PSR B0329+54: Statistics of Substructure Discovered within the Scattering Disk on RadioAstron Baselines of up to 235,000 km

2.2 MB·English

by  C. R. Gwinn

Astrophysical journal
PreprinttypesetusingLATEXstyleemulateapjv.05/12/14
PSR B0329+54: STATISTICS OF SUBSTRUCTURE DISCOVERED WITHIN THE SCATTERING DISK ON
RADIOASTRON BASELINES OF UP TO 235,000 KM
C. R. Gwinn1, M. V. Popov2, N. Bartel3, A. S. Andrianov2, M. D. Johnson4, B. C. Joshi5, N. S. Kardashev2,
R. Karuppusamy6, Y. Y. Kovalev1,6, M. Kramer6, A. G. Rudnitskii2, E. R. Safutdinov2, V. I. Shishov7,
T. V. Smirnova7, V. A. Soglasnov2, S. F. Steinmassl8, J. A. Zensus6, V. I. Zhuravlev2
(Received January 18, 2015; Accepted March 15, 2016)
ABSTRACT
6
We discovered fine-scale structure within the scattering disk of PSR B0329+54 in observations with
1
the RadioAstron ground-space radio interferometer. Here, we describe this phenomenon, characterize
0
2 itwithaveragesandcorrelationfunctions,andinterpretitastheresultofdecorrelationoftheimpulse-
response function of interstellar scattering between the widely-separated antennas. This instrument
r
includedthe10-mSpaceRadioTelescope,the110-mGreenBankTelescope,the14×25-mWesterbork
a
M SynthesisRadioTelescope,andthe64-mKalyazinRadioTelescope. Theobservationswereperformed
at 324 MHz, on baselines of up to 235,000 km in November 2012 and January 2014. In the delay
domain,onlongbaselinestheinterferometricvisibilityconsistsofmanydiscretespikeswithinalimited
5
1 rangeofdelays. Onshortbaselinesitconsistsofasharpspikesurroundedbylowerspikes. Theaverage
envelopeofcorrelationsofthevisibilityfunctionshowtwoexponentialscales,withcharacteristicdelays

] of τ1 = 4.1±0.3 µs and τ2 = 23±3 µs, indicating the presence of two scales of scattering in the
A interstellar medium. These two scales are present in the pulse-broadening function. The longer scale
G contains0.38timesthescatteredpoweroftheshorterone. Wesuggestthatthelongertailarisesfrom
highly-scattered paths, possibly from anisotropic scattering or from substructure at large angles.
.
h Keywords: scattering — pulsars: individual B0329+54 — radio continuum: ISM — techniques: high
p angular resolution
-
o
r
t 1. INTRODUCTION have already been studied extensively theoretically (see,
s e.g., Prokhorov et al. 1975; Rickett 1977; Goodman &
a All radio signals from cosmic sources are distorted by
Narayan1989;Narayan&Goodman1989;Shishovetal.
[ the plasma turbulence in the interstellar medium (ISM).
2003) and observationally with ground VLBI of SgrA∗
Understanding of this turbulence is therefore essential
3 (Gwinn et al. 2014) and pulsars (see, e.g., Bartel et al.
for the proper interpretation of astronomical radio ob-
v 1985; Desai et al. 1992; Kondratiev et al. 2007), as well
servations. The properties and characteristics of this
9 as with ground-space VLBI of PSR B0329+54 (Halca,
turbulence can best be studied by observing point-like
4 Yangalov et al. 2001) and the quasar 3C273 (RadioAs-
radio sources, where the results are not influenced by
4 tron, Johnson et al. 2016). Whereas the VSOP pulsar
the extended structure of the source, but instead are di-
4 observations were done at a relatively high frequency of
rectlyattributabletotheeffectoftheISMitself. Pulsars
0
1.7 GHz and with baselines of ≈25,000 km and less,
. are such sources. Dispersion and scattering affect ra-
1 dio emission from pulsars. Whereas dispersion in the ground-space VLBI with RadioAstron allows observa-
0 tions at one-fifth the frequency, where propagation ef-
plasma column introduces delays in arrival time that
5 fects are expected to be much stronger, and with base-
depend upon frequency and results in smearing of the
1 lines ∼10 times longer (Kardashev et al. 2013). Such
pulse, scattering by density inhomogeneities causes an-
: observations can resolve the scatter-broadened image of
v gular broadening, pulse broadening, intensity modula-
apulsarandrevealnewinformationaboutthescattering
i tion or scintillation, and distortion of radio spectra in
X medium (Smirnova et al. 2014).
the form of diffraction patterns. The scattering effects
Inthispaper, westudythescatteredimageofthepul-
r
a 1University of California at Santa Barbara, Santa Barbara, sar B0329+54 with RadioAstron. We demonstrate that
CA93106-4030,USA the pulsar is detected on baselines that fully resolve the
2Astro Space Center of Lebedev Physical Institute, Prof- scattering disk. The interferometric visibility on these
soyuznaya84/32,Moscow117997,Russia
long baselines takes the form of random phase and am-
3York University, 4700 Keele St., Toronto, ON M3J 1P3,
Canada plitudevariationsthatvaryrandomlywithobservingfre-
4Harvard-Smithsonian Center for Astrophysics, 60 Garden quency and time. In the Fourier-conjugate domain of
St,Cambridge,MA02138,USA delay and fringe rate, the visibility forms a localized,
5National Centre for Radio Astrophysics, Post Bag 3,
extended region around the origin, composed of many
Ganeshkhind,Pune411007,India
6Max-Planck-Institut fu¨r Radioastronomie, Auf dem Hu¨gel random spikes. We characterize the shape of this region
69,Bonn53121,Germany using averages and correlation functions. We argue the-
7PushchinoRadioAstronomyObservatory,AstroSpaceCen- oretically that its extent in delay is given by the average
ter of Lebedev Physical Institute, Pushchino 142290, Moscow
envelope of the impulse-response function of interstellar
region,Russia
8Physik-Department, Technische Universit¨at Mu¨nchen, scattering, sometimes called the pulse-broadening func-
James Franck-Strasse 1, Garching bei Mu¨nchen 85748, Ger- tion. Wefindthattheobserveddistributioniswell-fitby
many
2 Gwinn et al.
Time t Fringe Rate f as a function g(te) of time te at the observer. Here, te
is Fourier-conjugate to ν and varies at the Nyquist rate.
