MA Fynn Wawrzyniak - Revision history

Lothar.brendel: /* Relevant Papers */ kein Delimiter |

2026-07-23T05:03:11Z

Relevant Papers: kein Delimiter |

← Older revision		Revision as of 07:03, 23 July 2026
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	== Relevant Papers ==		== Relevant Papers ==

			* [https://www.aanda.org/articles/aa/pdf/2009/48/aa12879-09.pdf Zanni 2009] - Discusses the boundary condition used for magnetospheric truncation accretion.

	~~- [https://www.aanda.org/articles/aa/pdf/2009/48/aa12879-09.pdf\| Zanni 2009] - Discusses the boundary condition used for magnetospheric truncation accretion.~~		* [https://arxiv.org/pdf/1111.3068 Romanova 2018] - Look into magnetospheric truncation and boundary layer accretion. Discusses BCs for both cases.

	- [https://arxiv.org/pdf/1111.3068\| Romanova 2018] - Look into magnetospheric truncation and boundary layer accretion. Discusses BCs for both cases.



	= Initial Conditions =		= Initial Conditions =

Fynn.W at 15:07, 22 July 2026

2026-07-22T15:07:04Z

@@ Line 1: / Line 1: @@
 = Initial Conditions =
 The simulations are performed on a spherical grid in the coordinates \((r,\theta,\phi)\). However, for the description of the disk structure it is useful to additionally use the corresponding cylindrical coordinates \((R,z,\phi)\).
 == Disk ==

Fynn.W: M: WIP IC

2026-06-05T15:16:26Z

M: WIP IC

← Older revision		Revision as of 17:16, 5 June 2026
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	= Initial Conditions =		= Initial Conditions =

			The simulations are performed on a spherical grid in the coordinates \((r,\theta,\phi)\). However, for the description of the disk structure it is useful to additionally use the corresponding cylindrical coordinates \((R,z,\phi)\).

			== Disk ==

			The disk is initialized as a smooth and axisymmetric model. The initial conditions do not represent a fully self-consistent equilibrium. Instead, they serve as a physically reasonable and numerically stable starting configuration. For this purpose, several approximations of a thin disk are used in the following.

			Deviations from a complete equilibrium are adjusted during the first time steps. The aim is therefore to keep the transition from the prescribed initial condition to the physically evolved state as short and stable as possible.

			=== Temperature distribution ===

			Inside the disk, it is assumed that the initial temperature profile depends only on the cylindrical radius, \(T_{\mathrm{disk}} = T_{\mathrm{disk}}(R)\). This assumption is consistent with the thin-disk approximation. Close to the disk midplane, a geometrically thin disk satisfies \(\|z\| \ll R\). Therefore, from \(r = \sqrt{R^2+z^2}\), one obtains approximately \(r \approx R\). The temperature profile inside the disk can thus be described as a function of the cylindrical radius.

			For the initial temperature profile, a simple power law is used.

			\begin{align}
			T_{\mathrm{disk}}(R) = T_0 \left(\frac{R}{R_0}\right)^{-p_T}
			\end{align}

			Here, \(T_0\) is the temperature at the reference radius \(R_0\). For the exponent, \(p_T = \frac{1}{2}\) is chosen. This choice is motivated by the typical temperature decrease of a disk irradiated from the inside. For a central radiation source, the radiative flux \(F\) decreases approximately as \(F \,\propto\, R^{-2}\) for geometrical reasons. Combining this with the rough estimate \(F \,\propto\, T^4\) gives \(T \,\propto\, R^{-1/2}\).

			This estimate is strongly simplified, since the actual temperature structure additionally depends on the optical depth, the vertical structure of the disk, the irradiation geometry, and the local radiation transport. Nevertheless, it provides a useful qualitative approximation for a smooth initial temperature profile. In addition, the mean temperature profile inside the disk was analyzed from already evolved simulations. The comparison in Fig.~\ref{fig:initial_conditions_tgas_profile} shows that a power law with \(p_T = \frac{1}{2}\) approximately describes the radial temperature profile in the disk. Based on these simulation results, the normalization is chosen as \(T_0 = 1000\,\mathrm{K}\).

