Parallax distortion by the weak microlensing effect

Mon. Not. R. Astron. Soc. 323, 952±964 (2001) Parallax distortion by the weak microlensing effect M. V. Sazhin,w V. E. Zharov and T. A. Kalinina Sternberg Astronomical Institute, Moscow 119899, Russia Accepted 2000 December 11. Received 2000 November 9; in original form 2000 June 1 A B S T R AC T Key words: gravitational lensing ± reference systems ± stars: distances ± galaxies: active ± quasars: general. 1 INTRODUCTION Very long baseline interferometry (VLBI) has achieved a precision level of position measurements of tens of microarcseconds (Ma et al. 1998; Gontier et al. 1999). VLBI may soon achieve an accuracy of several microarcseconds, or the fundamental limit of accuracy of the position measurements being determined by the non-stationary curvature of space-time in our Galaxy (Sazhin 1996; Sazhin et al. 1998). Besides, the creation of space interferometers much exceeding the Earth's diameter is in sight (Andreyanov & Kardashev 1981; Andreyanov et al. 1986; Kardashev 1986) A 10±100 times increase in the interferometer baseline is likely to allow a precision of position measurements of the order of one microarcsecond (,1 mas), or even a hundred nanoseconds of arc (,100 nas) to be achieved. In optical ground-based astronomy, until the last decade, the positional accuracy amounted to ,0.1 arcsec, which was far worse than in radioastronomy. The great success of the space project Hipparcos lies in the achievement of a precision of ,1 milliarcsecond (mas) in optical astronomy for measuring stellar coordinates and parallaxes. Astronomers hope to develop this success in the space experiments being planned (Projects GAIA, SIM, DARWIN, FAME and DIVA). A precision of , 1±10 mas is planned. High-precision positional measurements would allow the stellar distance scale to be increased from 1 kpc to several tens or hundreds of kpc. The inclusion of general relativistic effects has become a necessary part of the observation reduction procedure when the accuracy of observations is close to the value of ,1 mas. The reduction procedure involves gravitational effects induced by the Sun and planets of the Solar system (IERS 1996). These w E-mail: bodies induce a non-stationary curvature of space-time in the Solar system, making a ray from a celestial source move along a curved trajectory. The positions, velocities and masses of the Sun and planets are known with a high degree of accuracy, which permits a precise estimate of the gravitational effects in positional measurements. As the accuracy increases, the astronomers will undoubtedly encounter new phenomena. One of them, the non-stationarity of space-time, is due to the motion of visible stars and invisible bodies in our Galaxy. A list of phenomena which will be important in microarcsecond and submicroarcsecond astrometry is discussed by Kopeikin & Gwinn (2000). The non-stationary curvature induced by the Sun and planets is a deterministic process. The non-stationary curvature created by moving stars in our Galaxy is a stochastic process, since the distances to most (in particular, invisible) stars as well as their masses are unknown. Thus it is impossible to reduce the observations with the same degree of definiteness as in the Solar system. For the observer a distant source will execute a stochastic motion under the action of this process around an average position which is the true position of the source in the sky. The value of this oscillation is of the order of 1 mas, and the characteristic time of motion is in the range from tens to hundreds of years. Since the reduction procedure becomes impossible, a value of ,1 mas has been called the fundamental limit of positional measurement accuracy, and the effect itself called the weak microlensing effect. In previous papers the weak microlensing effect has been discussed for a single observer. However, very important astronomical information is given by observations made simultaneously by two observers from different points of space, or one observer at different moments of time. Such observations allow the parallax of a celestial source to be measured and the distance q 2001 RAS Parallax measurements allow distances to celestial objects to be determined. Together with measurements of their position on the celestial sphere, they give a full three-dimensional picture of the location of the objects relative to the observer. The distortion of the parallax value of a distant source affected by weak microlensing is considered. This means that the weak microlensing leads to distortion of the distance scale. The gravitational deflection causes a retrograde apparent motion of the image, which is revealed as a negative parallax. It is shown that the distortions may significantly change the parallax values when they amount to several microseconds of arc. In particular, at this level many measured values of parallaxes should be negative. Parallax distortion by weak microlensing q 2001 RAS, MNRAS 323, 952±964 the parallax value, a `jump' of the image may occur from one side of the lens to the other. This effect will result in a considerable distortion of the source position and parallax. Of course, the weak microlensing effect observed from the barycentre of the Solar system is not added to the source parallax motion. An attempt to reduce the position of the source, observed from the Earth, to the barycentre following the standard equations will lead to a mistake, since in this case the parallactic displacement value itself will depend on the microlensing effect. In the problem related to the distortion of measured parallaxes there naturally appear some quantities having different orders of magnitude. We shall consider that the unit vectors indicating the direction to the light source are the leading term of zeroth order, and the quantities containing the parallax of the source as a factor are of first order. We shall also regard the proper shift of the light source or lens over the time of observation as first-order terms. These values are products of the proper motion of the object. We shall consider the angular distance between the lens and the light source to be a small parameter as well. Besides these small parameters, one more small parameter arises in the problem which is not related to the geometry of the problem under consideration, but is related to general relativistic effects. This is the squared ratio of two values. The first is the angular size of the Einstein cone, and the second value is the angular distance between the source and lens. As is seen from the foregoing, the small parameters take different values. Thus, for example, the lens parallax may be 10 mas, whereas the parallax of an extragalactic source may be 10 nas, which 106 times less. Nevertheless, the terms containing the lens parallax squared are less than the terms proportional to the first power of the source parallax. Hence in the equations we shall retain linear terms with respect to small paramete (...truncated)