`ISRN Astronomy and AstrophysicsVolume 2012 (2012), Article ID 906951, 21 pageshttp://dx.doi.org/10.5402/2012/906951`
Review Article

## Analytical Methods in Gravitational Microlensing

Astronomical Observatory, Taras Shevchenko National University of Kyiv, Observatorna Street 3, Kiev 04053, Ukraine

Received 30 April 2012; Accepted 17 June 2012

Academic Editors: J. Gallimore and M. Sazhin

Copyright © 2012 V. I. Zhdanov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. P. Schneider, J. Ehlers, and E. E. Falko, Gravitational Lenses, Springer, New York, NY, USA, 1992.
2. A. F. Zakharov, Gravitational Lenses and Microlenses, Yanus-K, Moscow, Russia, 1997.
3. A. O. Petters, H. Levine, and J. Wambsganss, Singularity Theory and Gravitational Lensing, Birkhäuser, Boston, Mass, USA, 2001.
4. R. Gil-Merino, J. González-Cadelo, L. J. Goicoechea, V. N. Shalyapin, and G. F. Lewis, “Is there a caustic crossing in the lensed quasar Q2237$+$ 0305 observational data record?” Monthly Notices of the Royal Astronomical Society, vol. 371, no. 3, pp. 1478–1482, 2006.
5. A. N. Alexandrov and V. I. Zhdanov, “Asymptotic expansions and amplification of a gravitational lens near a fold caustic,” Monthly Notices of the Royal Astronomical Society, vol. 417, pp. 541–554, 2011.
6. A. N. Alexandrov, V. I. Zhdanov, and E. V. Fedorova, “Asymptotic formulas for the magnification of a gravitational lens system near a fold caustic,” Astronomy Letters, vol. 36, no. 5, pp. 329–337, 2010.
7. C. R. Keeton, B. S. Gaudi, and A. O. Petters, “Identifying lenses with small-scale structure. II. Fold lenses,” Astrophysical Journal, vol. 635, no. 1, pp. 35–59, 2005.
8. V. I. Zhdanov and V. V. Zhdanova, “Analytical relations for time-dependent statistical microlensing,” Astronomy & Astrophysics, vol. 299, pp. 321–325, 1995.
9. V. I. Zhdanov, “The general relativistic potential of astrometric studies at microarcsecond level,” in Astronomical and Astrophysical Objectives of Sub-Milliarcsecond Optical Astrometry, E. Hog and P. K. Seidelmann, Eds., pp. 295–300, Kluwer, Dordrecht, The Netherlands, 1995.
10. V. I. Zhdanov, E. V. Fedorova, and A. N. Alexandrov, “Gravitational dragging of distant source images caused by Galaxy stars,” Kinematika i Fizika Nebesnykh Tel, vol. 20, pp. 422–429, 2004 (Russian).
11. V. I. Zhdanov and S. A. Salata, “Motion of the image of a distant object microlensed by stars in a foreground galaxy,” Kinematics and Physics of Celestial Bodies, vol. 14, pp. 156–161, 1998.
12. V. I. Zhdanov, S. A. Salata, and E. V. Fedorova, “Background-field effects in astrometric microlensing,” Astronomy Letters, vol. 27, no. 9, pp. 562–567, 2001.
13. E. Fedorova, V. I. Zhdanov, and A. N. Alexandrov, “Motion of source image in Chang-Refsdal lens,” Journal of Physical Studies, vol. 4, pp. 465–468, 2002.
14. A. N. Alexandrov, V. M. Sliusar, and V. I. Zhdanov, “Caustic crossing events and source models in gravitational lens systems,” Ukrainian Journal of Physics, vol. 56, no. 4, pp. 389–400, 2011.
15. C. J. Fluke and R. L. Webster, “Investigating the geometry of quasars with microlensing,” Monthly Notices of the Royal Astronomical Society, vol. 302, no. 1, pp. 68–74, 1999.
16. V. N. Shalyapin, “Caustic crossing in the gravitational lens Q2237$+$0305,” Astronomy Letters, vol. 27, no. 3, pp. 150–155, 2001.
