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

Pulsars J0737-3039A&B

A

BxB

yB0

• 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 VCM, so A and B follow spiral paths relative to the

ISM => Scintillation observations allow estimates of VCM

• Scintillation of A shows strong orbital modulation due to changing transverse velocity VISS. The timescale is tISS = sISS/VISS , where sISS is the spatial scale of the scintillation pattern

• Pulses from B are only visible for a narrow range of orbital phases

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:

IA(t,)

Eclipses barely visible at

PSR J0737-3039B

ISS Dynamic spectrum as above:

IB(t,)

Note the two narrow time windows in each orbit where the B pulsar is visible

We characterize the ISS by a timsescale - tiss by auto-correlating the ISS spectra IA(ta,) [deviations from the meanIA(ta)]

(ta,) = [IA(ta,) IA(ta+,)]/{

IA

2(ta,) }

We average over a range in ta and define (ta,tiss) = 0.5We plot 1/tiss

2 versus ta (or orbit phase ):

by

bx parallel to orbit

ISS model - 1

For diffractive Kolmogorov scattering: (bx,by) = exp[-(Q/siss

2)5/6]where Q is quadratic form of an ellipse

Q = a bx2 + b by

2 + c bxby

a = cos2/A+Asin2; b = Acos2+sin2/A ;

c = sin2(1/A-A)

for anisotropic turbulence with

Axial ratio A at orientation angle

siss is the geometric mean spatial scale of the pattern.

The ISS pattern has a spatial correlation function(bx,by) = < IA(x+bx,y+by) IA(x,y) >

Pulsar A

Intensity pattern IA(x,y,)

sD

(1-s)D

ISM screen

baseline bx,by

Orbital modulation of ISS Timescale for pulsar A

Observer samples the pattern due to velocity of the pattern (Viss at the pulsar due to velocities of Pulsar, Earth and ISM).

Characteristic timescale is where =1/e : ie b=Visstiss Hence

1/tiss2 = (aVax

2 + bVay2 + cVaxVay)/siss

2

where A pulsar’s velocity is

Vax = Voax + Vcmx ; Vay = Voay + Vcmy (par and perp to orbit plane)

With Voax, Voay the known orbital velocities and unknown center of mass velocity Vcm.

From timing we know the orbital velocities Voax and Voay in terms of the orbital phase relative to the line of nodes and find:

1/tiss2 = Ho + Hs sin + Hc cos + Hs2 sin2 + Hc2 cos2

In general these 5 coefficients describe the orbital modulation - including eccentricity terms.

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)

Vo mean orbital velocity, e orbit eccentricity, longitude of periastron, i inclination of orbit.

Unknown parameters :

Vcmx,Vcmy velocity of center of mass of A&B

VEx,VEy 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 Vcm. *** Allowing for A>1 changes conclusions about Vcm ***

Orbital modulation of ISS Timescale 4

Since both Hc and Hs2 are proportional to cosi they are negligible for J0737, leaving 3 coefficients H0, Hs, Hc2 .

Orbital modulation of ISS Timescale 5

The 3 coefficients depend on:Vcmx,Vcmy velocity of center of mass A axial ratio of ISS patternorientation of ellipsesiss scale of ISS diffraction pattern (measured at the pulsar)We eliminate siss by dividing by Hc2 and have two observable coeffs:

hs = Hs/Hc2 = [4Vcxe+ 2(c/a)Vcye]/Vo

h0 = H0/Hc2 = -[1+ 2V2cxe+ 2{ab/c2}[(c/a)Vcye]2

+ 2(c/a)VcyeVcxe]/V2

o

where Vo is the mean orbital velocity and

Vcxe = Vcmx- eVo sin

Vcye = Vcmy

where e is orbit eccentricity, is longitude of periastron, Note offsets due to motion of Earth (VE) and ISM (Vism)

