T z X T x Y T y Y T z Y T x Z T y Z T z Z (5.13 After some passages, it can be written: Y Γ xz ZΓ xy. F z ZΓ yy + Y Γ yz. F y ZΓ yz + Y Γ zz Ω lin,s = F z + ZΓ xx XΓ xz. ZΓ xy XΓ yz. F x + ZΓ xz XΓ zz F y + XΓ xy Y Γ xx. F x + XΓ yy Y Γ xy. XΓ yz Y Γ xz (5.14

53 5.8. Angular gravity-gradient matrix for the force 43 The variation of torque on the TM due to linear stiffness is then given, for each source, by: T s = Ω lin,s (dr M dr S This variation is expressed in the nominal TM frame. 5.8 Angular gravity-gradient matrix for the force The following observation is used. If the TM rotates by a small angle while it is subjected to a field of a point-like source, its rotation its equivalent to an opposite rotation of the source followed by a projection of the result to a set of axis that have been rotating following the source. In general, the angular derivative of any gravity action depending on the source coordinates can be written as: α = ( ˆα ˆα (r s (5.15 where α is the generic angle, ˆα is its versor, and the subscript s stands for the derivative with respect to the source coordinates. = By applying Eq to the force, the following result is obtained: Γ ang,s = F x ϑ F y ϑ F z ϑ F x η F y η F z η F x ϕ F y ϕ F z ϕ = Y Γ xz ZΓ xy. F z + ZΓ xx XΓ xz. F y + XΓ xy Y Γ xx F z + Y Γ yz ZΓ yy. ZΓ xy XΓ yz. F x + XΓ yy Y Γ xy F y + Y Γ zz ZΓ yz. F x + ZΓ xz XΓ zz. XΓ yz Y Γ xz It is worthy to notice that it is: (5.16 Γ ang,s = Ω t lin,s The variation of force on the TM due to angular stiffness is then given, for each source, by: F s = Γ ang,s dα M This variation is expressed in the nominal TM frame.

54 44 CHAPTER 5. THE EQUATIONS OF SELF-GRAVITY = 5.9 Angular gravity-gradient matrix for the torque The same procedure used for the force is used here for the torque. After some passages, it becomes: Ω ang,s = T x ϑ T y ϑ T z ϑ T x η T y η T z η T x ϕ T y ϕ T z ϕ Y Fy+ZFz+Y 2 Γzz 2Y ZΓyz+Z 2 Γyy = XFy+Y ZΓxz XY Γzz Z 2 Γxy+XZΓyz. ZFz+XFx+Z 2 Γxx 2XZΓxz+X 2 Γzz. Y Fx+Y ZΓxz XY Γzz Z 2 Γxy+ZXΓyz. ZFx+XY Γyz Y 2 Γxz XZΓyy+Y ZΓxy. ZFy+XZΓxy Y ZΓxx X 2 Γyz+XY Γxz XFz+XY Γyz Y 2 Γxz XZΓyy+ZY Γxy. Y Fz+XZΓxy Y ZΓxx X 2 Γyz+XY Γxz. XFx+Y Fy+X 2 Γyy 2XY Γxy+Y 2 Γxx The variation of torque on the TM due to angular stiffness is then given, for each source, by: T s = Ω ang,s dα M This variation is expressed in the nominal TM frame. ( Conclusions Once the stiffness matrices have been introduced, Equations 5.11, 5.12 can be written as: F s F DC s + Γ lin,s (dr M dr s + Γ ang,s dα M (5.18 T s T DC s + Ω lin,s (dr M dr s + Ω ang,s dα M (5.19 As shown in the preceding section, the stiffness matrices depend only on force, force linear gradient, and source position component. In conclusion, in order to calculate any gravitational action and stiffness, one only need to calculate the force and the force linear gradient by the analytical formulas. No explicit angular derivatives are needed.

