1 INTRODUCTION
Andersson (Reference Andersson1998) and Friedman & Morsink (Reference Friedman and Morsink1998) demonstrate that all rotating stars are unstable with respect to excitation of r-modes (similar to Earths Rossby waves controlled by the Coriolis force) at any rotation rate, if dissipation is neglected. It is a particular case of Chandrasekhar–Friedman–Schutz (CFS) instability (Chandrasekhar Reference Chandrasekhar1970; Friedman & Schutz Reference Friedman and Schutz1978a, Reference Friedman and Schutz1978b). Evolution equations for dissipative neutron star, affected by the r-mode instability, were derived by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) (see also Levin Reference Levin1999; Ho & Lai Reference Ho and Lai2000; Alford & Schwenzer Reference Alford and Schwenzer2014a; Gusakov, Chugunov, & Kantor Reference Gusakov, Chugunov and Kantor2014a), assuming the slow rotating Newtonian stellar model. These equations are widely applied in the literature and many observational consequences were predicted (see Haskell Reference Haskell2015; Chugunov, Gusakov, & Kantor Reference Chugunov, Gusakov and Kantor2017, for recent reviews). In particular, the r-mode instability can limit spin frequencies of neutron stars (Bildsten Reference Bildsten1998; Andersson, Kokkotas, & Stergioulas Reference Andersson, Kokkotas and Stergioulas1999), generate potentially observable gravitational waves (e.g., Owen Reference Owen2010), lead to anti-glitches in millisecond pulsars and neutron stars, accreting in low mass X-ray binaries (Kantor, Gusakov, & Chugunov Reference Kantor, Gusakov and Chugunov2016) and even formation of additional class of hot rapidly rotating neutron stars—HOFNARS—which could reveal itself by stable thermal emission from surface, but do not accrete (Chugunov, Gusakov, & Kantor Reference Chugunov, Gusakov and Kantor2014). Confronting observations with predictions of r-mode instability theory one can put important constraints on the physics of neutron stars, including properties of their depths (see e.g., Haskell, Degenaar, & Ho Reference Haskell, Degenaar and Ho2012; Haskell Reference Haskell2015; Chugunov et al. Reference Chugunov, Gusakov and Kantor2017).
To a great extent, the above results are based on the canonical energy formalism and Lagrangian perturbation theory formulated by Friedman & Schutz (Reference Friedman and Schutz1978a, Reference Friedman and Schutz1978b) and generalised for relativistic case by Friedman (Reference Friedman1978). This theory is, of course, mathematically strict, but rather complicated: Friedman & Schutz (Reference Friedman and Schutz1978a, Reference Friedman and Schutz1978b) reveal class of Lagrangian displacements—called trivials—which do not modify physical variables and introduce another class of Lagrangian displacements—called canonical—which are orthogonal to the trivials. They also introduce canonical energy functional for Lagrangian perturbations and demonstrate that perturbations described by canonical Lagrangian displacement with negative canonical energy are unstable with respect to gravitational radiation (in the absence of viscosity). It is worth to note that canonical energy can be equal to the physical change of energy under certain conditions (e.g., growth of normal mode under the action of radiation reaction force in the absence of viscosity), but generally it is not the case (Friedman & Schutz Reference Friedman and Schutz1978a, Reference Friedman and Schutz1978b). This feature leads to conceptual critique (Levin & Ushomirsky Reference Levin and Ushomirsky2001b, see also footnote 2) of the first derivation of evolution equations for CFS unstable star (Owen et al. Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998). Objections were rather convincingly replied to by Ho & Lai (Reference Ho and Lai2000), who suggest slightly modified evolution equations. Currently, both versions are applied by different authors (e.g., compare Haskell Reference Haskell2015 and Mahmoodifar & Strohmayer Reference Mahmoodifar and Strohmayer2013), but which of them is more precise?
