Advances in Spacecraft Technologies 230 0 50 100 150 200 250 -1 0 1 True and estimation quaternion time, sec q 1 true gyroscope star tracker 1 star tracker 2 estimation 0 50 100 150 200 250 -1 0 1 time, sec q 2 0 50 100 150 200 250 -1 0 1 time, sec q 3 0 50 100 150 200 250 -1 0 1 time, sec q 4 Fig. 15. Local and global estimates with FDI algorithm (star tracker fault) 0 50 100 150 200 250 -0.05 0 0.05 Error time, sec gyroscope 0 50 100 150 200 250 -1 0 1 time, sec star tracker 1 0 50 100 150 200 250 -0.02 0 0.02 time, sec star tracker 2 0 50 100 150 200 250 -0.02 0 0.02 time, sec master filter Fig. 16. Error between true quaternion and estimates with FDI algorithm (star tracker fault) Fault-Tolerant Attitude Estimation for Satellite using Federated Unscented Kalman Filter 231 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 3 fault detection time, sec Fig. 17. Fault detection index (star tracker fault) Federated UKF Sensor Fault Type Single UKF Without FDI With FDI Gyroscope failure 145.2083 12.6870 1.0022 Star tracker failure 89.4021 40.9636 1.2780 Table 2. Error sum of attitude estimates 5. Conclusion In this study, the federated UKF with the FDI algorithm is proposed for the estimation of the satellite attitude. The UKF gives the accurate estimates for nonlinear systems, and the federated UKF makes the system fault-tolerant and reliable. Since the FDI algorithm can detect and isolate the sensor failure immediately, the global estimate is not affected by the poor local estimate due to the faulty sensor. In this respect, the error of the global estimate using the federated UKF and the FDI algorithm is smaller than that using the federated UKF only. Numerical simulation results show that the proposed algorithm provides efficient and accurate attitude estimation of the satellite despite the fault of the attitude sensors. The proposed algorithm can be applied not only for the satellite systems but also for the ground mobile robots and aerial robot systems. 6. References Agrawal, B. N. & Palermo, W. J. (2002). Angular Rate Estimation For Gyroless Satellite Attitude Control. AIAA Guidance, Navigation, and Control Conference, Monterey, CA, Aug. 2002. Advances in Spacecraft Technologies 232 Bae, J. & Kim, Y. (2010). Attitude Estimation for Satellite Fault Tolerant System using Federated Unscented Kalman Filter. International Journal of Aeronautical and Space Science, Vol. 11, No. 2, 2010, pp. 80-86, ISSN: 1229-9626. Crassidis, J. L. & Markley, F. L. (2003). Unscented Filtering for Spacecraft Attitude Estimation. Journal of Guidance, Control, and Dynamics, Vol. 25, No. 4, 2003, pp. 536- 542, ISSN: 0731-5090. Edelmayer, A. & Miranda, M. (2007). Federated Filtering for Fault Tolerant Estimation and Sensor Redundancy Management in Coupled Dynamics Distributed Systems. Mediterranean Conference on Control and Automation, Athens, Greece, July 2007. Hwang, I.; Kim, S.; Kim, Y. & Seah, C.E. (2010). A Survey of Fault Detection, Isolation, and Reconfiguration Methods. IEEE Transactions on Control Systems Technology, Vol. 18, No. 3, May 2010, pp.636-653, ISSN: 1063-6536. Jayaraman, P.; Fischer, J.; Moorhouse, A. & Lauer, M. (2006). Star Tracker Operational Usage in Different Phases of the Mars Express Mission. SpaceOps 2006 Conference, Rome, Italy, June 2006. Jiancheng, F. & Ali, J. (2005). Multisensor Data Synthesis using Federated Form of Unscented Kalman Filtering. IEEE International Conference on Industrial Technology, Hong Kong, Dec. 2005. Jin, Y.; Liu, X. & Hou, C. (2008). Relative Attitude Determination for Fly-Around Based on UKF. 7th World Congress on Intelligent Control and Automation, Chongqing, China, Jun. 2008. Julier, S. J. & Uhlmann, J. K. (2004). Unscented Filtering and Nonlinear Estimation. Proceedings