Extrasolar Moons as Gravitational Microlenses Christine Liebig

More documents

Recommendations

Info

$µ MATH-PHYS-INFO - Fachschaft MathPhys an der Uni Heidelberg$

Chapter 3 Method The search for extrasolar planets via microlensing is well established today. See Gould (2008b) for a recent summary of successful (and some not so successful) planet findings. Microlensing probes deeply into the low-mass range, even Earthmass bodies are within the range of possible discoveries. Hence it seems natural to expand theoretical investigations to the effects that satellites of extrasolar planets have on microlensing light curves. The goal of this work is to evaluate and quantify under which conditions an extrasolar moon will be detectable in the observational situation. We propose that lunar effects will first show up as noticeable irregularities in light curves that have been initially observed and classified as light curves with planetary signatures. The simplest model is that of a binary lens 1 , consisting of the lensing star and a planetary companion. The simplest system incorporating an extrasolar moon is a triple-lens system of the lensing star, a planet and a moon in orbit around that planet (sketched in figure 4.1. To measure the detectability of triple-lens systems among binary lenses, we have to determine whether the resulting triple-lens light curves differ significantly from light curves from a corresponding binary-lens system. To obtain many independent light curves of a selected lens system with moon, we first produce a magnification pattern for the given parameter configuration. This will be explained in detail in section 3.1. Light curves are obtained as one dimensional cuts through this pattern (section 3.2). Analogously, light curves of a corresponding binary-lens system will be produced. We fit the latter to each examined triplelens light curve. For the best-fit binary light curve we employ χ 2 -statistics to see whether the triple-lens light curve could be explained as a normal fluctuation within the error boundaries of the binary case (section 3.3). If this is not the case, the moon is considered detectable. 1 Throughout this text, the term binary lens will always refer to a system consisting of a host star and a planet, i.e. a binary with a mass-ratio very different from unity. In this text it never denotes a binary of stars. 15
16 3.1. MAGNIFICATION PATTERNS 3.1 Magnification Patterns The three-mass lens equation, a tenth-order polynomial, can in principle be solved analytically. For a more detailed discussion refer to section 2.2.1. Han and Han (2002) reported that numerical noise in the polynomial coefficients caused by limited computer precision was too high (∼ 10 −15 ) when solving the polynomial for the very small mass ratios of satellite and star (∼ 10 −5 ). Therefore, we employ the inverse ray-shooting technique, which has the further advantages of being able to account for finite source sizes and non-uniform source brightness profiles more easily. It also gives us the option of incorporating additional lenses (further planets or moons) without a rise in complexity. This technique was developed by Schneider and Weiß (1986) and Kayser et al. (1986) and enhanced by Wambsganß (1990) and Wambsganß (1999). My starting basis was a January 2008 version of the Wambsganß code, subsequently referred to as microlens. 3.1.1 The microlens code Inverse ray-shooting means that light rays are traced from the observer back to the source plane. This is equivalent to tracing rays from the background source star to the observer plane. The influence of all individual masses in the lens plane on the path of light is calculated. In the thin lens approximation, the deflection angle is just the sum of the deflection angles of every single lens. After deflection, all light rays are collected in pixels of the source plane. Lengths in this map can be translated to angular separations or – assuming a constant relative velocity between source and lens – to time intervals. Any common plotting programme that is capable of handling two-dimensional data can be used to visualise the result, for example Gnuplot. Thus a magnification pattern (cf. figure 3.1) is produced. The number of collected light rays per pixel N is correlated to the magnification µ of a background source at the respective position in the source plane. It is scaled to the usual measure of luminosity in magnitudes as follows: magpixel = −2.5 log 10 µ = −2.5 log 10 N Naverage (3.1) To get the magnification of a certain source, the magnification pattern is convolved with the brightness profile of the source. A light curve is extracted as a one-dimensional cut through the convolved magnification pattern along the source path. microlens goes far beyond being a ray-shooting programme. It makes use of a two-dimensional, hierarchical tree code and a multipole expansion to decide which lenses have to be included exactly to determine the deflection angle for a given light ray, and which can be bundled and evaluated together as a single mass, because they are too distant from the light ray to add significantly to the total deflection angle. Thanks to this algorithm it is possible to account for a very large number of lenses within reasonable computing time. Another feature of microlens is its ability to
Page 1 and 2: Extrasolar Moons as Gravitational M
Page 4: Extrasolar Moons as Gravitational M
Page 8: Never know anything about it. Waste
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 and 20: 10 2.2. BASIC EQUATIONS We can inte
Page 21 and 22: 12 2.2. BASIC EQUATIONS −∆mag F
Page 23: 14 2.3. SEARCH FOR PLANETS VIA GALA
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
Page 33 and 34: 24 3.3. STATISTICAL ANALYSIS curve
Page 35 and 36: 26 3.3. STATISTICAL ANALYSIS detail
Page 37 and 38: 28 3.3. STATISTICAL ANALYSIS We use
Page 39 and 40: 30 3.3. STATISTICAL ANALYSIS
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
Page 51 and 52: 42 4.3. THE STANDARD SCENARIO trans
Page 54 and 55: Chapter 5 Results: Detectability of
Page 56 and 57: CHAPTER 5. DETECTABILITY OF EXTRASO
Page 62 and 63: Chapter 6 Conclusion We showed that
Page 64 and 65: Appendix A Analysed Magnification P
Page 66 and 67: APPENDIX A. ANALYSED MAGNIFICATION
Page 74 and 75:
Appendix B Numerical Results In thi
Page 76:
APPENDIX B. NUMERICAL RESULTS 67 (a
Page 79 and 80:
70 BIBLIOGRAPHY Bevington, Philip R
Page 81 and 82:
72 BIBLIOGRAPHY eprint arXiv:0803.4
Page 83 and 84:
74 BIBLIOGRAPHY Sartoretti, Paola a
Page 85 and 86:
76 BIBLIOGRAPHY
Page 87 and 88:
78 BIBLIOGRAPHY Gabriele Maier. I t
show all

Extrasolar Moons as Gravitational Microlenses Christine Liebig

Create successful ePaper yourself

Delete template?

Save as template?