SPEX Reference manual (PDF) - SRON

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118 Response matrices Figure 7.1: Displacement λ k as a function of the number of resolution elements R. A polynomial approximation is given in the text. Table 7.6: Coefficients for the approximations for λ k and c α i a i b i 0 0.1342 1.3596 1 -0.3388 4.0609 2 0.7994 -4.3522 3 -1.1697 5.2225 4 0.9053 -3.7881 5 -0.2811 1.1491 7.3.8 Examples Let us apply the theory derived before to some specific line spread functions. First we consider a Cauchy (or Lorentz) distribution, given by f(x) = a π(a 2 +x 2 ) , (7.30) where a is the width parameter. The FWHM equals 2a. Next we consider the Gaussian distribution, given by f(x) = 1 √ 2πσ e −x2 /2σ 2 , (7.31)
7.3 Data binning 119 Figure 7.2: Critical value c α as a function of the number of resolution elements R. A polynomial approximation is given in the text. with σ the standard deviation. The FWHM is given by √ ln256σ, which is approximately 2.3548σ. We have computed the value for δ ≡ λ k / √ N numerically for both distributions. We have taken into account also the effect of possible phase shifts, i.e. that the center of the distribution does not coincide with a grid point but with a certain phase of the grid. The maximum bin width ∆/FWHM as a function of the maximum absolute difference δ between the true and approximatedcumulative density distribution is plotted for both distributions in fig. 7.3. We can approximate both curves with sufficient accuracy by a polynomial in x ≡ 10 logδ as 10 log(∆/FWHM) = 10 log(∆/FWHM) = 4∑ 10 log(δ) c i ( ) i , (7.32) 10 i=0 4∑ 10 log(δ) g i ( ) i , (7.33) 10 for the Cauchydistribution (7.32)and Gaussiandistribution (7.33) respectively, with the coefficents given in table 7.7. i=0 Table 7.7: Coefficients for the approximations for the Cauchy and Gauss distribution. i c i g i 0 0.400 0.277 1 5.065 3.863 2 9.321 8.470 3 9.333 9.496 4 3.584 3.998 The rms deviation of these fits over the range 10 −9 < δ < 1 is 0.0086 for the Cauchy distribution and
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SPEX Reference Manual Jelle Kaastra
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4 CONTENTS 2.29 System . . . . . .
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6 CONTENTS 6.6 Rgsvprof . . . . . .
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8 Introduction 1.2.2 Sky sectors Fi
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10 Introduction
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12 Syntax overview Table 2.2: Abund
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14 Syntax overview tcl: the continu
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16 Syntax overview Examples calc -
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18 Syntax overview data delete inst
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20 Syntax overview dem mult #i - Do
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22 Syntax overview distance show -
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24 Syntax overview Examples elim 0.
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26 Syntax overview There is yet ano
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28 Syntax overview ignore 1:8 unit
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30 Syntax overview written to this
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32 Syntax overview eV, keV, Å, as
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34 Syntax overview (Z = 7), ... Ni
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36 Syntax overview plot view x1 #r
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38 Syntax overview plot model lw #i
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40 Syntax overview 2.25 Shiftplot:
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42 Syntax overview 2.27 Step: Grid
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44 Syntax overview system exe #a -
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46 Syntax overview Update: several
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48 Syntax overview Also the totals
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50 Spectral Models Table 3.1: Avail
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52 Spectral Models nr name chemical
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54 Spectral Models (cf. Rybicky & L
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56 Spectral Models 3.6.5 Abundances
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58 Spectral Models 3.8 dabs: dust a
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60 Spectral Models 3.11 Dem: Differ
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62 Spectral Models • ... • last
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64 Spectral Models f9 - Transmissio
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66 Spectral Models 3.22 Neij: Non-E
Page 68 and 69: 68 Spectral Models Warning: By defa
Page 70 and 71: 70 Spectral Models seeing equal con
Page 72 and 73: 72 Spectral Models • type=4: b 1
Page 74 and 75: 74 Spectral Models v - Velocity bro
Page 76 and 77: 76 Spectral Models For more informa
Page 78 and 79: 78 More about spectral models Slab
Page 80 and 81: 80 More about spectral models depth
Page 82 and 83: 82 More about spectral models grid
Page 84 and 85: 84 More about spectral models The e
Page 86 and 87: 86 More about plotting Table 5.1: A
Page 88 and 89: 88 More about plotting 5.5 Plot tex
Page 90 and 91: 90 More about plotting Table 5.4: S
Page 92 and 93: 92 More about plotting Table 5.5: x
Page 94 and 95: 94 More about plotting the plot wil
Page 96 and 97: 96 More about plotting
Page 98 and 99: 98 Auxilary programs Property Spect
Page 100 and 101: 100 Auxilary programs Property Resp
Page 102 and 103: 102 Auxilary programs 15. Optional:
Page 104 and 105: 104 Auxilary programs 6.3 RGS fluxc
Page 106 and 107: 106 Auxilary programs This redistri
Page 108 and 109: 108 Auxilary programs 6.6.1 Theory
Page 110 and 111: 110 Response matrices models. Since
Page 112 and 113: 112 Response matrices Table 7.1: We
Page 114 and 115: 114 Response matrices Table 7.2: Sc
Page 116 and 117: 116 Response matrices since the lar
Page 120 and 121: 120 Response matrices Figure 7.3: R
Page 122 and 123: 122 Response matrices 7.4 Model bin
Page 124 and 125: 124 Response matrices In this secti
Page 126 and 127: 126 Response matrices The resulting
Page 128 and 129: 128 Response matrices where w 1 and
Page 130 and 131: 130 Response matrices spectrumextra
Page 132 and 133: 132 Response matrices may be helpfu
Page 134 and 135: 134 Response matrices 1. The OGIP r
Page 136 and 137: 136 Response matrices Table 7.9: Se
Page 138 and 139: 138 Response matrices could be the
Page 140 and 141: 140 Response matrices Table 7.13: R
Page 142 and 143: 142 Response matrices Figure 7.8: B
Page 144 and 145: 144 Response matrices Figure 7.9: B
Page 146 and 147: 146 Installation and testing
Page 148 and 149: 148 Acknowledgements
Page 150 and 151: 150 BIBLIOGRAPHY [1989] Kaastra, J.
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