of the IEEE, Vol. 92, No. 3, 2004, pp. 401–422, ISSN: 0018-9219. Karlgaard, C. D. & Schaub, H. (2008). Adaptive Huber-Based Filtering Using Projection Statistics: Application to Spacecraft Attitude Estimation. AIAA Guidance, Navigation, and Control Conference, Honolulu, HI, Aug. 2008. Kerr, T. (1987). Decentralized Filtering and Redundancy Management for Multisensor Navigation. IEEE Transactions on Aerospace and Electronic Systems, Vol. 20, No. 1, 1987, pp. 83-119, ISSN: 0018-9251. Kim, Y. S. & Hong, K. S. (2003). Decentralized Information Filter in Federated Form. SICE Annual Conference, Fukui, Japan, Aug. 2003. Lee, D. (2008). Unscented Information Filtering for Distributed Estimation and Multiple Sensor Fusion. AIAA Guidance, Navigation, and Control Conference, Honolulu, HI, Aug. 2008. Mehra, R. & Bayard, D. (1995). Adaptive Kalman Filtering, Failure Detection and Identification for Spacecraft Attitude Estimation. 4th IEEE Conference on Control Application, Albany, NY, Sep. 1995. Nagendra, R. G.; Alex, T. K. & Seetharama, B. M. (2002). Incremental-Angle and Angular Velocity Estimation Using a Star Sensor. Journal of Guidance, Control, and Dynamics, Vol. 25, No. 3, 2002, pp. 433-441, ISSN: 0731-5090. Schaub, H. & Junkins, J. L. (2003). Analytical Mechanics of Space Systems, American Institute of Aeronautics and Astronautics, ISBN: 1-60086-721-9, Reston, VA. Simon, D. (2006). Optimal State Estimation, Wiley-Interscience, ISBN: 0471708585, Malden, MA. Wei, M. & Schwarz, K. P. (1990). Testing a Decentralized Filter for GPS/INS Integration. Position Location and Navigation Symposium, Las Vegas, NV, Mar. 1990. Xu, Y. (2009). Nonlinear Robust Stochastic Control for Unmanned Aerial Vehicles. Journal of Guidance, Control, and Dynamics, Vol. 32, No. 4, 2009, pp. 1308-1319, ISSN: 0731- 5090. 0 Nonlinear Electrodynamics: Alternative Field Theory for Featuring Photon Propagation Over Weak Background Electromagnetic Fields and what Earth Receivers Read off Radio Signals from Interplanetary Spacecraft Transponders Herman J. Mosquera Cuesta 1,2,3 1 Departmento de F´ısica Universidade Estadual Vale do Acara ´u Avenida da Universidade 850, Campus da Betˆania, CEP 62.040-370, Sobral, Cear´a 2 Instituto de Cosmologia, Relatividade e Astrof´ısica (ICRA-BR) Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, CEP 22290-180, Urca Rio de Janeiro, RJ 3 International Center for Relativistic Astrophysics Network (ICRANet) International Coordinating Center, Piazzalle della Repubblica 10, 065112, Pescara 1,2 Brazil 3 Italy 1. Introduction A few observational and/or experimental results have dramatically pushed forward the research program on gravity as those from the radio-metric Doppler tracking received from the Pioneer 10 and 11 spacecrafts when the space vehicles were at heliocentric distances between 20 and 70 Astronomical Units (AU). These data have conclusively demonstrated the presence of an anomalous, tiny and blue-shifted frequency drift that changes smoothly at a rate of ∼ 6 × 10 −9 Hz s −1 . Those signals, if interpreted as a gravitational pull of the Sun on each Pioneer vehicle, translates into a deceleration of a P =(8.74 ± 1.33) × 10 −10 ms −2 . This sunward acceleration appears to be a violation of Newton’s inverse-square law of gravitation, and is referred to as the Pioneer anomaly, the nature of which remains still elusive to unveil. Within the theoretical framework of nonlinear electrodynamics (NLED) in what follows we will address this astrodynamical puzzle, which over the last fifteen years has challenged in a fundamental basis our understanding of gravitational physics. To this goal we will first, and briefly, review the history of the Pioneers 10 and 11 missions. Then a synopsis of currently available