Estimating flux densities
The algorithm used to estimate a sky-model component's flux density depends on the flux-density type.
Note that in the calculations below, flux densities are allowed to be negative. It is expected, however, that a sky-model component with a negative flux density belongs to a source with multiple components, and that the overall flux density of that source at any frequency is positive. A source with a negative flux density is not physical.
Both power-law and curved-power-law sources have a spectral index (\( \alpha \)) and a reference flux density (\( S_0 \)) defined at a particular frequency (\( \nu_0 \)). In addition to this, curved power laws have a curvature term (\( q \)).
To estimate a flux density (\( S \)) at an arbitrary frequency (\( \nu \)), a ratio is calculated:
\[ r = \left(\frac{\nu}{\nu_0}\right)^\alpha \]
For power laws, \( S \) is simply:
\[ S = S_0 r \]
whereas another term is needed for curved power laws:
\[ c = \exp\left({q \ln\left(\frac{\nu}{\nu_0}\right)^2 }\right) \] \[ S = S_0 r c \]
\( S \) can represent a flux density for Stokes \( \text{I} \), \( \text{Q} \), \( \text{U} \) or \( \text{V} \). The same \( r \) and \( c \) values are used for each Stokes flux density.
To estimate a flux density (\( S \)) at an arbitrary frequency (\( \nu \)), a number of considerations must be made.
In the case that a list only has one flux density, we must assume that it is a power law, use a default spectral index (\( -0.8 \)) for it and follow the algorithm above.
In all other cases, there are at least two flux densities in the list (\( n >= 2 \)). We find the two list frequencies (\( \nu_i \)) and (\( \nu_j \)) closest to \( \nu \) (these can both be smaller and larger than \( \nu \)). If the flux densities \( S_i \) and \( S_j \) are both positive or both negative, we proceed with the power law approach: A spectral index is calculated with \( \nu_i \) and \( \nu_j \) (\( \alpha \)) and used to estimate a flux density with the power law algorithm. If \( \alpha < -2.0 \), a trace-level message is emitted, indicating that this is a very steep spectral index.
If the signs of \( S_i \) and \( S_j \) are opposites, then we cannot fit a spectral index. Instead, we fit a straight between \( S_i \) and \( S_j \) and use the straight line to estimate \( S \).
No estimation is required when \( \nu \) is equal to any of the list frequencies \( \nu_i \).
When estimating flux densities from a list, it is feared that the "jagged" shape of a component's spectral energy distribution introduces artefacts into an EoR power spectrum.
It is relatively expensive to estimate flux densities from a list type. For all these reasons, users are strongly encouraged to not use list types where possible.