To see the problem in a different light, I suggest another approach, which as I believe, is also useful from methodological point of view. Namely, I avoid to use Lagrangian perturbation theory and canonical energy and angular momentum of perturbations, but deal with global physical properties of the star (energy, angular momentum, and the thermal state) and consider instability of normal modesFootnote 1 . The key point is Equation (3), which follows from the multipolar expansion of gravitational wave emission (Thorne Reference Thorne1980) and couples the rates of change of the energy and the angular momentum. It naturally reveals physics of CFS instability: emission of gravitational waves should remove angular momentum from the star, thus releasing the rotation energy. The instability occurs if gravitational waves carry less energy than the released amount of the rotation energy (see first equality in Equation (4)).
Similar explanation was given, e.g., by Andersson (Reference Andersson1998); Andersson & Kokkotas (Reference Andersson and Kokkotas2001), but that was rather heuristic arguments than strict proof (e.g., Friedman & Stergioulas Reference Friedman, Stergioulas, Bičák and Ledvinka2014)Footnote 2 . Explicit reference to Equation (3) allows to formulate these arguments in a precise form and not only derive instability criterion, but also obtain general evolution equations for CFS unstable star (see Section 2). It is worth noting that this derivation does not require simplifying assumptions of slow rotation or Newtonian gravitation, but is valid in full general relativity framework. Additional advantage of derivation in Section 2 is that it allows straightforward generalization for superfluid neutron stars, because it exploits the global properties of the star (energy and angular momentum), but not complicated structure of internal perturbations.
To apply the evolution equations, derived in Section 2 one, of course, should deal with detailed description of internal perturbations to calculate efficiency of the gravitational radiation and dissipation. It is very complicated problem, which can be crucially affected by the neutron star core composition (e.g., Jones Reference Jones2001; Lindblom & Owen Reference Lindblom and Owen2002; Nayyar & Owen Reference Nayyar and Owen2006; Alford & Schwenzer Reference Alford and Schwenzer2014b), superfluidity (Yoshida & Lee Reference Yoshida and Lee2003a; Lee & Yoshida Reference Lee and Yoshida2003; Andersson, Glampedakis, & Haskell Reference Andersson, Glampedakis and Haskell2009; Haskell, Andersson, & Passamonti Reference Haskell, Andersson and Passamonti2009; Gusakov, Chugunov, & Kantor Reference Gusakov, Chugunov and Kantor2014b; Kantor & Gusakov Reference Kantor and Gusakov2017), crust-core coupling (Rieutord Reference Rieutord2001; Levin & Ushomirsky Reference Levin and Ushomirsky2001a; Glampedakis & Andersson Reference Glampedakis and Andersson2006a, Reference Glampedakis and Andersson2006b), in-medium effects (Kolomeitsev & Voskresensky Reference Kolomeitsev and Voskresensky2015), and, of course, general relativity (e.g., Kojima Reference Kojima1998; Lockitch, Andersson, & Friedman Reference Lockitch, Andersson and Friedman2001; Yoshida & Lee Reference Yoshida and Lee2002; Yoshida & Lee Reference Yoshida and Lee2003b; Ruoff & Kokkotas Reference Ruoff and Kokkotas2002; Lockitch, Friedman, & Andersson Reference Lockitch, Friedman and Andersson2003; Lockitch, Andersson, & Watts Reference Lockitch, Andersson and Watts2004; Krüger, Gaertig, & Kokkotas Reference Krüger, Gaertig and Kokkotas2010). However, I leave these problems beyond the scope of this paper, because they provide parameters for the evolution equations, but do not modify their form.
In Section 3, I illustrate the general evolution equations on the example of slow rotating Newtonian stellar model. In particular, it is shown that the gravitational radiation directly leads that the star spins down, being thus in agreement with arguments by Levin & Ushomirsky (Reference Levin and Ushomirsky2001b). In Section 4, these results are compared with widely applied equations by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000). At the first glance, my equations differ from those in both of the papers: latter attribute decrease of the effective spin frequency (as introduced by Owen et al. Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998 and Ho & Lai Reference Ho and Lai2000) to the dissipation of the unstable mode, but not with instability as it is. I demonstrate that this difference is associated with definition of the effective frequency and just as it is taken into account, the evolution equations agree in the leading order in mode amplitude. I also argue that the next-to-leading order terms, which differ for Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000), was not yet derived accurately and should be omitted. I conclude in Section 5.
