MA Emilio Schmidt/Lothars Notes: Difference between revisions
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= Characteristic Length Scale = | |||
Given that the mass/length | |||
\[ \lambda = \frac{2RT}{GM} \] | |||
is the only parameter appearing in the equilibrium ODE (the boundary conditions don't provide further parameters) and that the solution must be expressible as | |||
\[ \rho(r) = \tilde\rho\,f(r/\tilde r) \] | |||
with \(\tilde\rho\) and \(\tilde r\) being units for density and length, respectively, it already follows that \(\tilde r\) is an arbitrary length in | |||
\[ \rho(r) = \frac{\lambda}{\tilde r^2}f(r/\tilde r)~, \] | |||
which allows the solution to be scaled freely. | |||
Interestingly, integrating the 2D-solution over the plane yields | |||
\[ \lambda_\text{tot} = \lambda\cdot(F_{2D}(\infty)-F_{2D}(0)) \] | |||
while an integration of the 3D-solution over the whole space gives | |||
\[ m_\text{tot} = \lambda \tilde r\cdot(F_{3D}(\infty)-F_{3D}(0))~. \] | |||
That means that in the 2D case, the contained 2D-mass (i.e. mass/length) is uniquely determined by the given parameter \(\lambda\) while in the 3D case, the contained mass can be chosen freely by varying \(\tilde r\). | |||
Adding a magnetic pressure means introducing a pressure unit \(\tilde P\), which cannot be combined with \(\lambda\) to yield a length scale, either. | |||
= Pressure Truncation = | = Pressure Truncation = | ||
While it's true that space is not vacuum, in which sense and to what degree is it more accurate to ignore ''all'' mass beyond \(R_\text{max}\) (being defined via \(\rho(R_\text{max})=\rho_\text{background}\)) instead of letting the density vanish continuously and thus ignoring ''less'' of the mass? Is the background pressure (or density) supposed to have an ''effect'' or is it just a means to an end, namely to define a finite \(R_\text{max}\)? | While it's true that space is not vacuum, in which sense and to what degree is it more accurate to ignore ''all'' mass beyond \(R_\text{max}\) (being defined via \(\rho(R_\text{max})=\rho_\text{background}\)) instead of letting the density vanish continuously and thus ignoring ''less'' of the mass? Is the background pressure (or density) supposed to have an ''effect'' or is it just a means to an end, namely to define a finite \(R_\text{max}\)? | ||
Revision as of 21:24, 11 June 2026
Characteristic Length Scale
Given that the mass/length
\[ \lambda = \frac{2RT}{GM} \]
is the only parameter appearing in the equilibrium ODE (the boundary conditions don't provide further parameters) and that the solution must be expressible as
\[ \rho(r) = \tilde\rho\,f(r/\tilde r) \]
with \(\tilde\rho\) and \(\tilde r\) being units for density and length, respectively, it already follows that \(\tilde r\) is an arbitrary length in
\[ \rho(r) = \frac{\lambda}{\tilde r^2}f(r/\tilde r)~, \]
which allows the solution to be scaled freely.
Interestingly, integrating the 2D-solution over the plane yields
\[ \lambda_\text{tot} = \lambda\cdot(F_{2D}(\infty)-F_{2D}(0)) \]
while an integration of the 3D-solution over the whole space gives
\[ m_\text{tot} = \lambda \tilde r\cdot(F_{3D}(\infty)-F_{3D}(0))~. \]
That means that in the 2D case, the contained 2D-mass (i.e. mass/length) is uniquely determined by the given parameter \(\lambda\) while in the 3D case, the contained mass can be chosen freely by varying \(\tilde r\).
Adding a magnetic pressure means introducing a pressure unit \(\tilde P\), which cannot be combined with \(\lambda\) to yield a length scale, either.
Pressure Truncation
While it's true that space is not vacuum, in which sense and to what degree is it more accurate to ignore all mass beyond \(R_\text{max}\) (being defined via \(\rho(R_\text{max})=\rho_\text{background}\)) instead of letting the density vanish continuously and thus ignoring less of the mass? Is the background pressure (or density) supposed to have an effect or is it just a means to an end, namely to define a finite \(R_\text{max}\)?