Long-term Scintillation Observations of Five Pulsars at 1540 MHz

1.3 MB·English

by  N. Wang

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev1.4)
Long-term Scintillation Observations of Five Pulsars at
1540 MHz
⋆
N. Wang,1,2,3 R. N. Manchester,3 S. Johnston,2 B. Rickett,4 J. Zhang,1
A. Yusup,1 M. Chen1
1 National Astronomical Observatories, CAS, 40-5 South Beijing Road, Urumqi, 830011, China
2 School of Physics, Universityof Sydney, NSW 2006, Australia
3 Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia
5
4 Department of Electrical and Computer Engineering, University of California, San Diego, USA
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0
2

n
5February2008
a
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ABSTRACT
1
From 2001 January to 2002 June, we monitored PSRs B0329+54, B0823+26,

B1929+10, B2020+28 and B2021+51 using the Nanshan 25-m radio telescope of
1
Urumqi Observatory to study their diffractive interstellar scintillation (DISS). The
v
averageinterval between observations was about 9 days and the observationduration
8
1 ranged between 2 and 6 hours depending on the pulsar. Wide variations in the DISS
2 parameters were observed over the 18-month data span. Despite this, the average
1 scintillation velocities are in excellent agreement with the proper motion velocities.
0 The average two-dimensional autocorrelation function for PSR B0329+54 is well de-
5 scribed by a thin-screen Kolmogorov model, at least along the time and frequency
0 axes. Observed modulation indices for the DISS time scale and bandwidth and the
/
h pulsar flux density are greater than values predicted for a Kolmogorov spectrum of
p electron density fluctuations. Correlated variations over times long compared to the
- nominalrefractive scintillationtime are observed,suggesting that larger-scaledensity
o
fluctuationsareimportant.Forthesepulsars,thescintillationbandwidthasafunction
tr of frequency has a power-law index (∼ 3.6) much less than expected for Kolmogorov
s turbulence (∼ 4.4). Sloping fringes are commonly observed in the dynamic spectra,
a
: especially for PSR B0329+54. The detected range of fringe slopes are limited by our
v
observing resolution. Our observations are sensitive to larger-scale fringes and hence
i
X smaller refractive angles, corresponding to the central part of the scattering disk.
r Key words: − ISM:general − ISM:structure − pulsars:general
a
1 INTRODUCTION range from kHz to MHz, with distant pulsars scintillating
fasterin bothfrequencyandtime(Cordes,Pidwerbetsky &
Diffractive interstellar scintillation (DISS) occurs when ra-
Lovelace 1986; Gupta, Rickett & Lyne 1994; Bhat, Rao &
diation from a pulsar has phase fluctuations imposed on it
Gupta1999). Longer-term fluctuationsin pulsarfluxdensi-
duetosmallscalevariationsintheinterstellarelectronden-
ties result from a focussing and defocussing of theray bun-
sity.Thescatteredraysinterfereconstructivelyanddestruc-
dle due to larger-scale inhomogeneities in the interstellar
tively, resulting in a modulation of the pulse intensity as a
electron density. This phenomenon, known as refractive in-
function of frequency and position on the observer plane.
terstellar scintillation (RISS),is broad-band and shows less
PulsarsaregoodprobesofDISSbecauseoftheirverysmall
modulation in distant pulsars.
