An Autonomous Relativistic Positioning System Uroš Kostić Martin Horvat Andreja Gomboc University...
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An Autonomous Relativistic Positioning System Uroš Kostić Martin Horvat Andreja Gomboc University of Ljubljana, Faculty of Mathematics and Physics
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User Satellite tracking from Earth Constants of motion Coordinate transformation emission coordinates (Coll 2002, Rovelli 2002) (from Coll & Pozzo, CQG 23, 2006)
Transcript of An Autonomous Relativistic Positioning System Uroš Kostić Martin Horvat Andreja Gomboc University...
An Autonomous Relativistic Positioning System
Uroš KostićMartin Horvat
Andreja Gomboc
University of Ljubljana, Faculty of Mathematics and Physics
User
Satellite tracking
from Earth
Constantsof motion
Coordinatetransformation
4321 t,t,t,t oooo z,y,x,t
User
Satellite tracking
from Earth
Constantsof motion
Coordinatetransformation
oooo z,y,x,t 43,2,1, ττττ
1τ2τ
3τ
emission coordinates (Coll 2002, Rovelli 2002)
(from Coll & Pozzo, CQG 23, 2006)
32,1, τττ
User
Satellite tracking
from Earth
Constantsof motion
Coordinatetransformation
43,2,1, ττττ oooo z,y,x,t
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ 43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
ISL
User
Constantsof motion
Coordinatetransformation
43,2,1, ττττ oooo z,y,x,t
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ 43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
ISL
Data reduction+
Clock correction
ABC – Autonomous Basis of Coordinates
University of Ljubljana, ESA Advanced Concepts Team(Ariadna study 09/1301, 09/1301 CCN, PECS project Relativistic GNSS)
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ 43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
ISL
perturbed space-timeEarth multipolesocean tides, solid tidesMoon, Sun, Jupiter, Venusrelativity, Kerr
KerrrelativitytidesplanetsGM a+a+a+a+a+a=a
62
Kerrμν
tidesμν
planetsμνμνμνμν h+h+h+h+g=g 620
Schwarzschild Regge-Wheeler-Zerilli multipole expansion
User
Constantsof motion
Coordinatetransformation
43,2,1, ττττ oooo z,y,x,t
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττData reduction
+Clock correction
Positioning algorithm in GR
3110~ t s=T 0.042510~
zy,x,
User
Constantsof motion
Coordinatetransformation
43,2,1, ττττ oooo z,y,x,t
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττ
43,2,1, ττττData reduction
+Clock correction
ABC – Autonomous Basis of Coordinates
ABC – Autonomous Basis of Coordinates
)()( 12 ttT f
ABC – Autonomous Basis of Coordinates
212 ])[(]),[(|][(])[(]),[(|][(
])[],[(]),[],[(|][],[(
))0(),0((
kPkQktkPkQkt
kkPkkQkkT
PQS
ii
ii
ki
if
ii
212 ))(),(|())(),(|(
)),(),,(|,())0(),0((
ii
ii
ii
fii
PQtPQt
PQTPQS
12D minimization
ABC – Autonomous Basis of Coordinates
12D minimization
ABC – Autonomous Basis of Coordinates
221000 ii P,Q
12D minimization
ABC – Degeneracies
no perturbations
Δ ι=±0.01
ABC – Degeneracies
Earth multipolesSolid tidesOcean tides
Δ ι=±0.01
t
x
y
ABC – Autonomous Basis of Coordinates
ABC – Refinement of gravitational parameters
5
2,0
2,0 107 =MΔM
23
2,0
2,0 107 MΔM
ABC – Refinement of gravitational parameters
ABC – Refinement of gravitational parameters
Robustness of recovering constants of motion with respect to noise in the data
Consistency of description with redundant number of satellites
Possibility to use the constellation as a clock with long term stability
Its realization does not rely on observations from Earth No entanglement with Earth internal dynamics No Earth stations for maintaining of the frame
Stability and accuracy Based on well-known satellite dynamics Satellite orbits are very stable in time, and can be
accurately described Applications in science
geophysics, relativistic gravitation and reference frames determine/refine values of gravitational parameters
ABC – Autonomous Basis of Coordinates
