# 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 $T$:

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

At radio frequencies ($h\nu \ll k_B T$) this reduces to $T$,
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 $h\nu / k_B T < 10^{-10}$):

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

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

## Physics Constants

Two modes are supported for backward compatibility:

```{eval-rst}
.. 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
```

`KalibrateCompat` is the **default** (`PhysicsConstants::default()`).
It 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:

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

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

where $g_s x_s$ and $g_i x_i$ 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()`