Sensitivity of pulsar light curves to spacetime geometry and efficacy of analytic approximations

In order to examine the pulse profile from a pulsar, we derive the formula for describing the flux from antipodal hot spots with any static, spherically symmetric spacetime. We find that the pulse profiles are almost independent of the gravitational geometry outside the star when the compactness of neutron stars is low enough, e.g., the stellar mass and radius are $1.4M_{\odot }$ and 14 km, respectively. On the other hand, the pulse profiles depend strongly on the gravitational geometry when the compactness of neutron stars is so high, e.g., the stellar mass and radius are $1.8M_{\odot }$ and 10 km, respectively. Thus, one may probe the spacetime geometry outside the star and even distinguish gravitational theories via the observation of pulse profile with the help of another observations for the stellar compactness, if the compactness of the central object is high enough. We also derive the first and second order approximation of the flux with respect to a parameter defined by the ratio of the gravitational radius of considered spacetime to the stellar radius. Then, we find that the relative error from full order numerical results in the bending angle becomes $\sim 20\%–30\%$ with the first order and $\sim 5\%–10\%$ with the second order approximations for a typical neutron star, whose mass and radius are $1.4M_{\odot }$ and 12 km, respectively. Our results with the first order approximation for the Schwarzschild spacetime are different from those obtained in the literature, which suggests that the first order approximation has been misunderstood to yield a highly accurate prediction.

Figure 1

Image of the photon trajectory from the stellar surface with the dashed line, where $R$ and $b$ denote the stellar radius and impact parameter. The photon radiates from the stellar surface with the emission angle $\alpha$ (between the local radial direction and radiating photon direction), where the radiation point on the surface is located with the angle $\psi$ with respect to the direction of the observer. In addition, $\beta$ is the bending angle defined as $\psi -\alpha$.

Figure 2

Image for ${\psi }_{\mathrm{cri}}$, where $\alpha =\pi /2$, and invisible zone shown by the shaded region.

Figure 3

Image of the hot spots on the rotating star with the angular velocity $\omega$. Two hot spots are associated with the magnetic polar caps, where the magnetic axis is inclined to the rotational axis with the angle $\mathrm{\Theta }\in [0,\pi /2]$. The unit vector $\mathit{d}$ denotes the direction of the observer, while the unit vectors $\mathit{n}$ and $\overline{\mathit{n}}$ are the normals on the primary and antipodal spots, respectively. The angle between $\mathit{d}$ and the rotational axis is $i\in [0,\pi /2]$.

Figure 4

The classification whether the hot spots are observed or not depending on angles $\mathrm{\Theta }$ and $i$ for the neutron star model with $R_c=14\mathrm{km}$ and $M=1.4M_{\odot }$ in the Schwarzschild spacetime, where ${\psi }_{\mathrm{cri}}=0.635\pi$. The regions denoted by I, II, III, and IV, correspond to the situations of I, II, III, and IV explained in the text.

Figure 5

Pulse profile from the neutron star with $R_c=14\mathrm{km}$ and $M=1.4M_{\odot }$ in the Schwarzschild spacetime, where I–IV correspond to $(\mathrm{\Theta }/\pi ,i/\pi )=(0.1,0.05)$, (0.3, 0.2), (0.45, 0.4), and (0.45, 0.02), respectively. The dashed, dotted, and solid lines denote the flux from the primary hot spot $F$, the flux from the antipodal hot spot $\overline{F}$, and the observed flux $F_{\mathrm{ob}}≔F+\overline{F}$, which are normalized by the observed maximum flux $F_{\max }$.

Figure 6

${\psi }_{\mathrm{cri}}$ as a function of the stellar compactness $M/R_{\mathrm{c}}$ for different spacetime, i.e., the S spacetime (solid line), the RN spacetime with $Q/M=0.5$ and 1.0 (dotted lines), and the GHS spacetime with $Q/M=0.5$ and $\sqrt{2}$ (dashed lines). We remark that the results for the Reissner-Nordström spacetime with $Q/M=0.5$ are almost the same as the results for the Garfinkle-Horowitz-Strominger spacetime with $Q/M=0.5$. The shaded region denotes the possible value of $M/R_{\mathrm{c}}$ for the stellar models with $R_{\mathrm{c}}=10–14\mathrm{km}$ and $M=1.4–1.8M_{\odot }$.

Figure 7

Classification for the stellar models with $M/R_c=0.148$ and 0.266 for various spacetimes, where the solid, dotted, and dashed lines denote the boundary of the classification with the S spacetime, the RN spacetime with $Q/M=1.0$, and the GHS spacetime with $Q/M=\sqrt{2}$. The dots in the figure denote the specific angles of $\mathrm{\Theta }$ and $i$, with which the pulse profiles are shown in Figs. 8 and 9.

Figure 8

Pulse profiles for the stellar models with $M/R_c=0.148$ are shown as a function of $t/T$ for various angles $\mathrm{\Theta }$ and $i$ with different spacetimes. The upper, middle, and lower panels respectively correspond to the results for the Schwarzschild spacetime, the RN spacetime with $Q/M=1.0$, and the GHS spacetime with $Q/M=\sqrt{2}$. For each spacetime, the panels from left to right are results with $\mathrm{\Theta }/\pi =0.1$, 0.2, 0.3, and 0.4. In each panel, the different lines denote the results for $i/\pi =0.05$, 015, 0.25, 0.35, and 0.45. In addition, the different type of lines corresponds to the different class whether the two hot spots are observed or not as shown in Fig. 4, where the solid and dashed lines correspond to class II and IV, respectively, while the dotted lines correspond to class I or III. The amplitude of pulse profiles is normalized by the amplitude at $t/T=0$, denoted by $F_0$, and shifted a little in order to easily distinguish the different lines.