angular size. The relative motion of the pulsar, scattering
Byanalogywithneutralgasturbulencetheory,theden-
material and the observer transforms the spatial variation
sity fluctuations in the ionized interstellar medium (ISM)
intoatimevariationofthesignalintensity.Themodulation
can be described by a power-law spectrum. The three-
ischaracterizedbyatimescale(∆t )ofminutestohoursand
d dimensional spatial power-law spectrum is of the form:
adecorrelation frequencyscale(∆ν )overawidefrequency
d
P =C2q−β (1)
3n n
where C2 is a measurement of the mean turbulenceof elec-
n
⋆ Email:[email protected] tron density along the line of sight and q = 2π/s is the
(cid:13)c 0000RAS
wavenumberassociated with the spatial scale of turbulence Quasi-periodic fringes resulting from interference be-
s. The spatial scale s is in the range of s s s , tweenmultipleimagesofthepulsarbythescatteringscreen,
inn out
≪ ≪
wheres ands correspondtotheinnerandouterscales are often observed in dynamic spectra (Hewish, Wolszczan
inn out
of the density fluctuations. The index β is thought to lie & Graham 1985; Wolszczan & Cordes 1987; Bhat, Rao &
in the range 3 < β < 5. For propagation of turbulent en- Gupta1999),withbroadandnarrowfringesindicatingsmall
ergy from large scales to small scales, the well-known Kol- and large separations of the ray paths (or images) respec-
mogorov theory of turbulence gives a value of β = 11/3. tively.Thefringesaregenerallyslopedonthedynamicspec-
Observations show that this generally applies in the inter- tra, i.e., periodic in both frequency and time and crossed
stellar medium between scales of s 1017 1019 cm fringes (i.e., slopes of both signs) are common. Such sloped
out
∼ −
ands 108 1010 cm(Spangler&Gwinn1990;Kaspi& fringes result from motion of the fringe pattern across the
inn
∼ −
Stinebring1992;Armstrong,Rickett&Spangler1995;Bhat, observer plane due to relative motion of the pulsar, Earth
Gupta& Rao 1999). and the scattering screen. In the case of an equivalent thin
There are discrepancies in the observed scintillation screen located approximately midway between the pulsar
spectra compared to the predictions of Kolmogorov turbu- and theobserver, the maximum fringe slope is given by
lence (Gupta, Rickett & Coles 1993). For example, mod-
dt Dθ
ulation depths and time scales often do not scale accord- = r (9)
dν V ν
s
ing to the Kolmogorov model. Various modifications to the
where θ is the refraction angle and V is the velocity of
Kolmogorov modelhavebeenproposed,includingthesteep r s
thefringepatternacross theobserverplane,normally dom-
spectrummodel(β 4)(Blandford&Narayan1985;Good-
≥ inated by the pulsar velocity (Hewish 1980; Bhat, Gupta
man&Narayan1985)andtheinnerscalemodelarenotable.
& Rao 1999). The refractive angle θ is approximately pro-
Thesteepspectrummodeldoesnotrequireaturbulentcas- r
portional to ν−2 for a given gradient in refractive index.
cade; it could result from the superposition of noninteract-
Therefore, for large-scale gradients, the fringe slope dt/dν
ingdiscretestructuresinspace.Theinnerscalemodelhasa
might be expected to vary approximately as ν−3. For dif-
cutoff at the scale at which the turbulent energy dissipates
ferent projections of the velocity on the line between the
(Coles et al. 1987).
images, the slope varies in magnitude and sign, resulting
Thevariation inobservedpulsarfluxdensityasafunc-
in crossed fringes of varying slope. Earlier papers discussed
tion of time and frequency is known as the dynamic spec-
Equation 9 in terms of thefrequency dependenceof there-
trumandtheinterferencemaximainthedynamicspectrum
fraction angle (e.g. Gupta, Rickett & Lyne, 1994), but this
are called scintles. Decorrelation time scale and bandwidth
equationresultsdirectlyfromconsiderationofthegeometric
are parameters which quantify the scintle size. A simplified
delays.Forinterferencebetween astrongcentral image and
model is often used in which a thin scattering disk is as-
otherpointswithinthemidplacedscatteringdisk,thefringe
sumed to lie midway between the pulsar and observer. In
rates in frequency (f ) and time (f ) are related by
general this is in good agreement with observations. For a ν t
pulsaratdistanceD from theEarth,thedecorrelation time D
f = f2, (10)
scale for DISS has theform of ν 2cν2V2 t
s
∆td ∝ν2/(β−2)D−1/(β−2)Vs−1 (β <4), (2) where c is speed of light (Stinebring et al. 2001), and Vs
is dominated by thepulsar propermotion. This shows that
∆t ν(β−2)/(6−β)D−(β−3)/(6−β)V−1 (β>4), (3) crossedfringepatternstransformtoparabolicarcstructures
d ∝ s in the ‘secondary spectrum’, that is, the two-dimensional
where ν is the radio frequency (Rickett 1977; Goodman & Fouriertransform of thedynamicspectrum.Suchparabolic
Narayan 1985). In particular, for a Kolmogorov spectrum, arcsandothermorecomplexstructureshavebeenobserved
∆t ν6/5D−3/5V−1. (4) in a numberof pulsars (Hill et al. 2003).
d ∝ s In this paper, we describe observations of five pulsars
The decorrelation bandwidth for DISS is (PSRs B0329+54, B0823+26, B1929+10, B2020+28 and
B2021+51). InSection 2 weintroducetheobservations and
∆ν ν2β/(β−2)D−β/(β−2) (β <4), (5)
d ∝ data analysis. Section 3 presents the dynamic spectra and
inSection4&5weshowtheauto-correlationfunctionsand
∆νd ν8/(6−β)D−β/(6−β) (β>4) (6) describe the time variations of the scintillation parameters.
