BA Mihály Török

Natural Units

Originally the CGS system of units was used for the simulation. Instead I will use natural units. The advantage of the natural units is, that the most relevant physical constants or parameters can be used to express other physical quantities. For example if I would take the accretion radius \(\text{R}_\text{HL}\) as the natural unit for length then the radius of the accretor could be given as \(\text{R} = 0.1 \cdot \text{R}_\text{HL}\). This of course only makes sense if \(\text{R}_\text{HL}\) doesn't change during the project. And it also means that \(\text{R}_\text{HL}\) is given as \(\text{R}_\text{HL}\) = \(\text{R}_\text{HL}\cdot 1 \).

The most important physical quantities for the project are: Length, Velocity and Mass. The natural units for each of these are defined as follows:

The most important length scale for the simulation is the Hoyle-Lyttleton radius or accretion radius. Therefore it is reasonable to use it as the natural unit for the Length. It also doesn't change during the project.

The natural unit of the velocity will be the external velocity, the velocity of the gas defined at a point where the gas is unperturbed. This also won't change during the project, and is used to calculate the Mach number so it is also a reasonable choice for a natural unit.

For the mass the natural unit is defined through \(\rho_{ext}\), the density of the gas at a point where it is unperturbed. With that the unit for mass is given as \(\rho_\text{ext} \cdot R_\text{HL}^3 = 1\). This is also reasonable since \(\rho_{ext}\) doesn't change during the project.

We also set \(\text{G}\cdot \text{M} = \frac{1}{2}\). G is the gravitational constant and M is the mass of the black hole.

Grid

During the project we want to test if the boundary condition set at the edge of the accretor has any effect on the flow morphology. To do that we need the dependence of the number of cells in radial direction \(\text{N}_r\) on \(\text{R}\) and \(\text{R}_{max}\). For the radial component it is true, that \(r\in[R,R_\text{max}]\) and the coordinate of the inner cell wall of cell number \(k\) is given as follows: \[ r_k = R\left(\frac{R_\text{max}}{R}\right)^{k/N_r} \] The value of \(k\) starts at 0 and ends at \(N_r - 1 \)
With this and with the condition for quadratic cells we can calculate the dependence. The condition for quadratic cells is: \(\Delta \Theta \cdot r_k = \Delta r_k\). With \(\Delta r_k = r_{k+1} - r_k\) follows: \(\Delta \Theta = \frac{r_{k+1}}{r_k}-1\). Using the equation for \(r_k\) we get the following equation: \[ \left(\frac{R_\text{max}}{R}\right)^{1/N_r} = \Delta \Theta + 1 = \frac{\pi}{N_{\Theta}} + 1 \] After taking the natural logarithm of the equation and simplifying it we get: \[ N_{r} = \frac{ln(R_{max}/R)}{ln(1 + \pi / N_{\Theta})} \]

Steady State?

In this section it will be discussed if the simulation has reached steady state and when. The simulation is started with a homogenous gas that has physical parameters according to the starting parameters. Then the effects of the accretor will be applied, the black hole will be "switched on". After this the flow morphology will start to change immediately. When this change stops then the simulation is in steady state. To check for the steady state I used the kinetic energy calculated from the radial velocity. This will be plotted over the snapshot number and if there is a convergence the simulation is in steady state.
The first simulation was started with 100 Cells in \(\Theta\) direction and 550 cells in \(r\) direction. The Mach number was 1.5, the radius of the domain \(R_{max}\) was equal to \(10^4\) and \(t_{stop} = 1\cdot 10^4\) was chosen. The plot of the kinetic energy looks as follows:

Kinetic energy over the snapshot number

The energy hits a plateau first and then increases steadily. This means, that the simulation isn't in a steady state. The reason for this is, that the simulation added the accreted mass to the mass of the black hole and increased it's gravitational potential accordingly. Since the accretor will accrete more mass over time the system won't reach a steady state.
Therefore a new simulation was started with the same starting parameters but with a constant mass for the accretor. The same plot for this simulation looks as follows:

Kinetic energy over the snapshot number

This looks much more promising then the last one. To check if the energy really doesn't change I looked at the end of the graph more closely. The plot for that looks like this:

Closeup of the energy over the snapshot number

The picture shows a decrees at the start but this decrees converges to a constant line. This means the simulation is in steady state.
Next I will look at how the mass flux behaves over the time, and see what effects not being in steady state has on it. The mass flux is calculated as follows: \[ \dot{M} = - \rho \cdot v_r \] In the next graph \(\dot{M}\) is depicted for ten different cases. For each case I took \(t\) and only used data that had a snapshot number \(\leq t\). This way the graph shows the change in mass flux over time.

Mass flux over time (with artificial vertical shift)

The graph shows two things of interest. First, that there is more inflow for higher angles, this we will refer to as the anisotropy of the accretion. Second, that the curves are not smooth, they have small buckles which change over time. The anisotropy of the accretion is expected, the buckles are not. To understand why these buckles are formed we can calculate the Fourier series of the function.

Fourier Decomposition of the Mass Flow Density

To be able to do the Fourier decomposition a periodic function is needed. Since the function above is dependent on the polar angle \(\theta\in[0,\pi]\) it is not periodic. Although if a system that has cylindrical symmetry goes from it's south pole to the north again on the other side of the system (\(\theta\in[\pi,2\pi]\)), then the function will be periodic. The following plots show the Fourier series and the original curve for n = 1,12,20 where n is the order of the Fourier series.