Figure 9

Same as Fig. 8, but for the stellar model with $M/R_c=0.266$.

Figure 10

Relative error in the bending angle $\beta =\psi -\alpha$ with various sets of $(\alpha ,u)$ for the Schwarzschild spacetime. The left and right panels respectively correspond to $e_1$ and $e_2$ defined as $e_1=({\beta }_f-{\beta }_1)/{\beta }_f$ and $e_2=({\beta }_f-{\beta }_2)/{\beta }_f$, where ${\beta }_f$ is the bending angle calculated with Eq. (8), while ${\beta }_1$ and ${\beta }_2$ are calculated with Eq. (b2) up to the linear order of $u$ and Eq. (b2) up to the second order of $u$, respectively. In the figure, the lines denote the values of $e_1$ and $e_2$ with the fixed value of $u$, i.e., $u=0.01$, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.2, 0.3, 0.4, and 0.5 in order from the bottom.

Figure 11

The left and right panels are the value of ${\psi }_{\mathrm{cri}}$ and the visible fraction of the stellar surface $S_{\mathrm{cri}}/4\pi R_c^2$, respectively, as a function of $u$. In both panels, the solid, dotted, and dashed lines respectively correspond to the full order numerical values, approximate values obtained from Eq. (b2) up to the first order of $u$, and those from Eq. (b2) up to the second order of $u$. The shaded region corresponds to the possible value of $u$ for the stellar models with $R_c=10–14\mathrm{km}$ and $M=1.4–1.8M_{\odot }$.

Figure 12

The classification whether the two hot spots are observed or not is shown as a dependence on the angle $\mathrm{\Theta }$ and $i$ for the Schwarzschild spacetime, as in Fig. 4. The left, middle, and right panels correspond to the stellar models with $M/R_c=0.148$, 0.197, 0.266, respectively. In the figure, the solid, dotted, and dashed lines denote the boundaries of classification with ${\psi }_{\mathrm{cri}}$ obtained by the full order numerical integration, the first order approximation of $u$, and the second order approximation of $u$. The dots in the left and right panels denote the stellar models with which the pulse profiles are shown in Fig. 14.

Figure 13

The observed flux $F_{\mathrm{ob}}$ normalized by the maximum flux $F_{\max }$ is shown in the left panel, while the flux from the primary $F$ and the antipodal hot spots $\overline{F}$ normalized by $F_{\max }$ are shown in the right panel. The results are for the stellar model with $M/R_c=0.148$ and for $(\mathrm{\Theta }/\pi ,i/\pi )=(0.3,0.2)$. The solid, dotted, and dashed lines denote the results obtained from the full order numerical integration, the first order approximation of $u$, and the second order of approximation of $u$.

Figure 14

The observed flux normalized by the maximum flux with various angles of $\mathrm{\Theta }/\pi$ and $i/\pi$ for two stellar models. The upper and lower panels correspond to the pulse profiles from the stellar models with $M/R_c=0.148$ and 0.266. The panels from left to right correspond to the profiles with $(\mathrm{\Theta }/\pi ,i/\pi )=(0.1,0.05)$, (0.3,0.2), (0.45,0.4), and (0.45,0.02), which denote in Fig. 12 with the dots. The solid, dotted, and dashed lines denote the results obtained by the full order numerical integration, the first order approximation of $u$, and the second order approximation of $u$. In the panels, I–IV denote the classification as shown in Fig. 4 with the results obtained by the full order numerical integration.

Figure 15

Same as Fig. 10, but for the Reissner-Nordström spacetime. The upper and lower panels correspond to the cases for $Q/M=0.5$ and 1.0, respectively.

Figure 16

Same as Fig. 11, but for the Reissner-Nordström spacetime. The upper and lower panels correspond to the cases for $Q/M=0.5$ and 1.0, respectively.

Figure 17

Same as Fig. 12, but for the Reissner-Nordström spacetime. The upper and lower panels correspond to the cases for $Q/M=0.5$ and 1.0, respectively.

Figure 18

Same as Fig. 14, for the Reissner-Nordström spacetime with $Q/M=1.0$.

Figure 19

Same as Fig. 10, but for the Garfinkle-Horowitz-Strominger spacetime. The upper and lower panels correspond to the cases for $Q/M=0.5$ and $\sqrt{2}$, respectively.

Figure 20

Same as Fig. 11, but for the Garfinkle-Horowitz-Strominger spacetime. The upper and lower panels correspond to the cases for $Q/M=0.5$ and $\sqrt{2}$, respectively.

Figure 21

Same as Fig. 12, but for the Garfinkle-Horowitz-Strominger spacetime. The upper and lower panels correspond to the cases for $Q/M=0.5$ and $\sqrt{2}$, respectively.

Figure 22

Same as Fig. 14, for the Garfinkle-Horowitz-Strominger spacetime with $Q/M=\sqrt{2}$.