Precession of triaxially deformed neutron stars
RReceived 31 December 2020; Revised 31 December 2020; Accepted 31 December 2020DOI: xxx/xxxx
PROCEEDINGS
Precession of triaxially deformed neutron stars † Yong Gao ∗ , | Lijing Shao Department of Astronomy, School ofPhysics, Peking University, Beijing 100871,China Kavli Institute for Astronomy andAstrophysics, Peking University, Beijing100871, China National Astronomical Observatories,Chinese Academy of Sciences, Beijing100012, China
Correspondence *Email: [email protected]
Funding Information
National Natural Science Founda-tion of China, 11975027, 11991053,11721303. China Associationfor Science and Technology,2018QNRC001. Max Planck Soci-ety, Max Planck Partner Group.Chinese Academy of Sciences,XDB23010200.
A deformed neutron star (NS) will precess if the instantaneous spin axis and theangular momentum are not aligned. Such a precession can produce continuous grav-itational waves (GWs) and modulate electromagnetic pulse signals of pulsars. In thiscontribution we extend our previous work in a more convenient parameterization. Wetreat NSs as rigid triaxial bodies and give analytical solutions for angular velocitiesand Euler angles. We summarize the general GW waveforms from freely precess-ing triaxial NSs and use Taylor expansions to obtain waveforms with a small wobbleangle. For pulsar signals, we adopt a simple cone model to study the timing residualsand pulse profile modulations. In reality, the electromagnetic torque acts on pulsarsand affects the precession behavior. Thereof, as an additional extension to our pre-vious work, we consider a vacuum torque and display an illustrative example for theresiduals of body-frame angular velocities. Detailed investigations concerning con-tinuous GWs and modulated pulsar signals from forced precession of triaixal NSswill be given in future studies.
KEYWORDS: gravitational waves – pulsars: general – methods: analytical
A key challenge of modern astrophysics is to obtain the infor-mation on the internal structures of neutron stars (NSs). Theprecession of deformed NSs can modulate the precise pulsartiming signals (Jones & Andersson, 2001; Link & Epstein,2001) and produce continuous gravitational waves (GWs;Jones & Andersson, 2002; Zimmermann, 1980), which pro-vides a means to probe the structures of NSs.In this short contribution, we describe the dynamics offreely-precessing triaxial NSs in Sec. 2. The continuous GWs,electromagnetic timing residuals, and pulse-width modula-tions are studied in Secs. 3 and 4. As the electromagnetic torqueacts on the pulsars in reality, we discuss the vacuum electro-magnetic torque and give an illustrative example of the motionsin Sec. 5. Finally, we summarize our work in Sec. 6. † Poster presentation at the 9th International Workshop on Astronomy andRelativistic Astrophysics, 6–12 September, 2020. Abbreviations:
NS, neutron star; GWs, gravitational waves
