We develop a broad-band spectral model, AGNSLIM, to describe super-Eddington black hole accretion disc spectra. This is based on the slim disc emissivity, where radial advection keeps the surface luminosity at the local Eddington limit, resulting in L(r) ∝ r−2 rather than the r−3 expected from the Novikov-Thorne (standard, sub-Eddington) disc emissivity. Wind losses should also be important but these are expected to produce a similar radiative emissivity. We assume that the flow is radially stratified, with an outer standard disc, an inner hot Comptonizing region and an intermediate warm Comptonizing region to produce the soft X-ray excess. This gives the model enough flexibility to fit the observed data, but with the additional requirement of energy conservation to give physical constraints. We use this to fit the broad-band spectrum of one of the most extreme Active Galactic Nuclei, the Narrow Line Seyfert 1 RX J0439.6−5311, which has a black hole mass of (6 ∼ 9) × 106 M☉ as derived from the H β line width. This cannot be fit with the standard disc emissivity at this mass, as even zero spin models overproduce the observed luminosity. Instead, we show that the spectrum is well reproduced by the slim disc model, giving mass accretion rates around (5 ∼ 10) × Eddington limit. There is no constraint on black hole spin as the efficiency is reduced by advection. Such extreme accretion rates should be characteristic of the first Quasars, and we demonstrate this by fitting to the spectrum of a recently discovered super-Eddington Quasar, PSO J006 + 39, at z = 6.6.
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