Some improvements on relativistic positioning systems. (English) Zbl 1524.35653
Summary: We make some considerations about Relativistic Positioning Systems (RPS). Four satellites are needed to position a user. First of all we define the main concepts. Errors should be taken into account. Errors depend on the Jacobian transformation matrix. Its Jacobian is proportional to the tetrahedron volume whose vertexes are the four tips of the receiver-satellite unit vectors. If the four satellites are seen by the user on a circumference in the sky, then, the Jacobian and the tetrahedron volume vanish. The users we consider are spacecraft. Spacecraft to be positioned cannot be close to a null Jacobian satellites-user configuration. These regions have to be avoided choosing an appropriate set of four satellites which are not seen too close to the same circumference in the sky. Errors also increase as the user spacecraft separates from the emission satellite region, since the tetrahedron volume decreases. We propose a method to autonomously potion a user-spacecraft which can test our method. This positioning should be compared with those obtained by current methods. Finally, a proposal to position a user-spacecraft moving far from Earth, with suitable devices (autonomous), is presented.
MSC:
| 35Q85 | PDEs in connection with astronomy and astrophysics |
| 37M99 | Approximation methods and numerical treatment of dynamical systems |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
Keywords:
astronomy and astrophysics; mathematical physics; general relativity; relativistic positioning systems; numerical methodsReferences:
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