1. Interstellar Scintillation of the Double Pulsar J0737-3039: Effects of Anisotropy Barney Rickett and Bill Coles (UC San Diego) Collaborators: Maura McLaughlin, Andrew Lyne (Jodrell Bank), Ingrid Stairs (UBC), Scott Ransom (NRAO) International Colloquium "Scattering and Scintillation in Radio Astronomy" Pushchino June 2006

2. Pulsars J0737-3039A&B A B xBxB y B0 Pulsars (neutron stars) A and B orbit around each other in 2.45 hours. The orbit is small (0.003 AU); orbital speeds are fast ~300 km/s The orbit is 9% eccentric and its plane is nearly aligned with the Earth A is eclipsed by B for about 30 sec each orbit Center of mass moves at V CM, so A and B follow spiral paths relative to the ISM => Scintillation observations allow estimates of V CM Scintillation of A shows strong orbital modulation due to changing transverse velocity V ISS. The timescale is t ISS = s ISS /V ISS, where s ISS is the spatial scale of the scintillation pattern Pulses from B are only visible for a narrow range of orbital phases

3. Note fast and slow ISS timescales PSR J0737-3039A ISS Dynamic spectrum from GBT using 1024 x 0.8 MHz channels (SPIGOT) 10 sec time averages: I A (t, ) Eclipses barely visible at PSR J0737-3039B ISS Dynamic spectrum as above: I B (t, ) Note the two narrow time windows in each orbit where the B pulsar is visible

4. We characterize the ISS by a timsescale - t iss by auto-correlating the ISS spectra  I A (t a, ) [deviations from the mean  I A (t a )]  (t a,  ) =  [  I A (t a, )  I A (t a + , )]/{   I A 2 (t a, ) } We average over a range in t a and define  (t a,t iss ) = 0.5 We plot 1/t iss 2 versus t a (or orbit phase  ):

5. byby b x parallel to orbit  ISS model - 1  For diffractive Kolmogorov scattering:  (b x,b y ) = exp[-(Q/s iss 2 ) 5/6 ] where Q is quadratic form of an ellipse Q = a b x 2 + b b y 2 + c b x b y a = cos 2  /A+Asin 2  ; b = Acos 2  +sin 2  /A ; c = sin2  (1/A-A) for anisotropic turbulence with Axial ratio A at orientation angle  s iss is the geometric mean spatial scale of the pattern. The ISS pattern has a spatial correlation function  (b x,b y ) = Pulsar A Intensity pattern I A (x,y, ) sD (1-s)D ISM screen baseline b x,b y

6. Orbital modulation of ISS Timescale for pulsar A Observer samples the pattern due to velocity of the pattern (V iss at the pulsar due to velocities of Pulsar, Earth and ISM). Characteristic timescale is where  =1/e : ie b=V iss t iss Hence 1/t iss 2 = (aV ax 2 + bV ay 2 + cV ax V ay )/s iss 2 where A pulsar’s velocity is V ax = V oax + V cmx ; V ay = V oay + V cmy (par and perp to orbit plane) With V oax, V oay the known orbital velocities and unknown center of mass velocity V cm. From timing we know the orbital velocities V oax and V oay in terms of the orbital phase  relative to the line of nodes and find: 1/t iss 2 = H o + H s sin  + H c cos  + H s2 sin2  + H c2 cos2  In general these 5 coefficients describe the orbital modulation - including eccentricity terms.