We treat NSs as triaxial rigid bodies and ignore the complexinternal dissipation that may exist. In the body frame corotatingwith the NS, the equation of motion takes the following form(Landau & Lifshitz, 1960) ̇ 𝑳 + 𝝎 × 𝑳 = 0 , (1)where the dot denotes the derivative with respect to time 𝑡 , 𝝎 isthe angular velocity, and 𝑳 = 𝑰 ⋅ 𝝎 is the angular momentumwith 𝑰 represents the moment of inertia tensor. Let ̂𝒆 , ̂𝒆 , and ̂𝒆 denote the three unit eigenvectors along principal axes of 𝑰 with corresponding eigenvalues 𝐼 , 𝐼 , and 𝐼 . Then angularvelocity is expressed as 𝝎 = 𝜔 ̂𝒆 + 𝜔 ̂𝒆 + 𝜔 ̂𝒆 . For simplicity,at 𝑡 = 0 we set 𝜔 = 𝑎 , 𝜔 = 0 , 𝜔 = 𝑏 , and the magnitude ofthe angular velocity 𝜔 = 𝜔 .To describe the precession of NSs, we define 𝜖 ≡ 𝐼 − 𝐼 𝐼 , 𝛿 ≡ 𝐼 − 𝐼 𝐼 − 𝐼 , 𝜃 ≡ arctan 𝐼 𝑎𝐼 𝑏 , (2) a r X i v : . [ a s t r o - ph . H E ] N ov Y. Gao & L. Shao where 𝜖 is the oblateness, 𝛿 is the non-axisymmetry, and 𝜃 is the initial wobble angle between 𝑳 and ̂𝒆 . To give an esti-mation of 𝜖 , we consider two main causes of deformations:elasticity in the crust and the internal magnetic field. A smallfraction of rotational bulge can misalign with the instantaneousrotation axis due to the elastic stress in the crystallized crust ofNSs (Cutler, Ushomirsky, & Link, 2003; Melatos, 2000). Fora NS with mass 𝑀 , radius 𝑅 and a constant shear modulus 𝜇 in the crust, the oblateness from elastic deformation is (Baym& Pines, 1971; Gao et al., 2020) 𝜖 ela ≃ 4 . −8 ( 𝜔 𝜋 × 100 Hz ) 𝜇 𝑅 𝑀 −31 . , (3)where 𝑀 . , 𝑅 , and 𝜇 represent 𝑀 ∕(1 . 𝑀 ⊙ ) , 𝑅 ∕(10 cm) ,and 𝜇 ∕(10 erg cm −3 ) respectively. While the oblatenesscaused by internal magnetic field can be roughly estimated as(Lasky & Melatos, 2013; Zanazzi & Lai, 2015) 𝜖 mag ≃ 𝐵 𝑅 𝐺𝑀 = 2 × 10 −12 𝐵 𝑅 𝑀 −21 . , (4)where 𝐵 is the internal magnetic field and 𝐵 is 𝐵 ∕(10 G) . Ingeneral, the combination of the elastic field in the crust and theinternal magnetic field deform the NS into a triaxial shape andthe non-axisymmetry can be any positive value. The biaxialcase is a good approximation only if one of the deformationcauses can be ignored compared to the other ( 𝜖 ela ≪ 𝜖 mag or 𝜖 ela ≪ 𝜖 mag ), and the deformation is symmetric about a specificaxis instead of the rotational axis (Melatos, 2000).Eq. (1) can be solved analytically. With a more convenientdimensionless parameterization than that in Gao et al. (2020),we obtain the time evolution of 𝑢 𝑖 ≡ 𝜔 𝑖 ∕ 𝜔 ( 𝑖 = 1 , , ) interms of elliptic functions 𝚌𝚗 , 𝚜𝚗 , and 𝚍𝚗 (Landau & Lifshitz,1960; Zimmermann, 1980). This new parameterization is pre-sumed to have a more stable numerical behavior, in particularwhen a torque term is added (see Sec. 5). When 𝐿 > 𝐸𝐼 ,the solution takes the following form 𝑢 ( 𝑡 ) = 𝑢 𝚌𝚗 ( 𝜔 p 𝑡, 𝑚 ) , (5) 𝑢 ( 𝑡 ) = 𝑢 [ (1 + 𝛿 ) 𝛿 + 𝜖𝛿 ] 𝚜𝚗 ( 𝜔 p 𝑡, 𝑚 ) , (6) 𝑢 ( 𝑡 ) = 𝑢 𝚍𝚗 ( 𝜔 p 𝑡, 𝑚 ) , (7)for 𝑢 = 𝑢 ≡ 𝑎 ∕ 𝜔 , 𝑢 = 0 , and 𝑢 = 𝑢 ≡ 𝑏 ∕ 𝜔 at 𝑡 = 0 ,where the parameters 𝜔 p and 𝑚 are 𝜔 p = 𝑢 𝜔 𝜖 (1 + 𝛿 + 𝛿𝜖 ) −1∕2 , (8) 𝑚 = 𝛿 (1 + 𝜖 ) tan 𝜃 . (9)The dimensionless variables 𝑢 𝑖 are periodic with a period 𝑇 = 4 𝐾 ( 𝑚 )(1 + 𝛿 + 𝛿𝜖 ) 𝑢 𝜔 𝜖 , (10)where 𝐾 ( 𝑚 ) is the complete elliptic integral of the first kind(Landau & Lifshitz, 1960). In the biaxial case ( 𝛿 = 0 or ∞ ), theelliptic integral 𝐾 ( 𝑚 ) becomes 𝜋 ∕2 , leading to 𝑇 = 2 𝜋 ∕ 𝜔 p . The motion of the NS in the inertial frame can be describedby three Euler angles: 𝜙 , 𝜃 , and 𝜓 . We take the unit basis vec-tors of the coordinate system in the inertial frame as ̂𝒆 X , ̂𝒆 Y ,and ̂𝒆 Z . We let ̂𝒆 Z parallel to 𝑳 and define ̂𝑵 = ̂𝒆 Z × ̂𝒆 . Then,the Euler angles satisfy cos 𝜙 = ̂𝒆 X ⋅ ̂𝑵 , cos 𝜃 = ̂𝒆 ⋅ ̂𝒆 Z , cos 𝜓 = ̂𝒆 ⋅ ̂𝑵 . (11)The angles 𝜃 and 𝜓 are both periodic with period 𝑇 (Landau& Lifshitz, 1960) cos 𝜃 = cos 𝜃 𝚍𝚗 ( 𝜔 p 𝑡, 𝑚 ) , (12) tan 𝜓 = [
11 + 𝛿 + 𝛿𝜖 ] 𝚌𝚗 ( 𝜔 p 𝑡, 𝑚 ) 𝚜𝚗 ( 𝜔 p 𝑡, 𝑚 ) . (13)The angle 𝜙 equals to 𝜙 + 𝜙 , where (Landau & Lifshitz, 1960) exp [ 𝜙 ( 𝑡 ) ] = 𝜗 ( 𝜋𝑇 𝑡 + i 𝜋𝛼, 𝑞 ) 𝜗 ( 𝜋𝑇 𝑡 − i 𝜋𝛼, 𝑞 ) , (14) 𝜙 = 2 𝜋𝑇 𝑡 = ( (1 + 𝜖 ) 𝑢 𝜔 cos 𝜃 + 2 𝜋 i 𝑇 𝜗 ′4 (i 𝜋𝛼, 𝑞 ) 𝜗 (i 𝜋𝛼, 𝑞 ) ) 𝑡 . (15)Here 𝜗 is the fourth Jacobian theta functions with the nome 𝑞 = exp[− 𝜋𝐾 (1 − 𝑚 )∕ 𝐾 ( 𝑚 )] , and 𝛼 can be obtained via 𝚜𝚗 [2i 𝛼𝐾 ( 𝑚 )] = i cot 𝜃 . Once the values of 𝜖 , 𝛿 , 𝜃 , and 𝜔 aregiven, one gets the time evolution of the NS at any time fromEqs. (5–15). Free precession can be manifested in continuous GWs. If theprecessing triaxial NS rotates rapidly, the timing varying massquadrupole generates continuous GWs lie in the kilohertz(kHz) band, which is to be observed by ground-based GWdetectors like LIGO, Virgo, and KAGRA.The general GW waveforms for freely precessing triaxialNSs are (Van Den Broeck, 2005; Zimmermann, 1980) ℎ + = − 𝐺𝑟𝑐 [ ( 𝑖 cos 𝜄 + 𝑖 sin 𝜄 ) ( 𝑗 cos 𝜄 + 𝑗 sin 𝜄 ) − 𝑖 𝑗 ] 𝐴 𝑖𝑗 , (16) ℎ × = − 2 𝐺𝑟𝑐 ( 𝑖 cos 𝜄 + 𝑖 sin 𝜄 ) 𝑗 𝐴 𝑖𝑗 , (17)where ℎ + and ℎ × are “+” and “ × ” polarized GWs, 𝑟 is the lumi-nosity distance to the NS, 𝜄 is the inclination angle betweenthe line of sight and ̂𝒆 Z . Here 𝑖𝑗 is the rotation matrix, whichcan be represented by 𝜙 , 𝜃 , and 𝜓 . The tensor 𝐴 𝑖𝑗 is a functionof the moment of inertia tensor 𝑰 and the angular velocities 𝜔 𝑖 (see Eq. (21) in Zimmermann (1980)). In the frequencydomain, the emission of GWs mainly occurs at frequencies 𝑓 r + (2 𝑛 + 1) 𝑓 p , 𝑓 r + 2 𝑛𝑓 p , (18)where 𝑓 p = 1∕ 𝑇 is the free precession frequency, 𝑓 r equals to 𝑇 − 1∕ 𝑇 , and 𝑛 is an integer. . Gao & L. Shao In the case of small oblatenesses, small wobble angles andsmall non-axisymmetries, one can use Taylor expansions of 𝜃 and 𝛿 to obtain the waveforms. The first order contribu-tions occurs at 𝑓 r + 𝑓 p and 𝑓 r (Van Den Broeck, 2005), withamplitudes 𝐴 =2 × 10 −28 𝜃 sin 𝜄 ( 𝜖 −8 ) ( 𝑓 r
100 Hz ) ( 𝑟 ) , (19) 𝐴 =4 × 10 −28 𝛿 cos 𝜄 ( 𝜖 −8 ) ( 𝑓 r
100 Hz ) ( 𝑟 ) , (20)for “ × ” polarized GWs. The amplitudes for “+” polarized GWscan be obtained similarly but with a different dependence onthe inclination angle 𝜄 . The second order lines are too weak(Gao et al., 2020) for current observational interests and we donot discuss them here. For pulsars, the emission beam will rotates around the princi-pal axis ̂𝒆 during free precession and one can possibly observetiming residuals and pulse profile modulations with radio/X-ray