Abstract
As early as in 1937, Zwicky wrote about gravitational lenses acting as ‘space telescopes’, allowing the observation of faint and distant objects, the fluxes from which may be considerably enhanced due to the lensing. It is clear today that gravitational lensing may be helpful in performing another important task, one of the main purposes of telescopic observations, namely, increasing spatial resolution. The images of strongly lensed QSOs are affected by microlensing effects in the halo of the lensing galaxy. In contrast to the classical strong lensing, these effects are sensitive to the size and form of an object. The purpose of this chapter is to give a general introduction to quasar microlensing and to illustrate the capabilities of the method, with a review of the latest results in this field, concentrating especially on the results obtained in our three recent papers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For a fold caustic considered below, the amplification decreases inversely proportional to the distance to the caustic in the image plane, μ ∝ Δθ −1, or μ ∝ Δβ −1∕2 (see later this section). The integral of this expression over a restricted solid angle converges. For a more complex singularity (for example, a cusp), the finiteness of amplification follows from the asymptotic behaviour of amplification which decreases inversely proportional to the distance to this singularity (a point), or slower (Gaudi and Petters 2002).
- 2.
The photospheric size is proportional to \(R_1 \propto \varkappa \dot {M}\), while the product \(M\dot {M} \propto F_\nu ^{3/2}\) does not depend on opacity; the Thomson opacity ϰT, which is included in the Eddington luminosity normalization, is considered fixed.
References
Abolmasov P (2014) The thickness of a weakly magnetized accretion flow inside the last stable orbit of a Kerr black hole. Mon Not R Astron Soc 445:1269–1287. https://doi.org/10.1093/mnras/stu1753, 1408.6449
Abolmasov P (2017) Apparent quasar disc sizes in the “bird’s nest” paradigm. Astron Astrophys 600:A79. https://doi.org/10.1051/0004-6361/201628842, 1701.08957
Abolmasov P, Chashkina A (2015) On the Eddington limit for relativistic accretion discs. Mon Not R Astron Soc 454:3432–3444. https://doi.org/10.1093/mnras/stv2229, 1509.07261
Abolmasov P, Poutanen J (2017) Gamma-ray opacity of the anisotropic stratified broad-line regions in blazars. Mon Not R Astron Soc 464:152–169. https://doi.org/10.1093/mnras/stw2326, 1609.03350
Abolmasov P, Shakura NI (2012a) Microlensing evidence for super-Eddington disc accretion in quasars. Mon Not R Astron Soc 427:1867–1876. https://doi.org/10.1111/j.1365-2966.2012.21881.x, 1208.1678
Abolmasov P, Shakura NI (2012b) Resolving the inner structure of QSO discs through fold-caustic-crossing events. Mon Not R Astron Soc 423:676–693. https://doi.org/10.1111/j.1365-2966.2012.20904.x, 1203.2656
Abolmasov P, Shakura NI (2013) Erratum: microlensing evidence for super-Eddington disc accretion in quasars. Mon Not R Astron Soc 434:906–908. https://doi.org/10.1093/mnras/stt976
Agol E, Krolik JH (2000) Magnetic stress at the marginally stable orbit: altered disk structure, radiation, and black hole spin evolution. Astrophys J 528:161–170. https://doi.org/10.1086/308177, arXiv:astro-ph/9908049
Arnold VI, Wassermann GS, Thomas RK (2003) Catastrophe theory. Springer, Berlin/Heidelberg
Bardeen JM, Petterson JA (1975) The lense-thirring effect and accretion disks around Kerr black holes. Astrophys J 195:L65+. https://doi.org/10.1086/181711
Bardeen JM, Press WH, Teukolsky SA (1972) Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation. Astrophys J 178:347–370. https://doi.org/10.1086/151796