Frequency ν V!(ν,t) F−1 V!(ν, f ) The visibility is the product of Fourier transforms of g
f→t at the two antennas:
V˜ =g˜ g˜∗ (2)
F−1 F AB A B
ν→τ τ→ν
where g˜ is the Fourier transform of g(t ).
e
We denote the typical duration of g(t ) as τ , the
e sc
Delay τ V(τ,t) F V(τ, f ) broadeningtimeforasharppulse. Withinthistimespan,
t→f g(t ) has a complicated amplitude and phase. The func-
e
tion g(t ) changes over longer times, as the line of sight
e
2 shifts with motions of source, observer, and medium.
C(τ,t)= V(τ,t)
This change takes place on a timescale t , and over a
sc
spatial scale S . The shorter and longer timescales τ
K (Δτ,t)= C (τ,t)C (τ+Δτ,t) sc sc
RL R L τ and tsc lead to our use of dual time variables: te, of up
toafewtimesτ andFourier-conjugatetoν; andt, ofa
sc
= V (τ,t)2 V (τ+Δτ,t)2 fraction of tsc or more and Fourier-conjugate to f. This
R L τ duality is commonly expressed via the “dynamic spec-
trum” (see Section A.2). If the scattering material re-
Figure 1. RelationsamongtheinterferometricvisibilityV invar- mainsnearlyatrestwhilethelineofsighttravelsthrough
iousdomains,andfunctionsderivedfromit. Thefundamentalob-
it at velocity V , then one spatial dimension in the ob-
servable is the visibility in the domain of frequency ν and time t, ⊥
V˜(ν,t); this is known as the cross-power spectrum, or cross spec- server plane maps into time, and
trum. An inverse Fourier transform of ν to delay τ leads to the
t =S /V (3)
visibility V(τ,t); this is the cross-correlation function of electric sc sc ⊥
fields in the time domain (see Equation 21). A forward Fourier
transform of t to fringe rate f leads to V(τ,f). A forward trans- The averaged square modulus of g is the pulse-
formofτ backtoνproducesV˜(ν,f),andaninverseFouriertrans- broadening function G = (cid:104)g(te)g(te)∗(cid:105)S. Here, the sub-
form of f to t returns to V˜(ν,t) The square modulus of V(τ,t) is scripted angular brackets (cid:104)...(cid:105)S indicate an average over
C(τ,t). Thecross-correlationfunctioninτ ofCRforright-andCL realizations of the scattering. This function is the aver-
forleft-circularpolarizationisKRL(∆τ,t). WedenotetheFourier age observed intensity for a single sharp pulse emitted
transformbyF,andquantitiesinthedomainoffrequencyνbythe
at the source. An average over time is usually assumed
accent˜.
to approximate the desired average over an ensemble of
statistically-identical realizations of scattering.
a model that is derived from an impulse-response func-
We derive a number of representations of the visibil-
tionthathastwodifferentexponentialscales. Wediscuss
ity and quantities derived from it, and show that these
possible origins of the two scales.
provide straightforward means to extract the impulse-
2. THEORETICAL BACKGROUND response function. These functions are summarized in
Figure1,anddiscussedbrieflyhere,andindetailinSec-
Ourfundamentalobservableistheinterferometricvisi-
tion A of the Appendix. In particular, visibility in the
bilityV. Inthedomainoffrequencyν,thisistheproduct
domain of delay τ and time t is V(τ,t). This is the cor-
of electric fields at two antennas A and B:
relation function of electric field at the two antennas A
V˜ (ν,t)=E˜ (ν,t)E˜∗(ν,t). (1) and B (Equation 21), and is the inverse Fourier trans-
AB A B form of V˜(ν,t) from ν to τ. We are also concerned with
Thisrepresentationofthevisibilityisknownasthecross
the square modulus of V(τ,t) (see Section A.3.2):
spectrum, or cross-power spectrum. Because electric
fields at the antennas are complex and different, V˜AB C(τ,t)=|V(τ,t)|2 (4)
is complex. Usually visibility is averaged over multi-
ple accumulations of the spectrum, to reduce noise from We calculate C for right- and left-circular polarizations
background and the noiselike electric field of the source. separately, and then correlate them in delay τ to form
Thesecondargumenttallowsforthepossibilitythatthe KRL, the cross-correlation between polarizations:
visibility changes in time, as it does for a scintillating K (∆τ,t)= 1 (cid:80) C (τ,t)C (τ +∆τ,t) (5)
source, over times longer than the time to accumulate a RL N τ R L
single spectrum. Such a spectrum that changes in time Here, K is the correlation of a single measurement of
RL
is known as a “dynamic spectrum” (Bracewell 2000). C and C , and N is the number of samples in C and
R L R
The correlator used to analyze our data, as discussed in C .
L
Sections 3 and 4, calculates V˜ (ν,t) (Andrianov et al. When averaged over many realizations of the scat-
AB
2014). Hereafter, we omit the baseline subscript indicat- tering material, (cid:104)K (cid:105) is related to the statistics
RL S
ing baseline AB in this paper, except in sections of the of the pulse-broadening function G. Most commonly,
Appendix where the baseline is important. the average over many realizations of scattering ma-
Under the assumptions that the source is pointlike, terial is approximated by averaging over a time much
and that we can ignore background and source noise, longer than t ; for this reason we omit the time ar-
sc
the impulse-response function of interstellar scattering g gument for (cid:104)K (∆τ)(cid:105) . Equivalently, evaluation of
RL S
determines the visibility of the source. A single delta- (cid:104)K (∆τ,f )(cid:105) = (cid:104)C (τ,f )C (τ +∆τ,f )(cid:105) at
RL max R max L max τ
functionimpulseofelectricfieldatthesourceisreceived the fringe rate f of the maximum magnitude of K
max RL
RadioAstron discovery of scaterring substructure in PSR 0329+54 3
Table 1
Diaryofobservations
Epochof Time Ground Polarizations Scan
Observations Span Telescopes Length
2012Nov26through29 1hr/day GB RCP+LCP 570s
2014Jan1and2 12hr WB,KL RCP 1170s
(RDR). This type of recorder was also used at the KL,
Table 2 while the Mk5B recording system was used at the GB
ObservationsonEarth-SpaceBaselines
and WB. Table 1 summarizes the observations.
