Transcript Scintillation of the Double Pulsar

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
B
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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
PSR J0737-3039A
ISS Dynamic spectrum from GBT using
1024 x 0.8 MHz channels (SPIGOT)
10 sec time averages:
IA(t,n)
Eclipses barely visible at
Note fast and slow ISS timescales
PSR J0737-3039B
ISS Dynamic spectrum as above:
IB(t,n)
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 dIA(ta,n) [deviations from the mean IA(ta)]
n
n
r(ta,t) = S [dIA(ta,n) dIA(ta+t,n)]/{S dIA2(ta,n) }
We average over a range in ta and define r(ta,tiss) = 0.5
We plot 1/tiss2 versus ta (or orbit phase f):
ISS model - 1
Pulsar A
baseline bx,by
The ISS pattern has a spatial correlation function
r(bx,by) = < dIA(x+bx,y+by) dIA(x,y) >
sD
by
q
bx parallel to orbit
ISM screen
(1-s)D
For diffractive Kolmogorov scattering:
r(bx,by) = exp[-(Q/siss2)5/6]
where Q is quadratic form of an ellipse
Q = a bx2 + b by2 + c bxby
a = cos2q/A+Asin2q; b = Acos2q+sin2q/A ;
c = sin2q(1/A-A)
for anisotropic turbulence with
Axial ratio A at orientation angle q
Intensity pattern IA(x,y,n)
siss is the geometric mean spatial scale of the pattern.
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 r=1/e : ie b=Visstiss Hence
1/tiss2 = (aVax2 + bVay2 + cVaxVay)/siss2
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 f relative to the line of nodes and find:
1/tiss2 = Ho + Hs sinf + Hc cosf + Hs2 sin2f + Hc2 cos2f
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,
w 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
b angle of pulsar orbit in equatorial coords
a, b, c depend on axial ratio (A) and orientation of ISS pattern (q)
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 pattern
q orientation of ellipse
siss 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]/V2o
where Vo is the mean orbital velocity and
Vcxe = Vcmx- eVo sinw
Vcye = Vcmy
where e is orbit eccentricity, w 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 Spectra
PSR J0737-3039A
ISS Dynamic spectrum from GBT using
1024 x 0.8 MHz channels (SPIGOT)
10 sec time averages:
IA(t,n)
Eclipses barely visible at
PSR J0737-3039B
ISS Dynamic spectrum as above:
IB(t,n)
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.
n
n
dIa(ta,n)2 S
n
r(ta,tb) = S [dIa(ta,n) dIb(tb,n)]/{S
dIb(tb,n)2}0.5
Note normalization by each variance (over frequency)
Next slide shows r(ta,tb) for Dec 2003 (data at 1.4 GHz Ransom et al, 2004)
Rhoab all 52984
ta (10 sec units)
tb (10 sec units)
J0737-3039A&B Correlated ISS
= Origin at position of A at eclipse
B
b
A
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 =
but we introduced another yb0 .
hence we have one of the unknowns,
Model for A-B correlation
r(ta,tb) = r(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
dta = ta-tapk and dtb = tb- tbpk:
r(ta,tb) = exp[-{(c1 dta2 +c2 dtb2 + c3 dtadtb)}/siss2)5/6]
This definition of r(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}.
rab fit Mjd 52984 (Dec 2003)
Observation
Model
Residual
B profile
Evolution in the “on-windows” for 0737B
Burgay et al 2005
6/03
9/03
1/04
4/04
7/04
11/04
f=270 deg is near where the orbits cross
and we can see correlations in ISS from
A and B
Orbital Longitude f
Model for A-B correlation 2
Using the same model as before Q can be written as a quadratic form in
dta = ta - tapk and dtb = tb - tbpk:
r(ta,tb) = exp[-{(c1 dta2 +c2 dtb2 + c3 dtadtb)}/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 = 0.384
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.
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 3s 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 = 1.54
which gives an ellipse in the
Poincare circle:
[1-R2 cos2(2q)]/R2 sin2(2q) =1.54
where A = [(1+R)/(1-R)]0.5
Constraint on axial ratio:
Rmin = c/(2ab)0.5 = 0.81
Hence Amin = 3.1 (2.6 - 4.4)
R=(A2-1)/(A2+1)
2q
Refractive shifts
siss
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!
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:
2
2
2
2
H0= [a(0.5V o+ V cxe)+ b(0.5cos2i V o+V cye) + cVcxeVcye]/s2iss
Hs = -Vo (2aVcxe+ cVcye)/s2iss
Hc = Vocosi (2bVcye+ cVcxe)/s2iss
2
Hs2 = 0.5cV o cosi /s2iss
2
Hc2 = V o(b cos2i - a)/s2iss
siss scale of ISS diffraction pattern (not so interesting)
Vcxe = Vcmx- eVo sinw + VExs/(1-s) -Vismx/(1-s)
Vcye = Vcmy- eVo sinw cos i + VEys/(1-s) -Vismy/(1-s)
Known parameters (from pulsar timing):
Vo is the mean orbital velocity,
e is orbit eccentricity,
w 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
b angle of pulsar orbit in equatorial coords
a, b, c depend on axial ratio (A) and orientation of ISS pattern (q)
Vismx, Vism is velocity of ISM at distance s
r(ta,tb) inclination
Shapiro delay gives sin(i) = 0.9995±.0004 (Kramer et al. Texas Symp.)
ybo = 2a cos(i) (for a circular orbit of radius a) :
i (deg)
sin(i)
ybo (km) tA (sec)
90
1.0
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
0
tapk apparent was 33 sec !
r(ta,tb) model - 2
We can fit for the 3 coefficients c1, c2 and c3 and 2 times of peak correlation, which
depend 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 ratio
q orientation of ellipse (relative to line of nodes)
siss diffractive scintillation scale at J0737
i 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 from
J0737-3039A
2GHz July 17 2004
GBT04B11
16 x 10 min panels
Eclipse in #8
rab fit Mjd 53560 (July 2005)
ta (sec)
tb (sec)
Observation
tb (sec)
model
tb (sec)
residual
ISS model - AB
Pulsar A
baseline bx,by
Pulsar B
The ISS pattern has a spatial correlation function
r(bx,by) = < dIB(x+bx,y+by) dIA(x,y) >
sD
by
q
bx parallel to orbit
ISM screen
(1-s)D
Intensity pattern IB(x,y,n)
Intensity pattern IA(x,y,n)
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.
mjd53203
GBT 2 GHz
A-B cross
correlation
MJD 53203
eclipse of A
tapk has changed sign!
not yet corrected for B profile
B profile
10 sec time units
We fit Vcmx, Vcmy and siss. If we change A and q we get equally good
fits but with different Vcmx, Vcmy and siss.
Trade-Off for
Center of mass
velocity vs
Anisotropy angle q
with fixed A = 4
tiss data: J0737-3039A 820 MHz (Ransom 2005)