synthesizAR: A Python framework for forward-modeling optically-thin emission from field-aligned hydrodynamic models

Coronal Heating ISFM Meeting / NASA GSFC / 3 April 2024

Will Barnes

AU/NASA GSFC

Modeling Optically-thin Emission

• Modeling optically-thin intensity, \(I\), requires computing the following line-of-sight integral: \[ I = \int\mathrm{d}h\,\varepsilon(n,T), \]
• where:
  • \(h\) is a vertical coordinate along the line of sight (LOS)
  • \(\varepsilon\) is the emissivity
  • \(n\equiv n(h),T\equiv T(h)\) are the temperature and density along the LOS
• This requires knowing four things:
  1. What structures are emitting–geometry
  2. \(T,n\) of emitting structures as a function of space and time–loop model
  3. How the plasma is emitting as a function of \(n,T\)–emission model
  4. How is the emission being observed–instrument

Modeling Optically-thin Emission

\[ \begin{align} I &= \int\mathrm{d}h\,\varepsilon(n,T)\\ \int\mathrm{d}A\,I &= \int\mathrm{d}A\int\mathrm{d}h\,\varepsilon(n,T) = \int\mathrm{d}V\,\varepsilon(n,T)\\ I &\approx \frac{1}{A_{\mathrm{pix}}}\int\mathrm{d}V\,\varepsilon(n,T) \end{align} \]

• Emission confined to discrete loops intersecting LOS
• Loop segments have cross-section \(A_s\) and length \(\delta_s\) \[ \int\mathrm{d}V\,\varepsilon(n,T) = \sum_s\int\mathrm{d}V_s\varepsilon_s(n,T) \approx \sum_sA_s\delta_s\varepsilon_s(n,T)\\ I \approx \frac{1}{A_{\mathrm{pix}}}\sum_sA_s\delta_s\varepsilon_s(n,T) \]

The Emissivity kernel

• Compute per loop segment using field-aligned model–\(T\equiv T(s,t),n\equiv n(s,t)\)
• For a narrowband imager (e.g. AIA), \[ \varepsilon = K(T)n^2\quad[\mathrm{DN}\,\mathrm{pix}^{-1}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-1}] \]
• For a spectral line intensity with a transition at \(\lambda_{ij}\), \[ \begin{align} \varepsilon &= \frac{1}{4\pi}A_{ij}\frac{hc}{\lambda_{ij}}n_{ij}\quad[\mathrm{erg}\,\mathrm{cm}^{-3}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}] \\ &= \frac{1}{4\pi}A_{ij}\frac{hc}{\lambda_{ij}}\frac{n_{ij}}{n_{X^+}}\frac{n_{X^+}}{n_X}\frac{n_X}{n_H}\frac{n_H}{n_e}n_e = \frac{1}{4\pi}G_{ij}(n,T)n_e^2 \end{align} \]
• For the emission measure distribution on the interval \(T_a\le T<T_b\), \[ \varepsilon = n^2H(T-T_a)H(T_b-T)\quad[\mathrm{cm}^{-6}] \]

The synthesizAR Package

• synthesizAR = synthesis of Active Region emission (pronounced like “synthesizer”)
• Combine field-aligned models to produce spatially-resolved, time-dependent forward model
• Strengths:
  • Modular–geometry, field-aligned model, instrument all configurable
  • Modern–pure Python, leverages scientific Python and “PyAstro” software stack
  • Efficient–emissivity from each strand computed in parallel
  • Dynamic–forward-modeled emission is time-dependent
• Limitations:
  • All emission is assumed to be optically-thin
  • All emission is assumed to be thermal (i.e. no transport effects)
  • All emission confined to discrete field lines (i.e. not volume filling)

More details in Barnes et al. (2019)

Toy Loop Model

import synthesizAR
from synthesizAR.models import semi_circular_bundle

obstime = astropy.time.Time.now()
pos = SkyCoord(lon=0*u.deg, lat=0*u.deg, radius=1*u.AU, obstime=obstime, frame='heliographic_stonyhurst')
bundle_coords = semi_circular_bundle(50 * u.Mm, 1*u.Mm, 500, observer=pos)
print(bundle_coords[0][:2])

<SkyCoord (Heliocentric: obstime=2024-04-03 15:02:43.614007, observer=<HeliographicStonyhurst Coordinate (obstime=2024-04-03 15:02:43.614007, rsun=695700.0 km): (lon, lat, radius) in (deg, deg, AU)
    (0., 0., 1.)>): (x, y, z) in Mm
    [(-15.31694891, -0.13318253, 695.7      ),
     (-15.31687318, -0.13318253, 695.7481677)]>

strands = [synthesizAR.Loop(f'strand{i}', c) for i, c in enumerate(bundle_coords)]
bundle = synthesizAR.Skeleton(strands)
side_on_view = SkyCoord(lon=0*u.deg, lat=-90*u.deg, radius=1*u.AU, frame=pos.frame)
bundle.peek(observer=side_on_view)

Toy Loop Model

from synthesizAR.instruments import InstrumentSDOAIA
from synthesizAR.interfaces import RTVInterface

rtv = RTVInterface(heating_rate=1e-3*u.Unit('erg cm-3 s-1'))
bundle.load_loop_simulations(rtv)
print(bundle.loops[0].electron_temperature[0,:5])
print(bundle.loops[0].density[0,:5])

