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ISRN Astronomy and Astrophysics
Volume 2012 (2012), Article ID 906951, 21 pages
http://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.

Linked References

  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Scopus
  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. View at Scopus
  16. V. N. Shalyapin, “Caustic crossing in the gravitational lens Q2237+0305,” Astronomy Letters, vol. 27, no. 3, pp. 150–155, 2001. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Scopus
  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. View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Scopus
  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. View at Scopus
  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. View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  37. C. S. Kochanek, “Quantitative interpretation of quasar microlensing light curves,” Astrophysical Journal, vol. 605, no. 1, pp. 58–77, 2004. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Scopus
  55. M. Jaroszyński and B. Paczyński, “A possible planetary event OGLE-2002-BLG-055,” Acta Astronomica, vol. 52, no. 4, pp. 361–367, 2002. View at Scopus
  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. View at Scopus
  63. M. A. Walker, “Microlensed image motions,” Astrophysical Journal Letters, vol. 453, no. 1, pp. 37–39, 1995. View at Scopus
  64. V. Bozza, “Trajectories of the images in binary microlensing,” Astronomy and Astrophysics, vol. 374, no. 1, pp. 13–27, 2001. View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  66. M. Dominik and K. C. Sahu, “Astrometric microlensing of stars,” Astrophysical Journal, vol. 534, no. 1, pp. 213–226, 2000. View at Scopus
  67. A. Gould and C. Han, “Astrometric resolution of severely degenerate binary microlensing events,” Astrophysical Journal, vol. 538, no. 2, pp. 653–656, 2000. View at Scopus
  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. View at Scopus
  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. View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  76. S. A. Salata and V. I. Zhdanov, “Statistical astrometric microlensing of extended sources,” Astronomical Journal, vol. 125, no. 3, pp. 1033–1037, 2003. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  83. S. A. Klioner, “A practical relativistic model for microarcsecond astrometry in space,” Astronomical Journal, vol. 125, no. 3, pp. 1580–1597, 2003. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Scopus
  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. View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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. View at Publisher · View at Google Scholar · View at Scopus
  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.