Bastian N, Covey KR, Meyer MR (2010) A Universal stellar initial mass function? A critical look at variations. Annu Rev Astron Astrophys 48:339–389. https://doi.org/10.1146/annurev-astro-082708-101642, 1001.2965
Bate NF, Floyd DJE, Webster RL, Wyithe JSB (2008) A microlensing study of the accretion disc in the quasar MG 0414+0534. Mon Not R Astron Soc 391:1955–1960. https://doi.org/10.1111/j.1365-2966.2008.14020.x, 0810.1092
Berezhiani Z, Ciarcelluti P, Comelli D, Villante FL (2005) Structure formation with mirror dark matter. Int J Mod Phys D 14:107–119. https://doi.org/10.1142/S0218271805005165, arXiv:astro-ph/0312605
Blackburne JA, Pooley D, Rappaport S, Schechter PL (2011) Sizes and temperature profiles of quasar accretion disks from chromatic microlensing. Astrophys J 729:34. https://doi.org/10.1088/0004-637X/729/1/34, 1007.1665
Blaes O, Krolik JH, Hirose S, Shabaltas N (2011) Dissipation and vertical energy transport in radiation-dominated accretion disks. Astrophys J 733:110. https://doi.org/10.1088/0004-637X/733/2/110, 1103.5052
Blandford R, Narayan R (1986) Fermat’s principle, caustics, and the classification of gravitational lens images. Astrophys J 310:568–582. https://doi.org/10.1086/164709
Bogdanov MB, Cherepashchuk AM (2004) Analysis of a high-amplitude event in component A of the gravitational lens QSO 2237 + 0305. Astron Rep 48:261–266. https://doi.org/10.1134/1.1704671
Braibant L, Hutsemékers D, Sluse D, Anguita T, García-Vergara CJ (2014) Microlensing of the broad-line region in the quadruply imaged quasar HE0435–1223. Astron Astrophys 565:L11. https://doi.org/10.1051/0004-6361/201423633, 1405.5014
Braibant L, Hutsemékers D, Sluse D, Anguita T (2016) The different origins of high- and low-ionization broad emission lines revealed by gravitational microlensing in the Einstein cross. Astron Astrophys 592:A23. https://doi.org/10.1051/0004-6361/201628594, 1606.01734
Cao X (2009) An accretion disc-corona model for X-ray spectra of active galactic nuclei. Mon Not R Astron Soc 394:207–213. https://doi.org/10.1111/j.1365-2966.2008.14347.x, 0812.1828
Cassinelli JP, Hartmann L (1977) The effect of winds and coronae of hot stars on the infrared and radio continua. Astrophys J 212:488–493. https://doi.org/10.1086/155068
Chang K, Refsdal S (1979) Flux variations of QSO 0957+561 A, B and image splitting by stars near the light path. Nature 282:561–564. https://doi.org/10.1038/282561a0
Chang K, Refsdal S (1984) Star disturbances in gravitational lens galaxies. Astron Astrophys 132:168–178
Chwolson O (1924) Über eine mögliche Form fiktiver Doppelsterne. Astron Nachr 221:329
Collier S, Peterson BM (2001) Characteristic ultraviolet/optical timescales in active galactic nuclei. Astrophys J 555:775–785. https://doi.org/10.1086/321517
Collin S, Boisson C, Mouchet M, Dumont A, Coupé S, Porquet D, Rokaki E (2002) Are quasars accreting at super-Eddington rates? Astron Astrophys 388:771–786. https://doi.org/10.1051/0004-6361:20020550, arXiv:astro-ph/0203439
Dexter J, Agol E (2009) A fast new public code for computing photon orbits in a Kerr spacetime. Astrophys J 696:1616–1629. https://doi.org/10.1088/0004-637X/696/2/1616, 0903.0620
Doroshenko VT, Sergeev SG, Klimanov SA, Pronik VI, Efimov YS (2012) Broad-line region kinematics and black hole mass in Markarian 6. Mon Not R Astron Soc 426:416–426. https://doi.org/10.1111/j.1365-2966.2012.20843.x, 1203.2084
Eddington AS (1925) A limiting case in the theory of radiative equilibrium. Mon Not R Astron Soc 85:408
Eigenbrod A, Courbin F, Sluse D, Meylan G, Agol E (2008) Microlensing variability in the gravitationally lensed quasar QSO 2237+0305 = the Einstein cross. I. Spectrophotometric monitoring with the VLT. Astron Astrophys 480:647–661. https://doi.org/10.1051/0004-6361:20078703, 0709.2828
