Extrasolar Moons as Gravitational Microlenses Christine Liebig

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Chapter 2 Gravitational Lensing The phenomenon of gravitational lensing occurs when light rays of a background source are deflected by an intervening massive body. The light of the source appears magnified, if more light rays are deflected towards the observing point, than without the lens. Accordingly, the light can also be demagnified, if light rays are deflected away from the observer. This chapter aims at providing the reader with a short introduction to the history (section 2.1), the basic equations (section 2.2) and the application to planet searches (section 2.3) of gravitational lensing. Better and more comprehensive reviews are plentiful in literature, e.g. Meylan et al. (2006). 2.1 Historical Development The deflection of light by massive bodies must be concluded from Einstein’s general theory of relativity, which was first summarised in 1916 (Einstein, 1916). In this work Einstein first published the correct derivation of the deflection angle of a light ray passing the sun close to the surface. He had thought “ Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes” 1 before, such was the title of a paper published in the “Annalen der Physik” (Einstein, 1911), where he also references an earlier publication (Einstein, 1908). Even this is preceded by more than a century in form of a paper by Soldner (1801), who applied Newtonian gravity to compute a deflection angle of 0.84 arcsec for the sun, half of the correct value. Einstein had called for a test of his calculations in 19112 and, indeed, an expedition set out to measure precise star positions close to the limb of the sun during solar eclipse in 1914, but was hindered in their task by the outbreak of World War I. This was very fortunate for Einstein, as it gave him time to develop general relativity 1 “About the influence of gravity on the propagation of light” 2 “Es wäre dringend zu wünschen, daß sich Astronomen der hier aufgerollten Frage annähmen, auch wenn die im vorigen gegebenen Überlegungen ungenügend fundiert oder gar abenteuerlich erscheinen sollten. Denn abgesehen von jeder Theorie muß man sich fragen, ob mit den heutigen Mitteln ein Einfluß der Gravitationsfelder auf die Ausbreitung des Lichtes sich konstantieren laesst.” 7
8 2.2. BASIC EQUATIONS and publish the value of 1.7 arcsec, which was confirmed to within 20% in a 1919 expedition by Dyson et al. (1920). This result was recognised worldwide and was the basis of Einstein’s fame. In the following decade a number of people published their ideas concerning gravitational lensing. Chwolson (1924) investigated the geometry of star-on-star lensing, finding that the observer would see two images of the source star, one of which is inverted like a mirror image, except for the case of perfect alignment of source, lens and observer, when a ring around the lens star would appear (now usually referred to as Einstein ring). Then Einstein himself had worked out the magnification of the apparent brightness of a background source star, if a lensing star and the observer are appropriately positioned (Einstein, 1936). Zwicky (1937a) pointed out the great possibilities of lensing galaxies acting as a magnifying glass on the distant universe 3 and added later that it is “practically a certainty” to find galaxies acting as lenses (Zwicky, 1937b). The first time that the effects of a gravitational lens other than our sun were observed, was not until 40 years later, when Walsh et al. (1979) announced discovery of gravitationally lensed quasar QSO 0957+561. Paczyński (1986) was first to point out the possibility of gravitational lensing due to masses as small as 10 −11 M⊙ in the halo of the Milky Way with background stars in the nearby galaxies. If found, those masses would provide an explanation of the dark matter in the halo. The idea was eventually picked up and several searches for microlensing events were called into life in the early 1990s. 4 Alcock et al. (1993) and Aubourg et al. (1993) reported discoveries towards the Large Magellanic Cloud. Udalski et al. (1993) found events towards the Galactic bulge. As a general study of lensing behaviour Schneider and Weiß (1986) had investigated the case of binary point mass lenses. Paczyński (1991) provided details of lensing by Galactic bulge stars. On this foundation, Mao and Paczyński (1991) proposed to look for lensing signatures of binary stars and also, mentioned here for the first time in the context of gravitational lensing, extrasolar planets. 2.2 Basic Equations From the theory of general relativity we know that light is deflected by gravity. More precisely, light always follows the geodesics of spacetime, but spacetime is bent in the vicinity of a massive body. The mass acts as a lens, that focusses light rays that are passing near-by. We can work with the thin lens approximation for almost all practical applications, assuming that the action of deflection takes place in an imaginary plane at the location of the lensing mass, the so called lens plane. This simplification finds its justification in the fact, that in all astronomical settings the 3 “Any such extension of the known parts of the universe promises to throw very welcome new light on a number of cosmological problems.” 4 There is an interesting footnote in Zwicky (1937b), stating that as early as 1923 plans were made “for the search of such lens effects among stars”.
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APPENDIX A. ANALYSED MAGNIFICATION
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Appendix B Numerical Results In thi
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APPENDIX B. NUMERICAL RESULTS 67 (a
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70 BIBLIOGRAPHY Bevington, Philip R
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72 BIBLIOGRAPHY eprint arXiv:0803.4
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74 BIBLIOGRAPHY Sartoretti, Paola a
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78 BIBLIOGRAPHY Gabriele Maier. I t
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Extrasolar Moons as Gravitational Microlenses Christine Liebig

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