对等级树代码的一个应用

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case just 1M® stars where used, whereas for the first two cases a Salpeter mass function ( f ( m ) d m = m-2"35dm) with upper cutoff at 1 M® and lower cutoff at 0.1 M® was used. The lightcurves along the marked tracks are displayed in figures 3a,b and c. T h e results of raising a up to 0.95 can be summarized as follows: - total n u m b e r of (micro-.) caustics increases rapidly, overlapping of caustics gets more and more i m p o r t a n t "clustering tendency" of caustics (already known from lower values of a, see e.g. SW) increases dramatically - time scale of fixed variations (e.g. A m > 0.5mag) grows - lightcurves get smoother and flatter (for fixed source size); they do not show the U-shaped double peaks, that are well known for low surface mass densities - average and peak amplitudes of variations decrease (for fixed source size) - the typical slopes for our smallest sources decrease from about 2 mag within 3 normalized units (cr = 0.8) to about 1 mag within 8 time units (a = 0.95) individual caustics can not be resolved any more in the lightcurve for our source sat Macrocaustics We modelled a realistic macrocaustic of an isothermal sphere with a finite core and put part of the m a t t e r into stars. Without stars one gets a smooth decrease in magnification inversely proportional to the square root of the distance d between source and (macro-) caustic: A oc d -°'~ (see figs. 2a,3d). Figs. 2b,c show the "grainy" situation (corresponding lightcurves in figs. 3e,f) for different values of the scaling p a r a m e t e r s0/rio, the ratio of the deflection angles of the isothermal sphere and a single star:
Microlensing
calculations with a hierarchical New Results
tree
code:
Joachim Wambsganss Max-Planck-Institut ffir Astrophysik 8046 Garching, West-Germany (bitnet: J K W at DGAIPP1S) 1. Introduction Microlensing deals with the effects of individual compact objects inside the lensing galaxy on the (macro-) image of a lensed background quasar (for reviews see, e.g., Schneider 1990 or Watson 1989). Due to the enormous amount of computing time, microlensing calculations have been performed only for a comparatively small number of lensing stars and for moderate values of the (normalized) surface mass density a = E/Ecr, (see e.g. Schneider and Weiss 1987 (SW); Kayser, Refsdal and Stabell 1986 (KRS); Paczynski 1986; Young 1981) where the critical density is given by ~or = (c2/4~rG)(Ds/DLDLs) (here c is the velocity of light, G is the gravitational constant and DL,Ds, DLs are the angular diameter distances observer-lens, observer-source, lenssource, resp.). For QSO 2237+0305 this value is ~cr -~ 2.5g cm -2 _~ 1.2 * 104M®pc -2. With a new code based on the hierarchical tree method (Barnes and Hut 1986) the time consuming part of the computations, the calculation of the deflection angles of each star, can be reduced in an efficient and elegant way (Wambsganss et ai.1989, 1990), so that a much larger number of stars can be included, allowing, e.g., for a mass spectrum of the lensing stars or surface mass densities closer to the critical one. Here we present new results of microlensing calculations obtained with this hierarchical tree code in four different regimes: - surface mass densities a --~ 1 - microlensing at macrocaustics - effects of a mass spectrum with different lower cutoff masses (for constant a) - macroimages for a --* 1. We used the inverse ray shooting method (described, e.g., in SW, KRS) and obtained two-dimensional distributions of the magnification factors for extended sources. The side lengths of these maps are 20~0, where ~0 = X/(4GM®/c2)(DLsDL/Ds) is the Einstein radius of a 1M® star. The resolution is 10242 pixels for figs. la,b and 5002 pixels for figs. lc-f and 2a-c, so that one pixel length corresponds to about 0.02 ~0 or 0.04 ~0. The lightcurves are calculated along the three marked tracks for sources with Gaussian profiles of halfwidths 1 (thin line), 4 (medium) and 16 (thick) pixellengths. In the case of QSO 2237+0305 the smMlest effective source size is of order 10 -4 pc. The time unit is t = ~o(Ds/DLs)(1 + ZL)/V6oo, where ZL is the lens redshift and v600 is the lens velocity (relative to observer/source) in units of 600 km/sec. For QSO 2237+0305 one time unit is about 9 years. 2. Microlensing for surface mass densities a --* 1.0 In figs. la,b,c we show the magnification patterns for cr = 0.8, 0.9 and 0.95. The numbers of individual lenses used are 41820, 184723, 172616, respectively. For the latter