We developed a numerical method[-70pt] to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders’ algorithm. By using the new method, we show the rotation curves of some non-trivial discs: (i) truncated power-law discs, (ii) discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we present a couple of model fittings to the observed rotation curve of M33: (i) the standard deconvolution by assuming a spherical distributin of the dark matter and (ii) a direct fit of infinitely-thin disc mass with a double power-law distribution of the surface mass density.

The triennial meeting of Commission 4 was attended by 16 people. All of the presentations from the meeting are provided on the commission website at http://www.iaucom4.org/c4docs.html, so this report provides only summaries.

The Commission 4 Organizing Committee began its work for the 2009-2012 triennium by revising the commission's terms of reference, which serve as our “mission statement.” The new terms of reference are:

1. (a) Maintain cooperation and collaboration between the national offices providing ephemerides, prediction of phenomena, astronomical reference data, and navigational almanacs.

2. (b) Encourage agreement on the bases (reference systems, time scales, models, and constants) of astronomical ephemerides and reference data in the various countries. Promote improvements to the usability and accuracy of astronomical ephemerides, and provide information comparing computational methods, models, and results to ensure the accuracy of data provided.

3. (c) Maintain databases, available on the Internet to the national ephemeris offices and qualified researchers, containing observations of all types on which the ephemerides are based. Promote the continued importance of observations needed to improve the ephemerides, and encourage prompt availability of these observations, especially those from space missions, to the science community.

4. (d) Encourage the development of software and web sites that provide astronomical ephemerides, prediction of phenomena, and astronomical reference data to the scientific community and public.

5. Promote the development of explanatory material that fosters better understanding of the use and bases of ephemerides and related data.

At the 2006 General Assembly of the International Astronomical Union (IAU), a proposal was adopted to form the Working Group (WG) for Numerical Standards of Fundamental Astronomy (NSFA). The goal of the WG is to update the “IAU Current Best Estimates" conforming with IAU Resolutions, the International Earth Rotation and Reference Systems Service (IERS) Conventions, and the Système International d'Unités (SI). The need for changes to the numerical standards are due mainly to the adoption of a new precession model, the redefinition of Barycentric Dynamical Time (TDB), and the significant improvement of the accuracy of recent estimates for a number of constants. The work from the first triennium culminated in the acceptance at the 2009 IAU General Assembly of Resolution B2 which adopted the NSFA list of Current Best Estimates as the IAU (2009) System of Astronomical Constants.

As in the past, the primary activity of the IAU Working Group on Cartographic Coordinates and Rotational Elements has been to prepare and publish a triennial (“2009”) report containing current recommendations for models for Solar System bodies (Archinal et al. (2011a)). The authors are B. A. Archinal, M. F. A'Hearn, E. Bowell, A. Conrad, G. J. Consolmagno, R. Courtin, T. Fukushima, D. Hestroffer, J. L. Hilton, G. A. Krasinsky, G. Neumann, J. Oberst, P. K. Seidelmann, P. Stooke, D. J. Tholen, P. C. Thomas, and I. P. Williams. An erratum to the “2006” and “2009” reports has also been published (Archinal et al. (2011b)). Below we briefly summarize the contents of the 2009 report, a plan to consider requests for new recommendations more often than every three years, three general recommendations by the WG to the planetary community, other WG activities, and plans for our next report.

The IAU Commission 52 “Relativity in Fundamental Astronomy” (RIFA) has been established during the 26th General Assembly of the IAU (Prague, 2006) to centralize the efforts in the field of Applied Relativity and to provide an official forum for corresponding discussions.

Dr. George Kaplan, the current Vice-President of the Commission was nominated to be the new President. Dr. Catherine Hohenkerk was elected to be the next Vice-President of the Commission. As for the Membership of the Organizing Committee, Dr. Vondrak stepped down and Drs William Folkner of JPL and Steve Bell of HMNAO have been added. In the below, we present summaries of the reports from various institutions presented at the business session.

There were four 1.5-hour sessions of Division I business meetings during the XXVIIth IAU General Assembly. The first three were devoted to the reports of Commissions, Working Groups and services associated with the Division, discussion about plans for the next triennium and future structure of the Division. Scientific presentations on the future space astrometric mission Gaia were made at the fourth session.