2 CFS INSTABILITY AND EVOLUTION OF UNSTABLE STAR
In this section, I consider rotating relativistic star in its asymptotic rest frame; their total mass-energy E and angular momentum J are well defined (see, e.g., Section 19 in the textbook by Misner, Thorne, & Wheeler Reference Misner, Thorne and Wheeler1973). For given angular momentum J, uniform rotation corresponds to the minimal energy (at fixed baryon number), which is the rotational energy E rot (e.g., Boyer & Lindquist Reference Boyer and Lindquist1966; Hartle & Sharp Reference Hartle and Sharp1967; Stergioulas Reference Stergioulas2003). Corresponding spin frequency is (it follows, e.g., from variational principle by Hartle & Sharp Reference Hartle and Sharp1967)
$$\begin{equation}
\Omega =\frac{\partial E_\mathrm{rot}}{\partial J}.
\end{equation}$$
If star is perturbed, but the total angular momentum does not get changed, the energy E exceeds rotational energy (E > E rot), leading to positively defined excitation energy:
$$\begin{equation}
E_\mathrm{ex}=E-E_\mathrm{rot}.
\end{equation}$$
I would like to stress, that E ex should not be confused with non-positively-defined canonical energy, introduced by Friedman & Schutz (Reference Friedman and Schutz1978a, Reference Friedman and Schutz1978b).
As shown by Thorne (Reference Thorne1980), the rate of changes of the energy
$\dot{E}^\mathrm{GR}$
and angular momentum
$\dot{J}^\mathrm{GR}$
due to emission of gravitational waves can be expressed as sums over multipolar contributions, which comes from expansion of radiation field in the local wave zones. As it follows from Equations (4.16) and (4.23) by Thorne (Reference Thorne1980), for perturbations ∝eı(ωt + mϕ), these rates are coupled by the equationFootnote
3
$$\begin{equation}
-\frac{\omega }{m} \dot{J}^\mathrm{GR}=\dot{E}^\mathrm{GR}.
\end{equation}$$
Gravitational wave emission removes energy from the system, thus
$\dot{E}^\mathrm{GR}<0$
. Sign of rate of change of angular momentum is determined by ω/m. The rate of change of the excitation energy E
ex is
$$\begin{equation}
\dot{E}^\mathrm{GR}_\mathrm{ex}=\dot{E}^\mathrm{GR}-\dot{E}^\mathrm{GR}_\mathrm{rot} =\left(1+\frac{m\Omega }{\omega }\right) \dot{E}^\mathrm{GR}.
\end{equation}$$
Here,
$\dot{E}^\mathrm{GR}_\mathrm{rot}=\Omega \dot{J}^\mathrm{GR}$
. The excitation energy is increased by emission of gravitational waves if and only if (1 + mΩ/ω) < 0. This condition is equal to well known criterion of CFS instability (see e.g., Friedman & Schutz Reference Friedman and Schutz1978a; Friedman Reference Friedman1978; Andersson & Kokkotas Reference Andersson and Kokkotas2001; Friedman & Stergioulas Reference Friedman, Stergioulas, Bičák and Ledvinka2014): the prograde mode pattern in the inertial frame (−ω/m > 0), but retrograde mode pattern in the frame, corotating with the star (−Ω − ω/m < 0). Thus, the above discussion proves the CFS instability criterion for normal modes without appeal to the Lagrangian perturbation theory.