∝
Secondaryspectrafor PSRB0329+54 are presentedin Sec-
(Rickett 1977; Goodman & Narayan 1985). In the Kol-
tion6.InSection7wediscussourresultsandcomparethem
mogorov case,
with previous observations and the predictions of scintilla-
∆ν ν22/5D−11/5. (7) tion theories, and Section 8 summarises theresults.
d
∝
Equations 2 to 7 indicate that fluctuations are more rapid
andhavenarrowerbandwidthsformoredistantpulsarsand
2 OBSERVATIONS AND DATA ANALYSIS
at lower frequencies.
In the Kolmogorov case, the scaling factor Cn2 is given The observations were made by using the Nanshan 25-m
by telescope,operatedbyUrumqiObservatory,NationalAstro-
C2 0.002ν11/3D−11/6∆ν−5/6, (8) nomical Observatories of China. The pulsarswere regularly
n ≈ d monitoredfrom2001Januaryto2002Junewithanaverage
whereν isinGHz,D isin kpc,∆ν isin MHzandC2 isin intervalbetweenobservationsofninedays.Ourobservations
d n
unitsof m−20/3 (Cordes 1986). were centred at 1540 MHz, a relatively high frequency at
2
whichpulsarscintillationpropertieshavenotbeenwellstud- epochs over the 18-month interval, with a typical observa-
ied.Mostearlierobservations(e.g.,Cordes,Wesberg&Bori- tion time of three hours. Fig. 1 shows 12 of the dynamic
akoff1985;Gupta,Rickett&Lyne1994;Bhat,Rao&Gupta spectrawhichhavebeencorrectedforboththeshortbroad-
1999) were less frequent and centred on lower frequencies, bandinterferenceandthelonger-termnarrow-bandinterfer-
327,408and610MHz.Thereceiverwasadual-polarisation ence.
room-temperature system with noise temperature of 95 K, ThedynamicspectrashowninFig.1revealverydiffer-
equivalent to a system flux density of 1080 Jy. Frequency entpropertiescomparedtothelowerfrequencyobservations
resolution was provided by a filterbank system consisting reported by Gupta, Rickett & Lyne (1994) and Stinebring,
of 128 channels with channel bandwidths of 2.5 MHz giv- Faison & McKinnon (1996) which were at 408 MHz and
ing a total bandwidth of 320 MHz for each polarisation. 610 MHz respectively. There is a much wider variation in
This system was described in more detail by Wang et al. scintillation time-scales andbandwidthsfrom daytodayin
(2001). The data were folded online and saved to disk ev- the1540MHzdataandmanyspectrashowmultiplefringes,
ery four minutes. Total integration time was from three to which are usually sloping, for example, the dynamic spec-
six hours, depending on the pulsar scintillation time scale tra obtained on MJDs 52068.6, 52078.3, 52082.1, 52131.1
∆t . We made 6-hour observations for the closer pulsars, and52202.5. Broadandnarrowfringesareoftenseenatthe
d
PSRsB1919+10,B2020+28andB2021+51,whiletheobser- same time, and these often have opposite slopes. Examples
vations were threehours for the more distant pulsars PSRs arespectraforMJDs52027.4,52078.3and52082.1.Thedy-
B0329+54 and B0823+26. Between 38 and 65 observations namicspectrumforMJD51972.4isinterestinginthatithas
weremadeofeachpulsar.Wecalibratedthepulsarfluxden- narrow vertical fringes.
sity scale using ten strong, relatively distant pulsars which
allhavewell-measuredfluxdensitiesinthepulsarcatalogue.
