# Correction to: On Fields and Mass Constraints for the Uniform Propagation of Magnetic-Flux Ropes Undergoing Isotropic Expansion

**Part of the following topical collections:**

**Correction to: Solar Phys. (2013) 284:245 – 259** **https://doi.org/10.1007/s11207-012-0176-5**

*e.g.*Jackson 1963) gives an axial component that is

\(H\ (=+1\mbox{ or}-1)\) is the handedness of the flux rope (FR).

\(\mathbf{H}\ (= \mathbf{B}/ \mu _{0})\) is the magnetic field intensity, and \(\mathbf{B}\) the magnetic field density.

\(\mu _{0}\) is the magnetic permeability of the vacuum. (We assume no magnetization.)

\(R _{\mathrm{FRcore}}\) is the cross section radius of the FR modeled.

\(R _{\mathrm{cFR}}\) is the distance from the center of the Sun to the axis of the flux rope modeled.

\(A(t) = j _{0} / R _{\mathrm{FRcore}}\), where \(j _{0}\) is the first node of the grade 0 cylindrical Bessel function ‘\(\mathcal{J}_{0}(A(t,\rho,\varphi))\)’ of the first kind.

- \(B _{x} = B _{0} (t _{0} /t)^{2} \mathcal{J} _{0}\) (\(A(t,\rho,\varphi)\)) is the axial field of the flux-rope, where \(B _{0}\) is the magnetic field strength parameter in the MHD flux-rope solution at the initial time \(t _{0}\) in its evolution. Finally, the argument of the Bessel function \(\mathcal{J}_{0(1)}\) iswhich is given in Equation 7 of the paper. (\(\rho , \varphi\) are polar radial and angle coordinates.)$$A(t, \rho, \varphi) = A(t)\rho \bigl[1 + \rho/R_{\mathrm{cFR} } \bigl(\cos( \varphi) - |\sin(\varphi)|\bigr)\bigr], $$

The relevance of the previously missing term \(\Delta j _{cx}\) is that it dominates the axial current contribution within the FR plasma located in the region defined by \(0.93 R _{\mathrm{FRcore}} < \rho < 1.07R _{\mathrm{FRcore}}\).

It is straightforward to check that \(\mathbf{div}(\mathbf{J}_{\mathrm{c}}) = 0\) is well satisfied up to \(n=1\) in a perturbation expansion in \(\varepsilon ^{n}\) of our 3D MHD analytical solution of a FR evolution, where \(\varepsilon \equiv R _{\mathrm{FRcore}} / R_{ \mathrm{cFR} }\) is a small quantity.

## Notes

## References

- Jackson, D.: 1963,
*Classical Electrodynamics, Section 5.8*, 3rd edn. Wiley, New York. Google Scholar