55 Chapter 6 Self-gravity modelling for LISA Pathfinder In Chapter 5 the expressions for force, torque and stiffness matrices on a TM due to a point-like source have been derived. In this Chapter, these expression are applied to the actual source distribution of the LISA Pathfinder. Actually, up to date, the FEM model of the entire LISA Pathfinder spacecraft is not delivered, but only the LTP one is available. Therefore only the LTP is accounted for in the following analysis. Nevertheless once the FEM model of the entire S/C is defined, the formulation introduced in this section can be easily extended to the S/C as well, without big deal. For a given Test Mass, the sources can be considered belonging to: the other TM the LTP (without the Test Masses and the compensation masses system the compensation masses (CM system This subdivision is proposed as each previous item may change independently from the other ones. This subdivision is also used in the simulator. 45

56 46 CHAPTER 6. SELF-GRAVITY MODELLING Figure 6.1: LTP FE Model nodes. 6.1 Self-gravity generated by the LTP Force The gravitational force acting on one of the i th TM due to the LTP is given simply summing over the LTP mass distribution the formula written for a single point mass (Eq F Mi,LT P = F DC M i LT P + s LT P (Γ lin,s dr Mi + s LT P As dr Mi and dα Mi are independent of s, then it is: (Γ ang,s dα Mi F Mi,LT P = F DC M i,lt P + Γ lin dr Mi + Γ ang dα Mi s LT P s LT P (Γ lin,s dr s (6.1 (Γ lin,s dr s (6.2 where Γ lin = s LT P Γ lin,s, Γ ang = s LT P Γ ang,s

57 6.1. Self-gravity generated by the LTP 47 In the hypothesis that the no thermo-elastic deformations occur to the LTP, it holds dr s = 0 for any source s, and so the equation above reduces to: F Mi,LT P = F DC M i,lt P + Γ lin dr Mi + Γ ang dα Mi (6.3 The term s LT P (Γ lin,sdr s represents the variation in self gravity force on the TM due to thermo-elastic deformation. The computation of this term is not trivial. A simplified approach to cope with the influence of thermo-elastic deformations on self gravity is developed and proposed in Chapter Torque Performing the sum over the LTP, Equation 5.19 becomes: T Mi,LT P = T DC M i,lt P + (Ω lin,s dr Mi + (Ω ang,s dα Mi Then it can be written s LT P s LT P T Mi,LT P = T DC M i,lt P + Ω lin dr Mi + Ω ang dα Mi s LT P s LT P (Ω lin,s dr s (6.4 (Ω lin,s dr s (6.5 with obvious meaning of the symbols. Just as like as done with the force, the term s LT P (Ω lin,sdr s represents the effect of the LTP deformation on self-gravity and it will be modelled in Chapter 7 as well Results In this Section the results of the self gravity on the TMs due to LTP are shown. No thermo-elastic deformation is considered up to now, so the results refer only to the motion of the TMs in the unperturbed configuration of the LTP. The results relating to force and torque are divided by the TM mass and moment of inertia respectively. DC accelerations The linear DC acceleration on TM1 due to LTP is: a 1,LT P = [ ] 10 8 m/s 2

58 48 CHAPTER 6. SELF-GRAVITY MODELLING and it is: a 2,LT P = [ ] 10 8 m/s 2 The angular DC acceleration on TM1 due to CM is: ω 1,LT P = [ ] s 2 and it is: ω 2,LT P = [ ] s 2 Stiffness matrices The linear stiffness for the force on TM1 due to LTP is: Γ lin,1,lt P = s while it is Γ lin,2,lt P = s The linear stiffness for the torque on TM1 due to LTP is: Ω lin,1,lt P = m 1 s while it is Ω lin,2,lt P = m 1 s The angular stiffness for the force on TM1 due to LTP is given by: Γ ang,1,lt P = m/s 2 rad

