Figure 1

The polarization ellipticity of the photon mode as a function of density near the vacuum resonance. The two curves correspond to the two different modes. In this example, the parameters are $B={10}^{13}\mathrm{G}$, $E=5\mathrm{k}\mathrm{e}\mathrm{V}$, $Y_e=1$, and ${\theta }_{kB}=45°$. The ellipticity of a mode is specified by the ratio $K=-i{E}_x/E_y$, where $E_x$ ($E_y$) is the photon’s electric field component along (perpendicular to) the $\mathbf{k}$-$\mathbf{B}$ plane. The $O$ mode is characterized by $|K|\gg 1$, and the $X$ mode $|K|\ll 1$.Reuse & Permissions

Figure 2

A schematic diagram illustrating how vacuum polarization affects the polarization state of the emergent radiation from a magnetized NS atmosphere. This diagram applies to the “normal” field regime [$B\lesssim 7\times {10}^{13}\mathrm{G}$; see Eq. (3)] in which the vacuum resonance lies outside the photospheres of the two photon modes. The photosphere is defined where the optical depth (measured from outside) is $2/3$ and is where the photon decouples from the matter. At low energies (such as $E\lesssim 1\mathrm{k}\mathrm{e}\mathrm{V}$), the photon evolves nonadiabatically across the vacuum resonance (for ${\theta }_{kB}$ not too close to $0$), and thus the emergent radiation is dominated by the $X$ mode. At high energies ($E\gtrsim 4\mathrm{k}\mathrm{e}\mathrm{V}$), the photon evolves adiabatically, with its plane of polarization rotating by $90°$ across the vacuum resonance, and thus the emergent radiation is dominated by the $O$ mode. The plane of linear polarization at low energies is therefore perpendicular to that at high energies.Reuse & Permissions

Figure 3

The phase evolution of the observed spectral flux (upper panel, in arbitrary units), the linear polarization $F_Q/F_I$ (middle panel), and the polarization degree $P_L=(F_Q^2+P_U^2{)}^{1/2}/F_I$ (lower panel) produced by the magnetic polar cap of a rotating NS. The model parameters are the magnetic field $B={10}^{13}\mathrm{G}$, the effective temperature of the polar cap $T_{\mathrm{e}\mathrm{f}\mathrm{f}}=5\times {10}^6\mathrm{K}$, the angle between the spin axis and the line of sight $\gamma =30°$, and the angle between the magnetic axis and spin axis $\beta =70°$. The different curves are for different photon energies as labeled. In the upper panel, the flux at 5 keV has been multiplied by 10 relative to the other curves. The definition of the Stokes parameter is such that $F_Q/F_I=1$ corresponds to linear polarization in the plane spanned by the line-of-sight vector and the star’s spin axis (the zero phase corresponds to the polar cap in the same plane). Note that the polarization plane of high-energy ($E\gtrsim 4\mathrm{k}\mathrm{e}\mathrm{V}$) photons is perpendicular to that of low-energy photons ($E\lesssim 1\mathrm{k}\mathrm{e}\mathrm{V}$). Also note that for $E$ around $E_{\mathrm{a}\mathrm{d}}$, the polarization degree $P_L$ can be zero at certain phases such that $P_{\mathrm{c}\mathrm{o}\mathrm{n}}=1/2$.Reuse & Permissions

Figure 4

The phase-averaged linear polarization as a function of photon energy. The model parameters are $B={10}^{13}\mathrm{G}$ and $T_{\mathrm{e}\mathrm{f}\mathrm{f}}=5\times {10}^6\mathrm{K}$, and the three curves correspond to three different sets of angles ($\gamma ,\beta$) as labeled. The averaged $⟨F_U⟩=0$. The dashed curves depict the results when the vacuum polarization effect is turned off.Reuse & Permissions

Figure 5

Same as Fig. 3, except for $B=5\times {10}^{14}\mathrm{G}$ and $T_{\mathrm{e}\mathrm{f}\mathrm{f}}=5\times {10}^6\mathrm{K}$. The different curves are for different photon energies (1 keV dashed, 2 keV long-dashed, 5 keV solid lines). The thicker curves are for $\gamma =30°$, $\beta =70°$, and the thin curves for $\gamma =60°$, $\beta =40°$. In contrast to Fig. 3, 4, in the magnetar field regime, the planes of linear polarization at different photon energies coincide with each other.Reuse & Permissions