Nils Andersson

Missing pieces in the r-mode puzzle

Southampton Theory Astronomy Gravity Centre for Fundamental Physics

accreting systems Accreting neutron stars in LMXBs may be relevant gravitational-wave sources. Their rotation appears to be limited by some mechanism;

―  non-standard accretion torque

―  additional spin-down (mountains, r-modes, B-field)

Possible indirect evidence for a gravitational-wave component from spin vs orbit period in the observed systems.

Required deformation significantly below current LIGO limits, but may (just?) be within reach of the third generation Einstein Telescope.

(Ho, Maccarone & NA, ApJL 2011)

The r-modes belong to a large class of “inertial” modes, which are driven unstable by the emission of gravitational waves.

The l=m=2 r-mode grows due to current-multipole radiation on a timescale of a few tens of seconds. Instability window depends on uncertain core-physics, i.e. provides a probe of exotic physics

The simplest models account for damping due to shear- and bulk viscosity.

Leads to a very large instability window.

In principle, we should not observe any “usual” neutron stars inside the instability region. Can we use this to “rule out” theoretical models?

shear bulk

the r-mode instability

Young radio pulsars: Original r-mode window consistent with the inferred birth spin of the Crab PSR (19 ms), but not with the 16 ms X-ray PSR J0537-6910.

Recycled pulsars: Need to allow the formation of a cold 716 Hz PSR (presumably after recycling). This constrains the instability window at low temperatures.

LMXBs: Nuclear burning of accreted material provides a thermostat that sets the core temperature to 108 K. Fastest systems (around 640 Hz) require smaller instability region.

“constraints”

shear bulk

Consider temperature limits for systems in quiescence (Brown+Ushomirsky).

If a system is r-mode unstable, how would we know?

Are there systems that behave (in some sense) “funny”?

superfluid cores Revisit core temperature in accreting systems in light of recent evidence for neutron superfluidity in Cassiopeia A remnant.

Assume r-mode balances accretion spin-up torque, i.e. that observed systems are in spin equilibrium. (important caveats here!) Suppress neutrino processes involving superfluid nucleons. Enhance (just below Tc) emission due to Cooper pairing. Note: Inferred temperature may not be unique!

(Ho, NA & Haskell, PRL 2011)

superfluid cores Revisit core temperature in accreting systems in light of recent evidence for neutron superfluidity in Cassiopeia A remnant.

Assume r-mode balances accretion spin-up torque, i.e. that observed systems are in spin equilibrium. (important caveats here!) Suppress neutrino processes involving superfluid nucleons. Enhance (just below Tc) emission due to Cooper pairing.

Revisit core temperature in accreting systems in light of recent evidence for neutron superfluidity in Cassiopeia A remnant.

Assume r-mode balances accretion spin-up torque, i.e. that observed systems are in spin equilibrium. (important caveats here!) Suppress neutrino processes involving superfluid nucleons. Enhance (just below Tc) emission due to Cooper pairing.

superfluid cores

Demonstrates that our understanding of the r-modes is incomplete. Given the “best estimate” for the main damping mechanisms, many observed LMXBs should be unstable.

r-mode puzzle

rigid crust

no crust

with “slippage”

Rigid crust with viscous (Ekman) boundary layer would lead to sufficient damping… …but the crust is more like jelly, so the effect is reduced (“slippage”). Magnetic field is too weak to alter the nature of the boundary layer. Superfluid “mutual friction” (due to electrons scattered off vortices) has no effect.

Saturation amplitude due to mode-coupling is too large to allow evolution far into instability region.

resolutions?

i) Boundary layer redux:

―  pasta phase reduces/removes viscous boundary layer effect

―  magnetic field coupling/tension

― “resonances” between r-mode and toroidal crust modes?

―  transition to superconducting core?

4

8 90

200

400

600

800

1000! s (H

z)

8 9 log T (K)

8 9

superfluid mutual frictionsuperfluid hyperonscrust resonance

600 Hz

12.5 km

11 kmweak

strong

FIG. 3: Three scenarios that could explain r-mode stabilityin the observed LMXBs. Left panel: Crust mode resonanceat 600 Hz. Middle panel: Superfluid hyperons (based on [7]with χ = 0.1). Right panel: Strong vortex mutual friction(based on the strong/weak superfluidity models from [27] withB ≈ 0.01). The dashed lines indicate the break-up limit.

sition is likely to strongly affect the instability window,but the problem has not attracted real attention. Crustphysics may also be vital. There may be resonances be-tween the r-mode and torsional oscillations of the elasticcrust [9]. Such resonances would have a sizeable effect onthe slippage factor, leading to a complicated instabilitywindow. Figure 3 gives an example; the illustrated insta-bility window has a relatively broad resonance at 600 Hz,which is the typical frequency of the first overtone of purecrustal modes. Although our example is phenomenolog-ical (c.f., [9]), it suggests that this mechanism may ex-plain the stability of LMXBs. Realistic crust models areneeded to establish to what extent this is viable.