59 6.2. Self-gravity generated by the other TM 49 while it is Γ ang,2,lt P = m/s 2 rad The angular stiffness for the torque on TM1 due to LTP is given by: Ω ang,1,lt P = s 2 rad while it is Ω ang,1,lt P = s 2 rad Self-gravity generated by the other TM Force For the force acting on the i th TM due to the j th TM, it can be written: = F DC M i,m j + (Γ lin,s dr Mi + (Γ ang,s dα Mi (Γ lin,s dr s s M j s M j s M j F Mi,M j (6.6 As dr Mi and dα Mi are independent of s, then it is: F Mi,M j = F DC M i,m j + Γ lin dr Mi + Γ ang dα Mi (Γ lin,s dr s (6.7 s M j where the matrices Γ lin = s M j Γ lin,s Γ ang s M j Γ ang,s represents the stiffness on TM i due to TM j. Now, being the TM a rigid body, for any s M j it holds: dr s = dr Mj + dα Mj r s (6.8 where r s is the position vector that goes from the CoM of TM j to the source in its nominal position, while dr Mj gives the linear displacement of

60 50 CHAPTER 6. SELF-GRAVITY MODELLING the CoM of TM j with respect to its nominal position (see Figure 6.2. By definition, dr s must be expressed in the nominal frame of TM i. Actually the nominal frame of the two TMs differ only because a translation, so no further transformation matrix must be introduced. F Mi,M j Figure 6.2: Schematic the gravity of one TM on the other. With this assumption, Equation 6.7 becomes: = F DC M i,m j + Γ lin dr Mi + Γ ang dα Mi [ Γ lin,s (dr T M2 + dα ] T M2 r s As usual, being dr T M2 independent of s, then F Mi,M j s M j = F DC M i,m j + Γ lin (dr Mi dr Mj + Γ ang dα Mi which is the same as F Mi,M j s M j = F DC M i,m j + Γ lin (dr Mi dr Mj + Γ ang dα Mi + (6.9 ( Γ lin,s dαmj r s (6.10 s M j ( Γlin,s r sdα Mj (6.11 Now this expression can be rearranged in order to give a more compact and easy-to-handle formula. The first step is to add and subtract to Eq.6.11 the quantity: s M j ( Γ lin,s r ij + DC F s dα Mj (6.12

61 6.2. Self-gravity generated by the other TM 51 where r ij is the vector from the CoM of TM i to the CoM of TM j in their nominal position, so that r s = r ij + r s (6.13 Next it is noticed that for any source s it holds the relation So, it gives: Γ ang,s = F DC s Γ lin,s r s (6.14 F Mi,M j = F DC M i,m j + Γ lin (dr Mi dr Mj + Γ ang dα Mi + + ( Γ lin,s r DC s + Γ lin,s r ij + F s dα + Mj s M j }{{} Γ ang,s ( DC Γ lin,s r ij + F s dα Mj (6.15 s M j Equation 6.11 can be now written as F Mi,M j = F DC M i,m j + Γ lin (dr Mi dr Mj + + Γ ang (dα Mi dα Mj (Γ lin r ij + F DC dα Mj (6.16 Being Γ lin a diagonal matrix, r ij = [X ij 0 0] and F DC = [Fx DC 0 0] it becomes F Mi,M j = F DC M i,m j + [ [ ] ] drmi dr Γ lin Γ ang Mj + dα Mi dα Mj Γ lin,yy X ij Fx DC dα Mj ( Γ lin,yy X ij + Fx DC 0 }{{} CM force The underbraced matrix in the rhm is such as: CM force = 2Γ lin so that, finally, the linearized equation for the force on TM i due to the gravity exerted by TM j is given by: F Mi,M j = F DC M i,m j + [ Γ lin Γ ang ] [ drmi dr Mj dα Mi + dα Mj ] (6.18

