burgess_tully_descale#

fiasco.util.burgess_tully_descale(x, y, energy_ratio, c, scaling_type)[source]#

Convert scaled Burgess-Tully [BT92] parameters to physical quantities.

For a scaled temperature, \(x\) and scaled effective collision strength \(y\), the effective collision strength can be calculated as a function of the scaled energy \(U=k_BT_e/\Delta E_{ij}\) the ratio between the thermal energy and the energy of the transition \(ij\).

There are 6 different scaling types, depending on the type of transition. This scaling is explained in detail in section 5 of Burgess and Tully [BT92]. The scaled temperatures and collision strengths are related to \(U\) and \(\Upsilon\) by,

  • type 1

    \[x = 1 - \frac{\ln C}{\ln{(U + C)}},\quad y = \frac{\Upsilon}{\log(U + e)}\]

  • type 2

    \[x = \frac{U}{U + C},\quad y = \Upsilon\]

  • type 3

    \[x = \frac{U}{U + C},\quad y = (U + 1)\Upsilon\]

  • type 4

    \[x = 1 - \frac{\ln C}{\ln{(U + C)}},\quad y = \frac{\Upsilon}{\log(U + C)}\]

  • type 5

    \[x = \frac{U}{U + C},\quad y = \Upsilon U\]

  • type 6

    \[x = \frac{U}{U + C},\quad y = \log_{10}\Upsilon\]

where \(C\) is a scaling constant that is different for each transition. Note that Burgess and Tully [BT92] only considered scaling types 1 through 4. Types 5 and 6 correspond to dielectron and proton transitions, respectively.

To “descale” the scaled effective collision strengths that are stored in the database, a spline fit is computed to the new \(x\) as computed from \(U\) and then the relationship between \(\Upsilon\) and \(y\) is inverted to get \(\Upsilon\) as a function of \(U\).

Parameters:

  • x (array-like) – Scaled temperature. First dimension should have length n, the number of transitions. The second dimension will be the number of spline points, but may be different for each row. If each row has l spline points, x should have shape (n,l). If they are not all equal, x will have shape (n,).

  • y (array-like) – Scaled collision strength. Must have the same dimensions as x.

  • energy_ratio (array-like) – Ratio between the thermal energy and that of each transition with shape (n,m), where m is the dimension of the temperature array.

  • c (array-like) – Scaling constant for each transition with shape (n,)

  • scaling_type (array-like) – The type of descaling to apply for each transition with shape (n,). Must be between 1 and 6

Returns:

upsilon (array-like) – Descaled collision strength or cross-section with the same shape as energy_ratio.