17. O. Pejcha and D. Heyrovský, “Extended-source effect and chromaticity in two-point-mass microlensing,” Astrophysical Journal Letters, vol. 690, no. 2, pp. 1772–1796, 2009.
18. A. N. Alexandrov, S. M. Koval, and V. I. Zhdanov, “Asymptotic relations for high magnification events in presence of the dark matter,” Visnyk Kyivskogo Universytetu, Astronomiya. In press.
19. A. N. Alexandrov, V. I. Zhdanov, and E. V. Fedorova, “Analytical relations for gravitational lens mapping in the vicinity of a critical curve,” Visnyk Kyivskogo Universytetu, Astronomiya, vol. 39-40, pp. 52–59, 2003 (Ukrainian).
20. P. Schneider and A. Weiss, “A gravitational lens origin for AGN-variability? Consequences of micro-lensing,” Astronomy & Astrophysics, vol. 171, pp. 49–65, 1987.
21. P. Schneider and A. Weiss, “The gravitational lens equation near cusps,” Astronomy & Astrophysics, vol. 260, pp. 1–13, 1992.
22. B. S. Gaudi and A. O. Petters, “Gravitational microlensing near caustics. II. Cusps,” Astrophysical Journal Letters, vol. 580, no. 1, pp. 468–489, 2002.
23. A. B. Congdon, C. R. Keeton, and C. E. Nordgren, “Analytic relations for magnifications and time delays in gravitational lenses with fold and cusp configurations,” Monthly Notices of the Royal Astronomical Society, vol. 389, no. 1, pp. 398–406, 2008.
24. J. Diemand, M. Kuhlen, P. Madau et al., “Clumps and streams in the local dark matter distribution,” Nature, vol. 454, no. 7205, pp. 735–738, 2008.
25. B. Grieger, R. Kayser, and S. Refsdal, “Gravitational micro-lensing as a clue to quasar structure,” Astronomy & Astrophysics, vol. 194, pp. 54–64, 1988.
26. P. R. Woźniak, C. Alard, A. Udalski et al., “The optical gravitational lensing experiment monitoring of QSO 2237$+$0305,” Astrophysical Journal, vol. 529, no. 1, pp. 88–92, 2000.
27. A. Udalski, M. K. Szymański, M. Kubiak et al., “The optical gravitational lensing experiment. OGLE-III long term monitoring of the gravitational lens QSO 2237$+$0305,” Acta Astronomica, vol. 56, no. 4, pp. 293–305, 2006.
28. D. Alcalde, E. Mediavilla, O. Moreau et al., “QSO 2237$+$ 0305 VR light curves from gravitational lenses international time project optical monitoring,” Astrophysical Journal Letters, vol. 572, no. 2, pp. 729–734, 2002.
29. J. S. B. Wyithe, R. L. Webster, and E. L. Turner, “A measurement of the transverse velocity of Q2237$+$0305,” Monthly Notices of the Royal Astronomical Society, vol. 309, no. 1, pp. 261–272, 1999.
30. J. S. B. Wyithe, R. L. Webster, and E. L. Turner, “Interpretation of the OGLE Q2237$+$0305 microlensing light curve (1997–1999),” Monthly Notices of the Royal Astronomical Society, vol. 318, no. 4, pp. 1120–1130, 2000.
31. J. S. B. Wyithe, R. L. Webster, E. L. Turner, and D. J. Mortlock, “A gravitational microlensing determination of continuum source size in Q2237$+$0305,” Monthly Notices of the Royal Astronomical Society, vol. 315, no. 1, pp. 62–68, 2000.
32. A. Yonehara, “Evidence for a source size of less than 2000 AU in Quasar 2237$+$0305,” Astrophysical Journal Letters, vol. 548, no. 2, pp. L127–L130, 2001.
33. V. N. Shalyapin, L. J. Goicoechea, D. Alcalde, E. Mediavilla, J. A. Muñoz, and R. Gil-Merino, “The nature and size of the optical continuum source in QSO 2237$+$0305,” Astrophysical Journal Letters, vol. 579, no. 1, pp. 127–135, 2002.
34. M. B. Bogdanov and A. M. Cherepashchuk, “Reconstruction of the strip brightness distribution in a quasar accretion disk from gravitational microlensing data,” Astronomy Reports, vol. 46, no. 8, pp. 626–633, 2002.
