Calculating Time
----------------
The Exposure Time is calculated based on the user input SNR, instrument
configuration and input source spectrum. For spectroscopic modes, this value is
calculated at a particular wavelength which is specified by the user in Section
2 of the ETC forms.
**Please  make sure the wavelength entered in this section is covered by the selected mode.**
With this information, the count rate terms (in photons/sec) are retrieved from
*pysynphot*. Each photon count rate term is depicted below with an :sub:`r` subscript.
Taking formula (ii) from the previous page and expanding the count terms, we
get the following:


.. math::

    SNR = \frac{S_r * t}{sqrt{S_r * t  + Sky_r * t  + DC_r * t  + RN^2  + Thermal_r * t +  CS_r * t}}

This can be easily changed to:

.. math::

    S_r^2  *  t^2  -  SNR^2 * [ S_r + Sky_r + DC_r + Thermal_r + CS_r] * t  - SNR^2 * RN^2  = 0


Then solving for t gives us:

.. math:: 

    t = \frac{b + sqrt{b^2 + 4ac}}{2a}


*Where*:


:math:`b = (S_r + BG_r) * SNR^2`

:math:`c = RN^2 * SNR^2`

:math:`BG_r = ES_r + ZL_r + GL_r + DC_r + Thermal_r + CS_r`