Gravitational waves from pulsars in the context of magnetic ellipticity

In one of our previous articles we have considered the role of a time dependent magnetic ellipticity on pulsars' braking indices and on the putative gravitational waves these objects can emit. Since only nine of more than 2000 known pulsars have accurately measured braking indices, it is of interest to extend this study to all known pulsars, in particular to what concerns the gravitational waves generation. To do so, as shown in our previous article, we need to know some pulsars' observable quantities such as: periods and their time derivatives, and estimated distances to the Earth. Moreover, we also need to know the pulsars' masses and radii, for which, in here we are adopting current fiducial values. Our results show that the gravitational wave amplitude is at best $h \sim 10^{-28}$. This leads to a pessimistic prospect for the detection of gravitational waves generated by these pulsars, even for Advanced LIGO and Advanced Virgo, and the planned Einstein Telescope, whether the ellipticity has magnetic origin.


INTRODUCTION
It is well known that, besides compact binaries, rapidly rotating neutron stars are promising sources of gravitational waves (GWs) which could be detected in a near future by Advanced LIGO (aLIGO) and Advanced Virgo (AdV), and also by the planned Einstein Telescope (ET). These sources generate continuous GWs whether they are not perfectly symmetric around their rotation axis, i.e. if they present some equatorial ellipticity.
It is worth stressing that the equatorial ellipticity is an extremely relevant parameter since the GW amplitude is directly proportional to it. Therefore, whether the ellipticity be extremely small, i.e., 10 −5 , the GWs amplitude will be also extremely small, implying that the detection of such continuous GWs generated by pulsars may be unattainable (see de Araujo et al. 2016a,b) with the current technology. For a matter of comparison, some authors argue that an acceptable upper limit for the ellipticity would be around ∼ 10 −6 (see, e.g., Krastev et al. 2008). An important mechanism for producing asymmetries is the development of non-axisymmetric instabilities in rapidly rotating neutron stars driven by the gravitational emission reaction or by nuclear matter viscosity (see, e.g., Bonazzola et al. 1996, and references therein).
We explore, in the present paper, some consequences of an ellipticity generated by the magnetic dipole of the pulsars. It is well known that, for strong magnetic fields (∼ 10 12 − 10 15 G), the equilibrium configuration of a neutron star can be distorted due to the magnetic pressure. Therefore, both rotation and magnetic filed combined can produce a flattened equilibrium star. However, star rotation and strong magnetic field are not sufficient for GW emission, other effects must be associated, such as pulsar precession (see, e.g., Zimmermann & Szedenits 1979).
The main goal of the present paper is to extend our previous study into the role of ellipticity of magnetic origin ( B ) was considered (de Araujo et al. 2016c). Since in that paper we were also interested in braking indices, which are until now accurately measured for only nine pulsars, we had restricted that study exclusively for those very pulsars.
However, B does not depend on the pulsar braking index, maybe it is the other way around. In fact, B is mostly associated to both the pulsar period (P) and its time derivative (Ṗ), for a given value of mass, radius and momentum of inertia. Therefore, it is straightforward to extend the B calculation for all pulsars with known P andṖ. Consequently, with B in hands, we can calculate the GW amplitudes for all pulsars with known P,Ṗ, and their distances to the Earth.
To do so this paper is organized as follows. Section 2 is devoted to a brief procedure description which is conducted by a basic set of equations. In Section 3 we present how the calculations are done and discuss the obtained results. And, finally, in Section 4 we summarize the main conclusions and remarks about them.

