J Geod (2006) 80: 137–149 DOI 10.1007/s00190-006-0035-y ORIGINAL ARTICLE M. Sekido · T. Fukushima A VLBI delay model for radio sources at a finite distance Received: 22 June 2005 / Accepted: 22 February 2006 / Published online: 28 April 2006 © Springer-Verlag 2006 Abstract A relativistic delay model for Earth-based very long baseline interferometry (VLBI) observation of sources at finite distances is derived. The model directly provides the VLBI delay in the scale of terrestrial time. The effect of the curved wave front is represented by using a pseudo source vector K = (R 1 + R 2 )/( R 1 + R 2 ), and the variation of the baseline vector due to the difference of arrival time is taken into account up to the second-order by using Halley’s method. The precision of the new VLBI delay model is 1 ps for all radio sources above 100km altitude from the Earth’s surface in Earth-based VLBI observation. Simple correction terms (parallax effect) are obtained, which can also adopt the consensus model (e.g. International Earth Rotation and Ref- erence Frames Service conventions) to finite-distance radio source at R > 10 pc with the same precision. The new model may enable estimation of distance to the radio source directly with VLBI delay data. Keywords Very Long Baseline Interferometry · General Theory of Relativity · Spacecraft Navigation 1 Introduction Very long baseline interferometry (VLBI) is a powerful tool for astronomy and space geodesy, boasting the highest angu- lar resolution. In fact, the International Celestial Reference Frame is realized by VLBI observation of extragalactic ob- jects and its overall precision was reported as 250 μas for the M. Sekido (B ) Kashima Space Research Center, National Institute of Information and Communications Technology, 893-1 Hirai, Kashima, Ibaraki 314-0012, Japan E-mail: sekido@nict.go.jp, Tel.: +81-299-84-7146 Fax: +81-299-84-7159 T. Fukushima National Astronomical Observatory, 2-21-1 Osawa Mitaka, Tokyo 181-8588, Japan E-mail: Toshio.Fukushima@nao.ac.jp position of individual sources and 20 μas for the axis orien- tation of the reference frame (Charlot 2004). Furthermore, a phase-reference-dedicated VLBI project, VLBI Exploration of Radio Astrometry (VERA; Kobayashi 2004), has started and its target precision is set at the 10 μas level. On the other hand, the VLBI technique has been used for spacecraft navigation as its engineering application (Border et al. 1982). Recently, the technique was used for the angu- lar observation of some closer targets such as NOZOMI, a Japanese Mars exploration mission (Yamamoto and Tsuruda 1998), and Cassini-Huygens, which is a joint European Space Agency (ESA)–National Astronautics and Space Adminis- tration (NASA) probe to Saturn’s satellite Titan (Lebreton and Matson 2002). Soon VLBI will be applied to SELENE, a Japanese Lunar gravimetry mission (Matsumoto, et al. 1999; Heki et al. 1999). To utilize the full power of VLBI, the establishment of precision VLBI delay model is essential. For sources at prac- tically infinite distances, such as quasars, various models have been proposed (Hellings 1986; Shahid-Saless and Hellings 1991; Zhu and Groten 1988; Soffel et al. 1991). They were unified into the so-called consensus model (Eubanks 1991). It has a 1 ps precision for Earth-based VLBI observation of extra-galactic radio sources. Resolution B1.3 of the Interna- tional Astronomical Union (IAU) XXIV General Assembly (2000) recommends use of the Barycentric Celestial Ref- erence System (BCRS) and Geocentric Celestial Reference System (GCRS) for the barycentric and geocentric reference systems in the framework of general relativity. The consensus model has reviewed and adapted to the substantial frame- work of the IAU-recommended BCRS, and has been used in the International Earth Rotation and Reference Systems Service (IERS) conventions (McCarthy and Petit 2003) and in the world VLBI community as the standard VLBI delay model. Unfortunately, this model was designed for extra-galac- tic radio sources and was derived based on the plane-wave approximation by ignoring the effect of source’s distance (Eubanks 1991). Therefore, it is inaccurate if the radio sources are at finite distance, e.g. pulsars (maser sources in our