Vcm = Vc + VEs/(1-s) - Vism/(1-s). Where Vc is the true system velocity Annual variation in VE provides extra information

Dynamic SpectraPSR J0737-3039A

ISS Dynamic spectrum from GBT using 1024 x 0.8 MHz channels (SPIGOT)10 sec time averages:

IA(t,)

Eclipses barely visible at

PSR J0737-3039B

ISS Dynamic spectrum as above:

IB(t,)

Note the two narrow time windows in each orbit where the B pulsar is visible

A-B correlation

ISS of A and B are correlated near the time of the eclipse. Correlation averaged in frequency domain at times ta and tb relative to eclipse.

(ta,tb) = [Ia(ta,) Ib(tb,)]/{

Ia(ta,)2

Ib(tb,)2}0.5

Note normalization by each variance (over frequency)

Next slide shows (ta,tb) for Dec 2003 (data at 1.4 GHz Ransom et al, 2004)

Rhoab all 52984

tb (10 sec units)

ta (10 sec units)

J0737-3039A&B Correlated ISS

A

B

x

y

= Origin at position of A at eclipse

At times ta and tb after the eclipse the transverse projected baseline vector from B to A is

bx = Vaxta - Vbxtb

by = ybo + Vayta -Vbytb

where Vax,Vay , Vbx,Vby are net velocities of A & B at eclipse.

Maximum correlation is at times tapk, tbpk which we can measure and give independent info:

yb0/Vcmy = tapk- tbpk where yb0 is the impact parameter at A's eclipse

Vcmx = hence we have one of the unknowns, but we introduced another yb0 .

b

Model for A-B correlation

(ta,tb) = (bx,by) = exp[-(Q/siss2)5/6]

Where the baseline is bx = Vaxta-Vbxtb; and by = Vayta-Vbytb

Using the same model as before Q can be written as a quadratic form in ta = ta-tapk and tb = tb- tbpk:

(ta,tb) = exp[-{(c1 ta2 +c2 tb

2 + c3 tatb)}/siss2)5/6]

This definition of (ta,tb) 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 h0 and hs . But tapk, tbpk give independent information from which we can estimate Vcmx , [(c/a)Vcye] and {ab/c2}.

ab fit Mjd 52984 (Dec 2003)

Observation Model Residual

Evolution in the “on-windows” for 0737BBurgay 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 p

rofi

le

Orbital Longitude

Model for A-B correlation 2

Using the same model as before Q can be written as a quadratic form in ta = ta - tapk and tb = tb - tbpk:

(ta,tb) = exp[-{(c1 ta2 +c2 tb

2 + c3 tatb)}/siss2)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 h0 and hs . But tapk, tbpk give independent information from which we can estimate Vcmx , [(c/a)Vcye] and {ab/c2}.The velocities include the changing Earth’s velocity and so vary with epoch.But {ab/c2} 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/c2} and apply it to the remaining 8 epochs in which the tiss data were fitted by three harmonic coefficients. This gave 11 epochs with an estimate of Vcmx as shown next

Annual change in Vcmx

Vcmx derived from AB correlation estimate ab/c2 = 0.384There 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.

Annual change in Vcmx (2)

The annual fit Vcmx gives estimates of Vcx, Vcy 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 Vcy known we will be able to go back to the tapk,tbpk measurements and refine our estimate of the orbital inclination.

Our earlier analysis yielded yb0 = 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.

The Poincaré circle

We find 4ab/c2 = 1.54which gives an ellipse in the Poincare circle:[1-R2 cos2(2)]/R2 sin2(2) =1.54where A = [(1+R)/(1-R)]0.5

Constraint on axial ratio:Rmin = c/(2ab)0.5 = 0.81Hence Amin = 3.1 (2.6 - 4.4)

2

R=(A2-1)/(A2+1)

Refractive shifts

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

siss

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.

ISS Timescale (full equations skip at AAS!)

