============================
Rayleigh-Jeans Correction
============================

At submillimeter wavelengths the Rayleigh-Jeans approximation breaks
down, and the full Planck function must be used to relate physical
temperature to brightness temperature.

Physical Motivation
-------------------

The Rayleigh-Jeans temperature is the equivalent brightness temperature
that would produce the same spectral radiance as a blackbody at physical
temperature :math:`T`:

.. math::

   T_{RJ}(\nu, T) = \frac{h\nu / k_B}{\exp\!\left(\frac{h\nu}{k_B T}\right) - 1}

At radio frequencies (:math:`h\nu \ll k_B T`) this reduces to :math:`T`,
but at 4.7 THz (upGREAT) the correction is significant — a 300 K load
has an effective RJ temperature of only ~290 K.

Low-frequency limit (used when :math:`h\nu / k_B T < 10^{-10}`):

.. math::

   T_{RJ} \approx T - \frac{h\nu}{2 k_B}

**Implementation:** ``cal-core/src/math/rayleigh_jeans.rs`` →
``antenna_temperature_rj()``

Physics Constants
-----------------

Two modes are supported for backward compatibility:

.. list-table::
   :header-rows: 1
   :widths: 25 35 40

   * - Mode
     - :math:`h/k_B`
     - Precision
   * - ``Exact`` (SI 2019)
     - :math:`4.799\,243\,073\,366 \times 10^{-11}` K/Hz
     - f64 throughout
   * - ``KalibrateCompat``
     - :math:`0.047995 \times 10^{-9}` K/Hz
     - f32 truncation to match legacy code

The ``KalibrateCompat`` mode reproduces the exact numerical behaviour of
the legacy kalibrate C++/Fortran code, enabling parity validation.

**Implementation:** ``cal-core/src/config.rs`` → ``PhysicsConstants`` enum

Sideband-Weighted (Primed) Temperatures
----------------------------------------

Heterodyne receivers are sensitive to both sidebands. The effective
("primed") load temperatures weight each sideband by its gain:

.. math::

   T'_{hot} = \frac{g_s x_s \cdot T_{RJ}(\nu_{sig}, T_{hot})
   + g_i x_i \cdot T_{RJ}(\nu_{img}, T_{hot})}
   {g_s x_s + g_i x_i}

.. math::

   T'_{cold} = \frac{g_s x_s \cdot T_{RJ}(\nu_{sig}, T_{cold})
   + g_i x_i \cdot T_{RJ}(\nu_{img}, T_{cold})}
   {g_s x_s + g_i x_i}

where :math:`g_s x_s` and :math:`g_i x_i` are the signal and image
sideband gain-suppression products (typically both 0.5 for DSB receivers).

**Implementation:** ``cal-io/src/resolve.rs`` →
``resolve_calibration_load()``