1. eXTP and accreting millisecond pulsars
Juri Poutanen (Univ. of Turku, Finland)

2. Plan M-R from AMSP’s light curves Polarization properties
Combined constraints on M-R from light curves, polarization and PRE bursts
Conclusions

3. Accreting millisecond pulsars
Source
Porb (min)
Spin (Hz)
Mc,min/ M
When?
1. SAX J
120
401
0.043
April 1998
2. XTE J 42
435
0.014
April 2002
3. XTE J 44
185
0.083
4. XTE J 40
190
0.0066
Feb 2003
5. XTE J
258
314
0.17
June 2003
6. IGR J
148
599
0.039
Dec 2004
7. HETE J 84
377
0.016
June 2007
8. SWIFT J
54.7
182
0.0067
9. Aql X-1
1140
551 ?
August 2007
10. SAX J
526
442
0.1
11. in glob.clust. NGC6440
57.3
206
August 2009
12. IGR J
208
245
0.13
Sept 2009
13. Swift J
529
518
0.6
April 2010
14. IGR J
231
August 2011
15. IGR J18245–2452
662
254
March 2013

4. Disc eclipses and pulse profile variations in SAX J1808.4-3658
Ibragimov & Poutanen (2009)
Hartman et al. (2008)
Profiles vary during the outburst
Only one spot is visible in the beginning
Two spots are visible at low accretion rate.

5. Spectral energy distribution
kTe=50-90 keV, τT~1
Hard X-ray component is
pulsating and thus is produced probably at the shock.
The blackbody is also pulsating  neutron star surface
Low-energy `blackbody’ is not pulsating  accretion disk

6. Spectral properties spot XTE J1751-305 kTe = 25 keV kTe = 60 keV
tT = 2.2
kTe = 60 keV
tT = 0.9
kTe = 33 keV
tT = 1.7
XTE J
IGR J
kTe = 60 keV
tT = 0.8
kTe = 37 keV
tT = 1.7
kTe = 49 keV
tT = 1.12
Note: almost constant product tT x Te

7. Pulse profiles disc Gierlinski & Poutanen 2005 Kirsch et al. (2003)
XTE J
disc
Gierlinski & Poutanen 2005
XTE J
XTE J
kTdisc=0.43 keV
kTseed = 0.75 keV
Aseed = 26 km2
kTe = 37 keV
tT = 1.7
Solid – persistent emission; histogram - X-ray bursts. Strohmayer et al. (2003).
Kirsch et al. (2003)

8. Doppler effect, aberration and light bending
Light emitted from the neutron star surface can be deflected by degrees (for solar mass and km NS)
In ms pulsars, rotational
velocities are high (b=v/c ≥0.1), the Doppler effect plays an important role

9. Gravitational light bending

10. Doppler effect and aberration
Poutanen & Gierlinski (2003).

11. Effect of the emission pattern
Fbb(fast) = Fbb(slow) x δ5
Doppler boosting Iobs=δ4 Iem
Aberration: projectionproptoδ
slow pulsar (dashes)
fast pulsar, =401 Hz
Fsc,E(fast) ~ δ3+Γ Fsc,E(slow)
Phase
Poutanen & Gierliński (2003)

12. Constraints on the Neutron Star Equation of State
Fit the light curves in two energy bands simultaneously and obtain constraints on M and R.
Poutanen & Gierliński (2003)

13. Comptonization in a shock
burst 
Emission pattern from a shock depends on the optical depth and temperature distribution.
No self-consistent model exist yet.

14. Polarization in Thomson scattering
Cold electrons and soft photons (Thomson scattering approximation)
Unpolarized incoming photons
 is cosine of the scattering angle

15. Polarization from optically thin electron scattering-dominated atmosphere
Thomson scattering approximation
Parallel to normal
Perpendicular
to normal
n=1,2,…30
- scattering order
Sunyaev & Titarchuk 1985; Haardt & Matt 1993; Viironen & Poutanen 2004

16. Polarization from optically thin electron scattering-dominated atmosphere
Strong changes in polarization degree is expected across the eXTP energy range for seed photons of ~1keV:
likely small P below 10 keV.
From Poutanen & Svensson (1996): Exact Klein-Nishina cross-section, scattering matrix from Nagirner & Poutanen (1993)

17. Polarization in Compton scattering
Hot isotropic electrons
Analytical expressions for the scattering matrix averaged over arbitrary isotropic electron distribution have been obtained by Nagirner & Poutanen (1993).
Polarization becomes smaller for hotter electrons.
2. Relativistic electrons  >> 1 with power-law distribution (Bonometto, Cazzola, Saggion 1970; Nagirner & Poutanen 1993, Poutanen 1994). Scattering matrix is diagonal -> no polarization is produced !