35. M. J. Mortonson, P. L. Schechter, and J. Wambsganss, “Size is everything: Universal features of quasar microlensing with extended sources,” Astrophysical Journal, vol. 628, no. 2, pp. 594–603, 2005.
36. L. J. Goicoechea, D. Alcalde, E. Mediavilla, and J. A. Muñoz, “Determination of the properties of the central engine in microlensed QSOs,” Astronomy and Astrophysics, vol. 397, no. 2, pp. 517–525, 2003.
37. C. S. Kochanek, “Quantitative interpretation of quasar microlensing light curves,” Astrophysical Journal, vol. 605, no. 1, pp. 58–77, 2004.
38. V. G. Vakulik, R. E. Schild, G. V. Smirnov, V. N. Dudinov, and V. S. Tsvetkova, “Q2237$+$0305 source structure and dimensions from light-curve simulation,” Monthly Notices of the Royal Astronomical Society, vol. 382, no. 2, pp. 819–825, 2007.
39. T. Anguita, R. W. Schmidt, E. L. Turner et al., “The multiple quasar Q2237$+$ 0305 under a microlensing caustic,” Astronomy and Astrophysics, vol. 480, no. 2, pp. 327–334, 2008.
40. S. Poindexter, N. Morgan, and C. S. Kochanek, “The spatial structure of an accretion disk,” Astrophysical Journal Letters, vol. 673, no. 1, pp. 34–38, 2008.
41. S. Poindexter and C. S. Kochanek, “The transverse peculiar velocity of the Q2237$+$0305 lens galaxy and the mean mass of its stars,” Astrophysical Journal Letters, vol. 712, no. 1, pp. 658–667, 2010.
42. S. Poindexter and C. S. Kochanek, “Microlensing evidence that a type 1 quasar is viewed face-on,” Astrophysical Journal Letters, vol. 712, no. 1, pp. 668–673, 2010.
43. I. M. Gel'fand and G. E. Shilov, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964.
44. M. Dominik, “Revealing stellar brightness profiles by means of microlensing fold caustics,” Monthly Notices of the Royal Astronomical Society, vol. 353, no. 1, pp. 118–132, 2004.
45. N. I. Shakura and R. A. Sunyaev, “Black holes in binary systems. Observational appearance,” Astronomy & Astrophysics, vol. 24, pp. 337–355, 1973.
46. H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, NY, USA, 1953.
47. R. Schmidt, R. L. Webster, and G. F. Lewis, “Weighing a galaxy bar in the lens Q2237 $+$0305,” Monthly Notices of the Royal Astronomical Society, vol. 295, no. 2, pp. 488–496, 1998.
48. X. Dai, G. Chartas, E. Agol, M. W. Bautz, and G. P. Garmire, “Chandra observations of QSO 2237$+$0305,” Astrophysical Journal Letters, vol. 589, no. 1, pp. 100–110, 2003.
49. E. V. Fedorova, V. I. Zhdanov, C. Vignali, and G. G. C. Palumbo, “Q2237$+$ 0305 in X-rays: spectra and variability with XMM-Newton,” Astronomy and Astrophysics, vol. 490, no. 3, pp. 989–994, 2008.
50. V. Vakulik, R. Schild, V. Dudinov et al., “Observational determination of the time delays in gravitational lens system Q2237$+$0305,” Astronomy and Astrophysics, vol. 447, no. 3, pp. 905–913, 2006.
51. J. S. B. Wyithe and E. L. Turner, “Determining the microlens mass function from quasar microlensing statistics,” Monthly Notices of the Royal Astronomical Society, vol. 320, no. 1, pp. 21–30, 2001.
52. A. B. Congdon, C. R. Keeton, and S. J. Osmer, “Microlensing of an extended source by a power-law mass distribution,” Monthly Notices of the Royal Astronomical Society, vol. 376, no. 1, pp. 263–272, 2007.
53. Y. Tsapras, K. Horne, S. Kane, and R. Carson, “Microlensing limits on numbers and orbits of extrasolar planets from the 1998-2000 OGLE events,” Monthly Notices of the Royal Astronomical Society, vol. 343, no. 4, pp. 1131–1144, 2003.