Einstein A (1936) Lens-like action of a star by the deviation of light in the gravitational field. Science 84:506–507. https://doi.org/10.1126/science.84.2188.506
Elitzur M (2008) The toroidal obscuration of active galactic nuclei. New A Rev 52:274–288. https://doi.org/10.1016/j.newar.2008.06.010, 0805.3699
Ferreras I, Saha P, Williams LLR (2005) Stellar and total mass in early-type lensing galaxies. Astrophys J 623:L5–L8. https://doi.org/10.1086/429995, arXiv:astro-ph/0503168
Floyd DJE, Bate NF, Webster RL (2009) The accretion disc in the quasar SDSS J0924+0219. Mon Not R Astron Soc 398:233–239. https://doi.org/10.1111/j.1365-2966.2009.15045.x, 0905.2651
Gaudi BS, Petters AO (2002) Gravitational microlensing near caustics. I. Folds. Astrophys J 574:970–984. https://doi.org/10.1086/341063, arXiv:astro-ph/0112531
Gil-Merino R, González-Cadelo J, Goicoechea LJ, Shalyapin VN, Lewis GF (2006) Is there a caustic crossing in the lensed quasar Q2237+0305 observational data record? Mon Not R Astron Soc 371:1478–1482. https://doi.org/10.1111/j.1365-2966.2006.10782.x, arXiv:astro-ph/0607162
Grier CJ, Peterson BM, Horne K, Bentz MC, Pogge RW, Denney KD, De Rosa G, Martini P, Kochanek CS, Zu Y, Shappee B, Siverd R, Beatty TG, Sergeev SG, Kaspi S, Araya Salvo C, Bird JC, Bord DJ, Borman GA, Che X, Chen C, Cohen SA, Dietrich M, Doroshenko VT, Efimov YS, Free N, Ginsburg I, Henderson CB, King AL, Mogren K, Molina M, Mosquera AM, Nazarov SV, Okhmat DN, Pejcha O, Rafter S, Shields JC, Skowron J, Szczygiel DM, Valluri M, van Saders JL (2013) The structure of the broad-line region in active galactic nuclei. I. Reconstructed velocity-delay maps. Astrophys J 764:47. https://doi.org/10.1088/0004-637X/764/1/47, 1210.2397
Guerras E, Mediavilla E, Jimenez-Vicente J, Kochanek CS, Muñoz JA, Falco E, Motta V (2013a) Microlensing of quasar broad emission lines: constraints on broad line region size. Astrophys J 764:160. https://doi.org/10.1088/0004-637X/764/2/160, 1207.2042
Guerras E, Mediavilla E, Jimenez-Vicente J, Kochanek CS, Muñoz JA, Falco E, Motta V, Rojas K (2013b) Microlensing of quasar ultraviolet iron emission. Astrophys J 778:123. https://doi.org/10.1088/0004-637X/778/2/123, 1309.2603
Hamadache C, Le Guillou L, Tisserand P, Afonso C, Albert JN, Andersen J, Ansari R, Aubourg É, Bareyre P, Beaulieu JP, Charlot X, Coutures C, Ferlet R, Fouqué P, Glicenstein JF, Goldman B, Gould A, Graff D, Gros M, Haissinski J, de Kat J, Lesquoy É, Loup C, Magneville C, Marquette JB, Maurice É, Maury A, Milsztajn A, Moniez M, Palanque-Delabrouille N, Perdereau O, Rahal YR, Rich J, Spiro M, Vidal-Madjar A, Vigroux L, Zylberajch S (2006) Galactic bulge microlensing optical depth from EROS-2. Astron Astrophys 454:185–199. https://doi.org/10.1051/0004-6361:20064893, arXiv:astro-ph/0601510
Hilbert S, White SDM, Hartlap J, Schneider P (2007) Strong lensing optical depths in a ΛCDM universe. Mon Not R Astron Soc 382:121–132. https://doi.org/10.1111/j.1365-2966.2007.12391.x, arXiv:astro-ph/0703803
Hogg DW (1999) Distance measures in cosmology. ArXiv Astrophysics. e-prints 9905116. arXiv:astro-ph/9905116
Hubeny I, Hubeny V (1998) Non-LTE models and theoretical spectra of accretion disks in active galactic nuclei. II. Vertical structure of the disk. Astrophys J 505:558–576. https://doi.org/10.1086/306207, arXiv:astro-ph/9804288
Inada N, Oguri M, Morokuma T, Doi M, Yasuda N, Becker RH, Richards GT, Kochanek CS, Kayo I, Konishi K, Utsunomiya H, Shin MS, Strauss MA, Sheldon ES, York DG, Hennawi JF, Schneider DP, Dai X, Fukugita M (2006) SDSS J1029+2623: a gravitationally lensed quasar with an image separation of 22.5”. Astrophys J 653:L97–L100. https://doi.org/10.1086/510671, arXiv:astro-ph/0611275