Parameters for the observed pulsars are listed in Ta- 3.2 PSR B0823+26
ble 1. Column 3 gives the pulsar dispersion measure and
Intotalweobtained48dynamicspectraforPSRB0823+26,
column 4 gives the pulsar distances obtained from paral-
and Fig. 2 shows 4 of them. This pulsar has a distance of
lax observations. Uncertainties in the last quoted digit are 0.38kpcandalargetransversevelocityof194kms−1.These
given in parentheses. Column 5 gives the transverse veloc-
parameters result in a wide decorrelation bandwidth and a
ities based on measured proper motion and the parallax
short decorrelation time scale. As will be discussed in Sec-
distance. Columns 6 and 7 give the pulsar Galactic longi-
tion 7, the average scintillation scale of PSR B0823+26 is
tude and latitude respectively, and pulsar distances from
about13minandthedecorrelationbandwidthisabouthalf
theGalactic planeare given in last column.
thereceiverbandwidth.Asseenwithotherpulsarsthescin-
To get thedynamicspectrum,we firstdedispersed and
tletime-scaleisshorterthantheintervalbetweenscintlesat
summedthedataforeachobservationtodeterminethepulse
a given frequency.
phaseatthecentrefrequency.Pulseintensitieswerethende-
Despite the frequency difference between our observa-
termined foreach sub-integration and each frequencychan-
tionsandthoseofGupta,Rickett&Lyne(1994)at408MHz
nelbysummingthedatawithinphase 0.05ofthepredicted
± and Bhat, Rao & Gupta (1999) at 327 MHz, the dynamic
phase of thepulse peak and subtracting a baseline offset.
spectra shown in Fig. 2 are similar with large fractional
Persistent narrow-band interference was found in fre-
decorrelation bandwidths and slow drifting. The character
quency ranges 1465–1472 MHz, 1482–1490 MHz, 1522–
ofthedynamicspectraoftenchangedramaticallyovershort
1527 MHz, 1545–1557 MHz and 1620–1625 MHz; in total
timeintervals,forexample, from averybroad single scintle
about15%ofthefrequencychannelswereaffected.Weused
tomultipleslopingscintlesinobservationsonMJDs52068.5
alinearinterpolationacrossaffectedchannelstoremovethe
and 52078.4 respectively. Clear reversals in the direction of
interferencefrom thedynamicspectra.Occasionally,impul-
driftslopeareseeninobservationsmadejustoneweekapart,
sivebroad-bandinterferencewhichlastedafewminuteswas
e.g., those on MJDs 52249.0 and 52256.0.
observed; a similar interpolation was used to removeit.
3.3 PSR B1929+10
3 DYNAMIC SPECTRA We obtained dynamic spectra for PSR B1929+10 at 38
epochs. This pulsar, with distance of 0.33 kpc, scintillates
In this section we present dynamic spectra
slowly and its decorrelation bandwidth is obviously compa-
for PSRs B0329+54, B0823+26, B1929+10, B2020+28 and
rableto orwider thanthereceiverbandwidth.Asshown in
B2020+51. Spectra selected to show the range of observed
Fig.3,thesix-hourobservationstypicallywereabletocover
features for each pulsar are given in Fig. 1 to Fig. 5. The
onlyoneortwoscintlesinthetimedomain.Featuresofthe
observingstarttime,pulsarnameandMJDaregivenatthe
spectra at 1540 MHz are very similar to those observed at
top of theeach spectrum.
lower frequencies byGupta, Rickett & Lyne(1994).
3.1 PSR B0329+54 3.4 PSR B2020+28
PSR B0329+54 is the strongest northern pulsar with a dis- Weobserved52dynamicspectraforPSRB2020+28.Atear-
tanceof1.06kpc.At1540 MHz,thepulsarscintillateswith lier stages most of the observations were about 3 hours but
typical time scales of 10 – 30 min and decorrelation band- later we commenced 6-hour observations in order to cover
widths of 5 – 15 MHz. We observed PSR B0329+54 at 64 several scintles in each observation. In Fig. 4 we present 4
3
Table 1.Parametersfortheobservedpulsars.
PSRJ PSRB DM Parallaxdist Vpm l b z
(pccm−3) (kpc) (kms−1) (deg ) (deg) (kpc)
(1) (2) (3) (4) (5) (6) (7) (8)
0332+5434 0329+54 26.84 1.06(12)a 98(11)a 145.00 –1.22 –0.02
0826+2637 0823+26 19.45 0.38(8)b 194(41)c 196.96 31.74 0.20
1932+1059 1929+10 3.18 0.33(1)a 163(5)a 47.38 –3.88 –0.02
2022+2854 2020+28 24.64 2.7(9)a 307(103)a 68.86 –4.67 –0.22
2022+5154 2021+51 22.65 2.0(3)a 119(18)a 87.86 8.38 0.29
a Briskenetal.(2002)
b Gwinnetal.(1986)
c Lyne,Anderson&Salter,(1982)
dynamicspectra.DespiteitsDMof24.6pccm−3 andadis- meanapartfromlong-termvariations.Becauseofourhigher
tanceof2.7kpc,PSRB2020+28showsratherbroadscintles observation frequency, the number of scintles is often small
in both time and frequency. and subtracting the mean from each sub-integration would
affect thediffractive parameters.