The data were transferred via internet to the Astro
Epoch Projected RA
SpaceCenter(ASC)inMoscowandthenprocessedwith
BaselineLength ObservingTime
(103 km) (minutes) theASCcorrelatorwithgatinganddedispersionapplied
(Andrianov et al. 2014). To determine the phase of the
2012Nov26 60 60
gate in the pulsar period, the average pulse profile was
2012Nov27 90 60
2012Nov28 175 60 computed for every station by integrating the autocor-
2012Nov29 235 60 relation spectra obtained from the ASC correlator. The
2014Jan1 20 60 autocorrelationspectraV (ν,t)arethesquaremodulus
2014Jan2 70 100 AA
of electric field at a single antenna.
2014Jan2 90 120
InNovember2012theprojectedbaselinestothespace
radio telescope were about 60, 90, 175, and 235 thou-
yieldsthesametimeaverage. Forthistheoreticaldiscus-
sand kilometers for the four consecutive days, respec-
sion, f = 0; for practical observations, instrumental
max
tively. Data were recorded in 570-second scans, with
factors can offset the fringe rate from zero, so that f
max
30-second gaps between scans. In January 2014 the pro-
provides the most reliable time average.
jectedbaselineswereabout20,70,and90thousandkilo-
Forabaselinethatextendsmuchfurtherthanthescale
metersduringthe12-hoursession. Datawererecordedin
of scattering S (see Equation 32):
sc
1170-second scans. The RA operated only during three
(cid:104)KRL(τ)(cid:105)S =G(τ)⊗G−(τ)⊗G(τ)⊗G−(τ) (6) sets of scans of 60, 100 and 120 min each, with large
(cid:0) (cid:1) gaps in between caused by thermal constraints on the
+ 1 if τ =0
spacecraft. The auto-level (AGC), phase cal, and noise
Here,weintroducethesymbol⊗toindicateconvolution, diode were turned off during our observations to avoid
and denote the time-reverse of G as G (τ)=G(−τ). interference with pulses from the pulsar. Table 2 gives
−
Our analysis method differs somewhat from Smirnova parameters of the Earth-space baselines observed.
et al. (2014), who used structure functions of intensity,
visibility, and visibility squared to study scattering of 4. DATA REDUCTION
pulsar B0950+08 on an extremely long baseline to Ra-
4.1. Correlation
dioAstron. Thetwomethodsarecloselyrelatedtheoreti-
cally. Structurefunctionsareparticularlyvaluablewhen All of the recorded data were correlated with the ASC
the characteristic bandwidth approaches the instrumen- correlator using 4096 channels for the November 2012
tal bandwidth, and can be extended to cases where the session and 2048 channels for the January 2014 session,
signal-to-noise ratio is low, as they discuss. with gating and dedispersion activated. The ON-pulse
window was centered on the main component of the av-
3. OBSERVATIONS erageprofile,withawidthof5msintheNovember2012
session and 8 ms in the January 2014 session. These
The observations were made in two sessions: the first
compare with a 7-ms pulse width at 50% of the peak
for one hour each on the four successive days November
flux density (Lorimer et al. 1995). The OFF-pulse win-
26 to 29, 2012, and the second for a total of 12 hours
dow was offset from the main pulse by half a period and
on the two days January 1 and 2, 2014. The first ses-
had the same width as the ON-pulse window. The cor-
sion used the 10-m RadioAstron Space Radio Telescope
relator output was always sampled synchronously with
(RA) together with the 110-m Robert C. Byrd Green
the pulsar period of 0.714 s (single pulse mode). We
Bank Telescope (GB). The second session used the RA
used ephemerides computed with the program TEMPO
together with the 14×25-m Westerbork Synthesis Ra-
for the Earth center (Edwards et al. 2006). The results
dio Telescope (WB), and the 64-m Kalyazin Radio Tele-
of the correlation were tabulated as cross power spectra,
scopes(KL).Bothright(RCP)andleftcircularpolariza-
V˜(ν,t), written in standard FITS format.
tions (LCP) were recorded in November 2012, and only
onepolarizationchannel(RCP)wasrecordedinJanuary
4.2. Single-Dish Data Reduction
2014. Because of an RA peculiarity at 324 MHz, the
316–332 MHz observing band was recorded as a single Using autocorrelation spectra at GB, KL, and WB,
upper sideband, with one-bit digitization at the RA and we measured the scintillation time t and bandwidth
sc
withtwo-bitdigitizationattheGB,WB,andKL.Science ∆ν = 1/2πτ . The results are given in Table 3. Our
sc sc
data from the RA were transmitted in real time to the analysis using interferometric data, for which the noise
telemetry station in Pushchino (Kardashev et al. 2013) baselineisabsentandthespectralresolutionwashigher,
and then recorded with the RadioAstron data recorder is more accurate for the constants τ and τ as discussed
1 2
4 Gwinn et al.
Table 3
MeasuredScatteringParametersofPSRB0329+54
Epoch tsc ∆νsc wnτ wnf τ1=1/k1 τ2=1/k2
(s) (kHz) (ns) (mHz) (µs) (µs)
(1) (2) (3) (4) (5) (6) (7)
Nov2012 114±2 15±2 50±5 20±2 4.1±0.3 23±3
Jan2014 102±2 7±2 43±3 25±3 7.5±0.3 –
Note. — Columns are as follows: (1) Date of observations, (2)
Scintillationtimefromautocorrelationspectraasthehalfwidthat
1/e of maximum, (3) Scintillation bandwidth from single-dish au-
tocorrelationspectraasthehalf-widthathalfmaximum(HWHM),
(4)HWHMofasincfunctionfittothecentralspikeofthevisibility
distribution along the delay axis, (5) HWHM of a sinc function fit
to the central spike of the visibility distribution along the fringe
rateaxis,(6)Scaleofthenarrowcomponentof|K (∆τ)|(Section
RL
5.2.3), (7) Scale of the broad component of |K (∆τ)| (Section
RL
5.2.3).
below, so we quote those values in Table 3. for a useful analysis. However, we decided to ana-
lyze the data from the time series of multiple pulses.