To describe the evolution of CFS unstable star, I parametrise it state by three parameters: (i) total angular momentum J, (ii) the mode energy E ex, and (iii) thermal state. The latter can be characterised by temperature in the stellar centre T, because neutron stars are almost isothermal (e.g., Page et al. Reference Page, Lattimer, Prakash and Steiner2004; Gusakov et al. Reference Gusakov, Kaminker, Yakovlev and Gnedin2005) due to high thermal conductivity in their depths (see, e.g., Shternin, Baldo, & Haensel Reference Shternin, Baldo and Haensel2013 for recent results).
Evolution of angular momentum due to emission of gravitational waves is described by Equation (4):
$$\begin{equation}
\dot{J}^\mathrm{GR} = -\frac{m}{\omega +m\Omega } \dot{E}^\mathrm{GR}_\mathrm{ex}.
\end{equation}$$
This equation is applicable for any oscillation mode (in particular for r-modes) at any spin frequency and even for general relativistic (not Newtonian) stellar models. Here Ω = Ω(J) is given by Equation (1).
Evolution of the mode energy is associated with energy pumping by gravitational waves
$\dot{E}^\mathrm{GR}_\mathrm{ex}$
and energy losses due to dissipation
$\dot{E}^\mathrm{dis}_\mathrm{ex}$
$$\begin{equation}
\dot{E}_\mathrm{ex}=\dot{E}^\mathrm{GR}_\mathrm{ex}+\dot{E}^\mathrm{dis}_\mathrm{ex}.
\end{equation}$$
Finally, the thermal evolution of star is governed by:
$$\begin{equation}
C \dot{T}= - \dot{E}^\mathrm{dis}_\mathrm{ex} -L_\mathrm{cool}.
\end{equation}$$
Here, L cool and C are total cooling power (neutrino and thermal emission from surface) and heat capacity of the star, respectively. One can also add torques, which are not associated with r-modes (e.g., accretion spin up, e.g., Ghosh & Lamb Reference Ghosh and Lamb1979; Wang Reference Wang1995 or magnetic braking, e.g., Beskin, Gurevich, & Istomin Reference Beskin, Gurevich and Istomin1993) to the angular momentum evolution equation (5) and additional heating [e.g., accretion-induced deep crustal heating (Brown, Bildsten, & Rutledge Reference Brown, Bildsten and Rutledge1998) or internal heating in millisecond pulsars (Alpar et al. Reference Alpar, Pines, Anderson and Shaham1984; Reisenegger Reference Reisenegger1995; Gusakov, Kantor, & Reisenegger Reference Gusakov, Kantor and Reisenegger2015)] to the thermal evolution equation (7).
To apply Equations (5)–(7), one should specify properties of the mode: ω,
$\dot{E}^\mathrm{GR}_\mathrm{ex}$
and
$\dot{E}^\mathrm{dis}_\mathrm{ex}$
as function of E
ex and J, which is, of course, very complicated problem, especially for relativistic stellar models (see e.g., Lockitch et al. Reference Lockitch, Friedman and Andersson2003; Kastaun Reference Kastaun2011). However, for the slow rotating Newtonian stellar models these parameters can be easily extracted from the literature (yet, depending on the microphysical assumptions, see e.g., Haskell Reference Haskell2015) and in the next section they are applied to illustrate Equations (5)–(7).
3 EVOLUTION OF R-MODE UNSTABLE NEUTRON STAR WITHIN SLOW ROTATING NEWTONIAN STELLAR MODELS
Here, I restrict myself to the slow rotating Newtonian stellar models, which are commonly used to study CFS instability in neutron stars (see, e.g., Haskell Reference Haskell2015 for recent review). In this case, Equation (1) can be written as Ω = J/I, where moment of inertia I does not depend on J. The most unstable mode is r-mode with l = m = 2 (e.g., Lindblom et al. Reference Lindblom, Owen and Morsink1998), with the frequency
$$\begin{equation}
\omega =-\frac{(m-1)(m+2)}{m+1}\Omega =-\frac{4}{3}\Omega .