Lagrangian formulations of NLED is given. And finally, we present our solution of this enigma by invoking a special class of NLED theories featuring a proper description of electromagnetic phenomena taking place in environments where the strength of the (electro)magnetic fields in the background is decidedly low. 12 2 Advances in Spacecraft Technologies 2. What is the problem: The Pioneer anomaly In this short voyage to the Pioneer 10 and 11 missions our main guide will be the comprehensive and richly documented recent review on the Pioneer Anomaly by [Turyshev, S. G. & Toth, V. T. (2010). Living Rev. Rel. 13 (2010) 4. arXiv:1001.3686, v2, gr-qc] from which we retake some ideas and references. (The attentive readers are kinldy addressed to this invaluable article). The Pioneer 10 and 11 spacecrafts were the first two man-made space vehicles designed to explore the outer solar system. The trajectories of the spaceships were projected to passage nearby Jupiter during 1972-1973 having as objectives to conduct exploratory investigation of the interplanetary medium beyond the orbit of Mars, the nature of the asteroid belt, the environmental and atmospheric characteristics of Jupiter and Saturn (for Pioneer 11), and to investigate the solar system beyond the orbit of the Jovian planet. 1 The Pioneer missions were the first space probes to adventure over the asteroid belt, heading for close-up observations of the gaseous giant planets, and for performing in situ studies of the physical properties of the interplanetary medium in the outer solar system. The design of their missions was guided by the simplicity, having a powerful rocket-launching system to push the spacecrafts on an hyperbolic trajectory aimed directly at Jupiter, which the spacecrafts were expected to fly-by approximately 21 months after launch (see Fig. 1). By the late 1960’s, the aerospace engineering technology available to the designers of the Pioneer missions made it no longer practical to use solar panels for operating a spacecraft at large distances, as for instance that of Jupiter. A cause of this, a built-in nuclear power system, in the form of radioisotope thermoelectric generators (RTGs) powered by 238 Pu, was chosen as the means to provide electrical power to the spaceship. As even this was relatively new technology at the time the missions were designed, the power subsystem was suitably over-engineered, being the unique design requirement to have a completely functional space probe capable of performing all planned scientific tasks by running only three (out of four) RTGs. The entire design of these spacecrafts and their science missions was characterized by such conservative engineering, and for sure it was responsible for both the exceptional longevity of the two spacecrafts and their ability to deliver science results which by far exceeded the expectations of their designers. The original plan envisioned a primary mission of two to three years in duration. Nevertheless, following its encounter with Jupiter, Pioneer 10 remained functional for over 30 years. Meanwhile, Pioneer 11, though not as long lived as its engineering-copy craft, successfully navigated a path across the solar system for another encounter with Saturn, offering the first close-up observations of the ringed planet. After the encounters with Jupiter and Saturn (for Pioneer 11, see Fig. 1), the space ships followed, near the plane of the ecliptic, hyperbolic orbits of escape heading to opposite sides of the solar system, continuing their extended missions. The spacecrafts explored the outer regions of the solar system, studying energetic particles from the Sun (solar wind), and cosmic rays entering our neighborhood in the Milky Way. (Their cousin spacecrafts, the Voyager1 and 2, that where launched contemporarily, studied in the beginning of their mission, the interplanetary space, what resulted in a very accurate mapping of the interplanetary magnetic field and its strength, as one can see in Fig. 2 below). 