			\( \color{red} Image \color{black} \)


	\( \color{red} TODO \color{black} \)		\( \color{red} TODO \color{black} \)

Fynn.W: K1 = 0 Special Case

2026-06-02T14:26:57Z

K1 = 0 Special Case

← Older revision		Revision as of 16:26, 2 June 2026
Line 700:		Line 700:

	[[File:MA FW 05 disk limit x Tgas.png]]		[[File:MA FW 05 disk limit x Tgas.png]]

			=== No direct stellar EUV radiation: \(K_1=0\) ===

			A slightly more general analytic limit is obtained if only the direct stellar EUV term vanishes,

			\begin{align}
			K_1 = 0 .
			\end{align}

			This corresponds to a cell that does not receive direct stellar EUV photons, while diffuse recombination photons may still contribute through \(K_2\). The local equilibrium equation then becomes

			\begin{align}
			0
			=
			K_2 \, (1-x)
			+
			K_3 \, x(1-x)
			-
			K_4 \, x^2 .
			\end{align}

			Expanding the terms gives

			\begin{align}
			0
			=
			K_2
			+
			(K_3-K_2)x
			-
			(K_3+K_4)x^2 .
			\end{align}

			This is a quadratic equation in \(x\). The two roots are

			\begin{align}
			x_{\pm}
			=
			\frac{
			(K_3-K_2)
			\pm
			\sqrt{
			(K_3-K_2)^2
			+
			4K_2(K_3+K_4)
			}
			}{
			2(K_3+K_4)
			}.
			\end{align}

			The derivative in this limit is

			\begin{align}
			f'(x)
			=
			(K_3-K_2)
			-
			2(K_3+K_4)x .
			\end{align}

			Evaluating the derivative at the two roots gives

			\begin{align}
			f'(x_{\pm})
			=
			\mp
			\sqrt{
			(K_3-K_2)^2
			+
			4K_2(K_3+K_4)
			}.
			\end{align}

			Thus, \(x_+\) is the stable root, while \(x_-\) is unstable. So \(x_+\) is the physically relevant solution in the interval \(0\le x\le 1\). In the additional limit \(K_2=0\), this result reduces to the shielded collisional-ionization case,

			\begin{align}
			x_- = 0,
			\qquad
			x_+
			=
			\frac{K_3}{K_3+K_4}.
			\end{align}

Fynn.W: Special Case x=0 and x=1

2026-06-01T12:15:23Z

Special Case x=0 and x=1

Show changes

Fynn.W: Ionization Shielded Gas Case

2026-05-31T22:43:47Z

Ionization Shielded Gas Case

← Older revision		Revision as of 00:43, 1 June 2026
Line 445:		Line 445:
	0 \le x \le 1 .		0 \le x \le 1 .
	\end{align}		\end{align}

			== Solver ==

			\( \color{red} TODO \color{black} \)

			== Special Cases ==

			=== Shielded gas: collisional ionization and recombination only ===

			In very optically thick regions, for example inside the disk, the direct stellar EUV radiation can be completely shielded. The incoming photon density \(u_L\) is zero and therefore

			\begin{align}
			K_1 = 0 .
			\end{align}

			The diffuse recombination radiation term can also be neglected in this limit. If the on-the-spot approximation is used \(u_{\mathrm{rec}}=0\) and therefore

			\begin{align}
			K_2 = 0 .
			\end{align}

			Even without explicitly using the on-the-spot approximation, this limit is still physically plausible in the optically thick disk. The recombination photons have a very short mean free path and are absorbed close to where they are produced. Setting \(K_1=K_2=0\), the local rate equation reduces to

			\begin{align}
			\partial_t x
			=
			K_3 \, x(1-x)
			-
			K_4 \, x^2 .
			\end{align}

			The two stationary solutions are therefore

			\begin{align}
			x_1 = 0,
			\qquad
			x_2
			=
			\frac{K_3}{K_3+K_4}.
			\end{align}

			The solution \(x_1=0\) is an exact mathematical equilibrium because the collisional ionization term is proportional to \(x\). However, if a small finite ionized fraction is present and \(K_3>0\), the collisional ionization term increases \(x\) and the solution moves toward \(x_2\). In this sense, \(x_2\) is the relevant stable equilibrium solution. Inserting \(K_3\) and \(K_4\), the density cancels out:

			\begin{align}
			x_2(T_{\mathrm{gas}})
			=
			\frac{C(T_{\mathrm{gas}})}
			{C(T_{\mathrm{gas}})+\alpha^{(1)}(T_{\mathrm{gas}})} .
			\end{align}

			Therefore, in this shielded limit the stable ionization state depends only on the gas temperature.

