A Helicity-Based Method to Infer the CME Magnetic Field Magnitude in Sun and Geospace: Generalization and Extension to Sun-Like and M-Dwarf Stars and Implications for Exoplanet Habitability

Abstract

Patsourakos et al. (Astrophys. J. 817, 14, 2016) and Patsourakos and Georgoulis (Astron. Astrophys. 595, A121, 2016) introduced a method to infer the axial magnetic field in flux-rope coronal mass ejections (CMEs) in the solar corona and farther away in the interplanetary medium. The method, based on the conservation principle of magnetic helicity, uses the relative magnetic helicity of the solar source region as input estimates, along with the radius and length of the corresponding CME flux rope. The method was initially applied to cylindrical force-free flux ropes, with encouraging results. We hereby extend our framework along two distinct lines. First, we generalize our formalism to several possible flux-rope configurations (linear and nonlinear force-free, non-force-free, spheromak, and torus) to investigate the dependence of the resulting CME axial magnetic field on input parameters and the employed flux-rope configuration. Second, we generalize our framework to both Sun-like and active M-dwarf stars hosting superflares. In a qualitative sense, we find that Earth may not experience severe atmosphere-eroding magnetospheric compression even for eruptive solar superflares with energies ${\approx}\, 10^{4}$ times higher than those of the largest Geostationary Operational Environmental Satellite (GOES) X-class flares currently observed. In addition, the two recently discovered exoplanets with the highest Earth-similarity index, Kepler 438b and Proxima b, seem to lie in the prohibitive zone of atmospheric erosion due to interplanetary CMEs (ICMEs), except when they possess planetary magnetic fields that are much higher than that of Earth.

Appendices

Appendix

The appendix presents a short description of the geometrical CME model used in the current investigation (Appendix A). Moreover, it contains short descriptions and equations relating magnetic field magnitude with $H_{\mathrm{m}}$ and various geometrical parameters for various theoretical CME models (Appendices B – G).

Appendix A: Deducing CME Geometrical Parameters from the GCS Model

To obtain the geometrical parameters $R$ and $L$, we adopt the graduated cylindrical shell (GCS) forward-fitting model of Thernisien, Vourlidas, and Howard (2009). This is a geometrical flux-rope model routinely used to fit the large-scale appearance of flux-rope CMEs in multi-viewpoint observations acquired by coronagraphs onboard the Solar and Heliospheric Observatory (SOHO) and Solar Terrestrial Relations Observatory (STEREO) spacecraft. The GCS user modifies a set of free parameters (CME height, half-angular width $w$, aspect ratio $k$, tilt angle, and central longitude and latitude) to achieve a best-fit agreement between the model and observations. A detailed description can be found in Thernisien, Vourlidas, and Howard (2009).

In the framework of the GCS model, the CME radius $R$ at a heliocentric distance $r$ is given by the following equation:

$$ R(r)=k r. $$

(4)

To assess the flux-rope length $L$, it is assumed that the CME front is a cylindrical section (see Figure 1 of Démoulin and Dasso, 2009) with an angular width provided by the geometrical fitting. One may then write

$$ L=2 w r_{\mathrm{mid}}, $$

(5)

where $r_{\mathrm{mid}}$ is the heliocentric distance half-way through the model cross section, along its axis of symmetry. The half-angular width $w$ is given in radians.

Appendix B: Cylindrical Linear Force-Free Model

The Lundquist flux-rope model (Lundquist, 1950) is arguably the most commonly used flux-rope model and corresponds to a cylindrical force-free solution.