4.3. VLBI Data Reduction Fourier transform of the cross spectrum, V˜(ν,t), to the
The ASC correlator calculates the cross-power spec- delay/fringe-ratedomainyieldsV(ν,f)andconcentrates
trum, V˜(ν,t), as discussed in Sections 2 and A.3.1. The the signal into a central region, and thus provides a high
resolution of the resulting cross-power spectra is 3.906 signal-to-noise ratio. The sampling rate of individual
kHzforthe2012observationsand7.812kHzforthe2014 cross spectra in the time series was the pulse period of
observations. Because the scintillation bandwidth was 0.714 s, as noted in Section 4.1. The time span of cross
comparable to the channel bandwidth for the 2014 ob- spectra used to form V(τ,f) varied, ranging from 71.4 s
servations, as shown in Table 3, and because the single to 570 s, depending upon the application.
recorded polarization at that epoch prevented us from
correlating polarizations to form K , as discussed in 5.1. Distribution of visibility
RL
Section 5.2.3, we focus our analysis and interpretation InFigure2wedisplaythemagnitudeofthevisibilityin
on the 2012 observations. the delay/fringe-rate domain, |V(τ,f)|, for a 500-s time
span. The data were obtained on 29 November 2012 in
5. ANALYSIS OF INTERFEROMETRIC VISIBILITY
the RCP channel for a projected 200Mλ GB-RA base-
We investigated the scattering of the pulsar from the line. The cross spectra, V˜(ν,t), from which we obtained
visibility in the delay-fringe-rate domain, V(τ,f). We |V(τ,f)|weresampledwith4096spectralchannelsacross
studied the statistics of visibility V(τ,f) as a function the 16-MHz band, at the pulsar period of 0.714 s; con-
of delay, fringe rate, and baseline length. If there were sequently, the resolution was 0.03125 µs in delay, and
no scattering material between the pulsar and the ob- 2 mHz in fringe rate. As Figure 2 shows, no dominant
server, we would expect for |V(τ,f)| one spike at zero central spike is visible at zero delay and fringe rate, as
delay and fringe rate with magnitude that remains con- would be expected for an unresolved source. Our long
stant as a function of baseline length, and with width baseline interferometer completely resolves the scatter-
equal to the inverse of the observed bandwidth in delay, ing disk. Instead we see a distribution of spikes around
and the inverse of the scan length in fringe rate. Scat- zero delay and fringe rate that is concentrated in a rela-
teringmaterialinbetweenchangesthispicture. Firstwe tivelylimitedregionofthedelay-fringeratedomain. The
expectthespikeatzerodelayandfringeratetodecrease locationsofthevariousspikesappeartoberandom. Be-
in magnitude with increasing baseline length, perhaps cause the scattering disk is completely resolved on our
to the point where it would become invisible. Second, long baseline, we conclude that the spikes are a conse-
we expect additional spikes to appear around the spike quence of random reinforcement or cancellation of paths
at zero delay and fringe rate. The distribution of these tothedifferentlocationsofthetwotelescopes,andhence
spikesgiveusinvaluableinformationaboutthestatistics interferometer phase.
of the scattering material. InFigure2,thedistributionofthemagnitudeofvisibil-
As we discuss in this section, we fitted models to the ity is relatively broad along the delay axis and relatively
distribution of visibility, as measured by the correlation narrowalongthefringerateaxis. Theextentislimitedin
function KRL, and thus derived scintillation parameters delaytoabouttheinverseofthescintillationbandwidth,
thatdescribetheimpulse-responsefunctionforpropaga- τ = 1/2π∆ν ; and in fringe rate to about the inverse
sc sc
tionalongthelineofsightfromthepulsar. Wealsocom- of the diffractive timescale t . Within this region, the
diff
puted the maximum visibility as a function of projected visibility shows many narrow, discrete spikes. If statis-
baseline length, as we discuss in detail in a separate pa- tics of the random phase and amplitude of scintillation
per (Paper II: Popov et al., in preparation). are Gaussian, and the phases of the Fourier transform
Forstrongsinglepulsesthevisibilityinthecrossspec- randomizethedifferentsumsthatcomprisethevisibility
tra, V˜(ν,t), had signal-to-noise ratios sufficiently large inthedelay-fringeratedomain,thenthesquaremodulus
RadioAstron discovery of scaterring substructure in PSR 0329+54 5
Figure 2. Magnitude of visibility in the delay-fringe rate domain |V(τ,f)|, for a 500-s time span on 29 November 2012 in the RCP
channel,ontheRA-GBbaseline. Visibilityisnormalizedforautocorrelation: |V(0,0)|=1. Theaxesshowinstrumentaloffsets,including
about6µsindelay. Top: three-dimentionalrepresentation;bottom: two-dimentionalrepresentation.
6 Gwinn et al.
of the distribution of magnitude for a range of baseline
lengths, as a function of delay. The maxima lie near
zero fringe rate, as expected. Under the plausible and
usual assumption that the correct fringe rate lies at the
fringerate,f ,wherethedistributionpeaks,thecross-
max
section represents the visibility averaged over the time
span of the sample:
V(τ,f )=(cid:104)V(τ,t)(cid:105) (7)
max t
The top panel of Figure 3 shows this cross-section
throughFigure2. Thenextlowerpanelshowsthecross-
section for the slightly shorter KL-RA baseline. The
threelowerplotsgivetheequivalentcross-sectionsfor10
times and 100 times shorter projected baselines. These
three short-baseline cross-sections are qualitatively dif-
ferentfromthelongbaselinecrosssections: thevisibility
has a central spike resulting from the component of the
cross-spectrumthathasaconstantphaseoverfrequency,
Figure 3. Examplesofthefinestructureofthemagnitudeofvis-
ibility,|V(τ,fmax)|,asafunctionofdelayτ,withfringeratefixed as well as the broad distribution from the component
atthemaximumofthedelay-fringeratevisibilitynearzerofringe that has a varying phase over frequency. The central
rate,fmax . Fromlowermosttouppermost,thecurvescorrespond spike is strongest for the shortest baseline and weaker
toprogressivelylongerbaselines,withthetelescopesindicatedand
for the next longer baselines, as expected based on the
the approximate baseline projections given in Mλ in parentheses.
