Appendix: The Photon Model Terms

Notes:

1) The pivot energy ``normalizes'' the energy at which the model is evaluated and should always be held fixed in the fitting process. Choosing a pivot energy in the middle of the fit energy range will reduce the cross-correlations of the other model parameters.
2) The photon model f is photon number flux in photons s  cm  keV .
3) The total photon model is composed of the sum of the selected additive terms times the product of the selected multiplicative terms ,

4) You must select at least one additive term. If no multiplicative terms are selected, the product defaults to one.
5) All energies are in keV.

• Term 1, Power Law:
  1. A = amplitude in photons s  cm  keV ,
  2. = pivot energy in keV,
  3. = index.

  

• Term 2, Broken Power Law:
  1. A = amplitude in photons s  cm  keV ,
  2. = pivot energy in keV,
  3. = index below break,
  4. = break energy in keV,
  5. = index above break.

  

• Term 3, Broken Power Law with Two Breaks:
  1. A = amplitude in photons s  cm  keV ,
  2. = pivot energy in keV,
  3. = index below first break,
  4. = first break energy in keV,
  5. = index between first and second break,
  6. = second break energy in keV,
  7. = index above second break.

  

• Term 4: Smoothly Broken Power Law: This model is derived as follows: we want to be a horizonal line at y= for small E, then later a horizontal line at y= for large E. This is done by making a line function (y = mx + b) applied to the hyperbolic cosine.
  1. A = amplitude in photons s  cm  keV ,
  2. = pivot energy in keV,
  3. , lower index,
  4. , break energy in keV,
  5. , break scale in decades of energy,
  6. , upper index.

  

  

  

  

  

  

  

• Term 5: David Band's Gamma-Ray Burst function (ApJ, 413, 281, 1993), parameterization, a different parameterization than that given in the ApJ article. The relation is .
  1. A = amplitude in photons s  cm  keV ,
  2. in keV,
  3. low-energy index ,
  4. high-energy index .

  

  if

  

  and

  

  if

  

• Term 6: David Band's Gamma-Ray Burst function (ApJ, 413, 281, 1993), ``old'' parameterization:
  1. A = amplitude in photons s  cm  keV ,
  2. in keV,
  3. low-energy index ,
  4. high-energy index .

  

• Term 7, Comptonized, parameterization. The relation between the old and new parameterizations is .
  1. A = amplitude in photons s  cm  keV ,
  2. in keV,
  3. , index,
  4. = pivot energy in keV.

  

• Term 8, Comptonized, ``old'' parameterization:
  1. A = amplitude in photons s  cm  keV ,
  2. kT in keV,
  3. , index,
  4. = pivot energy in keV.

  

• Term 9, J. J. Brainerd's scattered power law (ApJ, 428, 21-27, 1994). The parameterization has been revised to use a parameter ratio involving and instead of . This reduces the cross-correlation between the parameters.
  1. A = amplitude in photons s  cm  keV ,
  2. Parameter ratio = , where is the index of the unscattered power law,
  3. , optical depth,
  4. z, cosmological redshift,
  5. M, metallicity (solar = 1).

  

  where

  

  is the Klein-Nishina cross-section, and is the photo-electric cross-section:

  

• Term 10, Log Normal:
  1. A = amplitude in photons s  cm  keV ,
  2. , of energy,
  3. .

  

• Term 11: Gaussian (log E):
  1. A = amplitude in photons s  cm  keV ,
  2. , centroid energy in keV,
  3. W, log FWHM, decades of energy.

  

  

• Term 12: Gaussian (log E) with linearly varying FWHM:
  1. A = amplitude in photons s  cm  keV ,
  2. , centroid energy in keV,
  3. W, log FWHM at , decades of energy.
  4. s, slope of W w.r.t. , decades per decade.

  

  

  

• Terms 13 and 14, Sunyaev-Titarchuk Comptonization spectra (references: Patrick Nolan's Ph.D thesis has the exact notation used herein; Sunyaev and Titarchuk, A&A, 86, 121, 1980).
  1. A = amplitude in photons s  cm  keV ,
  2. kT = electron energy in keV,
  3. = optical depth,
  4. G = geometry factor, which should be fixed at 3 for a spherical cloud and at 12 for a disk of electrons.

  

  where

  

  

• Terms 15 and 16, Optically-Thin Thermal Bremsstrahlung (OTTB):
  1. A = amplitude in photons s  cm  keV ,
  2. kT = electron energy in keV,
  3. = pivot energy in keV.

  

• Terms 17 and 18, Black Body:
  1. A = amplitude in photons s  cm  keV ,
  2. kT = temperature in keV.