\end{equation}$$
The first-order Eulerian perturbations of velocity can be written as follows (e.g., Provost, Berthomieu, & Rocca Reference Provost, Berthomieu and Rocca1981):
$$\begin{equation}
\delta ^{(1)} \bm v= \alpha R\Omega \left(\frac{r}{R}\right)^m \bm Y_{mm}^B \exp ^{i\omega t}
\end{equation}$$
Here, α is dimensionless mode amplitude and
$$\begin{equation}
\bm Y_{lm}^B=\frac{1}{l(l+1)} r \nabla \times (r\nabla Y_{lm})
\end{equation}$$
is magnetic-type vector spherical harmonic (see e.g., Varshalovich, Moskalev, & Khersonskii Reference Varshalovich, Moskalev and Khersonskii1988). The excitation energy for r-mode can be written in the form
$$\begin{eqnarray}
E_\mathrm{ex}&=&\int \frac{\rho \delta v^2}{2} \mathrm{d} ^3 \bm r =\int \frac{\rho [\delta ^{(1)} \bm v]^2}{2} \mathrm{d} ^3 \bm r \nonumber \\
&+& \int \rho \bm v_0 \delta ^{(2)} \bm v \mathrm{d} ^3 \bm r +\mathcal {O}(\alpha ^3),
\end{eqnarray}$$
Here,
$\delta v^2=(\bm v)^2-(\bm v_0)^2=2 \bm v_0\delta \bm v+(\delta \bm v)^2$
is perturbation of squared velocity, and
$\delta \bm v=\sum _i\delta ^{(i)} \bm v$
is total perturbation of velocity, presented as a sum over orders in α [i.e.,
$\delta ^{(i)} \bm v=\mathcal {O}(\alpha ^i)$
]. The integral is taken over stellar volume. I also neglect density pertrubations because they are of the second order in Ω (e.g., Lindblom et al. Reference Lindblom, Owen and Morsink1998). The second term in Equation (11) depends on the second order velocity perturbation
$\delta ^{(2)}\bm v$
, but because of the finite velocity at the unperturbed state
$\bm v_0$
, it contributes to the energy at the same order as first-order perturbations (Friedman & Schutz Reference Friedman and Schutz1978a)Footnote
4
. However, only axysimetric part
$\delta ^{(2)}_\mathrm{sym}\bm v$
can contribute to the integralFootnote
5
, but the definition of excitation energy (Equation (2)) supposes that the perturbed state has the same angular momentum as unperturbed star, constraining
$\delta \bm v$
:
$$\begin{equation}
\delta J=\int \rho \left[\delta \bm v\times \bm r\right]\mathrm{d}^3 \bm r=0.
\end{equation}$$
As far as unperturbed state is uniform rotation
$\bm v_0=\bm \Omega \times \bm r$
, the contribution of
$\delta ^{(2)}_\mathrm{sym} \bm v$
to the energy should vanish in Equation (11) at the second order in α. Thus, the second-order excitation energy is determined exclusively by the first order perturbations and equals to the kinetic energy in the system corotating with star. It can be written as follows (see, e.g., Lindblom et al. Reference Lindblom, Owen and Morsink1998):
$$\begin{equation}
E_\mathrm{ex}=\frac{1}{2} \alpha ^2 \Omega ^2 R^{-2m+2} \int _0^R \rho r^{2m+2}\mathrm{d}^3 \bm r
\end{equation}$$
The instability timescale
$$\begin{equation}
\tau ^\mathrm{GR}=-2 \frac{E_\mathrm{ex}}{\dot{E}^\mathrm{GR}_\mathrm{ex}}
\end{equation}$$
can be calculated via multipolar expansion of gravitational radiation for Newtonian sources (see Section V.C in Thorne Reference Thorne1980), as it was done by Lindblom et al. (Reference Lindblom, Owen and Morsink1998):
$$\begin{eqnarray}
\frac{1}{\tau ^\mathrm{GR}} &=&-\frac{32\pi G\Omega ^{2m+2}}{c^{2m+3}} \,\frac{(m-1)^{2m}}{[(2m+1)!!]^2} \,\left(\frac{m+2}{m+1}\right)^{2l+2} \nonumber \\
&\times & \int _0^R \rho r^{2m+2} \mathrm{d} r.
