Platform for probing radiation transport properties of hydrogen at conditions found in the deep interiors of red dwarfs

Red dwarfs (M dwarfs) are the lightest and coolest main sequence stars and make up $∼ 70 %$ of all stars in the Sun's neighborhood.^1,2 Prominent examples are our nearest neighbor Proxima Centauri (0.12M_⊙) or TRAPPIST-1 (0.089M_⊙), which is only slightly larger than Jupiter, but much more massive. The interiors of red dwarfs mainly consist of hydrogen–helium mixtures, which are progressively shaped by screening effects, ion–ion correlations, and degeneracy as temperature decreases and density increases.³ These many-particle effects are challenging to model, in particular, for calculations of radiation transport, which plays a major role in modeling of sub-stellar objects and stars. From the solar abundance problem, we know that $∼ 20 %$ changes in opacity have paramount impact on our understanding of stellar interiors.^4,5 Whether energy can effectively be transported via radiation or, if radiation is not sufficient, convection sets in, is a property that is particularly influenced by stellar opacity. In general, the physics of red dwarfs is poorly understood in comparison with the hotter interior of the Sun, which is much closer to the ideal plasma state.

Figure 1 shows simulated pressure–temperature profiles of stars on the main sequence, demonstrating the extreme plasma conditions present inside those celestial objects. The curves were obtained using the MESA code for stellar evolution^6–10 (see the appendix for details on these simulations). The inset illustrates the schematic interiors of stars from the core (⁠ $m / M = 0$⁠) to the photosphere (⁠ $m / M = 1$⁠) divided into radiative and convective zones for stars with a solar composition of elements. Red dwarfs are characterized by masses between 0.075 and 0.5 solar masses so that their typical mass–temperature ratios overall place red dwarfs in a regime, where convection dominates the outer regions. Depending on the size of the individual object, a more or less developed radiative core is present. The smallest red dwarfs (⁠ $M ≲ 0.2 M ⊙$⁠) are thought to be fully convective. In this case, the fusion reactions in the core are permanently re-fueled by hydrogen from the outer layers. Combined with the low fusion rates due to the relatively low core temperatures, convection possibly allows some red dwarfs to last trillions of years until all hydrogen fuel is exhausted. However, even a small radiative core can strongly change this behavior and its existence crucially depends on the effectiveness of radiation transport in highly compressed matter.

Moreover, the internal structure of a star has a major impact on the activity of its surface.¹² The boundary between a radiative core and a convective layer can lead to strong magnetic fields and a turbulent atmosphere,^1,13,14 including radiative and plasma outbursts that may threaten life on nearby planets. Therefore, understanding the radiative properties of the complex plasmas within a host star is crucial when judging the possibility of an exoplanet to host life—especially for red dwarfs where the habitable zone is thought to be found relatively close to the star itself due to the low surface temperature.¹⁵ 

For red dwarf stars, the thermodynamic conditions at the boundary between radiative core and convective envelope are estimated to be in a pressure regime of few Gbars and temperatures of few million Kelvin.^1,11 Corresponding free electron densities are in the range of few 10²⁵ cm⁻³, which results in Fermi energies of similar order as the thermal energy of the free electrons. The energy transport in this so-called warm dense matter regime¹⁶ is extremely difficult to calculate, which gives rise to significant uncertainties in modeling the energy transport inside red dwarfs.

For stellar interiors, the radiative opacity $κ rad$ is usually divided into three contributions:¹⁷ 

where κ_bf is the opacity contribution by bound–free absorption, κ_ff denotes the free–free contribution, and $κ T = Z σ T / m i$ the absorption due to Thomson scattering from free electrons, which is solely dependent on the Thomson scattering transport cross section σ_T,^18,19 the average ion charge state Z, and the average ion mass m_i of the plasma. While Rayleigh scattering might be of interest for the atmosphere of K and M class stars,²⁰ the high ionization in hotter photospheres and deep within even the smallest stars often justifies to neglect its contribution to $κ rad$⁠. For the solar abundance problem, the bound–free opacity of metals is probably most relevant, but deep in the solar radiation zone as well as for many red dwarfs, particularly those with low metallicity, hydrogen free–free opacity, i.e., absorption due to inverse bremsstrahlung, is the dominant absorption mechanism of radiation.¹⁷ For these red dwarfs, the absolute values for free–free absorption determine where convection or radiation will be the dominant energy transport mechanism.