Irwin MJ, Webster RL, Hewett PC, Corrigan RT, Jedrzejewski RI (1989) Photometric variations in the Q2237 + 0305 system - first detection of a microlensing event. Astron J 98:1989–1994. https://doi.org/10.1086/115272
Ivanov PB, Illarionov AF (1997) The oscillatory shape of the stationary twisted disc around a Kerr black hole. Mon Not R Astron Soc 285:394–402
Ivezić Ž, Menou K, Knapp GR, Strauss MA, Lupton RH, Vanden Berk DE, Richards GT, Tremonti C, Weinstein MA, Anderson S, Bahcall NA, Becker RH, Bernardi M, Blanton M, Eisenstein D, Fan X, Finkbeiner D, Finlator K, Frieman J, Gunn JE, Hall PB, Kim RSJ, Kinkhabwala A, Narayanan VK, Rockosi CM, Schlegel D, Schneider DP, Strateva I, SubbaRao M, Thakar AR, Voges W, White RL, Yanny B, Brinkmann J, Doi M, Fukugita M, Hennessy GS, Munn JA, Nichol RC, York DG (2002) Optical and radio properties of extragalactic sources observed by the FIRST survey and the Sloan digital sky survey. Astron J 124:2364–2400. https://doi.org/10.1086/344069, astro-ph/0202408
Jaroszynski M, Wambsganss J, Paczynski B (1992) Microlensed light curves for thin accretion disks around Schwarzschild and Kerr black holes. Astrophys J 396:L65–L68. https://doi.org/10.1086/186518
Keeton CR (2001) A catalog of mass models for gravitational lensing. ArXiv Astrophysics. e-prints arXiv:astro-ph/0102341
Kochanek CS (2006) Part 2: Strong gravitational lensing. In: Meylan G, Jetzer P, North P, Schneider P, Kochanek CS, Wambsganss J (eds) Saas-Fee advanced course 33: gravitational lensing: strong, weak and micro. Springer, Berlin, pp 91–268
Kochanek CS, Keeton CR (1997) Gravitational lensing limits on early-type galaxies. In: Arnaboldi M, Da Costa GS, Saha P (eds) The nature of elliptical galaxies, 2nd Stromlo symposium. Astronomical society of the pacific conference series, vol 116, p 21. arXiv:astro-ph/9611217
Kochanek CS, Schechter PL (2004) The hubble constant from gravitational lens time delays. Measuring and modeling the universe. Cambridge University Press, Cambridge, p 117. astro-ph/0306040
Kochanek CS, Keeton CR, McLeod BA (2001) The importance of Einstein rings. Astrophys J 547:50–59. https://doi.org/10.1086/318350, arXiv:astro-ph/0006116
Kofman L, Kaiser N, Lee MH, Babul A (1997) Statistics of gravitational microlensing magnification. I. Two-dimensional lens distribution. Astrophys J 489:508–+. https://doi.org/10.1086/304791, arXiv:astro-ph/9608138
Kollatschny W, Zetzl M (2013) Vertical broad-line region structure in nearby active galactic nuclei. Astron Astrophys 558:A26. https://doi.org/10.1051/0004-6361/201321685, 1308.1902
Kolykhalov PI, Sunyaev RA (1984) Radiation of accretion disks in quasars and galactic nuclei. Adv Space Res 3:249–254 https://doi.org/10.1016/0273-1177(84)90100-5
Korista K (1999) What’s emitting the broad emission lines? In: Ferland G, Baldwin J (eds) Quasars and cosmology. Astronomical society of the pacific conference series, vol 162. Astronomical Society of the Pacific, San Francisco, p 165, astro-ph/9812043
Kormann R, Schneider P, Bartelmann M (1994) Isothermal elliptical gravitational lens models. Astron Astrophys 284:285–299
Lawrence A (2012) The UV peak in active galactic nuclei: a false continuum from blurred reflection? Mon Not R Astron Soc 423:451–463. https://doi.org/10.1111/j.1365-2966.2012.20889.x, 1110.0854
Meyer F, Liu BF, Meyer-Hofmeister E (2000) Evaporation: the change from accretion via a thin disk to a coronal flow. Astron Astrophys 361:175–188. astro-ph/0007091
Mihalas D (1978) Stellar atmospheres, 2nd edn. W.H. Freeman and Co., San Francisco
Morgan CW, Kochanek CS, Morgan ND, Falco EE (2010) The Quasar accretion disk size-black hole mass relation. Astrophys J 712:1129–1136. https://doi.org/10.1088/0004-637X/712/2/1129, 1002.4160