1 See details on the Pioneer missions at http://www.nasa.gov/centers/ames/missions/ archive/pioneer.html. Be awared that another member of Pioneer spacecrafts family, Pioneer 6, remained operational for more than 35 years after launch. 234 Advances in Spacecraft Technologies Nonlinear Electrodynamics: An Alternative Field Theory for Featuring Photon Propagation Over Weak Background Electromagnetic Fields and the Performance of Receivers of Radio Signals from . . . 3 Fig. 1. Ecliptic pole view of the spacecrafts Pioneer 10 and 11 interplanetary trajectories (see also the trajectories of the vehicles Voyager 1 and 2). Credit: http://www.nasa.gov/centers/ames/missions/archive/pioneer.html In virtue of a combination of many factors, the Pioneers were excellent space sondes for pursuing experiments of high precision celestial mechanics. This includes the presence of a coherent mode transceiver on board, the attitude control (spin-stabilized, with a minimum number of attitude correction maneuvers using thrusters), the design of the power system (the RTGs being on extended booms aided the stability of the craft and also helped in reducing thermal effects), and Doppler tracking of high precision (with the accuracy of post-fit Doppler residuals at the level of mHz). The exceptional built-in sensitivity to acceleration of the Pioneer 10 and 11 spacecrafts naturally allowed them to reach a level of accuracy of ∼ 10 −10 m/s 2 . The result was one of the most precise spacecraft navigations in deep space since the early days of space exploration. That is the great legacy of the Pioneer missions. After having had a brief accounting of the Pioneers missions, one can proceed to review our current understanding of nonlinear electrodynamics and to settle down the foundations for its use in the search for a solution to the Pioneer anomaly. In this Section we shall briefly review the theoretical foundations of some theories of NLED, focusing essentially on the fundamental prediction concerning the way photons propagate through a vacuum space permeated by electromagnetic (EM) fields: The fact that photons travel along the effective metric, and not over the geometry in the background. It is this peculiar feature what makes the photon to 235 Nonlinear Electrodynamics: Alternative Field Theory for Featuring Photon Propagation Over Weak Background Electromagnetic Fields and what Earth Receivers Read off Radio Signals 4 Advances in Spacecraft Technologies “feel” itself being acted upon by a force, and consequently to undergo acceleration. 2 In our understanding, such effect is responsible for the drift in frequency undergone by the photon. Next we will show that any NLED, independently of the specific form of its Lagrangian, brings in such a frequency shift. And in our view, it is such acceleration what can account for the Pioneer anomaly. 3. Some Lagrangian formulations of nonlinear electrodynamics To start with, it is worth to recall that according to quantum electrodynamics (QED: see Delphenich (2003; 2006) for a complete review on NLED and QED) a vacuum has nonlinear properties (Heisenberg & Euler 1936; Schwinger 1951) which affect the photon propagation. A noticeable advance in the realization of this theoretical prediction has been provided by [Burke, Field, Horton-Smith , etal., 1997), who demonstrated experimentally that the inelastic scattering of laser photons by gamma-rays in a background magnetic ield is definitely a nonlinear phenomenon. The propagation of photons in