FINER system temperature
========================

The Far-Infrared Nebular Emission Receiver (`FINER <https://finerreceiver.github.io/>`__) is being built for the LMT Observatory. It covers the frequency ranges between 120 and 360 GHz. It is set-up to cover channel spacings from 88 kHz to 0.18 GHz, with two distinct receiver temperatures characterising the lower and upper frequency ranges the instrument is sensitive to (breaking at 210 GHz). See `Tamura et al. 2024 <https://arxiv.org/abs/2406.07975>`__ for more details.

Here in the AtLAST sensitivity calculator, we use it as an exemplar of a heterodyne instrument capable of working at these frequencies.

In this module, we calculate the system temperature used in the overall :doc:`sensitivity equation <sensitivity>`. For a FINER like system, the system temperature is calculated as:

.. math::
    T_{sys} = \frac{1}{\eta_\mathrm{eff} \mathfrak{t}} \times [T_\mathrm{rx} + (\eta_\mathrm{eff} T_\mathrm{sky}) + (1-\eta_\mathrm{eff}) O(\nu, T_\mathrm{amb})]

where

* :math:`\eta_\mathrm{eff}` is the forward efficiency of the telescope
* :math:`\mathfrak{t}` is the atmospheric transmittance, defined as :math:`\mathfrak{t} = \textrm{exp}^{(-\tau_{atm})}`
* :math:`T_\mathrm{rx}` is the receiver temperature
* :math:`T_\mathrm{sky}` is the sky temperature (in terms of a Rayleigh-Jeans brightness temperature) calculated from the model grid described in :doc:`Weather Calculations <weather>`
* :math:`T_\mathrm{amb}` is the ambient temperature

Here, :math:`O(\nu, T)` converts a physical temperature to a Rayleigh-Jeans brightness temperature

.. math::
    O(\nu, T) = T\frac{h\nu/kT}{\exp(h\nu/kT)-1}.

Through consultation with the developers of the FINER instrument, we assume a constant receiver temperature across each sub-band the receivers are sensitive to:

.. math::
    T_{rx} = \left\{ \begin{array}{rcl}
            45\,\mbox{K} & \mbox{for} & 120<\nu<210\,\mbox{GHz} \\ 
            75\,\mbox{K} & \mbox{for} & 210<\nu<360\,\mbox{GHz} 
            \end{array}\right.

These values are used in the :math:`T_{sys}` equation above, which is in turn used to calculate the System Equivalent Flux Density used in the overall sensitivity (or integration time) calculation.