Based on results from Dasso et al. (2006), we obtain for a Lundquist flux rope:

$$ H_{\mathrm{m}}=\frac{4\pi{B_{0}}^{2}L}{\alpha} \int_{0}^{R}{J_{1}}^{2}( \alpha R)\,\mathrm{d}r, $$

(6)

with $L$ and $R$ the flux-rope length and radius, respectively, $J_{1}$ the Bessel function of the first kind, $B_{0}$ the maximum axial field, and $\alpha$ the force-free parameter. With the common assumption of a purely axial (azimuthal) magnetic field at the flux-rope axis (edge), we obtain

$$ \alpha R = 2.405. $$

(7)

Appendix C: Cylindrical Nonlinear Force-Free Model

This cylindrical nonlinear force-free flux-rope model was proposed by Gold and Hoyle (1960). According to Dasso et al. (2006), we have

$$ H_{\mathrm{m}}=L\biggl(\frac{8\pi[\ln (1+U^{2}/4)]^{2}}{U^{2}}\biggr)B_{0}^{2}R^{4}{ \tau}_{0}, $$

(8)

with $U=2 \tau_{0} R$ and $\tau_{0}=\frac{1}{2}\alpha$.

Appendix D: Toroidal Linear Force-Free Model

This toroidal force-free model was proposed by Vandas and Romashets (2016):

$$ H_{\mathrm{m}}=(2\pi H)\frac{B_{0}^{2}\pi{H}^{3}}{4{\alpha }_{0}}\biggl[8-\biggl(1+\frac{1}{{\alpha}_{0}^{2}} \biggr)\frac{R^{2}}{H^{2}}\biggr]J_{1}^{2}( \alpha_{0}), $$

(9)

with $H$ and $R$ corresponding to the torus major and minor axis, respectively, and $\alpha_{0} = 2.405$. We take $H$ equal to $r_{\mathrm{mid}}$. Both $R$ and $r_{\mathrm{mid}}$ are defined in Appendix A.

Appendix E: Linear Force-Free Model Spheromak

This linear force-free spheromak model was proposed by Kataoka et al. (2009):

$$ H_{\mathrm{m}}=0.045{r_{\mathrm{mid}}^{4}}B_{0}^{2}, $$

(10)

and corresponds to a Sun-centered sphere with radius $r_{\mathrm {mid}}$ meant to approximate a spherical magnetic cloud. The value of $r_{\mathrm {mid}}$ is the same as in Equation (4).

Appendix F: Cylindrical Constant-Current Non-Force-Free Model

This cylindrical constant-current non-force model was proposed by Hidalgo et al. (2000) and was generalized by Nieves-Chinchilla et al. (2016).

According to Dasso et al. (2006), we have that

$$ H_{\mathrm{m}} = \frac{7 \pi}{30} \tau_{0} L R^{4} B_{0}^{2} , $$

(11)

where ${\tau}_{0}$ is the twist per unit length at the flux-rope axis. The twist parameter ${\tau}_{0}$ can be written as

$$ {\tau}_{0}=\frac{N_{\mathrm{turns}}}{L}, $$

(12)

with $N_{\mathrm{turns}}$ the total number of field turns along the flux-rope axis. To estimate ${\tau}_{0,}$ we use $L$ as calculated in Appendix A and assume that $N_{\mathrm{turns}}$ is equal to 0.5 and 10, corresponding to the extreme cases between a weakly and a strongly twisted (with multiple turns) flux-rope, respectively. The number of $N_{\mathrm{turns}}$ covering this interval can be deduced from solar imaging and magnetic field observations (via photospheric magnetic field extrapolations) (e.g. Vrsnak et al., 1993; Gary and Moore, 2004; Guo et al., 2013; Chintzoglou, Patsourakos, and Vourlidas, 2015) and MC fits at 1 AU (e.g. Hu et al., 2014; Wang et al., 2016).

Appendix G: Cylindrical Linear Azimuthal Current Non-Force-Free Model

This cylindrical linear azimuthal current non-force-free model was proposed by Cid et al. (2002) and was generalized by Nieves-Chinchilla et al. (2016). In this model the azimuthal current increases with distance from the flux-rope axis.

According to Dasso et al. (2006), we obtain

$$ H_{\mathrm{m}} = \frac{\pi}{3} \tau_{0} L R^{4} B_{0}^{2}. $$

(13)

Like in the cylindrical constant-current non-force-free case, we assume that $N_{\mathrm{turns}}$ is equal to 0.5 and 10.