1. Goddard Space Flight Center A Neutron Star Oscillation Mode During a Superburst An l=7, m=4 toroidal mode (Anna Watts) ( 2 t 0 ) = 29.8 (R 10 ) -1 Hz (1.71 - 0.71M 1.4 /R 10 ) 1/2 (0.87 - 0.13M 1.4 /R 10 2 ) ( l t 0 ) = ( 2 t 0 ) [ l(l+1)/6 ] 1/2 [1 + (B/B  ) 2 ] -1/2 Tod Strohmayer, NASA’s Goddard Space Flight Center with Simin Mahmoodifar (University of Maryland)

2. Goddard Space Flight Center Neutron Star Seismology Stellar oscillations are a powerful probe of internal structure (e.g. helioseismology). For example, p-modes sense the sound speed. Mode frequencies scale as 1/2 (thus M and R). Solid crust supports torsional shear modes, l t n for n = 0 modes, effective wavelength is R, for n > 0 it is  R, if V s = constant, then f n=0 / f n>0 ~  R/R, constrain crust thickness (magnetar QPOs). g-modes supported by buoyancy (thermal and density gradients), can probe R, and envelope structure. Brightness pattern for an l=2, m=1 g-mode (slow rotation limit).

3. Goddard Space Flight Center Rotating Neutron Stars: r-modes r-modes are a subset of the inertial modes supported by rotation (Coriolis force, analogous to Rossby waves on Earth). Retrograde in frame rotating with the star, prograde in inertial frame, thus unstable to CFS gravitational radiation instability. Modes damped by viscous processes (shear, bulk), depends on phases of dense matter in the interior. Spin-down torque on the star, viscous damping heats the star. ω r = 2mΩ/(l(l + 1)) Mode displacement for l=m=2 r-mode

4. Goddard Space Flight Center R-mode Patterns in Co-rotating and Inertial Frames Co-rotating frame  retrograde pattern. Inertial frame  prograde pattern. ΩΩ

5. Goddard Space Flight Center r-modes: Potential Probe of Dense Neutron Star Matter As spin rate increases r-mode frequencies diverge from slow spin limit of 2Ω/3 (in rotating frame). Spin dependence of frequency is sensitive to EOS and phase of dense matter.

6. Goddard Space Flight Center How Might Non-radial Oscillations be Observed? Pulsation modes can modulate the temperature (flux) across the neutron star’s surface – coupled with spin can produce flux modulation at mode’s inertial frame frequency (Lee & Strohmayer, Heyl 2005). Surface displacements generated by pulsation modes can periodically distort the X-ray emitting hot-spot (Numata & Lee 2010). Works for transverse (quasi- toroidal) displacements. Such modes include surface g-modes, and r-modes. Since hot-spot rotates with the star, the modulation frequency seen by a distant observer is the co-rotating frame frequency (Strohmayer & Mahmoodifar 2014, ApJ, 784, 72). 4U 1636-536 Superburst

7. Goddard Space Flight Center Coherent Searches for Pulsation Modes Use orbit model to remove time delays associated with neutron star’s orbital motion. Observer is effectively at the binary’s center of mass. “Coherence Recovery,” improves sensitive to weak, coherent signals. Narrow band signal is not “smeared” into many Fourier bins. “Targeted” search for r-modes and surface g-modes. Spin frequency is known, so search mode frequency range (relative to spin) that is expected theoretically. Keeps N trial low and thus improves overall sensitivity. Bin light curve using corrected times (4096 Hz sampling). Create power spectra using 4 energy ranges (focusing on superburst thermal component). 0.417 ≤ σ / Ω ≤ 0.757 (rotating frame) 1.243 ≤ σ / Ω ≤ 1.583 (inertial frame)

8. Goddard Space Flight Center 4U 1636-535: February 2001 Superburst Orbital period known (3.79 hr), fit for other parameters; ν spin, v ns and epoch of T 90 (Strohmayer & Markwardt 2002). Barycenter, then remove orbital time delays. Poster 122.27 (Keek et al.) 582 Hz spin pulsations detected for ~800 s near burst peak. Frequency drift consistent with orbital motion of the neutron star. pulsation interval

9. Goddard Space Flight Center 4U 1636-535 Superburst Light Curve N bins ≈ (Δν/ν) = (v ns /c) = (0.5/582)*835/(1/8000) = 5700!

10. Goddard Space Flight Center Coherent Search in the Superburst 1.43546 x spin Estimated significance: 2.5 x 10 -4

11. Goddard Space Flight Center 835 Hz Coherent Modulation Folded light curve at 835.644 Hz, profile well fit by a sinusoid, A + B sin ( ϕ - ϕ 0 ), modulation amplitude B/A = 0.19 ± 0.04 %, Noise power distribution consistent with expected. Extensive Monte Carlo simulations confirm analytic estimate.

