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114 | 114 | "If you're new to extinction, here is a brief introduction to the types of quantities involved.\n", |
115 | 115 | "The fractional change to the flux of starlight is \n", |
116 | 116 | "$$\n", |
117 | | - "\\frac{dF_\\lambda}{F_\\lambda} = -\\tau_\\lambda", |
| 117 | + "\\frac{dF_\\lambda}{F_\\lambda} = -\\tau_\\lambda\n", |
118 | 118 | "$$\n", |
| 119 | + "\n", |
119 | 120 | "where $\\tau$ is the optical depth and depends on wavelength. Integrating along the line of sight, the resultant flux is an exponential function of optical depth,\n", |
120 | 121 | "$$\n", |
121 | 122 | "\\tau_\\lambda = -\\ln\\left(\\frac{F_\\lambda}{F_{\\lambda,0}}\\right).\n", |
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125 | 126 | "$$\n", |
126 | 127 | "\\tau_\\lambda = -2.303\\log\\left(\\frac{F_\\lambda}{F_{\\lambda,0}}\\right),\n", |
127 | 128 | "$$\n", |
| 129 | + "\n", |
128 | 130 | "and define an extinction $A_\\lambda = 1.086 \\,\\tau_\\lambda$ so that\n", |
129 | 131 | "$$\n", |
130 | 132 | "A_\\lambda = -2.5\\log\\left(\\frac{F_\\lambda}{F_{\\lambda,0}}\\right).\n", |
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168 | 170 | "$$\n", |
169 | 171 | "R_V \\equiv \\frac{A_V}{E_{B-V}}\n", |
170 | 172 | "$$\n", |
| 173 | + "\n", |
171 | 174 | "where $E_{B-V}$ is differential extinction $A_B-A_V$. In this example, we show the $R_V$-parameterization for the Clayton, Cardelli, & Mathis (1989, CCM) and the Fitzpatrick (1999) models. [More model options are available in the `dust_extinction` documentation.](https://dust-extinction.readthedocs.io/en/latest/dust_extinction/model_flavors.html)" |
172 | 175 | ] |
173 | 176 | }, |
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557 | 560 | "$$\n", |
558 | 561 | "A = -2.5\\log\\left(\\frac{\\int W_\\lambda F_{\\lambda,0} 10^{-0.4A_\\lambda} d\\lambda}{\\int W_\\lambda F_{\\lambda,0} d\\lambda} \\right),\n", |
559 | 562 | "$$\n", |
| 563 | + "\n", |
560 | 564 | "where $W_\\lambda$ is the fraction of incident energy transmitted through a filter. See the detailed appendix in [Bessell & Murphy (2012)](https://ui.adsabs.harvard.edu/#abs/2012PASP..124..140B/abstract)\n", |
561 | 565 | " for an excellent review of the issues and common misunderstandings in synthetic photometry.\n", |
562 | 566 | "\n", |
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