telescopes (Ashton, Jones, & Prix, 2017; Gao et al., 2020;Jones & Andersson, 2001; Link & Epstein, 2001).To study the timing residuals, we assume that the emissionis along the magnetic dipole 𝒎 and one can observe the pulsarsignals once 𝒎 sweeps through the plane defined by the line ofsight and ̂𝒆 Z . We define cos 𝜒 = ̂𝒎 ⋅ ̂𝒆 , cos Φ = ̂𝒆 X ⋅ ̂𝑴 , cos Θ = ̂𝒎 ⋅ ̂𝒆 Z , (21)where ̂𝒎 is 𝒎 ∕ | 𝒎 | , 𝜒 is the angle between ̂𝒎 and ̂𝒆 , ̂𝑴 = ̂𝒆 Z × ( ̂𝒎 × ̂𝒆 Z ) is the unit vector along the projection of 𝒎 onthe ̂𝒆 X − ̂𝒆 Y plane, Φ and Θ are the azimuthal and the polarangles of ̂𝒎 in the inertial frame. The azimuthal angle Φ canbe expressed as (Jones & Andersson, 2001) Φ = 𝜙 − 𝜋 ( cos 𝜓 sin 𝜒 sin 𝜃 cos 𝜒 − sin 𝜓 sin 𝜒 cos 𝜃 ) , (22)and the phase residual due to precession is ΔΦ = Φ − ⟨ Φ ⟩ , (23)which depends on the relative values of the wobble angle 𝜃 and the angle 𝜒 (see Eqs. (45) and (48) in Jones & Anders-son (2001)). In Eq. (23), “ ⟨ ⋅ ⟩ ” means the time averaged values.The mean spin period of the pulsar is 𝑃 = 2 𝜋 ∕ ⟨ ̇ Φ ⟩ . Thus, thetiming residual of the spin period 𝑃 is Δ 𝑃 = 2 𝜋̇ Φ − 2 𝜋 ⟨ ̇ Φ ⟩ ≃ − 𝑃 𝜋 Δ ̇ Φ . (24)To the second order expansions of 𝜃 and 𝛿 with a small wobbleangle and a small non-axisymmetry, the period residual is (Gao et al., 2020) Δ 𝑃 ≈ 𝑃 𝑓 p 𝜃 (2 𝛿 + 1) cot 𝜒 cos ( 𝜋𝑓 p 𝑡 ) + 𝑃 𝑓 p 𝜃 ( 𝜒 ) cos ( 𝜋𝑓 p 𝑡 ) . (25)Note that the period derivative residual Δ ̇𝑃 can be obtaineddirectly by taking the time derivative of Δ 𝑃 .The polar angle Θ , cos Θ = sin 𝜃 sin 𝜓 sin 𝜒 + cos 𝜃 cos 𝜒 , (26)varies as a function of time. The line of sight cuts differentregion of the emission cone during free precession and onecould observe pulse width modulations. Adopting a simplecone model, the pulse width 𝑊 reads (Gil, Gronkowski, &Rudnicki, 1984; Lorimer & Kramer, 2005) sin ( 𝑊 ) = sin ( 𝜌 ∕2) − sin ( 𝛽 ∕2)sin(Θ + 𝛽 ) sin Θ , (27)where 𝜌 is the angular radius of the emission cone, 𝛽 isthe impact parameter corresponding to the closest approachbetween the magnetic dipole moment and the line of sight.Detailed results can be found in Gao et al. (2020). Pulse-widthmodulations can be used to infer the emission shape and pulsarradiation properties (Link & Epstein, 2001). In our previous work (Gao et al., 2020), we assume that theprecession is free, namely without a torque. However, in real-ity the existence of electromagnetic torque will modulate thefree precession and affect continuous GWs and pulsar signals.We take a vacuum electromagnetic torque as an example (Gol-dreich, 1970; Jones & Andersson, 2002; Zanazzi & Lai, 2015) 𝑻 = 𝑻 + 𝑻 = 2 𝜔 𝑐 ( 𝝎 × 𝒎 ) × 𝒎 + 35 𝑅𝑐 ( 𝝎 ⋅ 𝒎 )( 𝝎 × 𝒎 ) , (28)and ignore the complex magnetospheric processes (Arza-masskiy, Philippov, & Tchekhovskoy, 2015). The first term, 𝑻 , is the secular spin down torque, which has componentsboth parallel and perpendicular to the angular momentum 𝑳 .The component parallel to 𝑳 accounts for the usual spin downand the component perpendicular to 𝑳 is responsible for thechange of the wobble angles (Jones & Andersson, 2002). Thesecond term, 𝑻 , is the anomalous torque, which originatesfrom the moment of inertia of magnetic dipole field (Melatos,2000; Zanazzi & Lai, 2015). This torque is perpendicular tothe angular velocity 𝝎 and does not decrease the energy or theangular momentum. However, it changes the spin period andthe wobble angles for precessing NSs (Jones & Andersson,2002). Y. Gao & L. Shao − δ u [ − ] − . . . δ u [ − ] − . − . − . . . . . t [10 s] − δ u [ − ] FIGURE 1