12. Goddard Space Flight Center Possible Mode Identifications g-modes in envelope above the solid crust. Piro & Bildsten (2004) and Strohmayer & Lee (1996) computed modes in accreting, nuclear burning envelopes (  -mechanism). Mode frequencies 20 – 30 Hz, but modes modified by fast rotation (Bildsten et al. 1996; Piro & Bildsten 2004). latitudinal dependence of Coriolis force “squeezes” modes closer to equator. Observed frequency (835 Hz) is larger than spin, so, if |m|=1, need prograde (m=-1) mode (this is a Kelvin mode), but frequency is not high enough to reach observed (likely rules out l=1). So, |m|=2. Then l=2, m=2 or l=2, m=-1 appear plausible. Piro & Bildsten (2004)

13. Goddard Space Flight Center Implications for Neutron Star Mass in 4U 1636-536 If intrinsic mode frequency is constant then detection suggests v ns = 134.5 ± 0.2 km/sec. High velocity suggests ``light” neutron star, interesting as low mass end constrains core collapse physics (Lattimer 2012).

14. Goddard Space Flight Center Conclusions, Future Work Detection of global neutron star oscillation modes would open a new window on neutron star interiors and cold ultra-dense matter. Accreting Millisecond Pulsars (AMXPs) are good search candidates. Spin and orbits known. Mode-driven perturbations to hot spot. Several initial searches reported (Strohmayer & Mahmoodifar 2014), additional searches using all RXTE archival data for AMXPs underway (including SAX J1808). For interpretation, need better r-mode (and g-mode) computations for realistic NS models with fast rotation and solid crusts. Light curve calculations for g-modes, to explore “visibility” of modes squeezed closer to equator. Larger colleting areas needed to improve sensitivities, ESA’s LOFT, and NASA’s NICER (Special Session Wed 1:30) can contribute.

15. Goddard Space Flight Center backup

16. Goddard Space Flight Center An r-mode in the Superburst

17. Goddard Space Flight Center An r-mode in the Superburst: Amplitude Conundrum

18. Goddard Space Flight Center Neutron Star Oscillations: perturbing the hot-spot Surface displacements generated by pulsation modes can periodically distort the X-ray emitting hot-spot (Numata & Lee 2010). Distortion is maximized for modes with dominant transverse (quasi-toroidal) displacements. Such modes include surface g- modes, and r-modes. Since hot-spot rotates with the star, the modulation frequency seen by a distant observer is the co-rotating frame frequency. Numata & Lee (2010)

19. Goddard Space Flight Center Accreting Millisecond X-ray Pulsars (AMXPs) Accretion stream close to spin axis produces X-ray hot-spot, spin modulation generates pulsations. (Lamb et al. 2009, Patruno & Watts 2011). Romanova et al. (2009)

20. Goddard Space Flight Center Pulsation Searches in AMXPs: XTE J1751−305 Discovered in April 2002 with RXTE/PCA Galactic Bulge monitoring program (Markwardt et al. 2002), 435 Hz spin period. Pulse timing revealed a 42.4 min orbital period, with a very low mass dwarf (likely He) companion. Strohmayer & Mahmoodifar arXiv:1310.5147

21. Goddard Space Flight Center XTE J1751−305 Search con’t Can do targeted search in the frequency range where r-modes (and g-modes) are theoretically expected. Fast spin increases frequency (k 1 ), crust interactions decrease, k 2, (Yoshida & Lee 2000). P sig = 1.6 x 10 -3

22. Goddard Space Flight Center Pulsations: Light curves and Power Spectra r-mode frequency encoded in the light curve at co-rotating frame frequency, 2Ω/3 (Numata & Lee 2010). Light curve modulation amplitude proportional to mode displacement amplitude Observed modulation, a j, can constrain A.

23. Goddard Space Flight Center XTE J1751−305 Search con’t Implied modulation amplitude of 7.5 x 10 -4, signal ~coherent over ~5 day light curve. No other broader bandwidth signals detected. Amplitude limit at ~2 x 10 -4. Light curve calculations and modeling can connect observed amplitude, a j, to mode amplitude (Numata & Lee 2010).

24. Goddard Space Flight Center Search in XTE J1814−338 Discovered in June 2003, 314.4 Hz spin frequency, 4.275 hr orbital period. Pulsar has strong first harmonic. Carried out similar search as for J1751. No candidate signals found. Set upper limit on amplitude of ~8 x 10 -4.

25. Goddard Space Flight Center Results for NGC 6440 X-2 RXTE Bulge scans caught glimpses of faint, recurrent transient in NGC 6440 (X-2) in August 2009. Pulsations seen at 205.9 Hz. Pulsar observed in only 4 RXTE orbits (not as much data as for J1751 and J1814). No candidate signals detected, with upper limits on the modulation amplitude of ~3 x 10 -3.

26. Goddard Space Flight Center Future Capabilities: LOFT Sensitivity for coherent search scales ~ 1 / (N tot ) 1/2. LOFT/LAD 10-12 m 2 (3 – 15 keV) 20-30 x count rate of RXTE/PCA Likely can reach a amp ~ 1 – 2 x 10 -5 ESA M3 candidate mission, study report (“yellow book”) recently submitted, down-select in early 2014. Potential launch 2022.

27. Goddard Space Flight Center r-mode spin-down limits

28. Goddard Space Flight Center The Neutron Star Equation of State Demorest & Ransom 2011 High mass measurements, limit softening of EOS from hyperons, quarks, other exotic stuff. Most good mass measurements at low end, and systems not conducive to radius estimates, EOS not constrained strongly. Need either masses for higher mass systems (accretors), and/or possibility to get R (AMPs).