Relativistic Warning to Space Missions Aimed to Reach Phobos
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International Journal of Astronomy and Astrophysics, 2011, 1, 200-203 doi:10.4236/ijaa.2011.14025 Published Online December 2011 (http://www.SciRP.org/journal/ijaa) Relativistic Warning to Space Missions Aimed to Reach Phobos Alexander P. Yefremov Institute of Gravitation and Cosmology of Peoples’ Friendship University of Russia, Moscow, Russia E-mail: [email protected] Received October 5, 2011; revised November 15, 2011; accepted November 27, 2011 Abstract Disagreement in estimations of the observed acceleration of Phobos yields several theories empirically modifying classical description of motion of the satellite, but its orbital positions detected by Mars-aimed spacecraft differ from predictions. It is shown that the satellite’s orbital perturbations can be explained as manifestations of the relativistic time-delay effect ignored in classical models. So computed limits of Phobos’ acceleration essentially exceed the experimental values. The satellite’s expected orbital shift is calculated for the moment of contact with a landing module of the Phobos-Grunt project; the shift assessed in kilometers may prevent the mission success. Limits of the apparent relativistic accelerations are predicted for fast satel- lites of Jupiter. Keywords: Planet, Satellite, Earth, Mars, Phobos, Acceleration, Shift, Quaternion, Relativity 1. Introduction: Phobos’ Irregular Motion, found Phobos in kilometers ahead of its expected posi- Theory and Practice tion. A new space mission Phobos-Grunt (Russia) is planned soon [10]; if its computer program determines A satellite of the planet Mars, Phobos, discovered by A. the target position using old models, the project may Hall in 1877 still attracts great attention. In 1911 after have problems. years of observations H. Struve offered a classical theory Another reason of the Phobos’ motion irregularity is of Martian satellites’ motion taking into account the considered here on the base of relativity theory. Section planet’s oblate shape and solar gravity. In 1945 B. Shar- 2 comprises deduction of formulae for apparent accelera- pless discovered a secular increase of Phobos velocity [1] tion and for relativistic shift of a solar system planet’s and surmised that the moon was spiraling in toward Mars. satellite observed from the Earth, using methods of qua- I.Shklovski ascribing the orbit’s decay to atmospheric ternion model of relativity. In Section 3 calculated and friction concluded that the moon could be hollow [2], experimental values of the Phobos’ acceleration are maybe artificial, but the orbit’s evolution was also re- compared, and the shift value is assessed for the Pho- ferred to influence of tidal forces [3]. Later mathematical bos-Grunt space mission. A compact discussion is found models [4-6] were developed in attempts to better ex- in Section 4 with prediction of relativistic shift of fast plain observational results, some of conclusions though satellites of Jupiter potentially observed from the Earth. uncertain about the acceleration value and even sign. Cosmic era made Phobos a desired but hardly accessi- ble goal. In 1988 Russian Phobos-1 (said badly operated) 2. Relativistic Explanation passed by the target while Phobos-2 disappeared at 50 m from the moon’s surface. In 1999 Mars Climate Orbiter Let the Earth (frame of reference ) and a planet of the (NASA, also said badly operated) was lost near Phobos’ solar system ( , e.g. Mars) have circular trajectories orbit, and Mars Polar Lander vanished hardly touching (for simplicity) in ecliptic plane and revolve about the the Martian atmosphere. In 2003 Beagle-2 (UK) shared Sun with velocities of constant values VVE , P . A planet’s the destiny without any firm conclusion of the loss. Sur- satellite with orbital period T const (from viewpoint vivors showed deficiency of existing theories: Mariner 9 of ) can be regarded as a clock. The value of (NASA, 1971) [7,8] and Mars Express (ESA, 2004) [9] relative velocity is found as Copyright © 2011 SciRes. IJAA A. P. YEFREMOV 201 VVVVV222cos t, (1) The form-invariant vector-interval describing the rela- PE PE tivistic system “Earth-planet (satellite)” is chosen in the where is the difference of orbital angular velocities form automatically satisfying the space-time orthogonal- of the Earth and the planet, t is a VVE , P -angle, its ity condition zero initial value is chosen at the planets’ opposition V (4) point (where the Martian satellites are usually observed ddzqiс ti 11 с dt qq 2 с optically [6]). The relative velocity value is always dif- ferent from zero, V 0 , hence a relativistic time-delay the fundamental velocity с is constant, dt is a proper effect exists. A clock belonging to should be slow in time interval in , dt is respective time interval