The astrometric effect of microlensing should be considered simultaneously with the photometric effect of the same microlensing event. This was pointed out in a paper by hoeg.novi, hereafter called HNP. Three photometric observations were shown to be in principle sufficient to determine three parameters of a ``fly-by'' event: The moment of closest encounter between the dark body and the line of sight to the star; the characteristic duration, , of the event; and the impact parameter, i.e., the ratio of the distance to the line of sight and the Einstein radius. When these three parameters are known it is in principle sufficient to obtain three one-dimensional astrometric observations of the same event in order to derive the proper motion, the distance and the mass, m, of the dark body. This assumes that the proper motion and the distance of the background star are accurately known, as can indeed be assumed. --- For the sake of redundancy and a better signal-to-noise ratio it is of course useful to have a much larger number of observations than just three photometric and three astrometric.
HNP predicts the size of the photometric and astrometric effects for the ROEMER satellite project proposed by lindegren.roemer and hoeg.lindegren.
The photometric effect is more easily detected than the astrometric. HNP expects more than 300 light amplifications with a signal-to-noise ratio of at least 10 during a 5 year ROEMER mission. This expected number of significant amplifications depends on the total mass of the Galactic halo of dark bodies, but not on the distribution function of the masses of the individual dark bodies. If they all have masses of, e.g., these amplified photometric observations would belong to about 25 fly-by events. The dark body masses are, however, at present assumed to be about 0.1 which would mean that the 300 amplifications would be distributed on a larger number of events. The background stars are typically 16th magnitude at distances of 6 kpc. With GAIA, as described by LP, the number of such amplifications will be at least an order of magnitude larger. Such photometric observations would be related to MACHOs much closer to the sun than MACHO observations so far and can therefore provide information on the local space density of MACHOs, and of the density in all directions from the sun.
The expected time scales of events are given in HNP-Table 1; e.g., for stars of V= 16 mag we get = 8 or 25 days if all halo dark bodies have the mass m= 0.1 or 1.0 , respectively.
The detection of microlensing from a scanning astrometric satellite is characterized by an uneven distribution in time of the observations, due to the strict scanning law. This is very different from the rather uniform distribution of ground based MACHO observations. A discussion of this subject is given in HNP, but it can be said that the uneven distribution offers advantages and disadvantages; many possible time scales will be sampled, but the sampling will be incomplete.
The astrometric measurements with ROEMER and GAIA are one-dimensional, as assumed above. HNP shows that a satellite giving astrometric observational errors 20 times smaller than ROEMER -- and those of GAIA will be even smaller than that -- would give a significant determination of the proper motion of a MACHO of passing close to a 16th magnitude star. Proper motion and distance give the tangential velocity, and thus a clue to the space velocity, otherwise inferred only from models of the halo.
HNP considered the prospects for astrometric observations of MACHOs to be meagre. But the paper was written in May 1994, and since then the prospects for space astrometry have improved drastically through new technical ideas and owing to initiatives by the European Space Agency.