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Kepler /
RedshiftingHere we briefly detail the conversion adopted used for comparisons to observations. These allow the models to be compared to a $1.4\,{M}_\odot$ neutron star with a radius of $11.2\,\mathrm{km}$. The gravitational potential within a neutron star atmosphere is sufficiently intense that general relativistic effects become relevant. As the neutron star atmosphere is small compared to the stellar radius, Newtonian gravity approximates general relativity well within the domain of the simulations, and all values provided in the database presented herein are in the Newtonian neutron star frame. Due to this approximation the lightcurves and parameters we provide require redshifting to be compared with observations. A full derivation of all the possible scalings that can occur given a choice of stellar radius or mass is provided in the appendix of Keek (2011). The gravitational potential of our simulations is equivalent to the Newtonian potential that arises from body with a radius of $r = 10\,\mathrm{km}$ and ${M} = 1.4\,{M}_\odot$ (note that quantities calculated in the Newtonian frame are unsubscripted). There is a plurality of masses (${M}_\mathrm{GR}$) and radii ($r_{\mathrm{GR}}$) that could give rise to the same potential using general relativity. The locus of these points occurs where the Newtonian and GR potentials are equal: $ G{M}/r^2=G{M}_{\mathrm{GR}}/\left(r_\mathrm{GR}^2\sqrt{1-2G{M}_\mathrm{GR}/(c^2r_\mathrm{GR})}\right). $ We have conducted our analysis for the for a neutron star that has the same mass as the star in the Newtonian simulations, choosing ${M}_\mathrm{GR} = {M} = 1.4\,{M}_\odot$. Given this choice, the radius of the neutron star to give the same general relativistic potential increases, yielding $r_\mathrm{GR} = \xi r=11.2\,\mathrm{km}$, where $\xi=1.12$ (note that this is not the radius measured by a distant observer). The redshift can now be found using $ 1+z = 1/\sqrt{1-2G{M}_\mathrm{GR}/(c^2 r_\mathrm{GR})} = 1.258. $ The relevant conversions for radius, luminosity, accretion rate, and time in this case for an observer (subscripted $\infty$), in terms of the Newtonian reference frame values, are: $ r_\infty = \xi (1+z)r$ $L_\infty = \xi^2 \cdot L/(1+z)^2 $ $t_\infty = (1+z)t $ $\dot{{M}}_\infty = \dot{{M}}$ $\frac{L_{\mathrm{acc},\infty}}{L_{\mathrm{Edd},\infty}} = \frac{z}{\zeta(1+z)} \frac{L_{\mathrm{acc}}}{L_{\mathrm{Edd}}}$ $\alpha_\infty = \frac{z}{\zeta(1+z)} \alpha $ where $\zeta = G{M}/(c^2 r)$. A consequence of setting the GR mass equal to the Newtownian mass is that the mass accretion rate is identical in the Newtonian and observer frames. Additionally, this also yields $\xi = \sqrt{1+z}$, which simplifies a number of these relationships. The variation in $L_\mathrm{acc}/L_\mathrm{Edd}$ is attributable to the increased Eddington luminosity that arises from the strengthened GR potential. It is also worth commenting that the scaling for $\alpha$ assumes only a contribution to the persistent emission from accretion. The radiation from persistent emission is redshifted in the same manner as burst luminosity. This can introduce a sytematic uncertainty into the calculation of a redshifted $\alpha$ if the accretion component is not calculated separately from the thermal component. As $L_\mathrm{acc}\gg L_{th}$ this is minimal, especially given that ${z}/{[\zeta(1+z)]} = 0.9934$ given our choice of mass. |