Figure 2 shows a density–temperature diagram of dominating absorption mechanisms thought to be present for the composition of population I stars in comparison with red dwarf interiors and the Sun. While bound–free transitions dominate at low densities and temperatures, free–free absorption starts to outrun the bound–free opacity with the increase in density due to increasing electron–ion collision rates as well as pressure ionization of heavier elements.²¹ At the highest densities, conduction by degenerate electrons becomes more efficient than radiation transport, whereas for low densities and highest temperatures, photon scattering from electrons (Thomson or Compton, depending on photon energy) is most significant.

A classical treatment of the spectral absorption coefficient due to inverse bremsstrahlung, derived from the description of electron–ion collisions in a weakly coupled plasma environment, yields for the absorption coefficient α_ff,²² 

where ν denotes the x-ray frequency, ρ is the mass density, Z is the average degree of ionization, n_e is the free electron number density, n_i is the ion number density, T is the plasma temperature, h is Planck's constant, and k_B is Boltzmann's constant. Additional corrections due to quantum and correlation effects are accounted for in a frequency-dependent correction factor $g f f ( ν , T )$⁠, the so-called Gaunt factor.²³ For a weakly coupled ideal plasma, the Gaunt factor can be interpreted as the logarithm of the ratio of maximum impact parameter $b max$ and minimum impact parameter $b min$ in the corresponding electron–ion collision (the so-called Coulomb logarithm²⁴)

This formalism is equivalent to the classical treatment of several other important plasma effects that involve Coulomb collisions of electrons and ions, e.g., stopping power of ions or electron–ion temperature equilibration in dense plasmas.²⁵ The maximum impact parameter $b max$ is usually given by $min ( v e / 2 π ν , λ s )$⁠, where v_e is the average velocity of the electrons, ν is the x-ray frequency, and λ_s is the screening length due to the surrounding plasma. On the contrary, $b min$ can be expressed as $max ( b ⊥ , λ t h$⁠), where $b ⊥ = Z e 2 / ( 4 π ϵ 0 m e v e 2 )$ is the impact parameter for an electron being deflected perpendicular to its direction of incidence and λ_th denotes the thermal de Broglie wavelength of the electrons.

However, for conditions relevant to the interiors of red dwarfs [e.g., $n e ∼$ few $10 25 cm − 3 , T e ∼$ few $100 eV$ (Ref. 3)], we find $v e / 2 π ν < λ t h$⁠, i.e., a negative Coulomb logarithm for x-ray frequencies larger than the plasma frequency. Thus, this simple classical treatment assuming a weakly coupled plasma is not appropriate for such conditions. Indeed, more sophisticated approaches have been developed to accommodate these conditions.^26,27

Figure 3 shows Average Atom calculations with a Mean Force potential (AA-MF)²⁸ and state-of-the-art Density Functional Theory Molecular Dynamics (DFT-MD) simulations²⁹ compared to the analytical model for the free–free opacity with constant Gaunt factor and calculations by van Hoof et al.^30,31 The first two simulation methods have previously been applied successfully for calculating the equation-of-state (EOS) and transport properties of dense plasmas.^28,32–34 The DFT-MD simulations were performed with up to 256 hydrogen atoms using the program package VASP.^35–38 Our considered density range spans 20–150 g cm⁻³ at 100, 150, and 200 eV. The simulations use the Baldereschi mean value point³⁹ and the Coulomb potential with an energy cutoff of 10 000 eV. Each DFT-MD point was run for 20 000 time steps with a time step size between 3 as (attoseconds) and 8 as depending on the thermodynamic conditions. The temperature was controlled with a Nosé–Hoover thermostat.⁴⁰ Subsequently, 10–20 snapshots were selected from each trajectory to calculate the electrical conductivity and opacity applying the Kubo–Greenwood formalism^41,42 and the Kramers–Kronig relation. For details on the AA-MF calculations (which were performed for identical temperatures and pressures), we refer to previous publications.^28,43–46

While the DFT-MD simulation naturally includes many-body effects in the description of wavefunctions and the density of states due to the multiple ions included in the simulation, the AA-MF approach, which is strictly speaking also DFT-based, simplifies the calculation by exclusively relying on the atom-in-jellium model. Either of the two formulations calculates opacity from the real part of the electrical conductivity. Both approaches agree on the quantity of extinction remarkably well, supporting each other. This result is particularly noteworthy as the similarity of the AA-MF calculation with DFT-MD—for the specific case of hydrogen—is highly desired: While DFT-MD is generally more accurate, AA-MF is computationally significantly less expensive and should be favored if benchmarks can show good agreement between the two methods. At the same time, the consistency of the computed opacity illustrates impressively that extreme states of matter can be treated by DFT-MD nowadays with the increase in computational power and, hence, number of energy bands included in the calculation.