Morgan CW, Hainline LJ, Chen B, Tewes M, Kochanek CS, Dai X, Kozlowski S, Blackburne JA, Mosquera AM, Chartas G, Courbin F, Meylan G (2012) Further evidence that quasar X-ray emitting regions are compact: X-ray and optical microlensing in the lensed quasar Q J0158–4325. Astrophys J 756:52. https://doi.org/10.1088/0004-637X/756/1/52, 1205.4727
Mortonson MJ, Schechter PL, Wambsganss J (2005) Size is everything: universal features of quasar microlensing with extended sources. Astrophys J 628:594–603. https://doi.org/10.1086/431195, arXiv:astro-ph/0408195
Muñoz JA, Falco EE, Kochanek CS, Lehár J, McLeod BA, Impey CD, Rix H, Peng CY (1998) The castles project. Astrophys Space Sci 263:51–54. https://doi.org/10.1023/A:1002120921330, arXiv:astro-ph/9902131
Narayan R, Yi I (1995) Advection-dominated accretion: underfed black holes and neutron stars. Astrophys J 452:710–+. https://doi.org/10.1086/176343, arXiv:astro-ph/9411059
Novikov ID, Thorne KS (1973) Astrophysics of black holes. In: Black holes (Les Astres Occlus). Gordon&Breach, Paris, pp 343–450
O’Dowd MJ, Bate NF, Webster RL, Labrie K, Rogers J (2015) Microlensing constraints on broad absorption and emission line flows in the quasar H1413+117. Astrophys J 813:62. https://doi.org/10.1088/0004-637X/813/1/62, 1504.07160
Paczynski B (1986) Gravitational microlensing at large optical depth. Astrophys J 301:503–516. https://doi.org/10.1086/163919
Page DN, Thorne KS (1974) Disk-Accretion onto a black hole. Time-averaged structure of accretion disk. Astrophys J 191:499–506. https://doi.org/10.1086/152990
Peterson BM (2006) The broad-line region in active galactic nuclei. In: Alloin D (ed) Physics of active galactic nuclei at all scales. Lecture notes in physics, vol 693. Springer, Berlin, p 77. https://doi.org/10.1007/3-540-34621-X_3
Pietrini P, Krolik JH (1995) The inverse Compton thermostat in hot plasmas near accreting black holes. Astrophys J 447:526. https://doi.org/10.1086/175897, astro-ph/9501093
Pooley D, Blackburne JA, Rappaport S, Schechter PL (2007) X-ray and optical flux ratio anomalies in quadruply lensed quasars. I. Zooming in on quasar emission regions. Astrophys J 661:19–29. https://doi.org/10.1086/512115, arXiv:astro-ph/0607655
Poutanen J, Lipunova G, Fabrika S, Butkevich AG, Abolmasov P (2007) Supercritically accreting stellar mass black holes as ultraluminous X-ray sources. Mon Not R Astron Soc 377:1187–1194. https://doi.org/10.1111/j.1365-2966.2007.11668.x, astro-ph/0609274
Raychaudhury S, Saslaw WC (1996) The observed distribution function of peculiar velocities of galaxies. Astrophys J 461:514–+. https://doi.org/10.1086/177078, arXiv:astro-ph/9602001
Riffert H, Herold H (1995) Relativistic accretion disk structure revisited. Astrophys J 450:508–+. https://doi.org/10.1086/176161
Rybicki GB, Lightman AP (1986) Radiative processes in astrophysics. Wiley, Weinheim
Sadowski A (2011) Slim accretion disks around black holes. ArXiv e-prints 1108.0396
Sadowski A, Narayan R, Tchekhovskoy A, Abarca D, Zhu Y, McKinney JC (2014) Global simulations of axisymmetric radiative black hole accretion disks in general relativity with a sub-grid magnetic dynamo. ArXiv e-prints 1407.4421
Schneider P (2005) Weak gravitational lensing. astro-ph/0509252 astro-ph/0509252
Schneider P (2006) Part 1: Introduction to gravitational lensing and cosmology. In: Meylan G, Jetzer P, North P, Schneider P, Kochanek CS, Wambsganss J (eds) Saas-Fee advanced course 33: gravitational lensing: strong, weak and micro. Springer, Berlin, pp 1–89
Shakura NI (1972) Disk model of gas accretion on a relativistic star in a close binary system. Astron Rep 49:921
Shakura NI, Sunyaev RA (1973) Black holes in binary systems. Observational appearance. Astron Astrophys 24:337–355