			[[File:MA FW 05 disk limit x Tgas.png]]

Fynn.W: M: Rename

2026-05-20T10:56:03Z

M: Rename

← Older revision		Revision as of 12:56, 20 May 2026
Line 387:		Line 387:

	\begin{align}		\begin{align}
	~~A_{\mathrm{E}}~~ = \frac{c u_L}{n_{\mathrm{H}}\Delta x},		K_1 = \frac{c u_L}{n_{\mathrm{H}}\Delta x},
	\qquad		\qquad
	~~q_{\mathrm{E}}~~ = n_{\mathrm{H}}\sigma_{\mathrm{EUV}}\Delta x,		Q_1 = n_{\mathrm{H}}\sigma_{\mathrm{EUV}}\Delta x,
	\qquad		\qquad
	~~A_{\mathrm{rec}}~~ = \sigma_{\mathrm{rec}}u_{\mathrm{rec}}c,		K_2 = \sigma_{\mathrm{rec}}u_{\mathrm{rec}}c,
	\qquad		\qquad
	~~A_{\mathrm{C}}~~ = C(T_{\mathrm{gas}}) n_{\mathrm{H}},		K_3 = C(T_{\mathrm{gas}}) n_{\mathrm{H}},
	\qquad		\qquad
	~~A_{\mathrm{R}}~~ = \alpha^{(1)}(T_{\mathrm{gas}}) n_{\mathrm{H}} .		K_4 = \alpha^{(1)}(T_{\mathrm{gas}}) n_{\mathrm{H}} .
	\end{align}		\end{align}

Line 403:		Line 403:
	\partial_t x		\partial_t x
	=		=
	~~A_E~~		K_1
	\left[		\left[
	1-e^{-~~q_E~~(1-x)}		1-e^{-Q_1(1-x)}
	\right]		\right]
	+		+
	~~A_{\mathrm{rec}}~~ \, (1-x)		K_2 \, (1-x)
	+		+
	~~A_{\mathrm{C}}~~ \, x(1-x)		K_3 \, x(1-x)
	-		-
	~~A_{\mathrm{R}}~~ \, x^2 .		K_4 \, x^2 .
	\end{align}		\end{align}

Line 428:		Line 428:
	f(x)		f(x)
	=		=
	~~A_E~~		K_1
	\left[		\left[
	1-e^{-~~q_E~~(1-x)}		1-e^{-Q_1(1-x)}
	\right]		\right]
	+		+
	~~A_{\mathrm{rec}}~~ \, (1-x)		K_2 \, (1-x)
	+		+
	~~A_{\mathrm{C}}~~ \, x(1-x)		K_3 \, x(1-x)
	-		-
	~~A_{\mathrm{R}}~~ \, x^2 .		K_4 \, x^2 .
	\end{align}		\end{align}

Fynn.W: M: WIP – Local ionization equilibrium solver

2026-05-19T19:27:48Z

M: WIP – Local ionization equilibrium solver

Show changes

Fynn.W: Plots + Initial Ion Fraction

2026-05-12T19:05:53Z

Plots + Initial Ion Fraction

@@ Line 148: / Line 148: @@
 Therefore, the right cell face can be almost completely dark while the cell-averaged photon density remains finite. However, it decreases with increasing optical depth as \(1/\tau\), instead of approaching the constant value \(u_\mathrm{L}/2\) from the old arithmetic average.
 In the crashing cell, the optical depth was extremely large,
@@ Line 219: / Line 221: @@
 y = NeutralFractionRateEquation(   ustar, urec, Tgas, nH, y);
 </syntaxhighlight>

Fynn.W at 17:41, 12 May 2026

2026-05-12T17:41:11Z

← Older revision		Revision as of 19:41, 12 May 2026
Line 6:		Line 6:
	= Problems =		= Problems =

	== Mean Photon Density in Irradiation Update ( ==		== Mean Photon Density in Irradiation Update (belt/src/Makemake2.1/Irradiation.c ~ Line 570) ==

	During the iterative irradiation update, the local ionization update requires a representative photon density inside each cell for the photoionization term. The ray tracer knows the incoming photon density at the left cell face, \(u_\mathrm{L}\), and computes the outgoing photon density at the right cell face, \(u_\mathrm{R}\).		During the iterative irradiation update, the local ionization update requires a representative photon density inside each cell for the photoionization term. The ray tracer knows the incoming photon density at the left cell face, \(u_\mathrm{L}\), and computes the outgoing photon density at the right cell face, \(u_\mathrm{R}\).