54. S. Mao and B. Paczyński, “Gravitational microlensing by double stars and planetary systems,” Astrophysical Journal Letters, vol. 374, no. 2, pp. L37–L40, 1991.
55. M. Jaroszyński and B. Paczyński, “A possible planetary event OGLE-2002-BLG-055,” Acta Astronomica, vol. 52, no. 4, pp. 361–367, 2002.
56. F. Chollet, “A new method of measuring stellar masses,” Comptes Rendus de l'Académie des Sciences B, vol. 288, pp. 163–165, 1979 (French).
57. M. Hosokawa, K. Ohnishi, and T. Fukushima, “Astrometric microlensing and degradation of reference frames,” in Proceedings of the 25th General Assembly on Highlights of Astronomy (IAU '03), O. Engvold, Ed., p. 602, Astronomical Society of the Pacific, San Francisco, Calif, USA, 2005.
58. M. Hosokawa, K. Ohnishi, T. Fukushima, and M. Takeuti, “Parallactic variation of gravitational lensing and measurement of stellar mass,” Astronomy & Astrophysics, vol. 278, pp. L27–L30, 1993.
59. J. Kovalevsky, F. Mignard, and M. Froschle, “Space astrometry prospects and limitations,” in Proceedings of the International Astronomical Union (IAU '79), J. Kovalevsky and V. A. Brumberg, Eds., vol. 114, pp. 369–382, 1979.
60. K. Ohnishi, M. Hosokawa, and T. Fukushima, “Secular component of apparent proper motion of QSOs induced by gravitational lens of the galaxy,” ASP Conference Proceedings Series, vol. 289, pp. 461–464, 2003.
61. E. Hog, I. D. Novikov, and A. G. Polnarev, “MACHO photometry and astrometry,” Astronomy & Astrophysics, vol. 294, pp. 287–294, 1995.
62. M. Miyamoto and Y. Yoshii, “Astrometry for determining the MACHO mass and trajectory,” Astronomical Journal, vol. 110, no. 3, pp. 1427–1432, 1995.
63. M. A. Walker, “Microlensed image motions,” Astrophysical Journal Letters, vol. 453, no. 1, pp. 37–39, 1995.
64. V. Bozza, “Trajectories of the images in binary microlensing,” Astronomy and Astrophysics, vol. 374, no. 1, pp. 13–27, 2001.
65. D. P. Bennett, S. H. Rhie, A. C. Becker et al., “Discovery of a planet orbiting a binary star system from gravitational microlensing,” Nature, vol. 402, no. 6757, pp. 57–59, 1999.
66. M. Dominik and K. C. Sahu, “Astrometric microlensing of stars,” Astrophysical Journal, vol. 534, no. 1, pp. 213–226, 2000.
67. A. Gould and C. Han, “Astrometric resolution of severely degenerate binary microlensing events,” Astrophysical Journal, vol. 538, no. 2, pp. 653–656, 2000.
68. C. Han, “On the astrometric behavior of binary microlensing events,” Monthly Notices of the Royal Astronomical Society, vol. 325, pp. L1281–L1287, 2001.
69. C. Han, S. H. Park, and Y. S. Lee, “Distribution of caustic-crossing intervals for galactic binary-lens microlensing events,” Monthly Notices of the Royal Astronomical Society, vol. 314, no. 1, pp. 59–64, 2000.
70. C. Han, M. S. Chun, and K. Chang, “Astrometric properties of gravitational binary-microlens events and their applications,” Astrophysical Journal Letters, vol. 526, no. 1, pp. 405–410, 1999.
71. M. Treyer and J. Wambsganss, “Astrometric microlensing of quasars dependence on surface mass density and external shear,” Astronomy and Astrophysics, vol. 416, no. 1, pp. 19–34, 2004.
72. S. Mao and H. J. Witt, “Extended source effects in astrometric gravitational microlensing,” Monthly Notices of the Royal Astronomical Society, vol. 300, no. 4, pp. 1041–1046, 1998.
73. C. H. Lee, S. Seitz, A. Riffeser, and R. Bender, “Finite-source and finite-lens effects in astrometric microlensing,” Monthly Notices of the Royal Astronomical Society, vol. 407, no. 3, pp. 1597–1608, 2010.