Time ephemeris is the location-independent part of the transformation formula relating two time coordinates such as TCB and TCG (Fukushima 2009). It is computed from the corresponding (space) ephemerides providing the relative motion of two spatial coordinate origins such as the motion of geocenter relative to the solar system barycenter. The time ephemerides are inevitably needed in conducting precise four dimensional coordinate transformations among various spacetime coordinate systems such as the GCRS and BCRS (Soffel et al. 2003). Also, by means of the time average operation, they are used in determining the information on scale conversion between the pair of coordinate systems, especially the difference of the general relativistic scale factor from unity such as LC. In 1995, we presented the first numerically-integrated time ephemeris, TE245, from JPL's planetary ephemeris DE245 (Fukushima 1995). It gave an estimate of LC as 1.4808268457(10) × 10−8, which was incorrect by around 2 × 10−16. This was caused by taking the wrong sign of the post-Newtonian contribution in the final summation. Four years later, we updated TE245 to TE405 associated with DE405 (Irwin and Fukushima 1999). This time the renewed vale of LC is 1.48082686741(200) × 10−8 Another four years later, by using a precise technique of time average, we improved the estimate of Newtonian part of LC for TE405 as 1.4808268559(6) × 10−8 (Harada and Fukushima 2003). This leads to the value of LC as LC = 1.48082686732(110) × 10−8. If we combine this with the constant defining the mean rate of TCG-TT, LG = 6.969290134 × 10−10 (IAU 2001), we estimate the numerical value of another general relativistic scale factor LB = 1.55051976763(110) × 10−8, which has the meaning of the mean rate of TCB-TT. The main reasons of the uncertainties are the truncation effect in time average and the uncertainty of asteroids' perturbation. As a compact realization of the time ephemeris, we prepared HF2002, a Fortran routine to compute approximate harmonic series of TE405 with the RMS error of 0.446 ns for the period 1600 to 2200 (Harada and Fukushima 2003). It is included in the IERS Convention 2003 (McCarthy and Petit 2003) and available from the IERS web site; http://tai.bipm.org/iers/conv2003/conv2003_c10.html.

Time ephemeris is the location-independent part of the transformation formula relating two time coordinates such as TCB and TCG (Fukushima 1995). It is computed from the corresponding (space) ephemerides providing the relative motion of two spatial coordinate origins such as the motion of geocenter relative to the solar system barycenter. The time ephemerides are inevitably needed in conducting precise four dimensional coordinate transformations among various spacetime coordinate systems such as the GCRS and BCRS (Soffel et al. 2003). Also, by means of the time average operation, they are used in determining the information on scale conversion between the pair of coordinate systems, especially the difference of the general relativistic scale factor from unity such as LC. In 1995, we presented the first numerically-integrated time ephemeris, TE245, from JPL's planetary ephemeris DE245 (Fukushima 1995). It gave an estimate of LC as 1.4808268457(10) × 10−8, which was incorrect by around 2 × 10−16. This was caused by taking the wrong sign of the post-Newtonian contribution in the final summation. Four years later, we updated TE245 to TE405 associated with DE405 (Irwin and Fukushima 1999). This time the renewed vale of LC is 1.48082686741(200) × 10−8 Another four years later, by using a precise technique of time average, we improved the estimate of Newtonian part of LC for TE405 as 1.4808268559(6) × 10−8 (Harada and Fukushima 2003). This leads to the value of LC as LC = 1.48082686732(110) × 10−8. If we combine this with the constant defining the mean rate of TCG-TT, LG = 6.969290134 × 10−10 (IAU 2001), we estimate the numerical value of another general relativistic scale factor LB = 1.55051976763(110) × 10−8, which has the meaning of the mean rate of TCB-TT. The main reasons of the uncertainties are the truncation effect in time average and the uncertainty of asteroids' perturbation. The former is a natural limitation caused by the finite length of numerical planetary ephemerides and the latter is due to the uncertainty of masses of some heavy asteroids. As a compact realization of the time ephemeris, we prepared HF2002, a Fortran routine to compute approximate harmonic series of TE405 with the RMS error of 0.446 ns for the period 1600 to 2200 (Harada and Fukushima 2003). It is included in the IERS Convention 2003 (McCarthy and Petit 2003) and available from the IERS web site; http://tai.bipm.org/iers/conv2003/conv2003_c10.html.