\end{eqnarray}$$
This result agrees with analytic treatment of r-mode instability up to second order in oscillation amplitude by Friedman, Lindblom, & Lockitch (Reference Friedman, Lindblom and Lockitch2016).
The dissipation rate
$$\begin{equation}
\tau ^\mathrm{dis}=-2\frac{E_\mathrm{ex}}{\dot{E}^\mathrm{dis}_\mathrm{ex}}
\end{equation}$$
should be specified for each certain model of dissipation (shear viscosity, mutual friction, etc.). Note, internal dissipative processes cannot affect total angular momentum of the star; thus, the rotational energy is conserved and dissipation time scale can be estimated from dissipation rate of the total energy
$\dot{E}^\mathrm{dis}$
(i.e., one can substitute
$\dot{E}^\mathrm{dis}$
instead of
$\dot{E}^\mathrm{dis}_\mathrm{ex}$
in Equation (16)). For example, the contribution of the shear viscosity η to the dissipation rate is (Lindblom et al. Reference Lindblom, Owen and Morsink1998):
$$\begin{equation}
\tau ^\mathrm{S}=(m-1)(2m+1)\frac{\int _0^R \eta r^{2l} \mathrm{d} r}{\int _0^R \rho r^{2l+2} \mathrm{d} r}.
\end{equation}$$
Introduction of these timescales gives
$\dot{E}^\mathrm{GR}_\mathrm{ex}$
, and
$\dot{E}^\mathrm{dis}_\mathrm{ex}$
as functions of J and E
ex, allowing thus to rewrite Equations (5)–(7) in the form:
$$\begin{equation}
\dot{\Omega } = \frac{2\tilde{Q} \alpha ^2}{ \tau ^\mathrm{GR}(\Omega )}\Omega
\end{equation}$$\\
$$\begin{equation}
\dot{\alpha } = -\left(\frac{1}{\tau ^\mathrm{GR}}+\frac{1}{\tau ^\mathrm{dis}}\right) \alpha
\end{equation}$$\\
$$\begin{equation}
C\dot{T} = \frac{\tilde{J} M R^2 \Omega }{\tau ^\mathrm{dis}} \alpha ^2 -L_\mathrm{cool}
\end{equation}$$
$$\begin{equation}
\tilde{J} = \frac{1}{MR^{2m}}\int _0^R \rho r^{2m+2} \mathrm{d} r\approx 1.64\times 10^{-2},
\end{equation}$$\\
$$\begin{equation}
\tilde{I} = \frac{I}{MR^2}=\frac{8\pi }{3MR^2}\int _0^R \rho r^4 \mathrm{d} r\approx 0.261,
\end{equation}$$\\
$$\begin{equation}
\tilde{Q} = \frac{m(m+1)\tilde{J}}{4\tilde{I}}\approx 9.4\times 10^{-2}.
\end{equation}$$
The enhancement of the mode amplitude can be limited by nonlinear saturation (e.g., Bondarescu, Teukolsky, & Wasserman Reference Bondarescu, Teukolsky and Wasserman2007; Bondarescu & Wasserman Reference Bondarescu and Wasserman2013; Haskell, Glampedakis, & Andersson Reference Haskell, Glampedakis and Andersson2014), which can be described by substitution of the effective dissipation rate τdis eff = |τGR| instead of τdis into all Equations (18)–(20).