Another possibility is an instability window that in-creases with temperature [24]. If this is the case, thenLMXBs may evolve to a quasi-equilibrium where the r-mode instability is balanced (on average) by accretionand r-mode heating is balanced by cooling (as in our tem-perature estimates). This solution is interesting becauseit predicts persistent (low-level) gravitational radiation.Figure 3 shows a model using hyperon bulk viscosity sup-pressed by superfluidity. However, this explanation hasa major problem. We must be able to explain how theobserved millisecond radio pulsars emerge from the ac-creting systems. Once the accretion phase ends, the NSwill cool, enter the instability window, and spin down to∼ 300 Hz (see Fig. 3). In other words, it would be verydifficult to explain the formation of a 716 Hz pulsar [25].

A more promising possibility involves mutual frictiondue to vortices in a rotating superfluid. The standardmechanism (electrons scattered off of magnetized vor-tices) is too weak to affect the instability window [26].However, if we increase (arbitrarily) the strength of thismechanism by a factor ∼ 25, then mutual friction dom-inates the damping (see Fig. 3). Moreover, this wouldset a spin-threshold for instability similar to the highestobserved νs and would allow systems to remain rapidlyrotating after accretion shuts off. Enhanced friction mayresult from the interaction between vortices and proton

fluxtubes in the outer core, as proposed in a model forpulsar free precession [28]. This mechanism has not beenconsidered in the context of neutron star oscillations andinstabilities, but it seems clear that such work is needed.

In summary, we considered astrophysical constraintson the r-mode instability provided by the observedLMXBs. Having refined our understanding of the likelycore temperatures in these systems using recent super-fluid data, we showed that several systems lie well in-side the expected instability region. This highlights ourlack of understanding of the physics of the instability andthe associated evolution scenarios and at the same timepoints to several interesting directions for future work.

WCGH thanks Peter Shternin for the APR EOS.WCGH appreciates use of computer facilities at KIPAC.WCGH and NA acknowledge support from STFC in theUK. BH holds a EU Marie Curie fellowship.

[1] D. Chakrabarty et al., Nature, 424, 42 (2003); B.P. Ab-bott et al., Astrophys. J., 713, 671 (2010).

[2] L. Bildsten, Astrophys. J., 501, L89 (1998); N. Anders-son, K.D. Kokkotas, and N. Stergioulas, Astrophys. J.,516, 307 (1999).

[3] N. Andersson and K.D. Kokkotas, Int. J. Mod. Phys. D,10, 381 (2001).

[4] P. Arras et al., Astrophys. J., 591, 1129 (2003).[5] R. Bondarescu, S.A. Teukolsky, and I. Wasserman, Phys.

Rev. D, 76, 064019 (2007).[6] L. Lindblom, G. Mendell, and B.J. Owen, Phys. Rev. D,

60, 064006 (1999); M. Nayyar and B.J. Owen, Phys. Rev.D, 73, 084001 (2006).

[7] B. Haskell and N. Andersson, Mon. Not. R. Astron. Soc.,408, 1897 (2010).

[8] L. Bildsten and G. Ushomirsky, Astrophys. J., 529, L33(2000).

[9] Y. Levin and G. Ushomirsky, Mon. Not. R. Astron. Soc.,324, 917 (2001).

[10] D. Page, M. Prakash, J.M. Lattimer, and A.W. Steiner,Phys. Rev. Lett., 106, 081101 (2011).

[11] Shternin, P. S., Yakovlev, D. G., Heinke, C. O., Ho,W. C. G., Patnaude, D. J. Mon. Not. R. Astron. Soc.,412, L108 (2011).

[12] B.J. Owen et al., Phys. Rev. D, 58, 084020 (1998).[13] E.F. Brown and G. Ushomirsky, Astrophys. J., 536, 915

(2000).[14] S.L. Shapiro and S.A. Teukolsky, Black Holes, White

Dwarfs, and Neutron Stars (John Wiley & Sons, NewYork, 1983).

