Evaluation of the Interplanetary Magnetic Field Strength Using the Cosmic-Ray Shadow of the Sun

We analyze the Sun’s shadow observed with the Tibet-III air shower array and find that the shadow’s center deviates northward (southward) from the optical solar disk center in the “away” (“toward”) interplanetary magnetic field (IMF) sector. By comparing with numerical simulations based on the solar magnetic field model, we find that the average IMF strength in the away (toward) sector is $1.54\pm {0.21}_{\text{stat}}\pm {0.20}_{\text{syst}}$ ($1.62\pm {0.15}_{\text{stat}}\pm {0.22}_{\text{syst}}$) times larger than the model prediction. These demonstrate that the observed Sun’s shadow is a useful tool for the quantitative evaluation of the average solar magnetic field.

Figure 1

Two dimensional maps of $D_{\text{obs}}$ in the away (a) and toward (b) sectors in 2000–2009. Each panel shows a two dimensional contour of $D_{\text{obs}}$ deduced from AS events with $\sum {\rho }_{\mathrm{FT}}>10$ corresponding to the modal primary energy of $\sim 3\mathrm{TeV}$ and $△d=0.9°$. In each panel, a small circle centered on the origin indicates the optical solar disk. The significance at the maximum deficit point derived with the Li-Ma formula in the away (toward) sector is $-23.9\sigma$ ($-22.1\sigma$) [9]. The projections of $D_{\text{obs}}$ on the horizontal and vertical axes are also attached to each panel.

Figure 2

The rigidity dependences of the N-S [(a),(b)] and E-W [(c),(d)] displacements of the center of the Sun’s shadow in away and toward sectors. Black solid circles and red open triangles in each panel show the observed and simulated displacements, respectively, each as a function of the rigidity ($R$) on the horizontal axis. The error bar of each solid circle indicates the statistical error. Black solid and red broken curves display the function of $\alpha /(R/10\mathrm{TV})$ best fitting to black solid circles and red open triangles, respectively. The best-fit parameter $\alpha$ to the observed data is indicated in each panel with systematic errors estimated from the systematic error of the primary energy in our analyses of the Moon’s shadow [7].

Figure 3

Comparison between the observed and simulated IMF components at Earth. Black solid circles (red open triangles) in the upper panels display the observed (simulated) $B_x$ (a) and $B_y$ (b), respectively, each as a function of the Carrington longitude. The lower panels show the correlations between the observed and simulated $B_x$ (c) and $B_y$ (d). The regression coefficient (slope of the black solid line) is significantly larger than 1 (slope of the dashed line).

Figure 4

The parameter $\alpha$ obtained from simulations by changing the multiplication factor $f$ (see text). Red open triangles in (a) and (b) show $\alpha$ in the away and toward sectors, respectively, each as a function of $f$. The horizontal black line in each panel shows the observed $\alpha$, while the vertical blue line indicates $f$ deduced from the observed and simulated $B_{\mathrm{IMF}}$ at 1 AU. The shaded area in each panel indicates the $1\sigma$ region of $f$ allowed by the observed $\alpha$.