Both methods show a discrepancy to the calculation of the free–free opacity from the semi-classical formula [Eq. (2)], as it can be seen in Fig. 3. The simplest approach of setting the Gaunt factor to unity reproduces the classical result of Kramers.²² Introducing quantum-mechanical corrections, van Hoof et al.^30,31 provide easily applicable, tabulated values for g_ff by following the seminal work of Karzas and Latter.⁴⁷ However, this calculation is requiring more assumptions than the AA-MF or the DFT-MD model, e.g., the velocity distribution of the electrons (with is assumed to be Maxwellian) in order to calculate thermally averaged free–free Gaunt factors and from these opacities.^30,31 Other authors perform similar calculations but integrate over the Fermi distribution of a degenerate electron gas.²⁴ 

In fact, for dense plasma conditions comparable to the interiors of main sequence stars, even advanced calculations of the Gaunt factor vary by more than 50% for frequencies larger than the plasma frequency.²⁷ In particular, the specific treatment of dynamic screening, strong collisions, and re-normalization due to higher moments can make a significant difference in comparison with widely used Born approximation treatments of the Gaunt factor.²⁷ Deviations are particularly significant in the photon energy regime from $∼ 500 eV$ to few keV, which is the dominant contribution when calculating the Rosseland mean opacity for the Sun and smaller stars. Indeed, varying the opacity by 50% can change the radii of the boundaries between convection and radiation zone by up to 10%, which, given the underlying density and temperature gradients, would significantly impact our general understanding of stars.^5,48

Using the largest laser system in the world, namely, the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory,⁴⁹ it is now possible to create and probe matter states relevant to stellar interiors in the laboratory.^50,51 To address the questions raised above, we have developed a concept to leverage NIF's unique capabilities to create relevant conditions and obtain a very first test of free–free opacity models in this very important plasma regime via x-ray absorption measurements of highly compressed hydrogen during the stagnation phase of layered capsule implosions. In this way, not only the various existing models and resulting tables for the Gaunt factor will be tested, but also modern DFT-MD and AA-MF simulations, which provide the absorption coefficient, can be benchmarked. Finally, due to the equivalent physics involved (electron–ion collisions in dense plasma environments²⁶), our results on free–free absorption will inform models for swift ion stopping in warm dense matter as well as corresponding electron–ion equilibration times.

A sketch of the experimental setup is shown in Fig. 4. We will use 184 out of the 192 NIF laser beams to heat a gold Hohlraum creating a quasi-thermal radiation field that implodes a layered fuel capsule at the center of the Hohlraum. The capsule is comprised of a 57 μm thick beryllium ablator shell, containing a 83 μm thick layer of cryogenic solid hydrogen.

The temporally shaped radiation field created by the laser drive will ablate the Be shell and, hence, accelerate the payload inward. Upon stagnation, a high density hydrogen layer with $ρ > 100 g cm − 3$ is formed while most of the Be ablator has been ablated.

The implosion design is derived from inertial confinement fusion (ICF) implosions at the NIF⁵⁴ and applies a well-tested model that matched a variety of spherical DT implosion experiments in NIF's ICF program.^54,55 In contrast to ICF implosions aiming for high neutron yield,⁵⁶ the peak radiation temperature of our Hohlraum drive (⁠ $T rad = 170 eV$⁠) is significantly reduced, deliberately slowing down the implosion to $∼ 200 km s − 1$ with the goal of creating extreme densities (⁠ $∼ 150 g cm − 3$⁠) at moderate temperatures (⁠ $∼ 200 eV$⁠) while reducing x-ray and neutron-related background signals near stagnation that would affect the radiography measurement. These conditions are directly relevant to the interiors of red dwarfs.