The IAU Working Group (WG) on Numerical Standards for Fundamental Astronomy has been tasked with updating the IAU Current Best Estimates (CBEs), conforming with the IAU Resolutions, IERS Conventions and Système International d'Unités whenever possible. As part of its effort to achieve this, the WG is working in close cooperation with IAU Commissions 4 and 52, the IERS, and the BIPM Consultative Committee for Units.

The Working Group Virtual Observatories, Data Centers, and Networks was established under Commission 5 at the Prague General Assembly in 2006. The purpose of the WG is to provide IAU oversight of the activities of the International Virtual Observatory Alliance (IVOA, <www.ivoa.net/>), to encourage data centers and other data providers to archive and publish data according to IVOA standards, and to help assure that astronomical research facilities are electronically linked with current network technologies. The WG coordinates activities closely with the WG-FITS, as the IVOA uses FITS as its primary format for binary data exchange, and the WG on Astronomical Data.

The tremendous progress in technology which we have witnessed during the last 30 years has led to enormous improvements of observational accuracy in all disciplines of fundamental astronomy. Relativity has been becoming increasingly important for modeling and interpretation of high accuracy astronomical observations during at least these 30 years. It is clear that for current accuracy requirements astronomical problems have to be formulated within the framework of General Relativity Theory. Many high-precision astronomical techniques have already required the application of relativistic effects, which are several orders of magnitude larger than the technical accuracy of observations. In order to interpret the results of such observations, one has to construct involved relativistic models. Many current and planned observational projects can not achieve their goals if relativity is not taken into account properly. The future projects will require the introduction of higher-order relativistic effects. To make the relativistic models consistent with each other for different observational techniques, to formulate them in the simplest possible way for a given accuracy, and to formulate them in a language understandable for astronomers and engineers who have little knowledge of relativity are the challenges of a multidisciplinary research field called Applied Relativity.

JPL planetary ephemeris development has been very active assimilating measurements from current planetary missions and supporting future missions. The NASA Mars Science Laboratory (MSL) mission with launch in 2009 requires knowledge of the Earth and Mars ephemerides with 30m accuracy. By comparison, the accuracy of the Mars ephemeris in the widely used DE405 ephemeris was about 3 km. Meeting the MSL needs requires an ongoing program of range and very-long baseline interferometry measurements of Mars orbiting spacecraft. The JPL ephemeris DE421 was released three months before the landing of the Phoenix mission on Mars, and has met the 300m requirement. Continued measurements are planned to support the MSL landing.

Division I provides a focus for astronomers studying a wide range of problems related to fundamental physical phenomena such as time, the inertial reference frame, positions and proper motions of celestial objects and precise dynamical computation of the motions of bodies in stellar or planetary systems in the Universe.

Division I provides a focus for astronomers studying a wide range of problems related to fundamental physical phenomena such as time, the inertial reference frame, positions and proper motions of celestial objects, and precise dynamical computation of the motions of bodies in stellar or planetary systems in the Universe.

In his address to the Commission, the outgoing president A. Milani explained what he considers have been done well in the past triennium, what has been done only in part, and what has not been done at all. Among the things in which the performance was rated good, he mentioned the successful sponsorship and/or co-sponsorship of four meetings (IAU Colloquia 196 and 197, and Symposia 229 and 236) which have been held in the previous period, as well as of the Symposium on exoplanets to be held next year in China. The only failure in this respect was the proposed meeting in India, which failed already at the proposal definition stage. Also, Milani expressed his satisfaction with the triennial report which has been compiled for the occasion, and his gratitude to the collaborating authors.

The IAU Division I Working Group on Nomenclature for Fundamental Astronomy (NFA) was established by the IAU XXV General Assembly with the task of providing proposals for new nomenclature associated with the implementation of the IAU XXIV GA resolutions (2000) and to make related educational efforts for bringing the issue to the notice of scientists in the community.

This business meeting was held from 16:00 to 17:30. Toshio Fukushima and George Kaplan were welcomed as the next president and vice-president, respectively. The following, in no particular order, are the summary reports from the various offices. The full versions will be made available on the Commission 4 website at <http://iau-comm4.jpl.nasa.gov/>.