7. Orbital modulation of ISS Timescale 3 The equations linking the five H-coefficients to the physical parameters are quadratic and so have two solutions. This was already noted by Ord et al. in their pioneering analysis of the orbital modulation of millisecond binary PSR J1141-65. H-coeffs depend on pulsar parameters which are already known (through timing) V o mean orbital velocity, e orbit eccentricity,  longitude of periastron, i inclination of orbit. Unknown parameters : V cmx,V cmy velocity of center of mass of A&B V Ex,V Ey Earth’s vel - known except for dependence on  angle of pulsar orbit in equatorial coords a, b, c depend on axial ratio (A) and orientation of ISS pattern (  ) s fractional distance from pulsar to scattering region Ord et al assumed circular symmetry (A=1) and so had 2 fewer parameters and were able to constrain the inclination i to one of two solutions and to estimate V cm. *** Allowing for A>1 changes conclusions about V cm ***

8. Orbital modulation of ISS Timescale 4 Since both H c and H s2 are proportional to cosi they are negligible for J0737, leaving 3 coefficients H 0, H s, H c2.

9. Orbital modulation of ISS Timescale 5 The 3 coefficients depend on: V cmx,V cmy velocity of center of mass A axial ratio of ISS pattern  orientation of ellipse s iss scale of ISS diffraction pattern (measured at the pulsar) We eliminate s iss by dividing by H c2 and have two observable coeffs: h s = H s /H c2 = [4V cxe + 2(c/a)V cye ]/V o h 0 = H 0 /H c2 = -[1+ 2V 2 cxe + 2{ab/c 2 }[(c/a)V cye ] 2 + 2(c/a)V cye V cxe ]/V 2 o where V o is the mean orbital velocity and V cxe = V cmx - eV o sin  V cye = V cmy where e is orbit eccentricity,  is longitude of periastron, Note offsets due to motion of Earth (V E ) and ISM (V ism ) V cm = V c + V E s/(1-s) - V ism /(1-s). Where V c is the true system velocity Annual variation in V E provides extra information

10. Dynamic Spectra PSR J0737-3039A ISS Dynamic spectrum from GBT using 1024 x 0.8 MHz channels (SPIGOT) 10 sec time averages: I A (t, ) Eclipses barely visible at PSR J0737-3039B ISS Dynamic spectrum as above: I B (t, ) Note the two narrow time windows in each orbit where the B pulsar is visible

11. A-B correlation ISS of A and B are correlated near the time of the eclipse. Correlation averaged in frequency domain at times t a and t b relative to eclipse.  (t a,t b ) =  [  I a (t a, )  I b (t b, )]/{   I a (t a, ) 2   I b (t b, ) 2 } 0.5 Note normalization by each variance (over frequency) Next slide shows  (t a,t b ) for Dec 2003 (data at 1.4 GHz Ransom et al, 2004)

12. Rho ab all 52984 t b (10 sec units) t a (10 sec units)

14. J0737-3039A&B Correlated ISS A B x y = Origin at position of A at eclipse At times t a and t b after the eclipse the transverse projected baseline vector from B to A is b x = V ax t a - V bx t b b y = y bo + V ay t a -V by t b  where V ax,V ay, V bx,V by are net velocities of A & B at eclipse.  Maximum correlation is at times t apk, t bpk which we can measure and give independent info:  y b0 /V cmy = t apk - t bpk where y b0 is the impact parameter at A's eclipse  V cmx = hence we have one of the unknowns, but we introduced another y b0. b

15. Model for A-B correlation  (t a,t b ) =  (b x,b y ) = exp[-(Q/s iss 2 ) 5/6 ]  Where the baseline is b x = V ax t a -V bx t b ; and b y = V ay t a -V by t b  Using the same model as before Q can be written as a quadratic form in   t a = t a -t apk and  t b = t b - t bpk :  (t a,t b ) = exp[-{(c 1  t a 2 +c 2  t b 2 + c 3  t a  t b )}/s iss 2 ) 5/6 ] This definition of  (t a,t b ) is properly normalized by the two variances, but it does not include the effect of a varying signal-to-noise ratio due A’s eclipse and B’s profile. So we explicitly corrected for this in our fit. Since Q describes the spatial structure of the ISS pattern its three coefficients depend on our unknown parameters in the same way as for the orbital harmonics h 0 and h s. But t apk, t bpk give independent information from which we can estimate V cmx, [(c/a)V cye ] and {ab/c 2 }.