The dense plasma states will be probed by x-ray line emission, which is created by the eight remaining NIF beams that heat a backlighter tube. We will perform 2D-imaging radiography of the implosion and stagnation phase using two different photon energies by varying the backlighter tube material. Vanadium will result in 5.2 keV line emission from 1s2p → 1s² (He-α) transitions of helium-like ions. In the same way, helium-like cobalt ions will produce 7.2 keV photons.

NIF's Crystal Backlighter Imager (CBI) system⁵² will allow for high-resolution, nearly monochromatic radiography images of the imploding and stagnating sphere. A gated single-line-of-sight (SLOS) detector⁵³ will provide four images per implosion with a time delay of $∼ 100 ps$ between consecutive images. From the radiography images, we will infer radial profiles of the opacity through Abel inversion^57,58 and forward fitting methods.

The reduced implosion velocity minimizes the hot spot x-ray emission at stagnation to improve the signal-to-noise ratio of the physics measurements. For the same reason, we use a (25%/75%) HT mixture (which is hydrodynamically equivalent to 50%/50% DT) for the ice layer to minimize the neutron yield and the related background signals. Tritium is required to enable the hydrogen ice layer formation through beta layering, where self-heating from beta decay leads to a redistribution of HT ice with time.^60–62 Figure 5 shows an overview of the $∼ 11 ns$ implosion trajectory simulated with HYDRA-1D.⁵⁹ Compared to previous ICF experiments with Be shells,⁵⁴ the ablator thickness is reduced to 57 μm to decrease the remaining ablator mass to near zero, which will maximize the radiography contrast of the hydrogen layer. Figure 5(b) illustrates examples of radial density, ionization, and temperature profiles predicted by hydrodynamic simulations. The high-density portion of the profiles consists of hydrogen (HT) only. Our simulations predict that the material is compressed to mass densities exceeding $150 g cm − 3$⁠, which corresponds to the electron densities of $3.6 × 10 25 cm − 3$⁠, at a temperature of $∼ 200 eV$⁠. These parameters result in Fermi energies of $∼ 400 eV$ and, thus, $∼ 2 ×$ larger than the thermal energies. Hence, we will probe degenerate plasma conditions as expected in the interiors of red dwarf stars. In addition, these conditions are expected to limit the influence of temperature on the free–free absorption, since the $1 / T$ term in the expression for the inverse bremsstrahlung [Eq. (2)] can be replaced approximately by $1 / T F$⁠, where T_F denotes the Fermi temperature. This behavior is supported by the results from DFT-MD and the Average Atom model, which indicates that the opacity is more sensitive to the Fermi temperature than the electronic temperature under dense degenerate plasma conditions (see Fig. 3 and inset). The weak dependence on T is highly beneficial for the analysis and interpretation of our data, as we can determine opacity and from there infer density, while a direct measurement of temperature would require additional diagnostics—for example x-ray Thomson scattering.⁶³ 

Reducing the ablator shell thickness and decreasing the remaining mass at stagnation might come with increased risk of hydrodynamic instability at the ablator–ice interface and ablator material mixing into the compressed ice layer. For a more quantitative estimate of mixing, we have performed 1D capsule-only simulations using a buoyancy-drag mix model,⁶⁴ which was successful in explaining experimental performance observations of previous layered Be implosions.⁵⁴ In addition, more recently the buoyancy-drag mix model has been calibrated in a focused series of experiments using a thin ice layer of varying thickness and detecting neutronic signatures of deuterated plastic, originally located near the inside of a plastic shell, mixing through the ice layer into the hot spot.⁶⁵ Figure 6 shows the results for our significantly slower implosion design in terms of atomic Be fraction as function of radius and time. The demarcation line between the Be ablator and the HT ice is indicated as a dot-dashed line. We have labeled the time of minimum radius of this interface by $t min$⁠, which coincides with the time when the rebounding shock wave, which leads to the formation of the high-compression HT ice, passes this interface. Our simulations clearly show that Be does not mix into the high-compression HT ice layer. The radiography measurements are aimed to record transmission images between 400 and 200 ps before $t min$⁠, which, hence, is not affected by Be mixing into the high compression HT ice layer.

Synthetic radiography images (see Fig. 7 and the appendix for more details) show a clear limb feature at the boundary of the central hydrogen and the beryllium layer. Applying mass conservation, this feature will be used as constraint for the density of the encapsulated hydrogen at smaller radii as the total initial fuel mass can be accurately characterized. Toward the outer edges of the radiography field-of-view of $600 × 600 μ m 2$⁠, we expect the plasma to become fully transmissive to the probe radiation, which will be used to normalize the opacities measured at smaller radii and obtain absolute values of the radial absorption coefficient.