[15] P. Haensel and J.L. Zdunik, Astron. Astrophys., 227, 431(1990).

[16] P. Haensel, A.Y. Potekhin, and D.G. Yakovlev, NeutronStars 1. Equation of State and Structure (Springer, NewYork, 2007).

[17] C.O. Heinke and W.C.G. Ho, Astrophys. J., 719, L167(2010).

[18] D.G. Yakovlev and C.J. Pethick, Annu. Rev. Astron. As-trophys., 42, 169 (2004); D. Page, U. Geppert, and F.Weber, Nucl. Phys. A, 777, 497 (2006).

[19] D.G. Yakovlev, K.P. Levenfish, and Yu.A. Shibanov,Phys.-Uspekhi, 42, 737 (1999); D. Page, J.M. Lattimer,M. Prakash, and A.W. Steiner, Astrophys. J., 707, 1131

resolutions?

ii) More “physics”:

―  hyperon (or quark) bulk viscosity (problematic if you want a 716 Hz PSR)

―  various quark pairing phases (seem too weak, core essentially inviscid)

―  crust “viscosity”?

―  winding up magnetic field (issue with weak field ms PSRs?)

4

8 90

200

400

600

800

1000! s (H

z)

8 9 log T (K)

8 9

superfluid mutual frictionsuperfluid hyperonscrust resonance

600 Hz

12.5 km

11 kmweak

strong

FIG. 3: Three scenarios that could explain r-mode stabilityin the observed LMXBs. Left panel: Crust mode resonanceat 600 Hz. Middle panel: Superfluid hyperons (based on [7]with χ = 0.1). Right panel: Strong vortex mutual friction(based on the strong/weak superfluidity models from [27] withB ≈ 0.01). The dashed lines indicate the break-up limit.

sition is likely to strongly affect the instability window,but the problem has not attracted real attention. Crustphysics may also be vital. There may be resonances be-tween the r-mode and torsional oscillations of the elasticcrust [9]. Such resonances would have a sizeable effect onthe slippage factor, leading to a complicated instabilitywindow. Figure 3 gives an example; the illustrated insta-bility window has a relatively broad resonance at 600 Hz,which is the typical frequency of the first overtone of purecrustal modes. Although our example is phenomenolog-ical (c.f., [9]), it suggests that this mechanism may ex-plain the stability of LMXBs. Realistic crust models areneeded to establish to what extent this is viable.

Another possibility is an instability window that in-creases with temperature [24]. If this is the case, thenLMXBs may evolve to a quasi-equilibrium where the r-mode instability is balanced (on average) by accretionand r-mode heating is balanced by cooling (as in our tem-perature estimates). This solution is interesting becauseit predicts persistent (low-level) gravitational radiation.Figure 3 shows a model using hyperon bulk viscosity sup-pressed by superfluidity. However, this explanation hasa major problem. We must be able to explain how theobserved millisecond radio pulsars emerge from the ac-creting systems. Once the accretion phase ends, the NSwill cool, enter the instability window, and spin down to∼ 300 Hz (see Fig. 3). In other words, it would be verydifficult to explain the formation of a 716 Hz pulsar [25].

A more promising possibility involves mutual frictiondue to vortices in a rotating superfluid. The standardmechanism (electrons scattered off of magnetized vor-tices) is too weak to affect the instability window [26].However, if we increase (arbitrarily) the strength of thismechanism by a factor ∼ 25, then mutual friction dom-inates the damping (see Fig. 3). Moreover, this wouldset a spin-threshold for instability similar to the highestobserved νs and would allow systems to remain rapidlyrotating after accretion shuts off. Enhanced friction mayresult from the interaction between vortices and proton

fluxtubes in the outer core, as proposed in a model forpulsar free precession [28]. This mechanism has not beenconsidered in the context of neutron star oscillations andinstabilities, but it seems clear that such work is needed.

In summary, we considered astrophysical constraintson the r-mode instability provided by the observedLMXBs. Having refined our understanding of the likelycore temperatures in these systems using recent super-fluid data, we showed that several systems lie well in-side the expected instability region. This highlights ourlack of understanding of the physics of the instability andthe associated evolution scenarios and at the same timepoints to several interesting directions for future work.

WCGH thanks Peter Shternin for the APR EOS.WCGH appreciates use of computer facilities at KIPAC.WCGH and NA acknowledge support from STFC in theUK. BH holds a EU Marie Curie fellowship.

[1] D. Chakrabarty et al., Nature, 424, 42 (2003); B.P. Ab-bott et al., Astrophys. J., 713, 671 (2010).