Four examples of normalised two-dimensional ACFs
3.5 PSR B2021+51
and one-dimensional cuts at zero lag are plotted in Fig. 6
As for PSR B2020+28, PSR B2021+51 initially was ob- for each of the five pulsars. Following convention, the scin-
served for three hours each day; from mid-November 2001 tillation time scale ∆td is defined as the time lag at zero
observationtimeswereincreasedtosixhours.Altogetherwe frequencylagatthepointwheretheACFis1/eofthemax-
observed65dynamicspectraforthispulsar,and8examples imum,and thedecorrelation frequency scale ∆νd is defined
are shown in Fig. 5. Clear drifting bands were observed on asthehalf-widthathalf-maximumoftheACFalongthefre-
several occasions, for example, the first observed dynamic quency lag axis at zero time lag (Cordes 1986). The ACFs
spectrum at MJD 52016.2. This pulsar shows two types of show spikes at zero time and frequency lag. These spikes
dynamic spectrum. In some cases the scintles are broad in are due to noise in the dynamic spectra, and are more sig-
bothfrequencyandtime(e.g.,atMJD52302.3),whereasin nificant for weaker pulsars. Slightly wider spikes resulting
othercasesthescintlesarenarrower(e.g.,atMJD52269.5). from theinterferenceexcision process areoften also present
There appears to be a correlation between signal strength around zero frequency lag. To remove these spikes we used
andscintillationtype,withthepulsargenerallybeingweaker the neighbouring frequency-lag points to fit for a parabola
when the scintles are narrow. across zero frequency lag. The central points are interpo-
lated from this fit.
4 SCINTILLATION PARAMETERS AND 4.2 Scintillation parameters
FLUX DENSITIES
We estimated the diffractive scintillation parameters for
4.1 Auto-correlation functions
each observation by fitting a two-dimensional elliptical
Gaussian function (Bhat, Rao & Gupta 1999) to the
Theparameters ofDISScan beobtained byforming atwo-
smoothedACFusingthenon-linearleast-squaresfittingrou-
dimensionalauto-correlationfunction(ACF)ofthedynamic
tineLMM. The Gaussian function has theform:
spectra S(ν,t), i.e., the pulsar flux density at frequency ν
and time t. The ACFis definedby C (ν,t)=C exp(C ν2+C νt+C t2), (13)
g 0 1 2 3
F(∆ν,∆t)= where C is held fixed to unity as the ACF is normalised.
0
[N(∆ν,∆t)]−1 ∆S(ν,t)∆S(ν+∆ν,t+∆t), (11) Thescintillationparameters∆tdand∆νdarethengivenby:
Xν Xt 1 0.5 ln2 0.5
∆t = ,∆ν = . (14)
where∆S(ν,t)=S(ν,t) S¯,and S¯ isthemean pulsar flux d (cid:16)C3(cid:17) d (cid:16)C1(cid:17)
−
density over each observation. N(∆ν,∆t) is the number of Fig. 6 shows that the Gaussian function represents the
correlatedpixelpairs.ThenormalisedACFisthengivenby: observed ACFs well in most cases, although it may not
be very reliable when the decorrelation bandwidth is wider
ρ(∆ν,∆t)=F(∆ν,∆t)/F(0,0). (12)
thanthereceiverbandwidth,asinmostobservationsofPSR
We note that in Cordes (1986) the flux density B1929+10. Time variations of ∆t and ∆ν from all obser-
d d
was normalised by subtracting the mean over each sub- vationsofthefivepulsarsareshowninFig.7toFig.11and
integration to minimise the effect of long-term variations, averagevaluesaregivenincolumns2and3ofTable2.One-
i.e., ∆S(ν,t) = S(ν,t) S¯(t). However this method can sigma error estimates from the least-squares fit are plotted
−
only beapplied tospectrathat contain manyscintles, since on each point in the figure. Errors in the table are one-
in this case the sub-integration mean is close to the global sigma uncertainties in the mean values, assuming a normal
4
Figure1.DynamicspectraforPSRB0329+54atcenterfrequency1540MHz,withfrequencyandtimeresolutionof2.5MHzand4min
respectively.Intensity isrepresentedbyalineargreyscalebetween1%(white)and85%(black)ofthemaximum.