In general the 5 harmonic coefficients are related to the physical parameters by:

Vcxe = Vcmx- eVo sin+ VExs/(1-s) -Vismx/(1-s)

Vcye = Vcmy- eVo sincos i + VEys/(1-s) -Vismy/(1-s)

Known parameters (from pulsar timing):Vo is the mean orbital velocity, e is orbit eccentricity, is longitude of periastron, i is the inclination of orbit.

Vcmx,Vcmy velocity of center of mass

VEx,Vey Earth’s vel - known except for dependence on

angle of pulsar orbit in equatorial coordsa, b, c depend on axial ratio (A) and orientation of ISS pattern ()

Vismx, Vism is velocity of ISM at distance s

H0= [a(0.5V2o+ V2

cxe)+ b(0.5cos2i V2o+V2

cye) + cVcxeVcye]/s2iss

Hs = -Vo (2aVcxe+ cVcye)/s2iss

Hc = Vocosi (2bVcye+ cVcxe)/s2iss

Hs2 = 0.5cV2o cosi /s2

iss

Hc2 = V2o(b cos2i - a)/s2

iss

siss scale of ISS diffraction pattern (not so interesting)

(ta,tb) inclination

i (deg) sin(i) ybo (km) tA (sec)

90 1.0 0 0

89.7 0.999986 4,712 56

89.19 0.9999 12,700 150

88.19 0.9995 28,400 340

87.57 0.9991 38,159 460

ybo = 2a cos(i) (for a circular orbit of radius a) :

Shapiro delay gives sin(i) = 0.9995±.0004 (Kramer et al. Texas Symp.)

tapk apparent was 33 sec !

(ta,tb) model - 2

We can fit for the 3 coefficients c1, c2 and c3 and 2 times of peak correlation, whichdepend on the known orbital velocities and the 6 unknown model parameters:

Vcx,Vcy velocity of center of mass (inc terms in VE and VISM/(1-s) )A axial ratioorientation of ellipse (relative to line of nodes)siss diffractive scintillation scale at J0737i inclination of orbit

In particular the inclination is determined by ybo (projected separation at eclipse) through:

tapk = (ybo/Vcy)(Vcx+Vob)/(Voa+Vob) ~ 0.6(ybo/Vcy)tbpk = (ybo/Vcy)(Vcx-Voa)/(Voa+Vob) ~ -0.4(ybo/Vcy)

Since Voa and Vob are larger than Vcx, the values of tapk tbpk are largely determined by (ybo/Vcy) and so can change as VEy changes:

Vcx = Vcmx+ VExs/(1-s) - VISMx/(1-s)

Vcy = Vcmy + VEys/(1-s) -VISMy/(1-s)

Orbital modulation in the ISS Arcs fromJ0737-3039A

2GHz July 17 2004GBT04B11

16 x 10 min panels

Eclipse in #8

ab fit Mjd 53560 (July 2005)

tb (sec) tb (sec) tb (sec)

ta (sec)

Observation model residual

by

bx parallel to orbit

ISS model - AB

The ISS pattern has a spatial correlation function(bx,by) = < IB(x+bx,y+by) IA(x,y) >

Pulsar A

Intensity pattern IA(x,y,)

sD

(1-s)D

ISM screen

baseline bx,by

Pulsar B

Intensity pattern IB(x,y,)

Annual plot

We observed J0737 every 2 months in 2004-5 with GBT at 1.7-2.2 GHz.

tiss 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 Hc2 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.

mjd53203GBT 2 GHzA-B cross correlationMJD 53203

eclipse of A

10 sec time units

tapk has changed sign!

not yet corrected for B profileB profile

We fit Vcmx, Vcmy and siss. If we change A and we get equally good

fits but with different Vcmx, Vcmy and siss.

Trade-Off for Center of mass velocity vs Anisotropy angle with fixed A = 4

tiss data: J0737-3039A 820 MHz (Ransom 2005)