Update lineinteraction.rst

[TusharNaugain]

Beta Sobolev Value is introduced to explain its role in determining the escape chance for packets during line interactions. The Lucy (2000) paper is referenced as the primary source for the theory behind the beta Sobolev value and its applications. The Sobolev escape fraction is described in more detail, including its general non-isotropic nature and the exception in homologous expansion.

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Type: 🪲 bugfix | 🚀 feature | ☣️ breaking change | 🚦 testing | 📝 documentation | 🎢 infrastructure

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[tardis-bot]

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The docs workflow has succeeded ✔️

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[tardis-bot]

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Hi human,
I ran ruff on the latest commit (f0af930).
Here are the outputs produced.
Results can also be downloaded as artifacts here.
Summarised output:

1	E999	SyntaxError: Expected a statement

Complete output(might be large):

docs/physics/montecarlo/lineinteraction.rst:1:1: E999 SyntaxError: Expected a statement
Found 1 error.

[tardis-bot]

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Hi human,
I ran benchmarks as you asked comparing master (068d33d) and the latest commit (f0af930).
Here are the logs produced by ASV.
Results can also be downloaded as artifacts here.

Significantly changed benchmarks:

All benchmarks:

Benchmarks that have stayed the same:

| Change   | Before [068d33d8] <master>   | After [f0af930d]    | Ratio   | Benchmark (Parameter)                                                                                                               |
|----------|------------------------------|---------------------|---------|-------------------------------------------------------------------------------------------------------------------------------------|
|          | 26.4±7μs                     | 35.6±10μs           | ~1.35   | transport_montecarlo_packet_trackers.BenchmarkTransportMontecarloPacketTrackers.time_generate_rpacket_last_interaction_tracker_list |
|          | 2.68±0ms                     | 3.23±0.03ms         | ~1.20   | opacities_opacity_state.BenchmarkOpacitiesOpacityState.time_opacity_state_initialize('scatter')                                     |
|          | 2.42±2μs                     | 2.13±2μs            | ~0.88   | transport_montecarlo_estimators_radfield_estimator_calcs.BenchmarkMontecarloMontecarloNumbaPacket.time_update_line_estimators       |
|          | 47.1±20μs                    | 41.4±20μs           | ~0.88   | transport_montecarlo_interaction.BenchmarkTransportMontecarloInteraction.time_line_emission                                         |
|          | 7.35±0.9μs                   | 6.33±0.7μs          | ~0.86   | transport_montecarlo_vpacket.BenchmarkMontecarloMontecarloNumbaVpacket.time_trace_vpacket                                           |
|          | 71.2±30μs                    | 54.6±30μs           | ~0.77   | transport_montecarlo_interaction.BenchmarkTransportMontecarloInteraction.time_line_scatter                                          |
|          | 7.35±2μs                     | 8.04±2μs            | 1.09    | transport_montecarlo_vpacket.BenchmarkMontecarloMontecarloNumbaVpacket.time_trace_vpacket_volley                                    |
|          | 1.69±0.02ms                  | 1.74±0.01ms         | 1.03    | transport_montecarlo_main_loop.BenchmarkTransportMontecarloMontecarloMainLoop.time_montecarlo_main_loop                             |
|          | 38.7±0.03μs                  | 39.8±0μs            | 1.03    | transport_montecarlo_packet_trackers.BenchmarkTransportMontecarloPacketTrackers.time_generate_rpacket_tracker_list                  |
|          | 531±100ns                    | 541±100ns           | 1.02    | opacities_opacity.BenchmarkMontecarloMontecarloNumbaOpacities.time_compton_opacity_calculation                                      |
|          | 3.82±0ms                     | 3.89±0.01ms         | 1.02    | opacities_opacity_state.BenchmarkOpacitiesOpacityState.time_opacity_state_initialize('macroatom')                                   |
|          | 1.63±0.3μs                   | 1.66±0.5μs          | 1.02    | transport_geometry_calculate_distances.BenchmarkTransportGeometryCalculateDistances.time_calculate_distance_line                    |
|          | 2.69±0.4ms                   | 2.74±0.5ms          | 1.02    | transport_montecarlo_single_packet_loop.BenchmarkTransportMontecarloSinglePacketLoop.time_single_packet_loop                        |
|          | 203±0.2ns                    | 205±0.3ns           | 1.01    | spectrum_formal_integral.BenchmarkTransportMontecarloFormalIntegral.time_intensity_black_body                                       |
|          | 2.08±0m                      | 2.09±0m             | 1.00    | spectrum_formal_integral.BenchmarkTransportMontecarloFormalIntegral.time_FormalIntegrator_functions                                 |
|          | 40.9±0.5s                    | 40.5±0.06s          | 0.99    | run_tardis.BenchmarkRunTardis.time_run_tardis                                                                                       |
|          | 1.08±0.01m                   | 1.07±0m             | 0.99    | run_tardis.BenchmarkRunTardis.time_run_tardis_rpacket_tracking                                                                      |
|          | 611±100ns                    | 601±100ns           | 0.98    | opacities_opacity.BenchmarkMontecarloMontecarloNumbaOpacities.time_pair_creation_opacity_calculation                                |
|          | 66.8±0.09ms                  | 63.2±0.4ms          | 0.95    | transport_montecarlo_packet_trackers.BenchmarkTransportMontecarloPacketTrackers.time_rpacket_trackers_to_dataframe                  |
|          | 3.12±0.5μs                   | 2.96±0.6μs          | 0.95    | transport_montecarlo_vpacket.BenchmarkMontecarloMontecarloNumbaVpacket.time_trace_bad_vpacket                                       |
|          | 1.22±0μs                     | 1.15±0μs            | 0.94    | transport_geometry_calculate_distances.BenchmarkTransportGeometryCalculateDistances.time_calculate_distance_boundary                |
|          | 3.22±0.9μs                   | 3.04±0.6μs          | 0.94    | transport_montecarlo_vpacket.BenchmarkMontecarloMontecarloNumbaVpacket.time_trace_vpacket_within_shell                              |
|          | 752±0.5ns                    | 696±0.2ns           | 0.93    | transport_montecarlo_interaction.BenchmarkTransportMontecarloInteraction.time_thomson_scatter                                       |
|          | 571±200ns                    | 520±100ns           | 0.91    | opacities_opacity.BenchmarkMontecarloMontecarloNumbaOpacities.time_photoabsorption_opacity_calculation                              |

If you want to see the graph of the results, you can check it here