16.  ab fit Mjd 52984 (Dec 2003) Observation Model Residual

17. Evolution in the “on-windows” for 0737B Burgay et al 2005  =270 deg is near where the orbits cross and we can see correlations in ISS from A and B 6/03 9/03 1/04 4/04 7/04 11/04 B profile Orbital Longitude 

18. Model for A-B correlation 2  Using the same model as before Q can be written as a quadratic form in   t a = t a - t apk and  t b = t b - t bpk :  (t a,t b ) = exp[-{(c 1  t a 2 +c 2  t b 2 + c 3  t a  t b )}/s iss 2 ) 5/6 ]  Since Q describes the spatial structure of the ISS pattern its three coefficients depend on our unknown parameters in the same way as for the orbital harmonics h 0 and h s. But t apk, t bpk give independent information from which we can estimate V cmx, [(c/a)V cye ] and {ab/c 2 }. The velocities include the changing Earth’s velocity and so vary with epoch. But {ab/c 2 } should be a constant. The AB correlation is only possible while B is visible during A’s eclipse. Unfortunately, this occurred during only 3 out of 11 observations. So we take the measured {ab/c 2 } and apply it to the remaining 8 epochs in which the t iss data were fitted by three harmonic coefficients. This gave 11 epochs with an estimate of V cmx as shown next

19. Annual change in V cmx V cmx derived from AB correlation estimate ab/c 2 = 0.384 There are two solutions at each epoch the slower velocities are chosen, since the faster ones are inconsistent with VLBI limits on the system proper motion.

20. Annual change in V cmx (2) The annual fit V cmx gives estimates of V cx, V cy The transverse center of mass velocity relative to the scattering region in the ISM, which is at fractional distance s from the pulsar. It also give the absolute orientation of the pulsar orbital plane projected onto the sky. With V cy known we will be able to go back to the t apk,t bpk measurements and refine our estimate of the orbital inclination. Our earlier analysis yielded y b0 = 4000±2000 km/s which is about 3  smaller than the value obtained by Kramer et al. from the observed Shapiro delay in the timing of pular A.

21. The Poincaré circle We find 4ab/c 2 = 1.54 which gives an ellipse in the Poincare circle: [1-R 2 cos 2 (2  )]/R 2 sin 2 (2  ) =1.54 where A = [(1+R)/(1-R)] 0.5 Constraint on axial ratio: R min = c/(2ab) 0.5 = 0.81 Hence A min = 3.1 (2.6 - 4.4) 22 R=(A 2 -1)/(A 2 +1)

22. Refractive shifts

23. Date mjd x1e-4 Estimated spatial ISS scale over one year. It should be constant. The changes may be due to refractive modulations? Evidently there is more to learn! s iss

24. Conclusions The ISS timescale for PSR J0737-3039A has been measured near 2 GHz versus orbital phase at 8 epochs over a year. With the Earth lying in the orbit plane there are 3 harmonic coefficients which have been estimated at each epoch. We present theoretical analysis of the harmonic coefficients in the presence of anisotropic ISS and how they vary with the Earth’s velocity. Anisotropy has a strong influence on the derived center of mass velocity. Fits to these annual changes in the coefficients are not fully consistent with the model and so do not yet improve the estimate of the center of mass velocity of the pulsars nor of the anisotropy in the interstellar scattering. Correlation in the ISS between the A&B pulsars provides strong independent evidence for the axial ratio and orientation. It also provides an independent estimate for the orbital inclination which is very close to 90 deg. However, the drift in the on-times for B have reduced the A-B correlation after Dec 2003. We are working to dig the correlation out when B is weak.