NLED has been examined by several authors [Bialynicka-Birula & Bialynicki-Birula, 1970; Garcia & Plebanski, 1989; Dittrich & Gies, 1998; De Lorenci, Klippert, Novello, etal., 2000; Denisov, Denisova & Svertilov, 2001a, 2001b, Denisov & Svertilov, 2003]. In the geometric optics approximation, it was shown by [Novello, De Lorenci, Salim & etal., 2000; Novello & Salim, 2001], that when the photon propagation is identified with the propagation of discontinuities of the EM field in a nonlinear regime, a remarkable feature appears: The discontinuities propagate along null geodesics of an effective geometry which depends on the EM field on the background. This means that the NLED interaction can be geometrized. An immediate consequence of this NLED property is the prediction of the phenomenon dubbed as photon acceleration, which is nothing else than a shift in the frequency of any photon traveling over background electromagnetic fields. The consequences of this formalism are examined next. 3.1 Heisenberg-Euler approach The Heisenberg-Euler Lagrangian for nonlinear electrodynamics (up to order 2 in the truncated infinite series of terms involving F) has the form Heisenberg & Euler (1936) L H−E = − 1 4 F + ¯ αF 2 + ¯ βG 2 , (1) where F = F μν F μν , with F μν = ∂ μ A ν − ∂ ν A μ , and G = 1 2 η αβγδ F αβ F γδ = −4  E ·  B, with greek index running (0, 1, 2, 3), while ¯ α and ¯ β are arbitrary constants. When this Lagrangian is used to describe the photon dynamics the equations for the EM field in vacuum coincide in their form with the equations for a continuum medium in which the electric permittivity and magnetic permeability tensors  αβ and μ αβ are functions of the electric and magnetic fields determined by some observer represented by its 4-vector velocity V μ [Denisov, Denisova & Svertilov, 2001a, 2001b; Denisov & Svertilov, 2003; Mosquera Cuesta & Salim, 2004a, 2004b]. The attentive reader must notice that this first order approximation is valid only for B-fields smaller than B q = m 2 c 3 e ¯ h = 4.41 ×10 13 G (Schwinger’s critical B-field Schwinger (1951)). In curved spacetime, these equations are written as 2 Because of the special theory of relativity constraints regarding the propagation of any perturbation, it becomes clear that such effect must manifest itself as a change in one or both of their physical properties: its frequency or its wavelength. Hence, through the Pioneer spacecrafts radio Doppler tracking we might be observing the effect on the photon frequency. 236 Advances in Spacecraft Technologies Nonlinear Electrodynamics: An Alternative Field Theory for Featuring Photon Propagation Over Weak Background Electromagnetic Fields and the Performance of Receivers of Radio Signals from . . . 5 D α ||α = 0, B α ||α = 0 , (2) D α ||β V β c + η αβρσ V ρ H σ||β = 0, (3) B α ||β V β c −η αβρσ V ρ E σ||β = 0 . (4) Here, the vertical bars subscript “ || ” stands for covariant derivative and η αβρσ is the antisymmetric Levi-Civita tensor. The 4-vectors representing the electric and magnetic fields are defined as usual in terms of the electric and magnetic fields tensor F μν and polarization tensor P μν E μ = F μν V ν c , B μ = F ∗ μν V ν c , (5) D μ = P μν V ν c , H μ = P ∗ μν V ν c , (6) where the dual tensor X ∗ μν is defined as X ∗ μν = 1 2 η μναβ X αβ , for any antisymmetric second-order tensor X αβ . The meaning of the vectors D μ and H μ comes from the Lagrangian of the EM field, and in the vacuum case they are given by H μ = μ μν B ν , D μ =  μν E ν , (7) where the permeability and tensors are given as μ μν =  1 + 2α 45πB 2 q  B 2 − E 2   h μν − 7α 45πB 2 q E μ E ν , (8)  μν =  1 + 2α 45πB 2 q  B 2 − E 2   h μν + 7α 45πB 2 q B μ B ν . (9) In these expressions α is the EM coupling constant (α = e 2 ¯hc = 1 137 ). The tensor h μν is the metric induced in the reference frame perpendicular to the observers determined by the vector field V μ . Meanwhile, as we are assuming that E α = 0, then one gets  α β = h α β + 7α 45πB 2 q B α B β (10) and μ αβ = μh αβ . The scalars  and μ can be read directly from Eqs.