In the investigated density and temperature regime, opacity due to Thomson scattering is expected to reach similar values as free–free absorption. As absorption due to Thomson scattering scales linearly with n_e and free–free opacity with $n e 2$⁠, this effect can become particularly significant in the lower density regions of the imploded capsule. To disentangle both mechanisms, we will perform the absorption measurements at two photon energies (5.2 and 7.2 keV as described above). While disagreeing in the actual magnitude, all but the classical approach (g_ff = 1) presented in Fig. 3 find the same proportionality of the free–free opacity with $∼ 1 / ( h ν ) ( 7 / 2 )$ for high photon energies $h ν$⁠. Classically, the Thomson cross section is in first order independent of probing frequency and temperature, which would allow us to separate the two contributions to the signal. While this assumption might give a reasonable first estimate, our AA-MF simulations (see Fig. 3) indicate in accordance with previous calculations by Boercker¹⁸ that this description as a constant is not applicable at the extreme conditions we intend to probe. However, models of the Thomson scattering have to provide only the ratio between the cross-section at the two backlighter energies to enable us to differentiate the former from inverse bremsstrahlung. Regardless of the model applied, the Thomson scattering's contribution to the overall opacity might also be used as an additional density constraint next to measuring the ablator–hydrogen interface and applying mass conservation.

Further constraints on the implosion parameters will be deduced from measuring multiple absorption images at different times during the implosion in one shot. The stagnation phase is usually well modeled by a self-similar description of the conservation laws,⁶⁸ which only allows certain shapes and time evolutions of the density profiles.

In summary, we presented a concept to leverage NIF's unique capabilities to investigate the deep interiors of red dwarf stars in the laboratory and shed light on their internal energy transports mechanisms. The proposed experiment has been accepted within NIF's Discovery Science Program for upcoming shot days in 2022 and 2023. The resulting measurement of free–free absorption and opacity will provide a benchmark for numerical and analytical approaches, which will in turn yield an improved description of the Gaunt factor. Finally, interior structure models for massive hydrogen-rich astrophysical objects, such as red dwarfs, can be revisited on the basis of the new opacity constraint.

M.B. was supported by the European Horizon 2020 programme within the Marie Skłodowska-Curie actions (xICE Grant No. 894725). J.L. and D.K. acknowledge support by the Helmholtz Association under No. VH-NG-1141 and by GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt as part of the R&D project No. SI-URDK2224 with the University of Rostock. The work of S.S. and D.K. was supported by Deutsche Forschungsgemeinschaft (DFG—German Research Foundation) Project No. 495324226. The work of B.B., L.D., G.N.H., S.F.K., N.I., O.L.L., S.A.M. L.M., M.O.S., P.A.S. and T.D. was performed under the auspices of the DOE by Lawrence Livermore National Laboratory under Contract No. DE-AC52–07NA27344. R.R. and M.S. acknowledge support from the DFG via the Research Unit FOR 2440. The DFT-MD calculations were performed at the North-German Supercomputing Alliance (HLRN) facilities and at the IT and Media Center of the University of Rostock.

The authors have no conflicts to disclose.