Figure 2.AsinFig.1,dynamicspectraforPSRB0823+26.
distribution. There is an additional statistical error related shown byerror barsintheupperleft cornerofthe∆t and
d
to the finite number of scintles in the dynamic spectrum. ∆ν plots in Fig. 7 to11.
d
FollowingCordes,Weisberg&Boriakoff(1985),wetakethe
fractionaluncertaintiesin∆t and∆ν tobegivenbyN−0.5
d d Fitting for the two-dimensional Gaussian ellipse pro-
whereN =T ∆ν /(4∆t ∆ν )isthenumberofobserved
obs obs d d vides parameters for the sloping features of the dynamic
scintles. Typicalvaluesoftheseadditional uncertaintiesare
spectrum. Following Bhat, Rao & Gupta (1999), we adopt
5
Figure 3.AsinFig.1,dynamicspectraforPSRB1929+10.NotethedifferenttimescaleontheverticalaxiscomparedtoFig.1.
Figure 4.AsinFig.1,dynamicspectraforPSRB2020+28.
2ν
dt/dν as the drift rate; in terms of the fitted parameters t ∆t (18)
dt/dν is given by: r ≈ ∆νd d
dt C
= 2 . (15)
dν −(cid:16)2C3(cid:17) (Rickett 1990). This parameter is given in column 8 of Ta-
ble 2. We use Equation 8 to estimate mean values of the
The slope visibility r is given by:
fluctuation scaling parameter C2; derived values are given
n
r=C /√4C C . (16) in column 9 of Table 2.
2 1 3
Pulsarvelocitiescanbeestimatedfromthescintillation
Derivedvaluesforthedrift rateandabsolute valueofslope
parameters∆t and∆ν .Forathinscatteringscreenplaced
visibility (r) are plotted in Fig. 7 to Fig. 11 and average d d
| | midwaybetweenthepulsarandtheobserver,thetransverse
values are given in columns 5 and 6 of Table 2.
velocity of thepattern is given by:
The scattering strength u is defined to be the ratio of
Fresnel scale s to the DISS scale s (Rickett 1990; Gupta
F d
1995); u 1 corresponds to strong scattering and u 1 √∆ν D
to weak s≫cattering. In terms of observable quantities,≪u is Vs≈3.85×104 ν∆dt (19)
d
expressed as
u= sF 2ν 0.5 (17) whereVsisinkms−1,∆νd isinMHz,thepulsardistanceD
s ∼ ∆ν
d (cid:16) d(cid:17) isinkpc,ν isinGHz,and∆t isinseconds(Gupta,Rickett
d
(Bhat, Gupta & Rao 1999), where ν is the observing fre- &Lyne1994).V isacombinationofthetransversevelocity
s
quency.Observedmeanvaluesofuaregivenincolumn7of of the pulsar, the scattering medium and the Earth’s mo-
Table2.ScatteringisstrongerforPSRB0329+54compared tion. The latter two components are usually comparatively
toPSRsB0823+26,B2020+28,andB2021+51,despitetheir small, soV representsthetransversevelocityofthepulsar.
s
similar DMs. From Equation 19 and the parallax distances given in Ta-
Based on the simple thin screen model with power-law ble 1, we derived transverse velocities for the five pulsars.
densityfluctuations,thetimescaleforrefractivescintillation The resulting values are plotted in Fig. 7 to Fig. 11 and
t is given by average values are given in column 10 of Table 2.
r
6
Figure 5.AsinFig.1,dynamicspectraforPSRB2021+51.
4.3 Flux densities Theseplotsshowgoodgeneralagreementinshapealong
bothaxes.Particularlyevidentisthesteeperslopenearzero
Long-termobservationshaveshownthatpulsarfluxdensity
lag in the frequency domain than in the time domain for
(S)variationsaredominatedbyrefractivescintillation.Dis-
both the model and the observations. The minimum chi-
tantpulsarsshowlittlefluxdensityvariation,implyingthat
squaredvalueisabouttwicethenumberofdatapointsinthe
intrinsicpulsarradioluminositiesarestable(Kaspi&Stine-
fit, which maybe due to too low an estimate of the error in
bring1992). Observedfluxdensitiesareplottedin Fig.7to
themeancorrelationfunction.Wesuspectthatthetemporal
Fig. 11 and mean values are given in column 11 of Table 2.
variations in thetwo-dimensional correlations donot follow
normalstatisticswhichcausestheunderestimateoftheerror
in its mean.