Curvesareoffsetvertically,andtheupper2magnifiedasthever- results of Sections A.3.1 and A.3.2. At very long base-
tical scale indicates, for ease of viewing. All curves show 71.4 s lines the central spike is absent even after averaging the
of data. Uppermost curve is from 2012 November 29; the rest visibility over the whole observing period, and only the
arefrom2014January,whenmultiplegroundtelescopesprovided
broad component is present. As expected from Figure
shorterbaselines. Notevariationinscatteringtimebetweenepochs
as given in Table 3. The uppermost panel is the cross-section of 2, in the delay/time domain the broad component ap-
thedatashowninFigure2,butfor71.4sintegration. Visibilityis pears as spikes distributed over a range of about 10 µs
normalizedasinFigure2. Thebestestimateofinstrumentaldelay
in delay. These spikes keep their position in delay for
hasbeenremovedforeachcurve.
the scintillation time of about 100 to 115 s, as listed in
Table 3.
ofV(τ,f)shouldbedrawnfromanexponentialdistribu- The character of the broad component changes with
tion,multipliedbytheenvelopedefinedbythedetermin- baseline length as well: mean and mean square visibility
istic part of the impulse-response function, as discussed arethesameforshortandlongbaselines; butexcursions
in the Appendix. tosmallandlargevisibilitiesaremorecommonforalong
Along the delay axis, |V(τ,f)| takes the general form baseline (Gwinn 2001, Eq. 12).
suggestedbyFigures2and3: anarrowspikesurrounded
byabroaddistribution. Wefoundthatthecentralspike 5.2. Averages and Correlation Functions
takestheformofasincfunctioninbothdelayandfringe
Averages of the visibility, and averages of the correla-
ratecoordinates,asexpectedforuniformvisibilityacross
tion function of visibility, extract the parameters of the
a square passband (Thompson et al. 2007). The widths
broad and narrow components of visibility. Such aver-
are somewhat larger than values expected from observ-
ages approximate the statistical averages discussed in
ing bandwidth of 16 MHz and time span of 71.4 s, of
Sections 2 and A. They seek to reduce noise from the
w = 31.25 ns and w = 14 mHz respectively, proba-
nτ nf observing system and emission of the source, as well as
blybecauseofthenon-uniformityofreceiverbandpasses
variations from the finite number of scintillations sam-
andpulse-to-pulseintensityvariations,respectively. The
pled, while preserving the statistics of scintillation. The
broader part of the distribution takes an exponential
averages and correlation functions allow the inference of
form along the fringe-rate axis in this case; more gen-
parameters of the impulse-response function of propaga-
erally, the form can be complicated, particularly over
tion from the statistics of visibility.
times longer than 600 s. Traveling ionospheric distur-
bances may affect the time behavior of our 92-cm obser-
5.2.1. Square Modulus of Visibility C
vations; in particular, they may be responsible for the
20 to 25 mHz width of the narrow component in fringe The mean square modulus of visibility, (cid:104)C(τ)(cid:105)S =
rate,asnotedinTable3. Wedonotanalyzethebroader (cid:104)|V(τ)|2(cid:105) , provides useful and simple characterization
S
distribution in fringe rate further in this paper; we will of visibility. To approximate the average over realiza-
discuss this distribution, and the influence of traveling tions of scattering (cid:104)...(cid:105) , we average over many samples
S
ionospheric disturbances, in a separate publication (Pa- in time t and over bins in delay τ. We realize the av-
per III, Popov et al. in preparation). Because of the rel- erage over time by evaluating V(τ,f) at the fringe rate
atively small optical path length of the ionosphere, even of maximum amplitude f , as discussed in Section 2.
max
at λ = 92 cm, they cannot affect the cross spectrum We also average over 16 lags in delay τ. The result-
(Hagfors 1976). ing average shows a broad component surrounding the
The distribution of the magnitude of the visibility in origin; on shorter baselines, it shows a spike at the ori-
delay/fringe-rate domain changes with baseline length. gin. The broad distribution samples the properties of
Figure 3 displays cross-sections through the maximum the fine structure seen in Figures 2 and 3, and the spike
RadioAstron discovery of scaterring substructure in PSR 0329+54 7
Figure 4. Cross-section of the mean square visibility in the
delay/fringe-rate domain (cid:104)C(τ)(cid:105)S = |V(τ,fmax)|2 along the de- oFnig2u0r1e25N.ovA2n8e,xaavmerpalgeeodfotvheerc5o7rr0elsasttioanrtfiunngcattio2n1(cid:104):4K0(:0∆0τU,fTm.aTx)h(cid:105)et
lpaeyakasx,isc,loastetthoezferirnogmeHrazt.eTfmheaxviwsihbeilrietiethsefomratghneitGuBde-RoAfvbisaisbeilliintye data were normalized by the square root of KRR and KRR at
∆τ =0. Thebest-fittingparametersfora2-exponentialfitofthe
on2012Nov28at21:40UTareshownasopencircles. Thevisibil-
formofEquation35areasindicated.
itieswerecomputedbyaninverseFouriertransformofthespectra,
V˜(ν,t), over 71.4 s time spans, and then by averaging over 6 ob-
normalizedthembytheautocorrelationfunctionsateach
servingscans,each570slong. Theywerethenfurtheraveragedin
delay,over16pointsor0.5µs,tosmoothfluctuations. Thedashed antenna. From these we formed the un-averaged cor-
horizontal line shows the offset contributed by background noise. relation function K (∆τ,t). We then averaged K
RL RL
The solid gray line shows the reconstructed form given by Equa- over 570-sec scans to form (cid:104)K (∆τ)(cid:105) . Averaging in
tion34,offsetbythenoiselevel,withparameterstakenfromthefit RL t
the time domain approximated an average over realiza-
showninFigure5. Thelightdashedcurveshowsonlythenarrow
component of the two-exponential model. Units of visibility are tions of the impulse-response function for the scattering
correlatorunits. medium. Each 570-sec scan included 100 to 250 strong
pulses,yieldingoneaveragedsampleof(cid:104)K (∆τ,t)(cid:105) for
to those seen on the shorter baselines in Figure 3. We RL t
each scan. We obtained 22 measurements in total, with
argue in Section A that the spike in (cid:104)C(τ)(cid:105) is related
S
6 samples of (cid:104)K (∆τ,t)(cid:105) for November 26 , 28, 29 ob-
totheaveragevisibility,andthebroadcomponenttothe RL t
serving sessions. We obtained only 4 such samples for
impulse-response function.
November 27 because of no significant detections of V
Figure 4 shows an example of the broad component of
(cid:104)C(τ)(cid:105) . This is estimated as |V(τ,f )|2, by selecting for two scans on that date.