Shalyapin VN, Goicoechea LJ, Alcalde D, Mediavilla E, Muñoz JA, Gil-Merino R (2002) The nature and size of the optical continuum source in QSO 2237+0305. Astrophys J 579:127–135. https://doi.org/10.1086/342753, arXiv:astro-ph/0207236
Siuniaev RA, Shakura NI (1977) Disk reservoirs in binary systems and prospects for observing them. Pis’ma Astron Zh 3:262–266
Takeuchi S, Ohsuga K, Mineshige S (2013) Clumpy outflows from supercritical accretion flow. Publ Astron Soc Jpn 65:88. https://doi.org/10.1093/pasj/65.4.88, 1305.1023
Udalski A, Szymanski MK, Kubiak M, Pietrzynski G, Soszynski I, Zebrun K, Szewczyk O, Wyrzykowski L, Ulaczyk K, Wiêckowski T (2006) The optical gravitational lensing experiment. OGLE-III long term monitoring of the gravitational lens QSO 2237+0305. Acta Astron 56:293–305, arXiv:astro-ph/0701300
Vakulik V, Schild R, Dudinov V, Nuritdinov S, Tsvetkova V, Burkhonov O, Akhunov T (2006) Observational determination of the time delays in gravitational lens system <ASTROBJ>Q2237+0305</ASTROBJ>. Astron Astrophys 447:905–913. https://doi.org/10.1051/0004-6361:20053574, astro-ph/0509545
Vernardos G, Fluke CJ (2014) The effect of macromodel uncertainties on microlensing modelling of lensed quasars. Mon Not R Astron Soc 445:1223–1234. https://doi.org/10.1093/mnras/stu1833, 1409.1640
Vestergaard M, Peterson BM (2006) Determining central black hole masses in distant active galaxies and quasars. II. Improved optical and UV scaling relationships. Astrophys J 641:689–709. https://doi.org/10.1086/500572, arXiv:astro-ph/0601303
Walsh D, Carswell RF, Weymann RJ (1979) 0957 + 561 A, B - Twin quasistellar objects or gravitational lens. Nature 279:381–384. https://doi.org/10.1038/279381a0
Wambsganss J (2006) Part 4: Gravitational microlensing. In: G Meylan, P Jetzer, P North, P Schneider, C S Kochanek, & J Wambsganss (ed) Saas-Fee advanced course 33: gravitational lensing: strong, weak and micro. Springer, Heidelberg, pp 453–540
Webb W, Malkan M (2000) Rapid optical variability in active galactic nuclei and quasars. Astrophys J 540:652–677. https://doi.org/10.1086/309341
Witt HJ, Kayser R, Refsdal S (1993) Microlensing predictions for the Einstein cross 2237+0305. Astron Astrophys 268:501–510
Woźniak PR, Alard C, Udalski A, Szymański M, Kubiak M, Pietrzyński G, Zebruń K (2000) The optical gravitational lensing experiment monitoring of QSO 2237+0305. Astrophys J 529:88–92. https://doi.org/10.1086/308258, arXiv:astro-ph/9904329
Yan CS, Lu Y, Yu Q, Mao S, Wambsganss J (2014) Microlensing of sub-parsec massive binary black holes in lensed QSOs: light curves and size-wavelength relation. Astrophys J 784:100. https://doi.org/10.1088/0004-637X/784/2/100, 1402.2504
Zakharov AF (1997) Gravitational lenses and microlenses (in Russian). Yanus-K, Moscow
Zakharov AF, Sazhin MV (1998) Reviews of topical problems: gravitational microlensing. Phys Usp 41:945–982. https://doi.org/10.1070/PU1998v041n10ABEH000460
Zeldovich IB, Novikov ID (1975) Structure and evolution of the universe. Izdatel’stvo Nauka, Moscow (in Russian), 736 p
Zhuravlev VV, Ivanov PB (2011) A fully relativistic twisted disc around a slowly rotating Kerr black hole: derivation of dynamical equations and the shape of stationary configurations. Mon Not R Astron Soc 415:2122–2144. https://doi.org/10.1111/j.1365-2966.2011.18830.x, 1103.5739
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Abolmasov, P., Shakura, N., Chashkina, A. (2018). Structure of Accretion Discs in Lensed QSOs. In: Shakura, N. (eds) Accretion Flows in Astrophysics . Astrophysics and Space Science Library, vol 454. Springer, Cham. https://doi.org/10.1007/978-3-319-93009-1_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-93009-1_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93008-4
Online ISBN: 978-3-319-93009-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)