BA Mihály Török: Difference between revisions
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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 \(\text{R}_\text{HL}\) = \(\text{R}_\text{HL}\cdot 1 \). <br> | |||
The most important physical quantities for the project are: Length, Velocity and Mass. The natural units for each of these are defined as follows:<br> | |||
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.<br> | |||
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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. <br> | |||
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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. | |||
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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 = | = Grid = | ||
Revision as of 14:41, 12 June 2025
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
Für die Radialkoordinate gilt \(r\in[R,R_\text{max}]\), und die Koordinate der inneren Zellgrenze von Zelle Nr. \(k\) (Zählung beginnend bei \(0\)) liegt bei \[ r_k = R\left(\frac{R_\text{max}}{R}\right)^{k/N_r} \].
Steady State?
Fourier-Zerlegung der Massenstromdichte
Die Fourier-Zerlegung setzt einen periodischen Funktionsverlauf voraus. Eine Funktion des Polarwinkels \(\theta\in[0,\pi]\) ist i.A. nicht periodisch, aber wenn man in einem System mit Zylindersymmetrie vom Südpol aus "auf der Rückseite" wieder zum Nordpol läuft (\(\theta\in[\pi,2\pi]\)), dann ist die Gesamtfunktion periodisch (und spiegelsymmetrisch bzgl. \(\pi\)).