62 52 CHAPTER 6. SELF-GRAVITY MODELLING Torque The same procedure used for deriving the force equation on one TM on the other can be adopted for the torque. It has been shown that the torque acting on TM i due to TM j can be written as: T Mi,M j = T DC M i,m }{{} i + (Ω lin,s dr Mi + (Ω ang,s dα Mi (Ω lin,s dr s s M =0 j s M j s M j where, for evident reasons of symmetry, since now it is assumed that T DC M i,m i = 0. Then, straightforwardly T Mi,M i = Ω lin dr Mi + Ω ang dα Mi s M j (Ω lin,s dr s Recollecting Eq. 6.8, it gives T Mi,M j Now the term = Ω lin (dr Mi dr Mj + Ω ang dα Mi + s M j Ω lin,s r ij dα Mj s M j ( Ωlin,s r sdα Mj is added and subtracted to Eq and it is made the observation that (6.19 Ω lin,s r s = Ω ang,s T DC s Equation 6.19 becomes T Mi,M j = Ω lin (dr Mi dr Mj + Ω ang (dα Mi dα Mj + TDC s + s M j }{{} =0 s M j Ω lin,s r ij dα M j (6.20 and, finally, the linearized equation for the torque on TM i due to the gravity exerted by TM 2 is given by: T Mi,M j = [ [ ] ] drmi dr Ω lin Ω ang Mj Ω dα Mi dα lin r ij dα Mj (6.21 Mj

63 6.2. Self-gravity generated by the other TM Results DC accelerations The linear DC acceleration on TM1 due to TM2 is: a 12 = [ ] m/s 2 Obviously, the linear DC acceleration on TM2 due to TM1 is: zero: a 21 = [ ] m/s 2 Because of the symmetry, the angular DC accelerations are both equal to Stiffness matrices α 12 = α 21 = [ ] The linear stiffness for the force on TM1 due to TM2 is: Γ lin,12 = s and it is Γ lin,21 = Γ lin,12 The linear stiffness for the torque on TM1 due to TM2 is: Ω lin,12 = m 1 s and it is: Ω lin,21 = Ω lin,12 The angular stiffness for the force on TM1 due to TM2 is given by: Γ ang,12 = m/s 2 rad And it holds: Γ ang,21 = Γ ang,12

64 54 CHAPTER 6. SELF-GRAVITY MODELLING The angular stiffness for the torque on TM1 due to TM2 is given by: Ω ang,12 = s 2 rad And it holds: Comments Ω ang,21 = Ω ang,12 Mutual gravitational interaction of TMs proved to be much smaller that the gravitational effect of the LTP. Anyway, linear DC acceleration cannot be disregarded as they still are within the same order of magnitude of the requirement. Something different holds for the stiffness. For example, any motion x n of one TM w.r.t its nominal position will produce a noise force along x equal to Γ xx x n. This noise force is due to mutual TM stiffness. If the requirement on TM displacement is considered satisfied, an estimation of the maximum noise force on one TM due to the other TM. If fact, it is: f x m Γ xxx n ( s 2 ( m/ Hz = ms 2 / Hz which is negligible, at least compared to the top science requirement of LISA Pathfinder. Plausibly, the mutual stiffness of the TM will not be disregarded for LISA. 6.3 Self-gravity generated by the compensation masses system A compensation masses (CM system is needed as the DC accelerations (linear and angular and stiffness on a TM generated by the LTP and the other TM don t cope with the requirement shown in Chapter 4. A preliminary CM system is proposed in [4]. Figure 6.3 shows the CM set for each TM. Clearly, in order to save mass, the CM are arranged as close as possible to the TM. This implies that CM must feature a very fine mesh in order to guarantee a sufficient accuracy in the results.

65 6.3. Self-gravity generated by the compensation masses 55 Figure 6.3: Preliminary compensation masses system layout Results DC accelerations The linear DC acceleration on TM1 due to CM is: and it is: a 1,CM = [ ] 10 8 m/s 2 a 2,CM = a 1,CM The angular DC acceleration on TM1 due to CM is: and it is: Stiffness matrices ω 1,CM = [ ] 10 9 s 2 ω 1,CM = ω 2,CM The linear stiffness for the force on TM1 due to CM is: Γ lin,1,cm = s