Four-hair relations for differentially rotating neutron stars in the weak-field limit

Joseph Bretz, Kent Yagi, and Nicolás Yunes

Phys. Rev. D 92, 083009 – Published 19 October 2015

Abstract

The opportunity to study physics at supra-nuclear densities through x-ray observations of neutron stars has led to in-depth investigations of certain approximately universal relations that can remove degeneracies in pulse profile models. One such set of relations determines all of the multipole moments of a neutron star just from the first three (the mass monopole, the current dipole and the mass quadrupole moment) approximately independently of the equation of state. These three-hair relations were found to hold in neutron stars that rotate rigidly, as is the case in old pulsars, but neutron stars can also rotate differentially, as is the case for proto-neutron stars and hypermassive transient remnants of binary mergers. We here extend the three-hair relations to differentially rotating stars for the first time with a generic rotation law using two approximations: a weak-field scheme (an expansion in powers of the neutron star compactness) and a perturbative differential rotation scheme (an expansion about rigid rotation). These approximations allow us to analytically derive approximately universal relations that allow us to determine all of the multipole moments of a (perturbative) differentially rotating star in terms of only the first four moments. These new four-hair relations for differentially rotating neutron stars are found to be approximately independent of the equation of state to a higher degree than the three-hair relations for uniformly rotating stars. Our results can be instrumental in the development of four-hair relations for rapidly differentially rotating stars in full general relativity using numerical simulations.

Received 8 July 2015

DOI:https://doi.org/10.1103/PhysRevD.92.083009

Authors & Affiliations

Joseph Bretz, Kent Yagi, and Nicolás Yunes

Department of Physics, Montana State University, Bozeman, Montana 59717, USA

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Vol. 92, Iss. 8 — 15 October 2015

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Images

Figure 1
Left: 3D plot of the four-hair relation showing the normalized mass hexadecapole moment ${\bar{M}}_{4}$ as a function of the normalized mass quadrupole ( ${\bar{M}}_{2}$ ) and mass-current octopole ( ${\bar{S}}_{3}$ ) moments computed with a polytropic EoS with various polytropic indices $n$ in the slow-rotation limit. The light blue-green plane represents the fiducial $n = 0.65$ case. Observe that the $n = 0.3$ (green dots), 0.5 (blue dots), 0.8 (orange dots), 1 (red dots) results are all almost on the same plane. The white dashed line in the left panel represents the uniform rotation curve for an $n = 1$ polytrope. Right: contour and color gradient plot in the ${\bar{S}}_{3} − {\bar{M}}_{2}$ plane of the maximum fractional difference between ${\bar{M}}_{4}$ computed with a polytropic EoS with index $n \in [0.3, 1]$ and a fiducial index of $n = 0.65$ assuming $| γ | < 1 / 2$ . The red dashed lines demarcate the region of $| γ | < 0.1$ with $n = 1$ . Observe that the maximum fractional difference is always less than $\sim 6 %$ and 3% for all values of ${\bar{S}}_{3}$ and ${\bar{M}}_{2}$ within the $| γ | < 1 / 2$ and $| γ | < 0.1$ regions respectively.
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Figure 2
Stellar density (color gradient) and isodensity layers under uniform rotation in the elliptical isodensity approximation (solid contours) and for a typical NS (dashed contours) constructed with a realistic SLy EoS [66]. The color gradient legend corresponds to the ratio of elliptical radius $\tilde{r}$ to the stellar surface $R_{*}$ . Observe how the solid contours are close to the dashed contours, except close to the stellar core.
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Figure 3
${\bar{B}}_{n, ℓ}$ and ${\bar{C}}_{n, ℓ}$ are plotted above for the first four $ℓ$ ’s with $n \in [0.3, 1]$ . The solid lines represent the exact numerical results, while the dashed lines come from the perturbative approach. The bottom panels show the fractional difference of the semianalytic results using a fiducial $n = 0.65$ .
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Figure 4
Contour plot of $γ ({\bar{M}}_{2}, {\bar{S}}_{3})$ under the slow-rotation limit. Observe how ${\bar{M}}_{2}$ and ${\bar{S}}_{3}$ specify the amount of differential rotation, and thus, if $| γ | < 0.1$ then only a subset of ${\bar{M}}_{2}$ and ${\bar{S}}_{3}$ are allowed. The contours are generated with a polytropic EoS with $n = 1$ because values of $n < 1$ yield less restrictive contours and thus a wider range of allowed ${\bar{M}}_{2}$ and ${\bar{S}}_{3}$ . The white dashed zero contour corresponds to the uniform rotation case, which shows a linear dependence between ${\bar{M}}_{2}$ and ${\bar{S}}_{3}$ that agrees with the three-hair relations for an $n = 1$ polytrope. The green lines at ${\bar{M}}_{2} = 30$ and ${\bar{S}}_{5} = 80$ show the range of ${\bar{S}}_{3}$ and ${\bar{M}}_{2}$ used in Fig. 5 that is kept within $| γ | < 1 / 2$ , demarcated by the blue lines.
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Figure 5
Top panels: four-hair relations for differentially rotating stars. The normalized moments ${\bar{M}}_{4}$ and ${\bar{S}}_{5}$ are plotted against ${\bar{M}}_{2}$ with ${\bar{S}}_{3} = 80$ (left) and ${\bar{S}}_{3}$ with ${\bar{M}}_{2} = 30$ (right) for various polytropic indices $n$ in the slow-rotation limit. Magenta squares represent the uniform rotation case ( $γ = 0$ ), while the black diamonds show where $γ = 0.1$ for a given value of $n$ . The fiducial polytropic index of $n = 0.65$ is plotted as a solid black line. The inset zooms into a region around ${\bar{S}}_{3} = 71$ and ${\bar{M}}_{4} = 2.1 \times 1 0^{3}$ , where the vertical magenta (black) line corresponds to the difference in ${\bar{M}}_{4}$ between an $n = 0.65$ and $n = 1$ polytrope for uniformly (differentially) rotating stars with fixed ${\bar{M}}_{2}$ . Observe that the latter is smaller than the former when $| γ | < 0.1$ with an $n = 1$ polytrope. Bottom panels: the fractional difference of each of the four-hair relations relative to that of a fiducial polytrope ( $n = 0.65$ ). Observe that the maximum EoS variation is roughly 7% for ${\bar{M}}_{4}$ and 20% for ${\bar{S}}_{5}$ , but this goes down to 3% for ${\bar{M}}_{4}$ and 10% for ${\bar{S}}_{5}$ in the $| γ | < 0.1$ region marked by the horizontal black lines. The horizontal magenta lines show the maximum fractional difference in the three-hair relations for uniformly rotating stars.
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Figure 6
Same as Fig. 1 but for ${\bar{S}}_{5}$ as a function of ${\bar{M}}_{2}$ and ${\bar{S}}_{3}$ . The yellow (green) line in the right panel shows where the fractional difference of ${\bar{S}}_{5}$ computed with an $n = 0.3$ ( $n = 1$ ) polytrope vanishes with respect to the fiducial $n = 0.65$ one. Observe that the maximum fractional difference is always less than $\sim 20 %$ for all values of ${\bar{S}}_{3}$ and ${\bar{M}}_{2}$ that satisfy $| γ | < 1 / 2$ , while it is only between 3% and 7% in the $| γ | < 0.1$ region (dotted red lines).
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