  

• Term 19, power law parameterized by photon number flux:
  1. = Photon flux in photons s  cm between and ,
  2. = index,
  3. = lower bound of energy range over which flux is integrated: leave fixed,
  4. = upper bound of energy range over which flux is integrated: leave fixed.
  Amplitude A in photons s  cm  keV is

  

  so that

  

• Term 20, power law parameterized by energy flux:
  1. = Energy flux in keV s  cm between and ,
  2. = index,
  3. = lower bound of energy range over which flux is integrated: leave fixed,
  4. = upper bound of energy range over which flux is integrated: leave fixed.
  Amplitude A in photons s  cm  keV is

  

  or, for ,

  

  so that

  

• Term 21, Optically-Thin Thermal Bremsstrahlung (OTTB) parameterized by photon number flux:
  1. = Photon number flux in photons s  cm between & ,
  2. kT,
  3. = lower bound of energy range over which flux is integrated: leave fixed,
  4. = upper bound of energy range over which flux is integrated: leave fixed.
  Amplitude A in photons s  cm  keV is

  

  The integral must be evaluated numerically. Then

  

• Term 22, Optically-Thin Thermal Bremsstrahlung (OTTB) parameterized by energy flux:
  1. = energy flux in keV s  cm between and ,
  2. kT,
  3. = lower bound of energy range over which flux is integrated: leave fixed,
  4. = upper bound of energy range over which flux is integrated: leave fixed.
  Amplitude A in photons s  cm  keV is

  

  so that

  

• Term 23, Yang Soong's pulsar spectral form (references: Y. Soong, Ph.D thesis, Univ. of Calif., San Diego, 1988; Y. Soong, et al., ApJ, 348, 641, 1990):
  1. A = amplitude in photons s  cm  keV ,
  2. power law index ,
  3. break energy in keV,
  4. E-folding energy in keV,
  5. equivalent width of line, in keV,
  6. line centroid in keV,
  7. Full-Width at Half-Max of line, FWHM, in keV,
  8. pivot energy in keV,

  

  where

  

  

  and

  

• Term 24, Tanaka's pulsar model (reference: J. Groove, et al., ApJ, 438, L25-L28, 1995):
  1. A = amplitude in photons s  cm  keV ,
  2. ,
  3. kT,
  4. optical depth of first line, ,
  5. optical depth of second line, ,
  6. number of lines, ; fix at either 0, 1 or 2,
  7. line centroid of first line in keV,
  8. line width W, HWHM in keV,
  9. in keV.

  

  where

  

  

  

  and

  

• Terms 25, improved Titarchuk Comptonization spectra (reference: Hua and Titarchuk, ApJ, 1995).
  1. A = amplitude in photons s  cm  keV ,
  2. = electron energy in keV,
  3. = optical depth; Function is currently correct only for !
  4. G = geometry factor, which should be fixed at 3 for a spherical cloud and at 12 for a disk of electrons.
• Terms 34, 35 & 36, Gaussian Lines: These terms are actually evaluated using the error function (the complementary error function erfcc of Numerical Recipes by Press et al.) so that the mean of the function over the photon bin is used, rather than its value at the center. [For the other terms, if the photon bin is wide, the mean over the photon bin is evaluated numerically]. The energy edges of the photon bins are designated and .
  1. A = amplitude in photons/s-cm ,
  2. , centroid energy in keV,
  3. FWHM in keV.

  

  

  

  

• Term 40, (Relative) Effective Area Correction. This term is a crude model to empirically improve the detector response model. The detector response model is assumed to be incorrect only in the effective area assigned each detector. The detectors are brought into better agreement by dividing the input photon spectrum for each detector by an ad hoc correction factor; this is equivalent to multiplying the effective area by the same factor. Each detector has a different factor, which are all normalized with respect to the zeroth detector. The factors are assumed to be independent of energy.
  1. , correction factor 1st detector w.r.t 0th detector,
  2. , correction factor 2nd detector w.r.t 0th detector,
  3. , correction factor 3rd detector w.r.t 0th detector,
  4. , correction factor 4th detector w.r.t 0th detector,
  5. , correction factor 5th detector w.r.t 0th detector,
  6. , correction factor 6th detector w.r.t 0th detector,
  7. , correction factor 7th detector w.r.t 0th detector,
  8. , correction factor 8th detector w.r.t 0th detector,
  9. , correction factor 9th detector w.r.t 0th detector,
  10. , correction factor 10th detector w.r.t 0th detector,
  If there are N detectors numbered k=0, 1, ... N, then for the 0th detector

  

  while for the kth detector, k greater than 0,

  

• Term 41, low-energy cutoff:
  1. cutoff energy ,
  2. folding energy ,

  

• Term 42, high-energy cutoff:
  1. cutoff energy ,
  2. folding energy ,

  

• Term 43, Multiplicative Power Law:
  1. index ,
  2. pivot energy .

  

• Term 44, Interstellar absorption (reference: Morrison and McCammon, ApJ, 270, 119-122, 1983). Above 10 keV, function is an extrapolation of that of Morrison and McCammon; accuracy decreases due to neglect of Compton scattering. Scattering is by all elements according to solar (?) abundances even though function is specified in hydrogen column depth.
  1. Hydrogen column depth in units of Hydrogen atoms per cm .
• Terms 45 and 46, multiplicative Gaussian lines:
  1. Intensity I,
  2. Centroid in keV,
  3. FWHM in keV,

  

  

  

• Term 47, multiplicative Lorentzian line:
  1. Intensity I,
  2. Centroid in keV,
  3. width W in keV,

  

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