S max
thepeakfringeratef toaverageintimeforeachof6
max 5.2.3. Two Exponential Scales
scans, averaging the results for the scans, and averaging
over 16 lags of delay to smooth the data. These averag- Examinationoftheaveragedcrosscorrelationfunction,
ing procedures serve to approximate the average over an (cid:104)K (∆τ,t)(cid:105) , revealed a spike at the origin and two
RL t
ensemble of realizations of scattering. Background noise exponential scales for the broad component, a large one
adds complex, zero-mean noise to V(τ,f), with uniform and a small one. Figure 5 shows an example.
variance at all lags; this adds a constant offset to the Thespikeattheoriginarisesfromthefinestructureof
average (cid:104)C(τ)(cid:105) =|V(τ,f )|2. scintillation in the broad component of visibility, as seen
S max
in Figures 2 and 3. This structure is identical in right-
5.2.2. Correlation Function K and left-circular polarizations, so its correlation leads to
UsingEquation5,weestimated(cid:104)K (∆τ)(cid:105) ,theaver- the spike.
RL S
aged cross-correlation function between the square mod- The two exponential scales are apparent as the slopes
ulusofright-circularpolarized(RCP)andofleft-circular of the steeper and narrower parts of the distribution.
polarized (LCP) of visibility in the delay domain. (Note We see these two scales even for single pulses, which
that (cid:104)K (∆τ)(cid:105) is not the correlation function of the are strong enough to show the two-scale structure. We
RL S
average (cid:104)C(cid:105) , but rather the average of the correlation did not observe these scales without doubt in spectra
S
function (cid:104)C ⊗ C (cid:105) .) Because the background noise from single-dish observations, because the resulting cor-
R L S
in the two circular polarizations is uncorrelated, they do relation functions are more subject to noise, gain fluc-
not contribute an offset to (cid:104)K (∆τ)(cid:105) . This allows us tuations, and interference. The scales are both present
RL S
to follow the effects of the impulse-response function to for (cid:104)C(cid:105)S, but the longer scale is seen more clearly in
much lower levels than for (cid:104)C(cid:105)S. The correlation func- (cid:104)KRL(∆τ)(cid:105)S (ascomparisonofEquations34through36
tion (cid:104)K (∆τ)(cid:105) is thus less subject to effects of noise, shows).
RL S
and is more sensitive to the broad component of the dis-
5.2.4. Model Fit
tribution, than (cid:104)C(cid:105) .
S
Tocomputeanestimateof(cid:104)K (∆τ)(cid:105) ,wecalculated We formalized the two exponential scales seen for
RL S
the squared sum of real and imaginary components of (cid:104)K(∆τ)(cid:105) withamodelfit. Themodelassumedapulse-
S
V(τ,t), the inverse Fourier transform of the cross-power broadening function with two exponential scales. Under
spectrum. We formed these for each strong pulse, and this assumption, a short pulse appears at the observer
8 Gwinn et al.
with average shape:
G(τ)=(cid:26)A1k1e−k1τ +A2k2e−k2τ, τ ≥0 (8)
0, τ <0
The pulse rises rapidly, and falls as the sum of the two
exponentials.
The assumed form for G leads to predictions for the
forms of (cid:104)C(cid:105) and (cid:104)K(cid:105), as discussed in Section B. For
(cid:104)C(cid:105), we expect a cusp at the origin, and two exponen-
tialswithscalesk andk anddifferentweightsoneither
1 2
side. For (cid:104)K(cid:105), correlation smooths the cusp at the ori-
gin,producingasmoothpeak,withthesameexponential
scales appearing to either side.
Figure 5 shows the best-fitting model of this form for
the data shown there. This model has parameters:
A /A =0.33 (9)
1 2
k =1/4.3 µs (10)
1
k =1/23 µs. (11)
2
The model reproduces the two scales, and the smooth
peak,well. Themodelalsopredictsthemagnitudeofthe Figure 6. Upperpanel: Thedistributionoflongandshorttime
spike accurately, with zero average visibility ρ =0. scales for exponential scales of (cid:104)KRL(∆τ,fmax)(cid:105)t. Each pair of
AB scales was measured for a 570-s interval on one of the 4 consec-
The model shown in Figure 4 shows the model for C,
utive observing days in 2012. Lower panel: The distribution of
reconstructed using Equation 34 with parameters from magnitudesoflongandshorttimescales.
the fit to Figure 5. The two scales appear in the model,
although the offset from noise contributes at large delay lar polarization, as well as in the correlation function
τ. As the figure shows, a single exponential does not fit (cid:104)K (∆τ)(cid:105) averaged over 570 s shown in Figure 5. For
RL t
the model well: the narrow component is satisfactory at anassumedscreendistanceofhalfthepulsardistanceof
smallτ,butfallswellunderthedataatlargerτ. Ahigh- D = 1.03+0.13 kpc (Brisken et al. 2002), the two scales
winged function such as a Lorentzian can fit C well, but −0.12
correspond to diffractive scales of:
the rounded peak leads to a very wide peak for K that
cannot match the data, and the inversion to a G(t) that λ (cid:114) D
remainsfinite,andiszerofort<0ascausalitydemands, (cid:96)d1 = 2π cτ =2.3×109 cm (12)
is problematic. 1
(cid:114)
The best-fitting scales and the magnitudes of the two λ D
(cid:96) = =1.0×109 cm
contributions varied from scan to scan, but in a manner d2 2π cτ
2
that was consistent with our finite sample of the scin-
The diffractive scale is the lateral distance at the screen
tillation pattern, and the inhomogeneous averaging of
where phases decorrelate by a radian (Narayan 1992).
pulses with different intensities. We show a histogram
Therefractivescalegivesthescaleofthescatteringdisk:
of the results of our fits to 570-sec intervals in Figure
6. On 26 to 29 November 2012, the shorter scale av- (cid:112)
(cid:96) = cτ D =1.9×1013 cm (13)
eraged to τ = 4.1 ± 0.3 µs, and the longer scale to r1 1
1 (cid:112)
τ2 = 22.5 ± 2.9 µs. The scales had a relative power (cid:96)r2 = cτ2D =4.6×1013 cm
of A /A =0.38.
