Extrasolar Moons as Gravitational Microlenses Christine Liebig

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CHAPTER 2. GRAVITATIONAL LENSING 11 where the impact parameter u = β is used, the angular separation of the source from the point mass in units of the Einstein angle. The magnification of the image inside the Einstein ring is formally negative, because it has negative parity. It is a mirror-inverted image of the source. The sum of the two magnifications is unity, θE µ1 + µ2 = 1, but the net magnification of the source flux in the two images is obtained by adding the absolute magnifications µ = |µ1| + |µ2| = u2 + 2 u √ u 2 + 4 . If the source is located at an angular separation of one Einstein radius from the lens, the impact parameter will become u = 1 and the total magnification µ = 3 √ 5 = 1.34. If the source moves closer towards the lens, the magnification will increase (figure 2.3). For β → 0 it reaches infinity. In reality, this is prevented by the finite extent of the source. The magnification is converted into units of magnitudes via mag = −2.5 log 10 µ. (2.6) According to the astrophysical convention, more negative values denote brighter luminosity. 2.2.1 Triple-Lens Equation Planetary systems have the characteristic trait of being multiple lens systems. After having described the single-lens case in the last section, we ought to have a look at how multiple lenses manifest themselves in light curves and how they can be accounted for in the lens equation. Using complex coordinates, let us express the projected source position as η and the image position as ξ. For N point masses the lens equation then becomes η = ξ + N qi , (2.7) ξi − ξ i=1 following Witt (1990) and also referring to Gaudi et al. (1998). ξi denotes the position of lens i. qi is the mass ratio between lens i and the primary lens (qi = Mi M1 ).
12 2.2. BASIC EQUATIONS −∆mag Figure 2.3: If a source star passes behind a lensing star along the source trajectory depicted in the magnification map on the left, this gives rise to a point-lens, pointsource light curve, also called Paczyński curve. All lengths are in units of Einstein radii of the primary lens. The magnification µI of image I, which has been derived for a single lens in an earlier section 2.2, is now obtained as the inverse of the determinant of the Jacobian of the the mapping equation (2.7), µI = 1 det J(ξI) time ∂η ∂η with det J(ξ) = 1 − ∂ξ ∂ξ . The Jacobian J can be equal to zero, it that case a hypothetical point source would be infinitely magnified. The mapping of all image positions satisfying J = 0 to the source plane forms the continuous caustics. These are lines of formally infinite magnification that arise due to the lens astigmatism, compare figure 2.4. Choosing complex coordinates is of course just a question of preference, the same calculation can be performed with Euclidean coordinates (see e.g. Bozza, 1999), but one has to be aware that the Jacobian J becomes much more complex to explicitly express. In this work, we focus on the triple lens case with the three bodies of the host star, the planet and the moon in orbit of the planet. The equation gets the following form, if we place the primary lens in the origin of the lens plane, η = ξ − 1 ξ qPlanet − − ξ − ξPlanet qMoon . (2.8) ξ − ξMoon As pointed out in Rhie (1997), and explicitly calculated in Rhie (2002), the triplelens equation is a tenth-order polynomial equation in ξ. Equation (2.8) is therefore analytically solvable.
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Page 11 and 12: 2 CONTENTS 5 Results: Detectability
Page 13 and 14: 4 first confirmed detection by the
Page 15 and 16: 6 Figure 1.1: The solar system seen
Page 17 and 18: 8 2.2. BASIC EQUATIONS and publish
Page 19: 10 2.2. BASIC EQUATIONS We can inte
Page 23 and 24: 14 2.3. SEARCH FOR PLANETS VIA GALA
Page 25 and 26: 16 3.1. MAGNIFICATION PATTERNS 3.1
Page 27 and 28: 18 3.2. LIGHT CURVES Figure 3.1: A
Page 29 and 30: 20 3.2. LIGHT CURVES 3.2.2 Light cu
Page 31 and 32: 22 3.3. STATISTICAL ANALYSIS origin
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Page 37 and 38: 28 3.3. STATISTICAL ANALYSIS We use
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Page 41 and 42: 32 4.1. PARAMETERS RELEVANT FOR MAG
Page 47 and 48: 38 4.2. PARAMETERS RELEVANT FOR LIG
Page 49 and 50: 40 4.2. PARAMETERS RELEVANT FOR LIG
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Page 56 and 57: CHAPTER 5. DETECTABILITY OF EXTRASO
Page 62 and 63: Chapter 6 Conclusion We showed that
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APPENDIX A. ANALYSED MAGNIFICATION
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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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76 BIBLIOGRAPHY
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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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