74. V. I. Zhdanov, A. N. Alexandrov, and S. A. Salata, “Motion of images of microlensed extended sources: analytical relations and numerical estimates for moderate optical depths,” Kinematika i Fizika Nebesnykh Tel, vol. 16, pp. 336–345, 2000 (Russian).
75. G. F. Lewis and R. A. Ibata, “Quasar image shifts resulting from gravitational microlensing,” Astrophysical Journal Letters, vol. 501, no. 2, pp. 478–485, 1998.
76. S. A. Salata and V. I. Zhdanov, “Statistical astrometric microlensing of extended sources,” Astronomical Journal, vol. 125, no. 3, pp. 1033–1037, 2003.
77. J. Wambsganss, “Gravitational microlensing,” in Gravitational Lensing: Strong, Weak, and Micro, G. Meylan, P. North, and P. Jetzer, Eds., Saas-Fee Advanced Course 33, Part 4, pp. 453–540, Springer, Berlin, Germany, 2006.
78. J. Wambsganss, “Gravitational lensing in astronomy,” Living Reviews in Relativity, vol. 1, article 12, 1998.
79. A. L. Erickcek and N. M. Law, “Astrometric microlensing by local dark matter subhalos,” Astrophysical Journal Letters, vol. 729, no. 1, article 49, 2011.
80. S. Proft, M. Demleitner, and J. Wambsganss, “Prediction of astrometric microlensing events during the Gaia mission,” Astronomy & Astrophysics, vol. 536, article A50, 11 pages, 2011.
81. M. V. Sazhin, O. S. Sazhina, and M. S. Pshirkov, “Apparent motions of quasars due to microlensing,” Astronomy Reports, vol. 55, pp. 954–961, 2011.
82. V. A. Belokurov and N. W. Evans, “Astrometric microlensing with the GAIA satellite,” Monthly Notices of the Royal Astronomical Society, vol. 331, no. 3, pp. 649–665, 2002.
83. S. A. Klioner, “A practical relativistic model for microarcsecond astrometry in space,” Astronomical Journal, vol. 125, no. 3, pp. 1580–1597, 2003.
84. N. S. Kardashev, “Cosmological proper motion,” Astronomicheskii Zhurnal, vol. 63, pp. 845–849, 1986.
85. M. V. Sazhin, “A fundamental limit to the accuracy of astrometric measurements,” Astronomy Letters, vol. 22, no. 5, pp. 573–577, 1996.
86. M. V. Sazhin, V. E. Zharov, A. V. Volynkin, and T. A. Kalinina, “Microarcsecond instability of the celestial reference frame,” Monthly Notices of the Royal Astronomical Society, vol. 300, no. 1, pp. 287–291, 1998.
87. E. Fedorova, “Binary gravitational microlensing of extragalactic sources,” Visnyk Kyivskogo Universytetu. Astronomiya, vol. 45, pp. 33–39, 2009.
88. K. Chang and S. Refsdal, “Star disturbances in gravitational lens galaxies,” Astronomy & Astrophysics, vol. 132, pp. 168–178, 1984.
89. A. F. Zakharov and M. V. Sazhin, “Non-compact astronomical objects as microlenses,” Astronomy & Astrophysics, vol. 18, pp. 27–38, 1999.
90. J. Diemand, B. Moore, and J. Stadel, “Earth-mass dark-matter haloes as the first structures in the early Universe,” Nature, vol. 433, no. 7024, pp. 389–391, 2005.
91. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, New York, NY, USA, 1959.
92. K. A. Piragas, V. I. Zhdanov, V. V. Zhdanova, and I. T. Zhuk, “Light propagation in a weak gravitational field of a stochastic system of pointlike sources,” Soviet Physics Journal, vol. 29, no. 12, pp. 1019–1023, 1986.
93. A. A. Minakov and V. N. Shalyapin, “Effect of the gravitational field of the Galaxy on the apparent position, brightness and spatial density of remote radiation sources. I. A lens model of the Galaxy and deflection angles of rays,” Kinematics and Physics of Celestial Bodies, vol. 6, pp. 49–59, 1990.