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Once white dwarf spectra have been obtained using the techniques described above, they must be analysed to extract useful information such as the atmospheric parameters and metal abundances. This involves fitting model spectra to the data. The models used here have been developed over many decades by Detlev Koester. Here I summarise the physics behind them, based on Koester (2009, 2010); Koester et al. (2014a). Note that these concern 1-d model atmospheres, which are slowly being superseded by 3-d models (Tremblay et al., 2011b) that more accurately describe convection. However the 3-d models do not yet fully incorporate metal accretion and diffusion, so cannot be used here.

2.2.1 White dwarf model atmospheres

The objective of white dwarf atmospheric models is to take the observed data, usually in the form of spectra, and return atmospheric parameters such as the effective temperature (Teff), surface gravity (logg) and metal abundances, accomplished by fitting the appropri-

ate model spectrum to the data. To produce a grid of models for this purpose, the white dwarf atmosphere must be simulated. This is computationally expensive so a number of as- sumptions are made: Firstly, that the atmosphere is thin relative to the star and is comprised of homogeneous, plane, and parallel layers, where density, pressure and temperature only vary vertically; secondly that the atmosphere is in hydrostatic equilibrium; thirdly that there is no energy generation or loss within the atmosphere itself; and lastly, the atmosphere is in Local Thermal Equilibrium (LTE). Each of these approximations may have domains in

which it does not apply. Most notably the LTE approximation is invalid for white dwarfs withTeff > 50000 K, and convective atmospheres obviously vary in more than the radial

direction.

With these assumptions in place, the equation of state and absorption coefficients are calculated for each layer, which when combined determine how energy is transmitted through the layers to eventually emerge as a photospheric spectrum. The intensity as a function of wavelength can then be calculated for each layer in a radiative atmosphere. Convection zones cannot be realistically simulated in 1-d, so a mixing-length approximation is used where the energy flux due to convection is incorporated over a certain multiple of the pressure scale height, calibrated using observations of pulsating white dwarfs where the depth of the convection zone can be estimated.

With all of the appropriate physics taken into account, the model produces a spec- trum for the surface intensity of the white dwarf over all wavelengths at low resolution. This can then be used to calculate higher resolution spectra over the wavelength coverage of the observations, incorporating the contribution of up to thousands of weak metal absorption lines that would be computationally impractical to include at each stage of the model. For given atmospheric parameters, the strength of each line is proportional to the atmospheric abundance of the element, unless the line saturates at very high abundances. The exact method of retrieving the final abundances varies by user, but usually involves producing a grid of model spectra with different atmospheric parameters and metal abundances and finding which spectrum best fits the data. The abundances are then tweaked until a good fit is achieved.

2.2.2 The accretion-diffusion scenario

Fitting model atmospheres to spectra returns the atmospheric parameters and metal abun- dances of the white dwarf. However, the metal abundances are not necessarily those of the disrupted planetesimal that the white dwarf is accreting, as each metal diffuses out of the photosphere on a different timescale. The timescales are dependent on the mass and charge of the element and on the depth of the white dwarf envelope, as well as the white dwarfTeff

and, in hotter white dwarfs, radiative levitation. The depth at which an atom is no longer in the envelope (i.e., no longer affecting the spectrum) and therefore where the diffusion timescale can be calculated introduces some uncertainty. In convective white dwarfs it is simply the base of the convection zone (Koester, 2009), but there is no clear boundary in a radiative atmosphere, so the convention is to calculate the diffusion timescale at a Rosse- land optical depth of 2/3 (Koester et al., 2014a). In hydrogen atmosphere white dwarfs with

Teff & 20000 K, metals may also be suspended by transfer of momentum from photons,

Figure 2.5: Example of the accretion-diffusion model for two elements with different diffu- sion timescales. The plot shows the atmospheric abundances of each element as a function of time. After the initial switch-on, the metals accreting and diffusing take approximately five diffusion timescales to reach a steady state. After the accretion ends the metals are removed from the atmosphere in a few diffusion timescales. Figure taken from Koester (2009).

has the effect of suspending small amounts (log[X/H] ∼ −8.00) of some elements, in par-

ticular Si and C, in the photosphere (Koester et al., 2014b). White dwarfs with metal abun- dances at or below these levels do not require ongoing accretion (Chayer, 2014). Where the metal abundances are too high to be supported by radiative levitation alone (as in all of the objects discussed in this thesis), the levitation is treated as a negative component of the diffusion timescales. Diffusion timescales have been calculated independently by Detlev Koester and by the Montreal white dwarf group, taking into account radative levitation and/or convection where necessary, and show good agreement4_.

To convert the abundances measured in the photosphere into the actual composition of the debris, a steady-state approximation is used where each element undergoes a constant flow through the atmosphere. The accretion rate is then calculated as the mass of each element in the atmosphere divided by the diffusion timescale. Assuming that the accretion rate is constant, the accretion takes roughly five diffusion timescales to reach a steady-state (Figure 2.5). When the accretion event ends the metals take≈5–10 diffusion timescales 4_{Comparehttp://www1.astrophysik.uni-kiel.de/ koester/astrophysics/astrophysics.html}

to leave the atmosphere. For hydrogen atmosphere white dwarfs with Teff > 13000 K,

where the diffusion timescales are less than one year, the steady-state assumption is almost certainly valid. A handful of white dwarfs have high-resolution spectra taken a decade or more apart (see for e.g. Chapter 6 and Manser et al., 2016b), none of which show changes in absorption line strength, indicating that the accretion rates are constant over these timescales. The chances of randomly observing a white dwarf with short diffusion timescales in either the build-up or post-event stages is therefore unlikely, so the steady- state approximation is valid and the abundances of the accreting debris can be known with confidence.

However, the picture is less clear for white dwarfs with convection zones, where the diffusion timescales extend from hundreds to millions of years. Given that the orbiting debris discs show variation on a timescale of years (Xu & Jura, 2014) it would be surprising if the accretion rates remained constant for the five diffusion timescales required to reach a steady-state, and at the very longest diffusion timescales the mass that would have sunk into the atmosphere over this time can become unrealistically large. Furthermore, the re- quirement for the build-up to a steady-state requires the accretion rate to be constant from the start of the event. As the disruption of a planetesimal and circularisation into a disc may take many years (Veras et al., 2014b), the rate is likely to change with the circumstellar environment. Each of these factors combined means that the steady-state assumption does not necessarily apply to white dwarfs with long diffusion timescales. Unfortunately those same objects are often the most useful for measuring planetesimal chemistry, as even trace elements are retained in quantities large enough to be detectable. It has become convention to present multiple models for the debris chemistry at these objects, comparing (for exam- ple) the steady-state assumption with the instantaneous atmospheric abundances (Xu et al., 2014), or with the chemistry at several diffusion timescales after accretion switch-off(Raddi et al., 2015). Looking forward, the increasing sample size of observed white dwarfs with short diffusion timescales may allow those with long diffusion timescales to be excluded from general studies of planetary chemistry, whilst advances in theoretical modelling of accretion events may shed light on the appropriate conversion from atmospheric to debris metal abundances.