Finite size effects on the light curves of slowly-rotating neutron stars

An important question when studying the light curves produced by hot spots on the surface of rotating neutron stars is, how might the shape of the light curve be affected by relaxing some of the simplifying assumptions that one would adopt on a first treatment of the problem, such as that of a pointlike spot. In this work we explore the dependence of light curves on the size and shape of a single hot spot on the surface of slowly rotating neutron stars. More specifically, we consider two different shapes for the hot spots (circular and annular) and examine the resulting light curves as functions of the opening angle of the hot spot (for both) and width (for the latter). We find that the pointlike approximation can describe light curves reasonably well, if the opening angle of the hot spot is less than $\sim 5°$. Furthermore, we find that light curves from annular spots are almost the same as those of the full circular spot if the opening angle is less than $\sim 35°$ independently of the hot spot’s width.

Figure 1

Geometry used to describe the emission of photons from a neutron star. The dashed line indicates the trajectory of a photon, emitted with an angle $\alpha$ (measured with respect to the normal to the star’s surface) from the star’s surface $R_c$ [$R_c=C^{1/2}(R)$] from a point located at $\psi$ (measured with respect to the line of sight to the observer). The star’s gravitational field bends the trajectory by an amount $\psi -\alpha$ and the photon is observed arriving with an impact parameter $b$. The angle $\varphi$ denotes the azimuthal angle in the observer’s sky.

Figure 2

At any time, one can choose the coordinate where the center of the hot spot is located at $(\psi ,\varphi )=({\psi }_{*},0)$.

Figure 3

Illustration of the hot spot on a rotating star with angular velocity $\omega$. The angle between the direction to the observer and the rotation axis is $i$, while the angle between the center of the hot spot and the rotation axis is $\mathrm{\Theta }$. The shaded region denotes the invisible zone whose boundary is determined by the angle ${\psi }_{\mathrm{cri}}$.

Figure 4

Visibility classification of the single hot spot, as a function of the angles of $i$ and $\mathrm{\Theta }$. The top panel corresponds to the case with a pointlike approximation, where three situations exist, i.e., (A) the hot spot can be always observed, (B) the hot spot can enter the invisible zone for a fraction of the period, and (C) the hot spot cannot be observed in any time. The bottom panel corresponds to the case with the effect of hot spot size, where the boundary of the classification becomes blurred. This example is the case for $\mathrm{\Delta }\psi =10°$. The colored region denotes the situation that a part of the hot spot can enter the invisible zone.

Figure 5

Illustration of two different geometries for the hot spot. Left image: a circular hot spot with angular radius $\mathrm{\Delta }\psi$. Right image: an annular geometry with width $\mathrm{\Delta }\psi -\mathrm{\Delta }{\psi }_i$. In both cases, the center of the hot spot is chosen to be $\psi ={\psi }_{*}$.

Figure 6

For the neutron star model with $R=5M$, the light curves calculated considering the finite area of the hot spot are compared with that with the pointlike spot approximation. In the top panels, the light curves with various opening angle ($\mathrm{\Delta }\psi =5°$, 10°, 30°, 50°, and 70°) are shown with different lines, while the light curves with the pointlike approximation are shown with the open circles, where the all light curves are normalized by their maximum values, i.e., the flux at $t/T=0$. In the bottom panels, the relative deviations calculated by Eq. (21) are shown. The panels from left to right correspond to the case for $i=\mathrm{\Theta }=30°$, 45°, 60°, and 90°. For $i=\mathrm{\Theta }=90°$, the time when the observed flux becomes 0 depends on $\mathrm{\Delta }\psi$, which is shown by the filled circle, filled diamond, filled square, and filled triangle for $\mathrm{\Delta }\psi =5°$, 10°, 30°, and 50° in the rightmost lower panel.

Figure 7

The visible area of the hot spot (normalized by $R^2$) for the neutron star model with $R=5M$, for various combinations of $\mathrm{\Delta }\psi$ and $i=\mathrm{\Theta }$. The significant decrease in the visible area indicates when a part of the hot spot enters the invisible zone. When $\mathrm{\Delta }\psi =70°$, the visible hot area never drops to 0 regardless of the values of $i=\mathrm{\Theta }$ considered.

Figure 8

The critical value of the opening angle, $\mathrm{\Delta }{\psi }_{\mathrm{cri}}$, given by $\mathrm{\Delta }{\psi }_{\mathrm{cri}}=\pi -{\psi }_{\mathrm{cri}}$ is shown as a function of $M/R$. For the case when the opening angle $\mathrm{\Delta }\psi$ is larger than $\mathrm{\Delta }{\psi }_{\mathrm{cri}}$ (the region above the line) the hot spot cannot entirely enter into the invisible zone. Two vertical dashed lines correspond to the values of $M/R$ for the stellar models with $R=5M$ and $4M$.

Figure 9

Same as Fig. 6, but for the neutron star model with $R=4M$.

Figure 10

Same as Fig. 7, but for the neutron star model with $R=4M$. This more compact neutron star has a smaller invisible zone and therefore hot spot sizes which would become invisible in the $R=5M$ case ($\mathrm{\Delta }\psi =30°$ and 50°) are now always visible.

Figure 11

Light curves from the annular hot spot with $\mathrm{\Delta }\psi =70°$. The upper and lower panels correspond to the observed flux normalized by the maximum flux for the neutron star model with $R=5M$ and $4M$, respectively. The panels from left to right correspond to the results for $i=\mathrm{\Theta }=30°$, 45°, 60°, and 90°. In each panel, the different lines correspond to the results with different values of $\mathrm{\Delta }{\psi }_i$, i.e., $\mathrm{\Delta }{\psi }_i=0°$, 10°, 30°, 50°, and 65°.

Figure 12

The ratio of the difference between the maximum and minimum flux to the maximum flux is shown as a function of $\mathrm{\Delta }{\psi }_i/\mathrm{\Delta }\psi$ with $\mathrm{\Delta }\psi =70°$, where the circles, squares, diamonds, and triangles correspond to the results for $i=\mathrm{\Theta }=30°$, 45°, 60°, and 90°. The left and right panels correspond to the results for the neutron star model with $R=5M$ and $4M$.

Figure 13

Same as Fig. 11, but for the neutron star model with $\mathrm{\Delta }\psi =35°$.

Figure 14

Same as Fig. 12, but for the neutron star model with $\mathrm{\Delta }\psi =35°$.