Extrasolar Moons as Gravitational Microlenses Christine Liebig

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CHAPTER 3. METHOD 23 point of the planetary caustic through which a first horizontal track is cut, then 9 tracks are drawn in parallel on either side, after a rotation of 15 ◦ again 19 parallel tracks are drawn, and so on, until the magnification map is covered with the evenly spaced-grid. Figure 3.6: 228 individual light curves are extracted from each triple lens magnification pattern. To have an unbiased statistical sample, the source trajectories are chosen independently of the lunar caustic features, though all light curves are required to pass through or very close to the planetary caustic. This is realised through generating a grid of source trajectories that is only oriented at of the planetary caustic. In the standard case we use solar sized source, the size of which is indicated in the lower left corner of the pattern. Curve comparison We want to use the properties of the χ 2 -distribution for a test of significance of deviation between two simulated curves. We start in section 3.3.1 with describing the standard procedure for quantifying the goodness-of-fit for any fit of a theoretical
24 3.3. STATISTICAL ANALYSIS curve to experimental data. In section 3.3.2 a method for directly comparing two simulated (light) curves is presented. 3.3.1 Using the χ 2 -distribution as a measure for significance of deviation between data and a theoretical curve After observing a microlensing event, we are eventually left with light curve data. Each of the data points has an associated photometric error and in general the data are randomly scattered with a standard deviation of σi, including error sources such as the photon noise √ photon count of data i or the scintillator noise. For comparing the data with a physical explanation, one has to fix a model and then test the goodness-of-fit of the model light curve to the data light curve to determine the optimal parameter values of the model, or to reject the whole model if no satisfying fit can be found. To test the goodness-of-fit of some theoretical light curve, the cumulative distribution function of the χ 2 -distribution is used to quantify the agreement between observations and theory. Let us recall the general usage of the χ 2 -test. Let Xi (with i = 1, . . . , n) be n independent, normally distributed random variables that have the mean µi = 0 and the standard deviation σi = 1 for all i. The sum over all X 2 i χ 2 ∼ Q 2 = is again a random variable and follows the χ 2 -distribution and can be called χ 2 - distributed with n degrees of freedom. See any standard book with an introduction to calculus of probability, for example Bronstein et al. (2000) or Bevington (1969). To compare data xi that has an associated error, or standard deviation, of σi with a theoretical curve µi, we can create a new random variable Q 2 µ = i=1 n i=1 σi X 2 i n 2 xi − µi . Q 2 µ will follow the χ 2 -distribution, if the assumed theory is correct (⇔ 〈xi − µi〉 = 0) and each summand is weighted with the standard deviation σi of the distribution function of the data spread. As in all science, it is not possible to derive positive affirmation of a theory and ultimately verify it. We can only try to find a discrepancy and falsify it. With a given probability density function, it is possible to calculate the probability that a random variable X lies in an interval [a; b] P (a ≤ X < b) = b a f(x)dx.
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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 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: 22 3.3. STATISTICAL ANALYSIS origin
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
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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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