(8, 9) as  ≡ μ = 1 + 2α 45πB 2 q B 2 . (11) Applying conditions (62) and (63) (derived in the Appendix) to the field equations when E α = 0, we obtain the constraints e μ  μν k ν = 0 and b μ k μ = 0 and the following equations for the discontinuity fields e α and b α : 237 Nonlinear Electrodynamics: Alternative Field Theory for Featuring Photon Propagation Over Weak Background Electromagnetic Fields and what Earth Receivers Read off Radio Signals 6 Advances in Spacecraft Technologies  λγ e γ k α V α c + η λμρν V ρ c  μb ν k μ −μ  λ α B ν k μ  = 0 , (12) b λ k α V α c −η λμρν V ρ c  e ν k μ  = 0 . (13) Isolating the discontinuity field from (12), substituting in equation (13), and expressing the products of the completely anti-symmetric tensors η νξγβ η λαρμ in terms of delta functions Stephani (2004), we obtain b λ (k α k α ) 2 +  μ  μ l β b β k α B α + βB β b β B α k α μ − βB 2  k λ +  μ  μl α b α (k β V β ) 2 (k α k α ) 2 − βB α b α (k β k β ) 2 μ − βB 2  B λ −  μ  μ l μ b μ k α B α k β V β  V λ = 0 . (14) This expression is already squared in k μ but still has an unknown b α term. To get rid of it, one multiplies by B λ , to take advantage of the EM wave polarization dependence. By noting that if B α b α = 0 one obtains the dispersion relation by separating out the k μ k ν term, what remains is the (-) effective metric. Similarly, if B α b α = 0, one simply divides by B γ b γ so that by factoring out k μ k ν , what results is the (+) effective metric. For the case B α b α = 0, one obtains the standard dispersion relation g αβ k α k β = 0 . (15) whereas for the case B α b α = 0, the result is   1 + μ  B μ + ˜ βB 2 μ − ˜ βB 2  g αβ − μ  B μ V α V β c 2 +  μ  B μ + ˜ βB 2 μ − ˜ βB 2  l α l β  k α k β = 0 , (16) where (  ) stands for d dB , and we have defined ˜ β = 7α 45πB 2 q , and l μ ≡ B μ |B γ B γ | 1/2 (17) as the unit 4-vector along the B-field direction. From the above expressions we can read the effective metric g αβ + and g αβ − , where the labels “+” and “-” refers to extraordinary and ordinary polarized rays, respectively. Then, we need the covariant form of the metric tensor, which is obtained from the expression defining the inverse metric g μν g να = δ α μ . So that one gets from one side g − μν = g μν (18) and from the other g + μν =  1 + μ  B μ + βB 2 μ − βB 2  −1 g μν + ⎡ ⎣ μ  B μ(1 + μ  B μ + βB 2 μ−βB 2 )(1 + βB 2 μ−βB 2 ) ⎤ ⎦ V μ V ν c 2 + ⎛ ⎝ μ  B μ + βB 2 μ−βB 2 1 + μ  B μ + βB 2 μ−βB 2 ⎞ ⎠ l μ l ν . (19) 238 Advances in Spacecraft Technologies [...]... guaranteed in this theory 10 242 Advances in Spacecraft Technologies Advances in Spacecraft Technologies Physical motivations for bringing in this theory have been provided in Novello et al (2004) Besides of those arguments, an equally unavoidable motivation comes from the introduction in the 1920’s of both the Heisenberg-Euler and Born-Infeld nonlinear electrodynamics discussed above, which are valid in. .. radiometer is a sensor with six narrow bands in the shortwave infrared region (1.6-2.43 μm) for discriminating rock and minerals using the specific absorption signatures 260 Advances in Spacecraft Technologies Sub system Band No Spectral range (µm) Ground Sampling Distance VNIR 1 2 3N 3B 0.52 - 0.60 0.63 - 0.69 0 .76 - 0.86 0 .76 - 0.86 15 m 4 5 6 7 8 9 1.600 - 1 .70 0 2.145 - 2.185 2.185 - 2.225 2.235 - 2.285... 