MA Emilio Schmidt/Lothars Notes

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 \(R_\text{max}\)?