Julian Lütgert: Formal analysis (equal); Visualization (lead); Writing – original draft (equal); Writing – review and editing (equal). Otto L. Landen: Conceptualization (equal). Stephan A. MacLaren: Conceptualization (equal); Formal analysis (equal); Software (equal); Writing – review and editing (equal). Laurent Pierre Masse: Conceptualization (equal); Software (equal). Ronald Redmer: Supervision (supporting); Writing – original draft (supporting). Maximilian Schörner: Software (supporting). Markus O. Schölmerich: Formal analysis (supporting); Investigation (supporting). Samuel Schumacher: Formal analysis (supporting). Nathaniel R. Shaffer: Software (equal). Charles Starrett: Software (equal). Philip A. Sterne: Resources (supporting). Mandy Bethkenhagen: Conceptualization (equal); Software (equal); Visualization (supporting); Writing – original draft (equal); Writing – review and editing (equal). Tilo Döppner: Conceptualization (equal); Formal analysis (equal); Investigation (lead); Project administration (equal); Supervision (supporting); Writing – original draft (supporting); Writing – review and editing (equal). Dominik Kraus: Conceptualization (equal); Formal analysis (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review and editing (supporting). Benjamin Bachmann: Formal analysis (supporting). Laurent Divol: Conceptualization (equal); Software (equal). Dirk O. Gericke: Conceptualization (supporting); Writing – original draft (supporting). Siegfried Glenzer: Conceptualization (supporting). Gareth Neville Hall: Conceptualization (supporting); Formal analysis (equal); Investigation (equal). Nobuhiko Izumi: Investigation (supporting). Shahab Firasat Khan: Investigation (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Profiles of temperature, density, and pressure inside stars with various masses were calculated using the “Modules for Experiments in Stellar Astrophysics” code (MESA, release r15140).^6–10 We used the default MESA equation-of-state (EOS), which is a blend of the OPAL,⁶⁹ SCVH,⁷⁰ FreeEOS,⁷¹ HELM,⁷² and PC⁷³ EOSes. Radiative opacities are primarily from OPAL,^74,75 with low-temperature data from Ferguson et al.⁷⁶ and the high-temperature, Compton-scattering dominated regime by Buchler and Yueh.⁷⁷ Electron conduction opacities are from Cassisi et al.⁷⁸ Nuclear reaction rates are from JINA REACLIB⁷⁹ plus additional tabulated weak reaction rates.^80–82 Screening is included via the prescription of Chugunov et al.⁸³ Thermal neutrino loss rates are from Itoh et al.⁸⁴ 

A pre-main-sequence model has been calculated from initial parameters of helium mass fraction $Y i = 0.2744$⁠, metallicity $Z i = 1 − X i − Y i = 0.01913$ (with X_i being the initial mass fraction of hydrogen), mixing length parameter $α MLT = 1.9179$⁠, and including element diffusion, which has been found to reproduce the solar model well.⁶ The ratio of elements heavier than helium was taken from Grevesse and Sauval.⁸⁵ The time span of the simulation was chosen so that the star spent a considerable amount of time on the main sequence. For stars smaller than the Sun, ten times the age of the star when the main sequence was entered has been chosen; masses $M > M ⊙$ were evolved until $t nuc / 2$ was reached, where $t nuc = ( M / M ⊙ ) − 2.9 × 10 10$ a is an approximation for the lifetime of star on the main sequence.⁸⁶ 

The decision whether regions of a star were considered convective or radiative was based on the Schwarzschild criterion.

The relation between thermal conductivity through photon transport (radiation) $λ rad$ and the opacity $κ rad$ is given by

where T is the temperature, ρ is the density, $a = 7.5657 × 10 − 16 J m − 3 K − 4$ is the radiation density constant, and c is the speed of light.^17,86 Equation (B1) can be used to define a conductive opacity $κ c$ from the thermal conductivity due to electrons $λ c$ by analogy. This quantity—as calculated by Hayashi et al. who reproduce the work of Mestel⁸⁷ and Lee⁸⁸—has been plotted in Fig. 2 to compare it with radiative opacities.

As the two contributions to the full thermal conductivity are additive, the total opacity $κ tot$ is given by $1 / κ tot = 1 / κ rad + 1 / κ c$⁠, i.e., in order for $κ c$ being the dominant contribution to the total opacity (see the high density and low temperature corner of Fig. 2), the former quantity has to be small compared to $κ rad$⁠.⁸⁶ 

In order to generate the transmission profiles [see Fig. 7(b)] from extinction coefficient line-outs [Fig. 7(c)], we assumed a perfect, spherically symmetric implosion and a uniform and monochromatic backlighter emitting parallel x rays. We calculated

where the integral runs over the path of the light and might, therefore, probe different temperature and density conditions. To account for the limited temporal resolution of the detector, a series of one-dimensional transmission profiles at different times was convolved with a Gaussian gate-function (35 ps FWHM). The resulting line-out has been rotated, and a spatial blur in the form of a two-dimensional Gaussian with 10 μm FWHM in both directions was applied before the transmission has been multiplied with the expected backlighter photon flux (240 μm⁻² ps⁻¹ (⁠ $E ph = 5.2 keV$⁠) or 288 μm⁻² ps⁻¹ (⁠ $E ph = 7.2 keV$⁠) at the target's position), corrected for attenuators shielding various components and re-binned to the pixel-size of the detector. In a last step, noise proportional to the individual pixels' intensity has been applied to the data.