[2636831.60572369 2636831.60572369 2636831.60572369 2636831.60572369
 2636831.60572369] K
[1.71094708e+09 1.71094708e+09 1.71094708e+09 1.71094708e+09
 1.71094708e+09] 1 / cm3

aia = InstrumentSDOAIA([0, 1]*u.s, side_on_view, pad_fov=(10, 10)*u.arcsec)
maps = aia.observe(bundle, channels=aia.channels[2:3])
maps['171'][0].peek()

Toy Loop Model: Summary of Steps

1. What structures are emitting–synthesizAR.Skeleton, synthesizAR.Loop
2. \(T,n\) of these structures as a function of \(s,t\)–synthesizAR.interfaces.RTVInterface
3. How plasma emits as a function of \(T,n\)–synthesizAR.atomic.EmissionModel (optional)
4. How is the emission being observed–synthesizar.instruments.InstrumentSDOAIA

flowchart LR
  coords["SkyCoord
              SkyCoord
          ...
              SkyCoord"]
  loops("synthesizAR.Loop
         synthesizAR.Loop
         ...
         synthesizAR.Loop"):::synthesizar
  skeleton("synthesizAR.Skeleton"):::synthesizar
  modelparams["model parameters"]
  model("synthesizAR.interfaces.RTVInterface"):::synthesizar
  observer["SkyCoord"]
  instrument("synthesizAR.instruments.InstrumentSDOAIA"):::synthesizar
  ions["fiasco.Ion
        fiasco.Ion
        ...
        fiasco.Ion"]
  emmodel("synthesizAR.atomic.EmissionModel"):::synthesizar
  smap(["sunpy.map.Map"])
  coords --> loops
  loops --> skeleton
  model --> skeleton
  modelparams --> model
  skeleton --> instrument
  observer --> instrument
  emmodel -.-> instrument
  ions --> emmodel
  instrument --> smap
  classDef synthesizar fill:#FE7900

Toy Loop Model: Different Observer

top_down_view = SkyCoord(lon=0*u.deg, lat=0*u.deg, radius=1*u.AU, frame=pos.frame)
aia = InstrumentSDOAIA([0, 1]*u.s, top_down_view, pad_fov=(10, 10)*u.arcsec)
maps = aia.observe(bundle, channels=aia.channels[2:3])
maps['171'][0].peek()

Toy Loop Model: Different Observer

Toy Loop Model: Using a Different Loop Model

from synthesizAR.interfaces import MartensInterface

martens = MartensInterface(1e-3*u.Unit('erg cm-3 s-1'))
bundle.load_loop_simulations(martens)
maps = aia.observe(bundle, channels=aia.channels[2:3])
maps['171'][0].peek()

Application: Active Region Heating Diagnostics

See Barnes et al. (2019),Barnes et al. (2021)

Application: Active Region Heating Diagnostics

• Emission measure slope, \(a\), \[ \begin{align} \mathrm{EM}(T)&\sim n^2\tau_{rad} \sim T^{1-\alpha}n \sim T^{1-\alpha+1/\ell}\\ \mathrm{EM}(T)&\propto T^a \end{align} \]
  • For uninterrupted radiative and enthalpy-driven cooling, \(2\lesssim a\lesssim2.5\)
  • As heating frequency increases, \(a\) increases with more isothermal \(\mathrm{EM}\)
• Time lag \(\tau_{AB}\) between AIA channels \(A\) and \(B\) \[ \begin{align} \mathcal{C}_{AB}(\tau) &= \mathcal{I}_A(t)\star\mathcal{I}_B(t) = \mathscr{F}^{-1}\left\{\mathscr{F}\left\{\mathcal{I}_A(-t)\right\}\mathscr{F}\left\{\mathcal{I}_B(t)\right\}\right\} \\ \tau_{AB} &= \mathrm{argmax}_{\tau}\,\mathcal{C}_{AB}(\tau) \end{align} \]
  • Proxy for the cooling time between characteristic temperatures of \(A\) and \(B\)
  • \(\tau_{AB}>0\) indicative of cooling, \(\tau_{AB}<0\) suggests heating

Bradshaw & Cargill (2010),Cargill (2014),Viall & Klimchuk (2012),Barnes et al. (2019)

Application: Active Region Heating Diagnostics

See Barnes et al. (2019)

Application: Active Region Heating Diagnostics

See Barnes et al. (2021)

Application: Active Region Heating Diagnostics

See Barnes et al. (2021)

Application: Active Region Heating Diagnostics

See Barnes et al. (2021)

Application: EUV Flare Emission

Courtesy Brock Parker and Chunming Zhu (MSU)

Application: Observing Flows in the TR with ESIS-II

Courtesy Jake Parker (NASA GSFC)

Summary

• synthesizAR–pure Python package for modeling time-dependent emission from field-aligned models
• Models thermal, optically-thin emission confined to discrete loop structures
• Prioritizes modularity and flexibility–geometry, loop models, instrument
• Capabilities:
  • Works with any field-aligned loop model
  • Incorporate detailed atomic physics in emission modeling
  • Time-dependent, spatially-resolved emission
  • High-resolution, spectrally-resolved diagnostics

References

Barnes, W. T., Bradshaw, S. J., & Viall, N. M. 2019, ApJ, 880, 56

Barnes, W. T., Bradshaw, S. J., & Viall, N. M. 2021, ApJ, 919, 132

Bradshaw, S. J., & Cargill, P. J. 2010, ApJ, 717, 163

Cargill, P. J. 2014, ApJ, 784, 49

Viall, N. M., & Klimchuk, J. A. 2012, ApJ, 753, 35