2 1
In contrast, Britton et al. (1998) measured angular
6. DISCUSSION broadening for PSR B0329+54 of θH <3.4 mas at ν =
325MHz, whereθ isthefullwidthofthescatteredim-
On a long baseline that fully resolves the scattering H
age at half the maxium intensity. This corresponds to a
disk,asFigure3shows,weobservemultiplesharpspikes √
refractivescaleof(cid:96) =(θ / 8ln2)D/2<1.1×1013 cm.
in the visibility V(τ,f) is a consequence of the variation r H
Thisupperlimitissomewhatsmallerthanthevaluesob-
of the amplitude and phase of visibility. (See also Paper
tained from our observations, even if one takes into ac-
II, Popov et al. in preparation). The characteristic re-
count the facts that the larger scale contains only 0.38
gionofthatvariation,∆τ·∆f,reflectstheproductofthe
of the power of the shorter one, and that the scattering
inverses of the scintillation bandwidth ∆τ ≈ 1/2π∆ν
sc material may be somewhat closer to the pulsar than to
and the scintillation timescale ∆f ≈ 1/2πt . These
sc the observer.
quantities are the width in time of the impulse-response
function, and the time for the impulse-response func-
6.1. Previous Observations
tion to change as the line of sight to the observer moves
through the scattering material. Shishov et al. (2003) studied the scattering properties
Detailedexaminationofthecorrelationfunctionofvis- of PSR B0329+54 in detail, using single-antenna obser-
ibility K (∆τ,t) reveals the presence of two character- vations at 102 MHz, 610 MHz, 5 GHz, and 10.6 GHz
RL
istic, exponential scales. Both scales are visible in the to form structure functions of the scintillation in time
single-pulse correlation functions of right and left circu- and frequency on a wide range of scales. They con-
RadioAstron discovery of scaterring substructure in PSR 0329+54 9
cluded that the scattering material has a power-law spa- simple model for anisotropic scattering in a thin screen,
tial spectrum with index α+2=3.50±0.05, marginally we expect the ratio of power in the scales to be approxi-
consistent with the value of 11/3 expected for a Kol- mately (cid:112)1+α2/2π(2+α2)≈0.40, as shown in Section
mogorov spectrum, with an outer scale of 2×1011 m < C of the Appendix. This compares well with our ob-
L < 1017 m. Using VLBI, Bartel et al. (1985) ob- servedratioofA /A =0.38. However,ourobservations
0 1 2
served PSR B0329+54 at 2.3 GHz and set limits on forground-spacebaselinesatavarietyoforientationsdo
the separation on the emission regions corresponding not show anisotropy. A variety of models, involving ma-
to different components of the pulse profile. Yangalov terial with varying anisotropy distributed along the line
etal.(2001)observedPSRB0329+54at1.650GHzwith of sight, and strong anisotropy that slips between our
ground-space baselines to HALCA, and found that the long baselines, might match our data.
sourcevariedstronglywithtime. Theyascribedthisvari- A second explanation is the complicated structure ob-
ation to scintillation, with the scintillation bandwidth servedwithindynamicspectra: mostcommonlyobserved
comparable to the observing bandwidth at their observ- as “scintillation arcs” (Stinebring et al. 2001). Recently,
ing frequency. Self-calibration with timespans less than it has been suggested that this structure arises from in-
the scintillation time returned a pointlike image, as ex- terferenceamongsubimages,resultingfromrefractionby
pected. Semenkov et al. (2004) analyzed these data, interstellarreconnectionsheets(Pen&Levin2014). This
including ground-ground baselines. They studied both complicatedstructureproducestimeandfrequencyvari-
single-antenna autocorrelation functions V (∆τ) and ations on a wide range of scales. Of course, we are con-
AA
cross-correlationfunctionsV (∆τ). Theydetectedtwo sidering very long baselines, where the scintillation-arc
AB
timescales for the scintillation pattern, of 20 min and 1 patterns should be completely uncorrelated between an-
min. They found that the properties of scattering could tennas. This may lead to blurring, resulting in a 2-scale
notbeexplainedbyasingle,thinscreen,andfurtherthat correlation function without particularly strong struc-
velocitiesindicatedrelativemotionswithinthescattering ture corresponding to the discrete arcs seen on shorter
medium. baselines (Brisken et al. 2010). We do not see any direct
Popov&Soglasnov(1984)hadpreviouslyobservedtwo evidence of scintillation arcs, as such. The magnitude of
coexistingscalesofscatteringforPSRB0329+54. They the visibility shows a featureless decline with increase of
found scintillation bandwidths of ∆ν = 115 Hz and either of the 2 dimensions |τ| and |f|. The GB autocor-
1
∆ν = 750 Hz, measured as the 1/e point of the cor- relation functions do not show scintillation arcs either,
2
relation function of intensity at an observing frequency for our observations.
of 102 MHz, using the Large Cophase Array of Puschino
Observatory. The ratio of these scales, ∆ν /∆ν = 6.5
2 1 7. SUMMARY
is larger than the ratio of k /k = 5.5 that we observe.
1 2
Scaledtoourobservingfrequencyof324MHz, usingthe We made VLBI observations of PSR B0329+54 with
∆ν ∝ ν22/5 scaling appropriate for a Kolmogorov spec- RadioAstron at 324 MHz on projected baselines of up to
235,000 km. Our goal was to investigate scattering by
trum,andconvertingfrom∆ν toτ usingtheuncertainty
the interstellar medium. These properties affect radio
relation τ = 1/2π∆ν, we find that these values corre-
observations of all celestial sources. While the results of
spond to 1.3 and 8 µs, respectively, about a factor of 3
suchobservationsareingeneralinfluencedbytheconvo-
smaller that the scales we observe. Of course, interpo-
lution of source structure with the scattering processes,
lation over a factor of 3 in observing frequency and the
pulsars are virtually point-like sources and signatures in
differentobservingtechniquesmayintroducebiases, and
the observational results can be directly related to the
scattering parameters likely vary over the years between
scattering properties of the interstellar medium.
the two measurements. Two scales of scattering have
Onlongbaselines,inthedomainofdelayτ andfringe-
also been observed for other pulsars (Gwinn et al. 2006;
rate f, the correlation function of visibility V(τ,f) is a
Smirnova et al. 2014).
collection of narrow spikes, located within a region de-
finedbytheinversesofthescintillationbandwidth∆τ ≈
6.2. Origin of Two Scales
1/2π∆ν and the scintillation timescale ∆f ≈ 1/2πt .
sc sc
Two scales of scattering may be a consequence of a For shorter baselines, a sharp spike at the center of this
variety of factors. Non-Gaussian statistics of scattering regionrepresentstheaveragevisibility; onlongbaselines
can produce multiple scales, although this usually ap- wheretheaveragevisibilitydropstonearzero,thisspike
pears as a continuum of scales rather than two different is absent.