An illustrative example for the residuals 𝛿𝑢 , 𝛿𝑢 ,and 𝛿𝑢 due to electromagnetic torque 𝑻 . Here we take 𝜖 =10 −5 , 𝛿 = 1 , 𝜃 = 3 ◦ , 𝜔 = 1 rad s −1 , and 𝜒 = 89 . ◦ . The timescales 𝜏 c and 𝜏 p are set to be s and .
11 × 10 s .The Euler equation for the forced precession with 𝑻 is(Arzamasskiy et al., 2015) ̇ 𝒖 + 1 𝜏 p ( 𝛿 𝛿 ̇𝑢 ̂𝒆 + ̇𝑢 ̂𝒆 + 𝛿 𝛿 𝑢 𝒖 × ̂𝒆 + 𝑢 𝒖 × ̂𝒆 ) = 𝑢 𝜏 c [ ( 𝒖 ⋅ ̂𝒎 ) ̂𝒎 − 𝒖 ] + 1 𝜏 𝑎 [ ( 𝒖 ⋅ ̂𝒎 ) ( 𝒖 × ̂𝒎 ) ] . (29)The precession time scale 𝜏 p , the secular spin down time scale 𝜏 c (corresponding to 𝑻 ), and the time scale 𝜏 a (correspondingto 𝑻 ) are, respectively, 𝜏 p = 1∕ 𝜖𝜔 , 𝜏 c = 3 𝑐 𝐼 ∕2 𝑚 𝜔 , 𝜏 a = 10 𝑅𝜔 𝜏 c ∕9 𝑐 , (30)where the relation 𝜏 p ≪ 𝜏 a ≪ 𝜏 c is usually satisfied (Arza-masskiy et al., 2015). We show an illustrative example inFig. 1 for forced precession. One notices that 𝑢 𝑖 oscillates dur-ing the forced precession, which is mainly caused by the torque 𝑻 . In this small initial wobble angle case, the angular velocity 𝜔 ≃ 𝜔 𝑢 and 𝑢 decreases on a longer time scale caused bythe spin down torque 𝑻 . Because of the relation 𝜏 p ≪ 𝜏 a ≪ 𝜏 c ,one can actually treat torques as perturbation on the free pre-cession in multi-timescale analysis (Arzamasskiy et al., 2015;Link & Epstein, 2001).The continuous GWs and pulsar signals from forced-precessing triaxial NSs have some new features and we plan topresent them in detail in future studies. We gave the analytical solutions of freely-precessing triaxialNSs and studied their characteristics in continuous GWs andpulsar signals. These results are ready to be used for futuresearches of precession. We also discussed the effects of theelectromagnetic torque on the motions of precession, which deserve more studies in the future. From observations, preces-sion will give us important information on the equation of stateof NSs and related astrophysical properties (Gao et al., 2020).
ACKNOWLEDGMENTS
We are grateful to Rui Xu, Ling Sun, Chang Liu, and Ren-XinXu for discussions. This work was supported by the
NationalNatural Science Foundation of China under Grant Nos. , the Young Elite ScientistsSponsorship Program by the
China Association for Scienceand Technology under the Grant No. , and the
Max Planck Society through the
Max Planck Partner Group .It was partially supported by the Strategic Priority ResearchProgram of the
Chinese Academy of Sciences under the GrantNo.
XDB23010200 , and the High-performance ComputingPlatform of Peking University.
Conflict of interest
The author declares no potential conflict of interests.
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