6 07, 665 (2004)] Inserting the latter figures in relations (A3) and (A4) yields : B0 = (0.005 ± 0.002) μG (A5), B1 = (0.008 ± 0.002) μG (A6) Since CMB is pure radiation (i e., E = Bc not equal to zero on average), we consider that relations (A4) and (A6) give a better estimate of B1 than the one put forward in Novello et al (2004) 18 250 Advances in Spacecraft Technologies Advances in Spacecraft Technologies. .. P (2005) Int J Mod Phys A 20, 2304 Mbelek, J P., Mosquera Cuesta, H J., Novello, M & Salim, J M (20 07) Nonlinear electrodynamics and the Pioneer 10/11 spacecraft anomaly Europhys Lett 77 , 19001 Mendonca, J T., (2000) Theory of photon acceleration (Taylor & Francis 2000) Print ISBN: ¸ 978 -0 -75 03- 071 1-6 eBook ISBN: 978 -1-4200-33 27- 4 See also Mendonca et al (2006) e-Print Archive: hep-ph/06 071 95 Milgrom,... Rev D 69, 1 273 01 e Pagels, H.& Tomboulis, E (1 978 ) Nuc Phys B 143, issue 3, 485 P´ rez Mart´nez, A., P´ rez Rojas, H., & Mosquera Cuesta, H J (2003) Eur Phys Journ C, 29, e ı e 22 254 Advances in Spacecraft Technologies Advances in Spacecraft Technologies 111 Plebanski, J (1 970 ) Lectures on nonlinear electrodynamics (Nordita, Copenhagen, 1 970 ) Pioneer Explorer Collaboration (2005) e-print gr-qc/0506139... − 4 L FF Fμα Fαν (see Novello et al LF (2000),Novello & Salim (2001)) Thus, following the usual analysis on the gravitational frequency shift 16 248 Advances in Spacecraft Technologies Advances in Spacecraft Technologies k μ ( k μ ) |0 = − ω c ˙ ω c (55) By inserting relation (55) in (54), and then expanding and arranging, one finds ˙ ν ν =− ˙ a a Q + 2FQ F 1−Q (56) where we have set Q = F L FF and... gravitational fields and there is a lot of space for speculation 14 246 Advances in Spacecraft Technologies Advances in Spacecraft Technologies other hand, in using a time dependent potential Iorio (2006a;b;c); Tangen (2006) to explain the Pioneer 10/11 data one may be pointing out to the need of an effective metric for the photons In fact, what is needed is just a time variation of the 4-momentum of the... equation of the discontinuity (that defines the effective metric, see the Appendix) is conformal invariant, one can multiply this last equation by the conformal factor −4 1 + F b2 3/2 to obtain μν geff = 1 + F b2 gμν − 2B2 μν [h + l μ l ν ] b2 (24) Then, by noting that F = Fμν F μν = −2( E2 − B2 ) , (25) 8 240 Advances in Spacecraft Technologies Advances in Spacecraft Technologies and recalling our assumption... boundary effect in the frequency domain, a twodimensional Hanning window is applied to the input image A low-pass filter is also applied in the frequency domain to eliminate the high-frequency components of each image The subpixel displacement between the images is obtained by sinc function fitting to the neighboring correlation values (Nagashima et al., 2006) Although the size of the correlation window must... today 12 244 Advances in Spacecraft Technologies Advances in Spacecraft Technologies As stressed above, unlike other spacecrafts like the Voyagers and Cassini which are three-axis stabilized (hence, not well-suited for a precise reconstitution of trajectory because of numerous attitude controls), the Pioneer 10/11, Ulysses and the by-now destroyed Galileo are attitude-stabilized by spinning about an . Read off Radio Signals 16 Advances in Spacecraft Technologies k μ (k μ ) |0 = − ω c  ˙ ω c  . (55) By inserting relation (55) in (54), and then expanding and arranging, one finds  ˙ ν ν  =. 4 Advances in Spacecraft Technologies “feel” itself being acted upon by a force, and consequently to undergo acceleration. 2 In our understanding, such effect is responsible for the drift in. that individual orbits corresponding to minima of the classical action dominate the Euclidean functional integral.” In view of this drawback, of the at the time understanding of ground states in