Circular stable orbits in $f(R)$ realistic static and spherically-symmetric spacetimes

Abstract

We investigate the geodesic structure of realistic static and spherically symmetric spacetimes embedding neutron stars in metric $f(R)$ gravity, focusing on the quadratic Starobinsky model $f(R)=aR^2$ with $a<0$. Neutron-star solutions are obtained by numerically solving the modified Tolman-Oppenheimer-Volkoff system for several realistic equations of state. Such solutions are then matched consistently to the exterior vacuum geometry by enforcing the full set of junction conditions required in metric $f(R)$ theories. Using an effective potential approach, we show that stable circular orbits appear in discrete radial bands separated by forbidden regions, with a dominant principal band of stability that depends sensitively on the stellar central pressure, the equation of state, and the magnitude of the parameter $|a|$. Outside the stable bands, massive particles can have bound but unstable precessing trajectories as well as unbounded motions. On the other hand, for null geodesics, we find no evidence for photon spheres outside the neutron star within the parameter range studied.

Figures (13)

• Figure 1: "Stiff", "middle" and "soft" EOS based on potential models' data as in Hebeler:2013nza that describe neutron matter. Soft possesses an EOS in which pressure increases most slowly with density, and stiff is the one in which this growth is the most rapid.
• Figure 2: Metric coefficients $A(r)$ and $B(r)$, as well as the Ricci scalar curvature $R(r)$, compared for GR and $f(R)=-0.05R^2$, using the Middle EoS for a star with central pressure $P_c=7\cdot10^{-4}$ km$^{-2}$ and a radius of 11.589 km. The main feature is an oscillation of the metric functions and the Ricci scalar outside the star around their GR counterparts, most visible on $R(r)$ (which is hence non-vanishing outside the star) and somewhat in $B(r)$.
• Figure 3: Radial condition functions $C_{i=1,2,3}(r)$, along with the SCO existence rings they produce (light green regions) for the Soft EoS and a central pressure $P_c=6\cdot10^{-4}$ km$^{-2}$, both for GR (darker) and $f(R)=-0.00062R^2$ (lighter and oscillating). The oscillations are only perceptible for $C_3^{f(R)}(r)$ in the depicted scales. The principal ring is highlighted in light violet, and the ISCO is illustrated as a red vertical line at $r^\text{GR}_\text{ISCO}=6M=17,273$ km. The left border of the figure corresponds to the stellar radius $r_*=9,731$ km, equal for both GR and $f(R)$. Since only their signs concern us, all condition functions have been rescaled by adequate positive constants to ensure visual comparability, without any loss of validity.
• Figure 4: Radial condition functions $C_{i=1,2,3}(r)$, along with the SCO existence rings they produce (light green regions) for the Soft EoS, both for GR (darker) and $f(R)=-0.005R^2$ (lighter and oscillating). The oscillations are only perceptible for $C_3^{f(R)}(r)$ at this scale. The principal ring is highlighted in light violet, and the ISCO is illustrated as a red vertical line. The vertical red and grey lines on the left of each panel correspond to the ISCOs and the stellar radii respectively. Upper left:$P_c=2\cdot10^{-4}$ km$^{-2}$, $r_*=9,742$ km, $r^\text{GR}_\text{ISCO}=12,972$ km, $r^{f(R)}_\text{ISCO}=11,649$ km; the oscillations on $C_3^{f(R)}(r)$ pass through zero repeatedly everywhere further than the ISCO, so there is no principal ring. Upper right:$P_c=3\cdot10^{-4}$ km$^{-2}$, $r_*=9,865$ km, $r^\text{GR}_\text{ISCO}=15,048$ km, $r^{f(R)}_\text{ISCO}=14,320$ km; $C_3^{f(R)}(r)$ is constantly positive in an intermediate region, resulting in a principal ring much wider than all the others. For higher values of $r$ rings continue appearing. Lower left:$P_c=6\cdot10^{-5}$ km$^{-2}$, $r_*=r^{f(R)}_\text{ISCO}=8,806$ km; the ISCO exactly coincides with the stellar radius and lies within the principal ring. Lower right:$P_c=9\cdot10^{-5}$ km$^{-2}$, $r_*=r^{f(R)}_\text{ISCO}=9,132$ km; the stellar radius is small enough for the maximum of $C_3(r)$ to show.
• Figure 5: Principal ring width $W_p$ versus the coefficient $|a|$ and central pressure $P_c$ / bare stellar mass $M_0$ for the EoS Soft, Middle and Stiff, respectively. The areas in black correspond to an absence of principal ring. A maximum color gradient limit has been set for the ring width representation, in order to avoid excessive distortion of the gradient and ensure that variations within the lower width range are clearly perceptible. Marked in red are the points studied in Fig. \ref{['fig:condiciones']}. Contour lines for $W_p=1,10,20,30$ km are shown for the EoS Middle.