[2] L. Bildsten, Astrophys. J., 501, L89 (1998); N. Anders-son, K.D. Kokkotas, and N. Stergioulas, Astrophys. J.,516, 307 (1999).

[3] N. Andersson and K.D. Kokkotas, Int. J. Mod. Phys. D,10, 381 (2001).

[4] P. Arras et al., Astrophys. J., 591, 1129 (2003).[5] R. Bondarescu, S.A. Teukolsky, and I. Wasserman, Phys.

Rev. D, 76, 064019 (2007).[6] L. Lindblom, G. Mendell, and B.J. Owen, Phys. Rev. D,

60, 064006 (1999); M. Nayyar and B.J. Owen, Phys. Rev.D, 73, 084001 (2006).

[7] B. Haskell and N. Andersson, Mon. Not. R. Astron. Soc.,408, 1897 (2010).

[8] L. Bildsten and G. Ushomirsky, Astrophys. J., 529, L33(2000).

[9] Y. Levin and G. Ushomirsky, Mon. Not. R. Astron. Soc.,324, 917 (2001).

[10] D. Page, M. Prakash, J.M. Lattimer, and A.W. Steiner,Phys. Rev. Lett., 106, 081101 (2011).

[11] Shternin, P. S., Yakovlev, D. G., Heinke, C. O., Ho,W. C. G., Patnaude, D. J. Mon. Not. R. Astron. Soc.,412, L108 (2011).

[12] B.J. Owen et al., Phys. Rev. D, 58, 084020 (1998).[13] E.F. Brown and G. Ushomirsky, Astrophys. J., 536, 915

(2000).[14] S.L. Shapiro and S.A. Teukolsky, Black Holes, White

Dwarfs, and Neutron Stars (John Wiley & Sons, NewYork, 1983).

[15] P. Haensel and J.L. Zdunik, Astron. Astrophys., 227, 431(1990).

[16] P. Haensel, A.Y. Potekhin, and D.G. Yakovlev, NeutronStars 1. Equation of State and Structure (Springer, NewYork, 2007).

[17] C.O. Heinke and W.C.G. Ho, Astrophys. J., 719, L167(2010).

[18] D.G. Yakovlev and C.J. Pethick, Annu. Rev. Astron. As-trophys., 42, 169 (2004); D. Page, U. Geppert, and F.Weber, Nucl. Phys. A, 777, 497 (2006).

[19] D.G. Yakovlev, K.P. Levenfish, and Yu.A. Shibanov,Phys.-Uspekhi, 42, 737 (1999); D. Page, J.M. Lattimer,M. Prakash, and A.W. Steiner, Astrophys. J., 707, 1131

resolutions?

iii) Superfluidity:

―  enhanced superfluid vortex mutual friction (comment P.B. Jones mechanism)

―  vortex turbulence?

―  crust pinning

―  fluxtube “cutting”

4

8 90

200

400

600

800

1000! s (H

z)

8 9 log T (K)

8 9

superfluid mutual frictionsuperfluid hyperonscrust resonance

600 Hz

12.5 km

11 kmweak

strong

FIG. 3: Three scenarios that could explain r-mode stabilityin the observed LMXBs. Left panel: Crust mode resonanceat 600 Hz. Middle panel: Superfluid hyperons (based on [7]with χ = 0.1). Right panel: Strong vortex mutual friction(based on the strong/weak superfluidity models from [27] withB ≈ 0.01). The dashed lines indicate the break-up limit.

sition is likely to strongly affect the instability window,but the problem has not attracted real attention. Crustphysics may also be vital. There may be resonances be-tween the r-mode and torsional oscillations of the elasticcrust [9]. Such resonances would have a sizeable effect onthe slippage factor, leading to a complicated instabilitywindow. Figure 3 gives an example; the illustrated insta-bility window has a relatively broad resonance at 600 Hz,which is the typical frequency of the first overtone of purecrustal modes. Although our example is phenomenolog-ical (c.f., [9]), it suggests that this mechanism may ex-plain the stability of LMXBs. Realistic crust models areneeded to establish to what extent this is viable.

