Laboratory analogue of a supersonic accretion column in a binary star system

Astrophysical flows exhibit rich behaviour resulting from the interplay of different forms of energy—gravitational, thermal, magnetic and radiative. For magnetic cataclysmic variable stars, material from a late, main sequence star is pulled onto a highly magnetized (B>10 MG) white dwarf. The magnetic field is sufficiently large to direct the flow as an accretion column onto the poles of the white dwarf, a star subclass known as AM Herculis. A stationary radiative shock is expected to form 100–1,000 km above the surface of the white dwarf, far too small to be resolved with current telescopes. Here we report the results of a laboratory experiment showing the evolution of a reverse shock when both ionization and radiative losses are important. We find that the stand-off position of the shock agrees with radiation hydrodynamic simulations and is consistent, when scaled to AM Herculis star systems, with theoretical predictions.

: List of the different numerical simulations performed in this study as seen in supplementary figures 5 and 6 (for cases B-E, the time listed corresponds to the time the shock reaches the same position as in case A). Colours refer to those seen in supplementary figure 6.

Radiograph comparison
Supplementary figure 1 shows a comparison of two different experimental X-ray radiographs taken at the same time delay, 55 ns, after the main drive lasers have fired. In both cases the same goldplastic layer pusher foil was used as the plasma source. The drive laser intensity was similar in both cases (7.0 x 10 14 W cm -2 and 7.4 x 10 14 W cm -2 respectively).
The dashed lines show the spatial position of the shock feature. The obstacle edge and shock front are very similar in both cases. The slight difference in the position of the obstacle plug at the end of the tube between the two images is as a result of target to target variation in obstacle orientation, within the allowed tolerances.

Compression Calculation
Compression was calculated using the following assumptions: 1) The spatial extent of the region of interest was the tube diameter.
2) The opacity was calculated as a weighted sum of the individual component opacities: 0.39 * gold opacity + 0.61 * plastic opacity, and plasma was considered fully mixed (see below). 3) The opacity was considered not to change across the shock front. See supplementary discussion -X-ray opacity.

where (I/I0) is the transmission,
κ is the weighted opacity as defined above, ρu is the unshocked density, is the chord length the X-ray travels through, set here to be the tube diameter.

5) In the post-shock region we have: ( )
, where is the spatial extend of the shock front, such that + = , and ρs is the shocked density.
6) The equations in 4) and 5) can be solved using the experimental radiograph values in PSL, to give the individual densities, and their ratio.

Mixing of Gold-Plastic Layers in Pusher Foil
We have performed simulations of mixing of plastic and gold layers, as a result of Rayleigh-Taylor instability, laser imprint etc. These are shown in supplementary figure 2 and they indicate that significant mixing has already occurred at 8 ns. This suggests that the assumption of uniformity at times around 55 ns is valid. (The effect of roughness between the plastic-gold layer interfaces is imposed by a multi-mode spectrum in the simulation. The simulation in supplementary figure 3 represents a best case for smoothness: it is likely that the actual target had a much rougher interface, which simulations show increase the rate of mixing between the two layers.)

Laser Intensity in Simulation
The nominal intensity of the laser (7.0 x 10 14 W cm -2 ), reported in figure 1 caption, does not include losses occurring in the laser beam path and/or due to various laser-plasma instabilities. Those are typically dependent on the angle of incidence and notoriously difficult to model. To estimate such losses, it is easier instead to look at the resultant flow behaviour. For the simulation, an intensity of 3.5 x 10 14 W cm -2 was used. If the simulation accurately captures the absorption of the laser, then values of the flow properties obtained from the code, most notably the velocity, should agree with the experimental ones. Indeed, this is the case, as seen under the Optical self-emission of the Results section, thus we have reasonable confidence that the simulation is modelling the laser absorption correctly.

X-ray Opacity
As a result of changes in temperature, density and ionisation state, the mean (frequency integrated) opacity will change across the shock front. However, it is the frequency dependent opacity, at the probe X-ray energy, which is of importance when estimating the density. The X-ray energy is much greater than the plasma temperature (3.75 keV X-rays compared to the ~30 eV plasma temperature) and therefore we do not expect the X-ray opacity to vary greatly across the shock front. This is confirmed in the supplementary information where supplementary figure 2 shows that the X-ray transmission does not appreciably change between the different runs

Numerical Simulations
We have run a range of 1-D and 2-D simulations in order to assess the sensitivity of the calculations with different equations-of-state (EoS) and opacities of the materials (see supplementary table 1). In all cases the results were very similar to those shown in the main manuscript (case A in supplementary table 1), including the reverse-shock compression factor.
In 1-D the reverse-shock compression factor varies from 5.2 to 5.9 for the above examples. And in 2-D the flow dynamics evolve very similarly to the original simulation run -shown in figure 2 (top panel) of the main manuscript. This is referred to as case A in supplementary table 1, and for this particular run the reverse shock is predicted to arrive at the position seen in the experimental images about 54 ns after the start of the laser drive. For runs B to E, the shock reaches the same position as the one of run A at very similar times -as shown in supplementary table 1 Sensitivities to the laser parameters were not investigated as the simulations are tuned to match the shock propagation and position in the experimental data and so the laser drive is constrained.

X-ray Energy
The X-ray energy, for each shot, was determined by fitting the measured x-ray signal transmitted through a step wedge placed on the image plate. By this method, it was found that the "effective" Xray energy is between 3.5 keV and 4.0 keV, and this was the range used in the estimates done in the main paper. This energy range was further confirmed by a measurement of the x-ray spectrum of the backlighter target alone.
To estimate the effective x-ray energy by the step wedge method, layers of 50 μm, 100 μm and 150 μm thick polypropylene were placed on the image plate. The transmission onto the image plate was recorded and compared to the expected transmission for different X-ray energies. This can be seen in supplementary figure 3. Also overlaid is the expected transmission through the step wedge for the X-ray spectrum that was recorded from the backlighter target only. The transmission from the experimental image and the expected transmission from the recorded X-ray spectrum of the backlighter foil are indeed in very good agreement.
The spectrum of the X-ray backlighter alone is given in supplementary figure 4. This shows emission from He-α at 2.78 keV, He-β at 3.27 keV, Ly-α at 2.96 keV and Ly-β at 3.50 keV, as expected, as well as continuum radiation.