5 MODELLING THE ACFS Whereasalong theaxesthemodelagrees with thethe-
ory, away from the axes the theory contours bulge outward
TheobservationsofPSRB0329+54gaveus64independent
morethantheobservedcontours.Theoutwardbulgingmay
estimatesofthefrequency-timecorrelationfunctionofinten-
berelatedtothearcphenomenoninthesecondaryspectrum
sity.Eventhoughtheseoftenshowdriftingfeaturestheaver-
(seeSection6).Wealsonotethatscatteringextendedalong
agecorrelation functionshouldstillbeestimatedwithgood
the line of sight causes a reduction in the bulging effect, as
confidence and we compared it with its theoretical shape
can beseen in Figure 5 of Lambert & Rickett (1999).
(assumingstrongscattering)fromathinscatteringmedium
with a Kolmogorov spectrum.
The theoretical intensity correlation versus spatial and
frequency offsets, assuming a Kolmogorov spectrum and a
6 SECONDARY SPECTRA
thin screen, has been derived by Lambert & Rickett (1999;
seetheirFigure5).InFig.12wecomparetheirmodelwith Secondary spectra, that is, two-dimensional Fourier trans-
the mean of the observed auto-correlation functions, by as- forms of dynamic spectra, give additional insight into scin-
sumingthattimeoffsetstranslatesimplyintospatialoffsets. tillation and scattering in the interstellar medium. Fig. 13
Weestimated∆ν byfittingthetheoreticalshapeofthecut shows secondary spectra, for several observations of PSR
d
alongthefrequencyaxisand∆t fromthecutalongthetime B0329+54 where significant fringing was observed. These
d
axis.Theobservedone-dimensionalcorrelationfunctionsare spectra are much more amorphous and centrally concen-
shown by the error bars (three times the nominal standard trated than those shown by Hill et al. (2003) for other pul-
error in themean) with a solid line for themodel. sars at a similar frequency.Wenote that theirobservations
7
Figure 6. Samples of normalised two-dimensional auto-correlation functions of the dynamic spectra shown in Fig. 1 to Fig. 5. The
contour interval is 0.1, with dashed lines representing negative contour levels. The solid line in the two smaller plots at the bottom
and right are the one-dimensional ACFs at zero lag in time and frequency respectively. The dotted lines are zero-lag cuts through the
two-dimensionalGaussianfitstotheobserveddata.
8
Table 2.Averagescintillationparametersoftheobservedpulsars.
PSRB ∆td ∆νd σN dt/dν |r| u tr logCn2 Vs S
(min) (MHz) (min/MHz) (day) (kms−1) (mJy)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
0329+54 16.9(5) 14(1) 0.11 0.07(7) 0.358(3) 15 2.5 −3.0 97(5) 199(11)
0823+26 13.6(8) 82(5) 0.24 0.02(2) 0.616(6) 6 0.4 −2.8 187(9) 24(1)
1929+10 32(2) 268(24) 0.52 0.02(1) 0.442(7) 3 0.3 −3.1 136(9) 65(5)
2020+28 31(2) 70(5) 0.29 –0.01(3) 0.302(4) 7 1.0 −4.3 225(16) 40(4)
2021+51 36(3) 52(3) 0.26 –0.11(5) 0.452(4) 8 1.5 −4.0 138(8) 59(6)
havenarrowerbandwidthsandshorterobservationtimesbut
much higher time and frequency resolution (10 – 30 s and
0.1MHzrespectively)comparedto4minand2.5MHzfor
∼
ourobservations.Ourobservationsthereforearesensitiveto
larger-scalefringesandhencesmallervaluesofθ thanthose
r
of Hill et al. (2003), i.e., we are observing the central part
of the scattering disk with high spatial resolution, whereas
the higher-resolution observations are senstive to large re-
fraction angles and very fine fringing structure. Although
there are some indications of rudimentary arc structure in
Fig. 13, the model of interference between a strong central
imageandsurroundingdiscreteimagesisclearlylessappro-
priate—weareinfactobservingthestructureofthecentral
image.