4 COMPARISON WITH PREVIOUS WORKS
At first glance, the evolution equations (18–20) differ from equations derived by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000) applied in vast majority of the papers dealing with the evolution of r-mode unstable NSs. Namely, in the leading order in mode amplitude their equations can be written in the form:Footnote 6
$$\begin{equation}
\dot {\hat \Omega} = -\frac{2\tilde Q \alpha^2}{ \tau^\mathrm{dis}}\hat\Omega
\end{equation}$$\\
$$\begin{equation}
\dot{\alpha } = -\left(\frac{1}{\tau ^\mathrm{GR}}+\frac{1}{\tau ^\mathrm{dis}}\right) \alpha
\end{equation}$$\\
$$\begin{equation}
C\dot{T} = \frac{\tilde{J} M R^2\hat{\Omega }}{\tau ^\mathrm{dis}} \alpha ^2 -L_\mathrm{cool}.
\end{equation}$$
$\hat{\Omega }$
is the effective spin frequency, as introduced by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000), which differs from Ω, given by Equation (1). Note, the rate of change of
$\hat{\Omega }$
is associated with dissipation timescale, but not with instability timescale as in Equation (18).
The reason of this difference originates from uncertainties in definition of mean spin frequency in CFS unstable neutron star arising from differential rotation, which can be generated in the star as a result of CFS instability (see, e.g., Spruit Reference Spruit1999; Rezzolla, Lamb, & Shapiro Reference Rezzolla, Lamb and Shapiro2000; Levin & Ushomirsky Reference Levin and Ushomirsky2001b; Friedman et al. Reference Friedman, Lindblom and Lockitch2016). Namely, Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000) assume that physical angular momentum associated with r-mode is equal to the canonical angular momentum
$$\begin{equation}
J_\mathrm{c}=-(3/2)\hat{\Omega }\tilde{J} MR^2 \alpha ^2,
\end{equation}$$
and write total angular momentum as
$$\begin{equation}
J=I\hat{\Omega }+J_\mathrm{c}.
\end{equation}$$
As a result, the effective spin frequency is
$$\begin{equation}
\hat{\Omega }= (1+\tilde{Q}\alpha ^2)\Omega .
\end{equation}$$
After this change of variables, our equations agree with results by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000) in the leading order in α.
Equations derived by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000) also contain terms, which are of the next order in α2. However, I suppose that these terms (which do not agree for Owen et al. Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998 and Ho & Lai Reference Ho and Lai2000) include only part of the required next-order corrections. To be brief, their equations are based on the linear (leading order) perturbation theory and hence cannot predict next-to-leading order corrections accurately. For example, both Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000) neglect corrections to the r-mode frequency associated with differential rotation (assuming r-mode frequency to be
$\omega =4\hat{\Omega }/3$
in all orders in α). However, as discussed by Chirenti, Skákala, & Yoshida (Reference Chirenti, Skákala and Yoshida2013), the differential rotation δΩ can affect the spin frequency at order of
$\mathcal {O}(\delta \Omega /\Omega )$
. As a result, the second-order differential rotation, associated with excitation of r-modes (Spruit Reference Spruit1999; Rezzolla et al. Reference Rezzolla, Lamb and Shapiro2000; Levin & Ushomirsky Reference Levin and Ushomirsky2001b; Friedman et al. Reference Friedman, Lindblom and Lockitch2016) leads to corrections to the r-mode frequency
$\mathcal {O}(\alpha ^2)$
, and consequently to corrections of the same order for the energy pumping rate, given by Equation (5)
Footnote
7
. Note, the numerical value of these corrections depend on the differential rotation profile (Chirenti et al. Reference Chirenti, Skákala and Yoshida2013), which can be dramatically modified by the magnetic field in the star (see Chugunov Reference Chugunov2015; Friedman et al. Reference Friedman, Lindblom, Rezzolla and Chugunov2017, who also demonstrate that magnetic field windup by differential rotation do not suppress r-mode instability thanks to back reaction of magnetic field) or viscosity. To be accurate with the next order effects in the evolution equations, one needs additional modelling of the differential rotation profile. Fortunately, these effects should be negligible in any case because current models of nonlinear saturation (e.g., Bondarescu et al. Reference Bondarescu, Teukolsky and Wasserman2007; Bondarescu & Wasserman Reference Bondarescu and Wasserman2013; Haskell, Glampedakis, & Andersson Reference Haskell, Glampedakis and Andersson2014) predict low saturation amplitude α ≲ 10−3 and actual observations constrain the r-mode amplitude in many potentially unstable neutrons stars even stronger (e.g., Mahmoodifar & Strohmayer Reference Mahmoodifar and Strohmayer2013; Alford & Schwenzer Reference Alford and Schwenzer2015; Schwenzer et al. Reference Schwenzer, Boztepe, Güver and Vurgun2017; Chugunov et al. Reference Chugunov, Gusakov and Kantor2017). Thus, I suggest to use leading order equations (18)–(20) (or equivalent equations 24–26) to model the evolution of r-mode unstable neutron stars and omit next-order terms.