18. Effect of electron temperature
XTE J 
Poutanen (1994)

19. Phase dependence of polarization degree
Polarization degree is Lorentz invariant.
Should be computed in the spot comoving frame accounting
for (1) light bending and (2) relativistic aberration

20. Phase dependence of polarization angle
Rotating vector model, RVM (Radhakrishnan & Cooke 1969)

21. Phase dependence of polarization angle
Relativistic RVM (Ferguson 1973, 1976) incorporates effects of special relativity.
Relativistic RVM of Viironen & Poutanen (2004) includes GR (light bending) and special relativity effects which rotate polarization plane by angle c

22. AMSP: pulse profiles and polarization
Comptonized emission from two small spots
Viironen & Poutanen (2004)

23. AMSP: pulse profiles and polarization
Comptonized emission from two small spots
Adapted from Viironen & Poutanen (2004)

24. Light curves and polarization of SAX J1808.4-3658
Black body
Comptonized emission.
Expected polarization.
Doppler factor.
Dashed –
slowly rotating pulsar

25. eXTP constraining neutron star EoS
+ RXTE
+ eXTP
i=70o
i=60o
i=50o
Black dots: constraints on EOS from 2 weeks of RXTE data.
Contour plots: constraints from 3 hours of eXTP data (with 3m2 of “Loft panels”) in the 3-5 keV band for fixed inclination obtained from the polarization data.

26. Constraints from PRE bursts using cooling tail method
Spectra close to thermal. Fitted by a diluted black body. 4U
Suleimanov et al. 2011

27. M-R and EoS constraints from the hard state bursts
Nättilä et al. 2015,
see also
Suleimanov et al. 2011,
Poutanen et al. 2014

28. Constraints on SAX J1808.4-3658 from PRE bursts
Three PRE bursts observed by RXTE showing similar evolution.
The cooling tail method gives:
FEdd= (2.09±0.05) x 10-7 erg cm-2 s-1
A=0.130±0.001 (km/10kpc)-1/2

29. Combined constraints on SAX J1808.4-3658
Combining pulse profiles modeling with inclination obtained from polarization and spectral evolution in PRE bursts gives very accurate determination of M-R.

30. Summary
AMPs pulse profiles contain information about M-R but the number of unknown parameters is too large to get accurate constraints.
AMPs should be polarized. Together with pulse profiles and oscillation amplitude, the phase dependence of PD and PA provide strong geometrical constraints and break degeneracy in inclination and magnetic inclination.
Polarization is expected to vary across eXTP band and also with the accretion rate as the visibility of the secondary pole changes.
Combination of pulse profiles, polarization and PRE bursts gives the most accurate constraints on the EoS.

31. Doppler effect, aberration and light bending
Fbb(fast) = Fbb(slow) x δ5
δ=1/Γ (1–β cos ξ) – Doppler factor
Γ=1/√1–β – Lorentz factor
Doppler boosting Iobs=δ4 Iem
Aberration: projection area is propto δ
Bending + Doppler
+ time delays for
pulsar =600 Hz
slow pulsar
+ bending
slow pulsar,
no bending

32. Effect of electron temperature
XTE J
Nagirner & Poutanen (1994)

33. Spectral energy distribution
Patruno et al. (2009)

34. SAX J1808 in October 2008

35. Polarization from Compton scattering
Cold electrons and high-energy photons
For unpolarized incoming photons, the polarization becomes smaller compared to Rayleigh/Thomson scattering case
In X-rays, this effect is unimportant.

36. Constraints on the geometry from oscillation amplitude
XTE J
Poutanen & Gierliński (2003)

37. Constraints on the Neutron Star Equation of State
Leahy et al. (2007) used pulse profiles from 1998 and 2002 outbursts integrated over all energies.