Across our wide observed bandwidth (∆ν =
obs
320 MHz), the frequency dependenceof the slope (dt/dν
ν−3), see Equation 9) corresponds to a 2:1 variation wit∝h
larger slopes (more vertical fringes) at lower frequencies.
ThereissomeevidenceforthisinFig.1,forexample,inthe
narrow fringing visible in the spectrum for MJD 52202.5.
This slope variation will smear the corresponding feature
in the secondary spectrum, contributing to the amorphous
appearance of thesecondary spectra (Fig. 13).
7 DISCUSSION
7.1 Dynamic spectra
For PSR B0329+54, the character of the dynamic spectra
is very different from that observed at lower frequencies.
Sloping fringes of different frequencies and slopes are very
common in the 1540 MHz observations, whereas at lower
frequencies they are uncommon (e.g., Stinebring, Faison &
McKinnon, 1996). The frequency dependence of decorrela-
tion bandwidth is directly observable within our receiver
bandwidth in many dynamic spectra. Taking the observa-
MJD-51900 tionatMJD52294.5 forPSRB0329+54 asanexample,the
widthofthescintleat 1400MHzisabout 13MHz,whereas
Figure 7. Time variations of ∆td, ∆νd, main axial angle,
slope visibility, transverse velocity and the flux density for PSR at 1640 MHz it is about 27 MHz, consistent with the pre-
B0329+54. Errorbars (±1σ) plotted oneach point arefromthe diction from Equation 7.
2D-ellipticalGaussianfitting. Thestatistical uncertainty related Despite PSRs B0329+54, B0823+26, B2020+28 and
tothefinitenumberofobservedscintlesisindicatedbythe ±1σ B2021+51 having very similar dispersion measures, their
error bar in the mean value shown at the top-left corner of the scintillation propertiesarequitedifferent.Asiswellknown,
∆td and∆νd plots. Galactic latitude appears to be important with stronger
scattering for PSR B0329+54. Distance seems less impor-
tantwithverydifferentDISSpropertiesforPSRB0823+26
and PSR B1929+10.
9
MJD-51900 MJD-51900
Figure 8.AsinFig.7,timevariationsof ∆td,∆νd,mainaxial Figure 9.AsinFig.7, timevariations of∆td,∆νd,mainaxial
angle,slopevisibility,transversevelocityandthefluxdensityfor angle,slopevisibility,transversevelocityandthefluxdensityfor
PSRB0823+26. PSRB1929+10.
7.2 Time variations
dicted modulation indices for thecase of a thinscreen with
Observeddynamicspectravarygreatly from observation to aKolmogorovdensityspectrum(β=11/3)andstrongscat-
observation, in all respects: decorrelation time scale, band- tering(Romani,Narayan&Blandford1986;Bhat,Gupta&
width,brightnessandthecharacteristicsofslopingfeatures. Rao1999).Inallcases,theobservedmodulationindicesare
Fig. 7 to Fig. 11 show that scintillation parameters have greater than the predicted values. With our relatively high
a wide range of variation. For example, for PSR B0329+54 observingfrequencyandrelatively nearbypulsars,thescat-
(Fig.7),theobserveddiffractivetimescaleanddecorrelation tering strength u is only modestly greater than one, so the
bandwidth vary over the ranges 12–30 min and 5–34 MHz strongscatteringassumptionisnotreallysatisfied.Also,the
respectively. Similar variations are observed for the other form of the variations suggests that large-scale fluctuations
four pulsars. The interval between observations is typically in the screen are important, so the fluctuation spectrum is
severaltimestherefractivetimescale t (Table2)andsowe likely to be steeper than in theKolmogorov case.
r
wouldexpectuncorrelatedvariations.However,theexpected Fig. 7 to Fig. 11 show that there are significant cor-
modulation due to refractive scintillation can only account related changes in decorrelation time-scale, bandwidth and
for a small amount of the observed variations. Modulation drift rate over intervals of several weeks and in some cases
indices(ratioofrmsdeviation tomean value)forthevaria- many months, much larger than the predicted refractive
tions in ∆t , ∆ν and flux density, m , m and m respec- timescalet .Forexample,inFig.7forPSRB0329+54,there
d d t b r r
tively,aregiveninTable3.AlsogivenintheTablearepre- are high values of decorrelation bandwidth at MJD 52035
10