Another possibility is an instability window that in-creases with temperature [24]. If this is the case, thenLMXBs may evolve to a quasi-equilibrium where the r-mode instability is balanced (on average) by accretionand r-mode heating is balanced by cooling (as in our tem-perature estimates). This solution is interesting becauseit predicts persistent (low-level) gravitational radiation.Figure 3 shows a model using hyperon bulk viscosity sup-pressed by superfluidity. However, this explanation hasa major problem. We must be able to explain how theobserved millisecond radio pulsars emerge from the ac-creting systems. Once the accretion phase ends, the NSwill cool, enter the instability window, and spin down to∼ 300 Hz (see Fig. 3). In other words, it would be verydifficult to explain the formation of a 716 Hz pulsar [25].

A more promising possibility involves mutual frictiondue to vortices in a rotating superfluid. The standardmechanism (electrons scattered off of magnetized vor-tices) is too weak to affect the instability window [26].However, if we increase (arbitrarily) the strength of thismechanism by a factor ∼ 25, then mutual friction dom-inates the damping (see Fig. 3). Moreover, this wouldset a spin-threshold for instability similar to the highestobserved νs and would allow systems to remain rapidlyrotating after accretion shuts off. Enhanced friction mayresult from the interaction between vortices and proton

fluxtubes in the outer core, as proposed in a model forpulsar free precession [28]. This mechanism has not beenconsidered in the context of neutron star oscillations andinstabilities, but it seems clear that such work is needed.

In summary, we considered astrophysical constraintson the r-mode instability provided by the observedLMXBs. Having refined our understanding of the likelycore temperatures in these systems using recent super-fluid data, we showed that several systems lie well in-side the expected instability region. This highlights ourlack of understanding of the physics of the instability andthe associated evolution scenarios and at the same timepoints to several interesting directions for future work.

WCGH thanks Peter Shternin for the APR EOS.WCGH appreciates use of computer facilities at KIPAC.WCGH and NA acknowledge support from STFC in theUK. BH holds a EU Marie Curie fellowship.

[1] D. Chakrabarty et al., Nature, 424, 42 (2003); B.P. Ab-bott et al., Astrophys. J., 713, 671 (2010).

[2] L. Bildsten, Astrophys. J., 501, L89 (1998); N. Anders-son, K.D. Kokkotas, and N. Stergioulas, Astrophys. J.,516, 307 (1999).

[3] N. Andersson and K.D. Kokkotas, Int. J. Mod. Phys. D,10, 381 (2001).

[4] P. Arras et al., Astrophys. J., 591, 1129 (2003).[5] R. Bondarescu, S.A. Teukolsky, and I. Wasserman, Phys.

Rev. D, 76, 064019 (2007).[6] L. Lindblom, G. Mendell, and B.J. Owen, Phys. Rev. D,

60, 064006 (1999); M. Nayyar and B.J. Owen, Phys. Rev.D, 73, 084001 (2006).

[7] B. Haskell and N. Andersson, Mon. Not. R. Astron. Soc.,408, 1897 (2010).

[8] L. Bildsten and G. Ushomirsky, Astrophys. J., 529, L33(2000).

[9] Y. Levin and G. Ushomirsky, Mon. Not. R. Astron. Soc.,324, 917 (2001).

[10] D. Page, M. Prakash, J.M. Lattimer, and A.W. Steiner,Phys. Rev. Lett., 106, 081101 (2011).

[11] Shternin, P. S., Yakovlev, D. G., Heinke, C. O., Ho,W. C. G., Patnaude, D. J. Mon. Not. R. Astron. Soc.,412, L108 (2011).

[12] B.J. Owen et al., Phys. Rev. D, 58, 084020 (1998).[13] E.F. Brown and G. Ushomirsky, Astrophys. J., 536, 915

(2000).[14] S.L. Shapiro and S.A. Teukolsky, Black Holes, White

Dwarfs, and Neutron Stars (John Wiley & Sons, NewYork, 1983).

[15] P. Haensel and J.L. Zdunik, Astron. Astrophys., 227, 431(1990).

[16] P. Haensel, A.Y. Potekhin, and D.G. Yakovlev, NeutronStars 1. Equation of State and Structure (Springer, NewYork, 2007).

[17] C.O. Heinke and W.C.G. Ho, Astrophys. J., 719, L167(2010).

[18] D.G. Yakovlev and C.J. Pethick, Annu. Rev. Astron. As-trophys., 42, 169 (2004); D. Page, U. Geppert, and F.Weber, Nucl. Phys. A, 777, 497 (2006).

[19] D.G. Yakovlev, K.P. Levenfish, and Yu.A. Shibanov,Phys.-Uspekhi, 42, 737 (1999); D. Page, J.M. Lattimer,M. Prakash, and A.W. Steiner, Astrophys. J., 707, 1131

what else?

Insert missing pieces here…