Resolution low medium high
n 35 50 60
θcells 37 53 63
φcells 113 161 193
rcells 69 98 118
cells per grid 288 489 836 234 1 392 768 total cells 576 978 1 672 468 2 785 536 interpolation points 62 928 120 736 171 808
∆θ 5.58×10−2 3.86×10−2 3.24×10−2 Minimum ∆φat r= 2M 6.01×10−2 4.21×10−2 3.50×10−2 Maximum ∆φatr= 2M 1.12×10−1 7.84×10−2 6.56×10−2 Minimum ∆r 5.87×10−2 4.08×10−2 3.37×10−2
Maximum ∆r 2.77 1.96 1.63
Table 7.2: Grid cell statistics for the resolutions used. Points removed by the overlapping grid routines are still included in these counts. These statistics are given for the case when a = 0. In this case, the inner grid radius is1.4M, and the outer grid radius is 50M. For spins a6= 0, the inner radius changes as in Table 7.1, thereby changing the radial spacing, but the numbers of cells in all dimensions and the angular resolutions remain fixed.
= − Z
S
√−gρuidSi
= −
Z
S
α√γρW µ
vi−βi α
¶ dSi.
In order to evaluate this numerically, we interpolate the integrand at evenly spaced points in theθ and φ coordinates around a sphere with some radius r and sum them weighted by the infinitesimal area dSi=γijnjdAat those points. We use 150 points in the θ(latitudinal) direction and 300 points in the φ(longitudinal) direction. This is sufficient as we show in§7.5.2.
We carry out the calculation in Kerr-Schild coordinates, as opposed to Font and Ib´a˜nez, who calculated in Boyer-Lindquist coordinates. As the result is a scalar, it should be the same in both coordinate systems.
In [83], however, the density ρin the above is replaced by n =√ρ, the baryon density, and so for comparisons with their exact solution, we use this slightly altered expression when dealing with a stiff fluid.
We also approximate the rate of accretion of linear momentum onto the black hole in the manner suggested by Petrichet al.[82]:
˙ pi=−
Z
∂V
Tij√
−gdSj. (7.14)
This is only an approximation since it neglects long-range gravitational effects, and also needs to be evaluated at a large radius in flat space-time, whereas we evaluate it at the same radius as the mass accretion rate.
7.4.2 Velocities
In order to provide a suitable comparison with previous work, we transform the velocities we obtain into those in Boyer-Lindquist coordinates, using the transformation (3.72). This is only possible outside the horizon of the black hole, and so results derived from these cannot extend inside the black hole. We can calculate the total velocityv=p
γijvivj and, in some cases, locate the stagnation point of the flow, which lies downstream of the black hole.
We can also extract some more information about the flow structure by calculating streamlines. These are found from a single snapshot of the flow, and so would not be streamlines in the strict sense, except that, since we reach steady state, the flow has no time dependence. When plotting streamlines, we use the vector
w=
sγijvivj
δijvivj (vx, vy, vz), (7.15) which satisfies
w2=δijwiwj =γijvivj =v2, (7.16)
and then requires no further alterations to Overture’s streamline plotter, which is designed for Newtonian flows.
7.4.3 Shock angles
In §8.2 we shall see that under some circumstances, including the ones we investigate, a shock-cone forms. Far from the black hole, the sides of the cone become straight, and we use this part of the flow to calculate the opening angle of the cone.
In order to find the shock location, we plot the velocity p
γijvivj in Boyer-Lindquist coordinates around circles centred on the origin, at various radii. We can then take the centre of the jump in velocity as the shock location, since conserved schemes preserve the shock location (see§4.2). The circles are in the plane containing the wind direction and they-axis (or the perturbation direction, if different). We show one of these plots in Figure 7.2, where we see that, although the shock is spread over about 11◦, the centre of the shock is fairly easy to find. In particular, we note that the centre of the shock in the angular direction corresponds to the centre of the shock in the velocity component, as we would expect from the conservation property of finite-volume methods. The data points are interpolated; however, we interpolate to the same number of points as there are grid cells, so there is a close correspondence between the data points and the actual values computed.
However, since the shock-cone is not centred at the origin, we then have to calculate the actual angle of the shock with the wind direction as
θ= tan−1
µr1sinθ1−r2sinθ2
r1cosθ1−r2cosθ2
¶
, (7.17)
where (r1, θ1) and (r2, θ2) are two points on the shock. This formula is derived to find the angle of the line passing through these two points to the horizontal axis. The radii used here should be sufficiently large that the shock has become straight. In practice, we user1= 20M andr2= 30M. We calculate the angle of the shock-cone both above and below the axis. This is illustrated in Figure 8.1 in Chapter 8.
0 45 90 135 180 225 270 315 360 0.65
0.7 0.75 0.8 0.85 0.9
d
d
Figure 7.2: Example of how to calculate the shock angle from a plot of velocity. The upper and lower horizontal lines mark the upper and lower bounds of the shock, the central horizontal line is midway between these, and the shock location is taken to be where this crosses the numerical solution, at 45◦ in this case. The plot is of v against angle around a circle at radius r = 30M for the model UB1 (see Chapter 8).
7.4.4 Observable effects
In practice, it is hard to resolve the fluid flow of real black hole systems at anywhere near the required resolution to distinguish the above effects, using current technology. However, two effects may be observable:
Firstly, the high densities and temperatures which occur near the black hole horizon lead to very high fluid temperatures, and therefore to the fluid emitting X-rays, which can be detected using current technology. An approximate procedure for calculating the X-ray luminosity can be found in [101]. An extension to include relativistic effects can be found in [100]. Since this makes a significant difference, of around 50%, to the luminosity, and would take substantial effort to calculate, including a generalisation of the argument in [100] to non-axisymmetric effects, we did not perform the calculation, but it could be performed in theory using the results we have obtained.
Secondly, when applied to the problem of EMRI, outlined in §7.1.3, the different drag rates could affect the rate of inspiral, although the overall impact will be low. This inspiral rate could be detected, although gravitational wave detectors may be required to provide accurate data.