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. No unit which contains the dimension time can do that.

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 \(R_\text{max}\)?