BASIC EQUATIONS
In de Araujo et al. (2016c) we consider in detail how to relate B to P andṖ. In addition, the basic equations used for calculating the amplitude of the putative GWs generated by pulsars are also presented. All those equations are used to calculate the relevant quantities on this present paper. Therefore, here we are only providing the main steps for deriving these relevant equations.
Recall that the equatorial ellipticity is given by (see e.g., Shapiro & Teukolsky 1983) where I xx , I yy , I zz are the moment of inertia with respect to the rotation axis, z, and along directions perpendicular to it. Regarding the ellipticity of magnetic origin, it was shown by different authors (Bonazzola & Gourgoulhon 1996;Konno et al. 2000;Regimbau & de Freitas Pacheco 2006) that it is given by where B 0 is the dipole magnetic field, R and M are the radius and the mass of the star respectively, φ is the angle between the rotation and magnetic dipole axes, whereas κ is the distortion parameter, which depends on both the star equation of state (EoS) and the magnetic field configuration (see e.g. Regimbau & de Freitas Pacheco 2006). Regarding the GW amplitude, one finds in the literature the following equation (see, e.g., Aasi et al. 2014) where one is considering that the whole contribution toḟ rot is due to GW emission. This equation must be modified to take into account the magnetic braking (see de Araujo et al. 2016a,b). This can be done by writinġ whereḟ rot can be interpreted as the part ofḟ rot related to the GW emission brake. Consequently, the GW amplitude is given byh |ḟ rot | f rot η. (5) On the other hand, the GWs amplitude can also be written as follows h = 16π 2 G c 4 I f 2 rot r (6) (see, e.g., Shapiro & Teukolsky 1983). By combining the two equations above one can obtain B in terms of P,Ṗ (observable quantities), η and I, namely = 5 512π 4 c 5 GṖ P 3 I η.
Still concerning η, as discussed in detail by de Araujo et al. (2016c), it can be also interpreted as the fraction of the rotation power (Ė rot ) emitted in the form of GWs (Ė GW ), or yet, the efficiency for GWs generation. Obviously, part of the rotation power is emitted in the form of electromagnetic radiation through magnetic dipole emission (Ė d ).
Also, it is shown in de Araujo et al. (2016c) an useful equation relating η thw pulsar dipole magnetic field, which is derived by recalling that the magnetic brake is related to P ansṖ, i.e.B 0 sin 2 φ = 3Ic 3 4π 2 R 6 PṖ whereB 0 would be the magnetic field whether the breaking is purely magnetic. Since pulsars might also emit GWs, B 0 <B 0 is a reasonable assumption. Thus, the equation relating η to the pulsar dipole magnetic field reads (see de Araujo et al. 2016c, for details) By substituting this last expression into equation 2 one obtains B = 3Ic 3 4π 2 GM 2 R 2 PṖ (1 − η) κ.
In addition, by combining this equation with equation 7, one has η = 288 5 Since and η 1, one can readily obtain the following useful equations B 3Ic 3 4π 2 GM 2 R 2 PṖκ (12) and η 288 5 Now, we are ready to calculate B , η and the GW amplitudes for all pulsars with known P,Ṗ, and estimated distances to the Earth, for given values of M, R, I and κ. The next section is devoted to such an issue as well as the corresponding discussion of the results.

CALCULATIONS AND DISCUSSIONS
The table with the necessary pulsars' parameters used for the calculation of B , η and h as well as the adopted criteria for the selection can be found in the Appendix. Fig.. 1 is made with columns 2 and 3 of this very table, in which the pulsars' periods and their corresponding time derivatives (spin-down rate) are shown. Although being a well known diagram, it will be very useful for the discussions of the subsequent results. It is easily distinguishable two pulsar population: the millisecond pulsars in the lower left (with periods P < 10 −2 s), composed by young or recently formed pulsars, and another class of pulsars with periods 10 −1 < P < 10 1 s, composed probably by older pulsars which have already depleted a respectful fraction of their rotation power. On the last three columns of Table 1, we present the results of our calculations, namely, B , η and h. In order to perform these calculations, we also need to provide values for M, R, I and κ. For the first three parameters, we are adopting fiducial values, namely, M = 1.4M , R = 10 km, and I = 10 38 kg m 2 . Regarding the distortion parameter κ, as already mentioned, it depends on the EoS and on the magnetic field configuration. In particular, we choose κ = 10, but values as high as κ = 1000 could be considered, although they are probably unrealistic (see e.g., Regimbau & de Freitas Pacheco 2006, for a brief discussion).
From equation 12, we find that B is extremely small (∼ 10 −19 −10 −15 ) for the millisecond pulsars of Table 1, even when an extremely optimistic case in which κ ∼ 1000 is considered. Moreover, the ellipticity distribution assumes values of the order of ∼ 10 −10 for the slowest pulsars. In fig. 2 we present log B versus P for κ = 10 for all pulsars of Table 1. The ellipticity for different values of κ, since B ∝ κ, can be readily obtained.
An interesting histogram can also be made from Table, namely, the number of pulsars for log B bin (see fig. 3). It worth noticing the high number of pulsars concentrated around ∼ 10 −10 (10 −8 ) for k = 10 (1000).
A similar analysis can be made for η by means of Eq. 13. In fig. 4 we present log η versus P for κ = 10 for all pulsars of Table 1. Notice that η is also extremely small, even if one consider κ = 1000. Also, an histogram for the number of pulsars for log η bin can be seen in fig. 5. One may notice a peak in the η histogram at 10 −16 − 10 −15 for the pulsars of Table 1. As for B , η for different values of κ can be easily obtained, since η ∝ κ 2 . Thus, even for κ = 1000 the peak in the histogram would be around 10 −12 − 10 −11 .
Before proceeding, it is worth stressing that the two pulsar populations aforementioned also clearly appear in figs 2 -5.
These extremely small values of B and η have very important consequences as regards the detectability of GWs generated by the pulsars, whether the ellipticity is mainly due to the magnetic dipole of the pulsars themselves.
Our calculations show that the GW amplitudes for most of these pulsars are at best seven orders of magnitude smaller than those obtained by assuming the spindown limit, see Fig. 6. Notice that, even considering an extremely optimistic case, the value of the ellipticity is at best B ∼ 10 −5 (for PSR J1846-0258) and the corresponding efficiency η ∼ 10 −8 . Thus, the GW amplitude even in this case would be four orders of mag-   nitude lower than the amplitude obtained by assuming the spindown limit (η = 1).