5 SUMMARY AND CONCLUSIONS
Using multipolar expansion of gravitational waves, formulated by Thorne (Reference Thorne1980), I derive criterion of CFS instability for normal modes and evolution equations (5)–(7) for a star, affected by this instability. The derivation does not appeal to the canonical energy formalism by Friedman & Schutz (Reference Friedman and Schutz1978a, Reference Friedman and Schutz1978b). Equations (5)–(7) describe evolution of angular momentum, mode energy and temperature and can be applied for relativistic star. They are illustrated by r-mode instability in the slowly rotating Newtonian stellar model (Equations (18)–(20)). In a latter case the evolution equations were earlier derived by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000). At first glance, the equations differ from those derived here. However, it is shown that it is spurious: in the leading order in mode amplitude, the only difference is definition of the (effective) spin frequency, entering into the equations (Ω or
$\hat{\Omega }$
, see Equation (29)). Formulas, suggested by Owen et al. (Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998) and Ho & Lai (Reference Ho and Lai2000), contain terms of the next order in the mode amplitude. However, I argue that they do not include all of the required terms (as a result, these terms do not agree for Owen et al. Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998 and Ho & Lai Reference Ho and Lai2000) and thus it would be more self-consistent to restrict consideration to the leading order evolution equations (Equations (18)–(20)).
The difference in definition of the spin frequency (Ω and
$\hat{\Omega }$
, see Equation (29), is associated with differential rotation, which can be generated during excitation of r-modes by CFS instability (e.g., Spruit Reference Spruit1999; Ho & Lai Reference Ho and Lai2000; Rezzolla et al. Reference Rezzolla, Lamb and Shapiro2000; Levin & Ushomirsky Reference Levin and Ushomirsky2001b; Friedman et al. Reference Friedman, Lindblom and Lockitch2016). As far as observed frequency is associated with specific point at the surface of neutron star (e.g., magnetic pole), the observed spin frequency Ωobs can differ from both frequencies Ω and
$\hat{\Omega }$
. Important advantage of the formalism, suggested in this work, that it can be naturally extended to include effects of differential rotation (because it explicitly deals with second-order axisymmetric velocity perturbations, see Section 3), and I plan to analyse them at the subsequent paper. As far as the spin down rate, given by Equation (18), qualitatively agrees with arguments by Levin & Ushomirsky (Reference Levin and Ushomirsky2001b), i.e. radiation of gravitation waves directly leads to spin down of the star (but not via dissipation of mode energy as it follows from Owen et al. Reference Owen, Lindblom, Cutler, Schutz, Vecchio and Andersson1998 and Ho & Lai Reference Ho and Lai2000, see Equation (24)), I believe that Ωobs would be closer to Ω, than to
$\hat{\Omega }$
. However, this conclusion should be checked by accurate calculations.
ACKNOWLEDGEMENTS
I am grateful to Misha Gusakov and Elena Kantor for valuable comments and discussions. This study was supported by the Russian Science Foundation (Grant no. 14-12-00316).