of , i.e. the satellite, as a point of -clock’s arrow, the observer. The cinema tic situation described by (4) is should be seen in at earlier position on its orbit than equivalent to the transformation of the type (2) it is in . Emphasize two features of the effect. First, it with the matrix is accumulated with time since the satellite’s apparent coshi sinh 0 shift increases, so the effect is potentially detected. Sec- Oi sinh cosh 0 (5) ond, the -observer will find the satellite’s motion kn non-uniform, since the relative velocity is variable, 001 VV ()t, the frames , being non-inertial. This what leads to standard expression for relative velocity as hampers computation of the shift-effect by means of a function of hyperbolic parameter Vc tanh , and to Special Relativity (SR) valid for inertial frames of refer- the time-delay relation ence, though SR can be applied locally as it is done in [11] in the deduction of formula for the Thomas preces- dtt dcosh (6) sion. But here assessment of the relativistic shift is done apparently the same as in SR but valid for the with the help of a more “technological” approach based non-inertial case. Now let the time-interval dtT be on quaternion square root from SR space-time interval, period of the satellite’s revolution measured in (in the method admitting computation of relativistic effects fact, a physically real period, in this case is small c om- for arbitrary frames without addressing tensor calculus of pared to t , time of observation), and dt T be the general relativity. The quaternion model of relativity similar “pe riod” (here a variable magnitude) o bserved theory is described in detail in Ref. [12], below its very from . Then (6) acquire the form short description is given. 22 It is straightforwardly verified that multiplication of TT cosh T 1 Vtc (7) quaternions, the hypercomplex numbers built on one so the period observed from the Earth is always greater scalar (ordinary) unit and three non-commutative vector that the real one TT . But search for the period’s dif- units q , is invariant under rotations of the vector units k ference, whatever desira ble it could be, is of no use since by matrices belonging to special orthogonal group with relativistic corrections, important as will be shown below, complex parameters would be slurred over by uncertainty of our knowledge qnnk O qk (2) of the involved magnitudes: gravitational constant [14], (summation in the repeating indices is assumed here and the planet’s and the satellite’s physical parameters [15]. So only the limits of the satellite’s acceleration value and further on), OSOnk (3,С ) , the group being 1:1 iso- morphic to the Lorentz group. It is also proved that simi- an integral apparent shift will be assessed with possible lar rotations keep form of the vector-quaternion accuracy. Making future expressions compact, denote ddxidtezq, (3) kkk 22 2 2 AVV PE 21 c , BVVc PE 1 under the space-time orthogonality condition dxkk e 0 , where dxk , dt are differentials of a particle’s space- and using (1, 7) find (up to the small A, B) the difference time coordinates in a frame qk , ek is any unit between the values of satellite’s real orbital velocity VS vector. The square of the vector-interval (3) yields estimated in , and observed velocity VtS () esti- ddtz 2 2dr2, the Minkowski-type space-time inter- mated in val of SR, so instead of invariance of this interval one 11 can analyze form-invariance of dz thus obtaining all VVtSS () 2π r VABS(cos t) (8) TT cinematic effects of SR with an additional advantage to consider non-inertial frames of reference [12,13]. Apply r being the radius of the satellite’s orbit. Differentiation the method for computation of characteristics of the sat- of (8) with respect to time of observation leads to for- ellite’s motion estimated by an Earth’s observer. mula for apparent satellite’s acceleration Copyright © 2011 SciRes. IJAA 202 A. P. YEFREMOV d()Vt 2011, and it is expected to reach Phobos within 1.25 yr aVBS sin t (9) dt S of flight. The satellite’s orbital parameters for the space- craft’s computer program could be obtained at the last Now using (8) find the satellite’s orbital shift, the dif- opposition in Jan. 2010 (i.e. plus 1.75 yr to the flight ference between its real position and that observed from time) or, for higher accuracy, in the last perihelion oppo- the Earth sition in Aug. 2003 (plus 8.25 yr ), thus the time inter- vals between the observations and the spacecraft-moon lVVtVAtBtSS ds S in (10) contact are t1 3 yr or t2 9.5 yr . So if the relativistic the integration constant is chosen zero assuming no shift effect is ignored, the mission c an find Phobos in at the beginning of the observation. As expected the l1 1.55 km or l2 5.18 km ahead of its expected shift’s value monotonously grows linearly in time with position, as earlier missions did.