individualscales,asinapower-lawdistributionoraLevy The mean square visibility, (cid:104)C(τ)(cid:105) = (cid:104)|V(τ)|2(cid:105) , is
S S
flight (Boldyrev & Gwinn 2003). A Kolmogorov model well fit with a smooth model, indicating that the vis-
for scattering in a thin screen does not fit as well as our ibility spikes are the result of random interference of
model based upon a two-exponential impulse-response many scattered rays. To form a quantity less subject
function, or even as one based upon one exponential. A toeffectsofnoise,weconvolvethemean-squareleft-and
model with two discrete scales appears to fit our data right-circular polarized visibility to form (cid:104)K (∆τ)(cid:105) =
RL S
better. (cid:104)|V (τ)|2⊗ |V (τ)|2(cid:105) . The average correlation
L τ→∆τ R S
One explanation is anisotropic scattering. This can function (cid:104)K (∆τ)(cid:105) shows two exponentials with dif-
RL S
produce two scales, corresponding to the major and mi- ferentcharacteristictimescales. Theformsof(cid:104)C(τ)(cid:105)and
nor axes of the scattering disk, as discussed in Section C (cid:104)K (τ)(cid:105) are well fit with a simple model, that assumes
RL
of the Appendix. The ratio of the scales of k2/k1 = 5.5 that the average pulse-broadening function G is the sum
corresponds to √the parameter α2 = 57, and an axial ra- of two exponentials with different timescales.
(cid:112)
tio of θ /θ = 1+α2 = 2(k /k )2−3 = 7.4. In a On 2012 Nov 26 to 29, the shorter timescale was
2 1 2 1
10 Gwinn et al.
4.1±0.3 µs, and the longer timescale was 23±3µs, with sian Federal Space Agency, in collaboration with part-
the longer-scale exponential containing approximately ner organizations in Russia and other countries. The
0.38 times the power of the shorter-scale exponential. National Radio Astronomy Observatory is a facility of
Thisdoubleexponentialmayarisefromanisotropicscat- the National Science Foundation operated under coop-
tering;orfromscatteredradiationatlargeangle,perhaps erative agreement by Associated Universities, Inc. This
corresponding to the subimages seen in single-dish and study was supported by Russian Foundation for Basic
shorter-baseline observations. Further investigation of Researchgrant13-02-00460andBasicResearchProgram
the properties of the image of the scattered pulsar on P-7 of the Presidium of the Russian Academy of Sci-
long and short baselines, using these data, will help to ences. C.R.G. acknowledges support of the US National
clarify the origin of the two scales. ScienceFoundation(AST-1008865). N.B.wassupported
by NSERC. We thank the referee for constructive com-
ments which helped to improve the manuscript.
The RadioAstron project is led by the Astro Space
Facilities: RadioAstron Space Radio Telescope
Center of the Lebedev Physical Institute of the Russian
(Spektr-R), GBT, WSRT, Kalyazin radio telescope.
Academy of Sciences and the Lavochkin Scientific and
Production Association under a contract with the Rus-
APPENDIX
A. IMPULSE-RESPONSE FUNCTION AND VISIBILITY
A.1. Introduction
Undergeneralassumptions, refractionandscatteringconvolvetheelectricfieldofasourcewithanimpulse-response
function g. This function varies with position in the observer plane, decorrelating over some lateral scale; and with
time, as the line of sight to the source moves with respect to the scattering material, and as the scattering material
evolves. Thetaskofthissectionistorelatetheimpulse-responsefunctiontothestatisticsofvisibility, asgivenbythe
functions C and K introduced in Section 2 above.
A.1.1. Notation
The visibility V is the conjugated product of electric fields at two antennas (Equation 1). We usually omit the
AB
subscriptsindicatingbaselineAB onV,unlesstheyareimportantfortheimmediateargument. WedenotetheFourier
transform from the time or delay domain (t or τ) to the frequency or fringe rate domain (ν or f) by F, and its inverse
by F−1. We accent symbols with tilde “˜” to denote quantities that depend on observing frequency ν, and the same
symbolswithoutaccentfortheFourier-conjugatedomainofdelayτ ortimet . Weassumethatthevariablesdescribing
e
time and frequency t ,ν,τ,t,f, are discrete. They range from −N/2 to N/2−1, where N is the number of samples in
e
the time or frequency span. For τ and t one sample is the inverse of the Nyquist rate, and they can span the time to
e
accumulate a single realization of the spectrum; for t and its Fourier conjugate f one sample is the averaging time for
one spectrum, and they can span one observation.
Our convention for normalization of the Fourier transform is that a function h(τ) normalized to unit area in the
delay domain has value unity at zero frequency: h˜(ν = 0) = 1. Conversely, if h˜(ν) is normalized to unit area in the
frequency domain, h(τ = 0) = 1/N. This is the “{1,−1}” convention of Wolfram Mathematica (Weisstein 2014).
With this convention, Parseval’s Theorem takes the form:
N/2−1 N/2−1
(cid:88) h(τ)h∗(τ)= 1 (cid:88) h˜(ν)h˜∗(ν) (14)
N
τ=−N/2 ν=−N/2
A.2. Impulse-Response Function
As noted above, the observed electric field of a pulsar E (t ) is the convolution of the electric field emitted at the
obs o
source with a kernel g that depends on scattering:
few(cid:88)×τsc
E (t )= g(t )E (t −t )=g⊗E (15)
obs o e src o e src
te=0
where we introduce the symbol ⊗ for convolution. The kernel g is the impulse-response function; in other words, if
the pulsar emits a sharp spike, then the observed electric field of the pulse is simply a copy of g. Because of this
convolution, g is also known as the propagation kernel; it is also known as the Green’s function, and the S matrix
(Gwinn & Johnson 2011, and references therein). Both E and g vary at the Nyquist rate: the inverse of the total
src
observedbandwidth. Usually,weassumethattheintrinsicelectricfieldofthesourceiswhitenoiseattheNyquistrate:
it is drawn from a Gaussian distribution in the complex plane at each instant (Rickett 1975). The impulse-response
function extends over a time span of a few times τ , representing the time over which a sharp pulse at the source
sc
would be received. It is zero outside this relatively narrow time window.
If the statistics of the scattering material are stationary, the characteristic shape and scales of g will remain fixed,
while details of amplitude and phase vary on the timescale t (Equation 3). An average of the squared electric field
sc