K. El Dayem, N. Aimar, A. Berdeu, J.-P. Berger, G. Bourdarot, W. Brandner, V. Cardoso, C. Correia, R. Davies, D. Defrère, A. Drescher, A. Eckart, F. Eisenhauer, M. Fabricius, H. Feuchtgruber, S. Flesch, A. Foschi, Q. Fournier, N.M. Förster Schreiber, P. Garcia, R. Garcia Lopez, R. Genzel, S. Gillessen, F. Gonté, X. Haubois, M. Houlle, S. F. Honig, S. Joharle, J. Kammerer, A. Kaufer, P. Kervella, L. Kreidberg, L. Labadie, S. Lacour, O. Lai, R. Laugier, J.-B. Le Bouquin, J. Leftley, R. Li, B. Lopez, D. Lutz, F. Mang, F. Millour, M. Montargès, N. Morujão, A. Mérand, H. Nowacki, M. Nowak, J. Osorno, T. Ott, C. Paladini, S. Pappert, T. Paumard, K. Perraut, G. Perrin, R. Petrov, P.O. Petrucci, N. Pourre, S. Rabien, D. C. Ribeiro, S. Robbe-Dubois, M. Sadun Bordoni, D. Santos, J. Sauter, J. Scigliuto, J. Shangguan, T. Shimizu, F. Soulez, C. Straubmeier, E. Sturm, M. Subroweit, C. Sykes, J. Sanchez Bermudez, L.J. Tacconi, P. Thevenet, I. Urso, F. Vincent, J. Woillez, G. Zins
High-precision astrometric observations at the microarcsecond level require a consistent relativistic framework for modelling both the dynamics of test particles and the propagation of light in gravitational fields. We aim to formulate the post-Newtonian (PN) expansion in the test body limit in a manner that yields simple and broadly applicable expressions for the motion of massive and massless particles in generic, spherically symmetric spacetimes. In particular, we intend to formulate it such that the formalism suits the modelling of observations of stars orbiting the Galactic centre (GC). We employed the PN expansion scheme to derive equations of motion for test particles and applied a perturbative solution method to solve the boundary value problem of identifying the geodesic connecting a given emitter--observer pair. Results are validated against numerical integration of the photon geodesic equation. We derive the PN equations of motion for test particles in a generic, spherically symmetric spacetime. We derive analytical expressions for the apparent position of stars as measured by an observer on Earth and show the importance of this formalism for the GC science case. We compare our results with other approaches taken in the literature and discuss its implications for the modelling of orbital motion around Sgr A*. The proposed framework is well suited for the analysis of astrometric observations of stars orbiting the GC with microarcsecond accuracy. In this context, measurements performed with the GRAVITY+ instrument at the Very Large Telescope Interferometer can be used to place constraints on metric theories of gravity through the combined modelling of stellar dynamics and relativistic light propagation near Sgr A*.