Radiative Models of Coronal Condensation
at Extreme Resolution:

DexRT

Chris Osborne

University of Glasgow

Contents

• Where are we now?
• Radiance Cascades ✨
• Application
• Outlook

Promotion…

Non-LTE Radiative Transfer

• Coupling between radiation field and atomic transitions.
• Extremely computationally demanding, but necessary for many spectral lines.
• Needed for models, diagnostics, and inversions.
• Also, ionisation.

HINODE BFI

Most Important Quantities

• Radiative rates depend on intensity.
• Intensity depends on radiative rates.

\begin{align*} J_\nu(\vec{p}) &= \frac{1}{4\pi}\oint_\mathbb{S^2} I_\nu(\vec{p},\hat{\omega}) \mathop{d \hat{\omega} } \\ &\vec{p} \in \mathbb{R}^3, \hat{\omega} \in \mathbb{S}^2\\ \end{align*}

• Specific intensity and its first moment.
• 4 and 6 dimensional functions!

Non-LTE Radiative Transfer

• RT codes designed around the idea of smooth atmospheres…

• But ours look like this

Jenkins & Keppens 2021

Carlsson+ 2016

Przybylski+ 2022

Prominence Albedo

Jenkins+ 2024

Ray Effects

Desired Result

How do we build this?

Observations

\begin{cases} A < B,\\ \alpha > \beta. \end{cases}

with some small angles approximations…

\begin{cases} \Delta_s < F(D) \propto D\quad&\mathrm{(spatial)}\\ \Delta_\omega < G(1/D) \propto 1/D\quad&\mathrm{(angular)} \end{cases}

• We can exploit this!
• Represent contributions from different shells separately.

How do we use this?

• Encode radiance field as a set of cascades, each describing I_\nu in annuli around \vec{p}.
• If penumbra criterion is satisfied, cascade is linearly interpolateable!
• Simple (maybe not optimal) definition of cascade i covering [t_i, t_{i+1}], \alpha \geq 1:

\begin{cases} \Delta_s \propto 2^i,\\ \Delta_\omega \propto \cfrac{1}{2^{\alpha i}},\\ t_i \propto 2^{\alpha i} \end{cases}

• Exponential scaling
• A single radiation field sample \mathcal{R}_{t_i, t_{i+1}}(\vec{p},\hat{\omega}) termed radiance interval.
• But what do these cascades look like?

Probe Interpolation

• This construction sparsely encodes the radiation field throughout the domain…
• But not in a form we can use directly.
• Solution:

Interpolation. But with minimal error.

Interpolated Ray Analysis

• Each cascade i+1 has 1/2 the rays of cascade i (denoted N_{C_i})
• By interpolation we construct \mathcal{R}_{t_{i+1},t_{i+2}}(\vec{p},\hat{\omega}) for each probe of cascade i \left([t_i, t_{i+1}]\right).
• Merging cascades 0 and 1:

\begin{cases} \text{Rays computed} = N_{C_0} + N_{C_1} = N_{C_0} + \cfrac{N_{C_0}}{2},\\ \text{Rays constructed} = 2N_{C_0}. \end{cases}

• In general:

\begin{cases} \text{Rays computed} = \sum_I N_{C_i} = \left(1 + \cfrac{1}{2} + \cdots + \cfrac{1}{2^I} \right) < 2N_{C_0},\\ \text{Rays constructed} = 2^I N_{C_0}. \end{cases}

Sublinear (bounded) scaling in compute, exponential scaling in constructed rays!

Interpolated Ray Analysis cont’d

• For a branching factor \alpha=2 in 2D
• Each cascade has the same number of rays.
• Merging cascades 0 and 1:

\begin{cases} \text{Rays computed} = N_{C_0} + N_{C_1} = 2N_{C_0},\\ \text{Rays constructed} = 4N_{C_0}. \end{cases}

• In general:

\begin{cases} \text{Rays computed} = \sum_I N_{C_i} = I N_{C_0},\\ \text{Rays constructed} = 2^{2I} N_{C_0}. \end{cases}

Linear scaling in compute, exponential scaling in constructed rays!

Somewhat better apparent quality (small angle approximation)

Tower of Optimisations

• Sparsity and cache-friendliness
• Lower resolution proxies for distant rays: mipmaps with directionality and adaptive error bounds (variance of log emissivity and opacity)

• In practice, upper cascades (longer rays) sample upper MIPs (fewer texels) when variance is low enough.

MIP Levels

Machine Learning

Myriad uses for hybrid models

1. Efficient wavelength integrals for \bar{J}, see Dias Baso+ 2023.

2. Direct evaluation of PRD scattering integral from J and velocity field, similar to hybrid treatment of Leenaarts+ 2012

3. Initial guess at departure coefficients from atmospheric structure, (Vicente Arévalo+ 2022), or “smarter” acceleration of Rate Equation convergence…

Playtime

Application

Application

Jenkins & Keppens 2021

In particular…

• 5.7 km resolution!
• 3072 x 2048

• 5+1 level H
• 5+1 level Ca ɪɪ
• Charge & pressure conservation

Short Characteristics Comparison

Spectra – Ca ɪɪ K

Spectra – Ly β

Ionisation Fraction

Outlook

Ca ɪɪ K in 3D (Donné & Keppens)

Ca ɪɪ K COCOPLOT

Ly β COCOPLOT

Outlook

• Radiative transfer effects are necessary for realistic solar models
  • Observational interpretation
  • Model synthesis
  • Ionisation effects
• Radiance cascades present viable route to non-LTE RMHD
  • SAMS Project
• And also to enhance radiation treatment across astrophysics
• GPUs are really powerful, and graphics programming has lots to offer us

Thanks!

Christopher.Osborne@glasgow.ac.uk

Paper

1.5D Comparison – Prominence

1.5D Comparison – Filament

Line Formation – Contribution Function

Line Formation – J_\nu