Formal integral documentation (#1921)

* creating file * rough draft minus I_nu(p) calculation * typo fix * adding ggb file * adding images * typo fix * changing images * very rough draft * fixing link * making edits * fixing links * fixing equation * fix docstring
tardis-sn · May 23, 2022 · 9f6d855 · 9f6d855
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diff --git a/docs/physics/est_and_conv/estimators.ipynb b/docs/physics/est_and_conv/estimators.ipynb
@@ -36,7 +36,7 @@
     "\n",
     "</div>\n",
     "\n",
-    "These estimators allow us to calculate the [radiative temperature](../setup/model.ipynb#Radiative-Temperature) $T_\\mathrm{rad}$ and [dilution factor](../setup/model.ipynb#Dilution-Factor) $W$ in each shell using\n",
+    "These estimators allow us to calculate the [radiative temperature](../setup/model.ipynb#Temperatures) $T_\\mathrm{rad}$ and [dilution factor](../setup/model.ipynb#Dilution-Factor) $W$ in each shell using\n",
     "\n",
     "$$T_{\\mathrm{rad}} = \\frac{h}{k_{\\mathrm{B}}} \\frac{\\pi^4}{360 \\zeta(5)} \\frac{\\bar \\nu}{J}$$\n",
     "\n",
@@ -86,7 +86,7 @@
     "\n",
     "</div>\n",
     "\n",
-    "If set to detailed mode (see [plasma configuration](../../io/configuration/components/plasma.rst)), the `J_blue` plasma property will will be replaced with the value of the $J^b_{lu}$ estimator (the raw estimator times the factor of $\\frac{ct_\\mathrm{explosion}}{4\\pi V \\Delta t}$). Otherwise, the `J_blue` in the plasma are calculated as they typically are in the plasma calculations, and the $J^b_{lu}$ estimator is only used for the [formal integral](../spectrum/sourceintegration.rst)."
+    "If set to detailed mode (see [plasma configuration](../../io/configuration/components/plasma.rst)), the `J_blue` plasma property will will be replaced with the value of the $J^b_{lu}$ estimator (the raw estimator times the factor of $\\frac{ct_\\mathrm{explosion}}{4\\pi V \\Delta t}$). Otherwise, the `J_blue` in the plasma are calculated as they typically are in the plasma calculations, and the $J^b_{lu}$ estimator is only used for the [formal integral](../spectrum/formal_integral.rst)."
    ]
   },
   {

diff --git a/docs/physics/images/formal_integral_2d.ggb b/docs/physics/images/formal_integral_2d.ggb
diff --git a/docs/physics/images/formal_integral_2d_above.png b/docs/physics/images/formal_integral_2d_above.png
diff --git a/docs/physics/images/formal_integral_2d_below.png b/docs/physics/images/formal_integral_2d_below.png
diff --git a/docs/physics/images/formal_integral_sphere.ggb b/docs/physics/images/formal_integral_sphere.ggb
diff --git a/docs/physics/images/formal_integral_sphere.jpg b/docs/physics/images/formal_integral_sphere.jpg
diff --git a/docs/physics/montecarlo/propagation.rst b/docs/physics/montecarlo/propagation.rst
@@ -155,6 +155,7 @@ When a packet is moved into a new cell, as mentioned before, it is moved to the
 boundary, the plasma properties are recalculated, and the propagation direction of the packet is updated (using
 :math:`\mu_f = \frac{l + r_i \mu_i}{r_f}`).
 
+.. _physical-interactions:
 
 Physical Interactions
 ---------------------

diff --git a/docs/physics/spectrum/basic.ipynb b/docs/physics/spectrum/basic.ipynb
@@ -422,7 +422,7 @@
    "source": [
     "You may notice that the bins are not uniformly spaced with respect to wavelength. This is because we use the same bins for both wavelength and frequency, and thus the bins which are uniformly spaced with respect to frequency are not uniformly spaced with respect to wavelength. This is okay though, since luminosity density allows us to alter the bins without significantly altering the graph!\n",
     "\n",
-    "Another thing you may notice in these graphs is the lack of a smooth curve. This is due to **noise**: the effects of the random nature of Monte Carlo simulations. This makes it very difficult to get a precise spectrum without drastically increasing the number of Monte Carlo packets. To solve this problem, TARDIS uses [virtual packets](virtualpackets.rst) and [the formal integral method](sourceintegration.rst) to generate a spectrum with less noise."
+    "Another thing you may notice in these graphs is the lack of a smooth curve. This is due to **noise**: the effects of the random nature of Monte Carlo simulations. This makes it very difficult to get a precise spectrum without drastically increasing the number of Monte Carlo packets. To solve this problem, TARDIS uses [virtual packets](virtualpackets.rst) and [the formal integral method](formal_integral.rst) to generate a spectrum with less noise."
    ]
   }
  ],

diff --git a/docs/physics/spectrum/formal_integral.rst b/docs/physics/spectrum/formal_integral.rst
diff --git a/docs/physics/spectrum/index.rst b/docs/physics/spectrum/index.rst
@@ -17,4 +17,4 @@ below.
 .. toctree::
     basic
     virtualpackets
-    sourceintegration
+    formal_integral
diff --git a/docs/physics/spectrum/sourceintegration.rst b/docs/physics/spectrum/sourceintegration.rst
diff --git a/tardis/montecarlo/montecarlo_numba/formal_integral.py b/tardis/montecarlo/montecarlo_numba/formal_integral.py
@@ -58,6 +58,7 @@ def numba_formal_integral(
         intensities at each p-ray multiplied by p
         frequency x p-ray grid
     """
+
     # todo: add all the original todos
     # Initialize the output which is shared among threads
     L = np.zeros(inu_size, dtype=np.float64)