25. ISS Timescale (full equations skip at AAS!) In general the 5 harmonic coefficients are related to the physical parameters by: V cxe = V cmx - eV o sin  + V Ex s/(1-s) -V ismx /(1-s) V cye = V cmy - eV o sin  cos i + V Ey s/(1-s) -V ismy /(1-s) Known parameters (from pulsar timing): V o is the mean orbital velocity, e is orbit eccentricity,  is longitude of periastron, i is the inclination of orbit. V cmx,V cmy velocity of center of mass V Ex,V ey Earth’s vel - known except for dependence on  angle of pulsar orbit in equatorial coords a, b, c depend on axial ratio (A) and orientation of ISS pattern (  ) V ismx, V ism is velocity of ISM at distance s H 0 = [a(0.5V 2 o + V 2 cxe )+ b(0.5cos 2 i V 2 o +V 2 cye ) + cV cxe V cye ]/s 2 iss H s = -V o (2aV cxe + cV cye )/s 2 iss H c = V o cosi (2bV cye + cV cxe )/s 2 iss H s2 = 0.5cV 2 o cosi /s 2 iss H c2 = V 2 o (b cos 2 i - a)/s 2 iss s iss scale of ISS diffraction pattern (not so interesting)

26.  (t a,t b ) inclination i (deg)sin(i) y bo (km) t A (sec) 901.000 89.70.9999864,71256 89.190.999912,700150 88.190.999528,400340 87.570.999138,159460 y bo = 2a cos(i) (for a circular orbit of radius a) : Shapiro delay gives sin(i) = 0.9995±.0004 (Kramer et al. Texas Symp.) t apk apparent was 33 sec !

27.  (t a,t b ) model - 2 We can fit for the 3 coefficients c 1, c 2 and c 3 and 2 times of peak correlation, which depend on the known orbital velocities and the 6 unknown model parameters: V cx,V cy velocity of center of mass (inc terms in V E and V ISM /(1-s) ) A axial ratio  orientation of ellipse (relative to line of nodes) s iss diffractive scintillation scale at J0737 i inclination of orbit In particular the inclination is determined by y bo (projected separation at eclipse) through: t apk = (y bo /V cy )(V cx +V ob )/(V oa +V ob ) ~ 0.6(y bo /V cy ) t bpk = (y bo /V cy )(V cx -V oa )/(V oa +V ob ) ~ -0.4(y bo /V cy ) Since V oa and V ob are larger than V cx, the values of t apk t bpk are largely determined by (y bo /V cy ) and so can change as V Ey changes: V cx = V cmx + V Ex s/(1-s) - V ISMx /(1-s) V cy = V cmy + V Ey s/(1-s) -V ISMy /(1-s)

28. Orbital modulation in the ISS Arcs from J0737-3039A 2GHz July 17 2004 GBT04B11 16 x 10 min panels Eclipse in #8

29.  ab fit Mjd 53560 (July 2005) t b (sec) t a (sec) Observation model residual

30. byby b x parallel to orbit  ISS model - AB The ISS pattern has a spatial correlation function  (b x,b y ) = Pulsar A Intensity pattern I A (x,y, ) sD (1-s)D ISM screen baseline b x,b y Pulsar B Intensity pattern I B (x,y, )

31. Annual plot We observed J0737 every 2 months in 2004-5 with GBT at 1.7-2.2 GHz. t iss vs orbit were estimated for each epoch and the two harmonic coefficients are shown together with a model fit. The fit is reasonable - not excellent. But the 2nd harmonic coefficient H c2 varies with epoch, which is not consistent with the model. So we are not satisfied with the result. We are attempting to resolve this via the correlation of the ISS between A & B pulsars.

32. mjd53203 GBT 2 GHz A-B cross correlation MJD 53203 eclipse of A 10 sec time units t apk has changed sign! not yet corrected for B profile B profile

33. We fit V cmx, V cmy and s iss. If we change A and  we get equally good fits but with different V cmx, V cmy and s iss. Trade-Off for Center of mass velocity vs Anisotropy angle  with fixed A = 4 t iss data: J0737-3039A 820 MHz (Ransom 2005)