Table Of ContentProceedingsofthe363.WE-HeraeusSeminaron:“NeutronStars and Pulsars”(Postersand contributedtalks)
Physikzentrum BadHonnef, Germany,May.14−19,2006, eds.W.Becker,H.H.Huang, MPEReport291,pp.216-218
Finding Binary Millisecond Pulsars with the Hough
Transform
Carsten Aulbert
Albert-Einstein-Institute
Am Mu¨hlenberg 1, Potsdam, Germany
7
0
0
2
n Abstract. TheHoughtransformationhasbeenusedsuc- Over the past decades quite a few search algorithms
a cessfully for more than four decades. Originally used for have been devised to counter this problem, e.g. the very
J tracking particle traces in bubble chamber images, this successful“accelerationsearch”1orthesidebandsearch2.
4 work shows a novel approach turning the initial idea into However, these approaches work purely on coherent data
a powerful tool to incoherently detect millisecond pul- sets, thus they are ultimately limited by the maximum
1
sars in binary orbits. possible length of an observation
v
7 This poster presents the method used, a discussionon We would like to present an alternative approach to
9 how to treat the time domain data from radio receivers search for weak pulsars in a binary orbit, which utilizes
0 and create the input ”image” for the Hough transforma- data taken in several observation runs. In the following
1
tion, details about the advantages and disadvantages of sectionwewillbrieflyexplainhowthe Houghtransforma-
0
this approach,and finally some results from pulsars in 47 tionworks,andhowitcanbeappliedtosearchforbinary,
7
0 Tucanae. millisecondpulsarsbeforesummarizingsomeoftheresults
/ obtained for 47 Tucanae.
h
p
-
o Introduction
The Hough transformation
r
t
s The problem of discovering millisecond pulsars in binary
PaulHoughdevelopedandpatentedthealgorithmaround
a orbits is apparent. Not only the parameter range for iso-
: 1960. The original algorithm was designed to search for
v lated pulsars has to be addressed, but — at least — five
straightparticle traces in bubble chamber images by sim-
i Keplerian parameters have to be included additionally. A
X ply turning the usual line equation y = ax + b into a
r simpleFourieranalysisisnotsufficientforthisclassofpul- parameter space equation b = y−ax.
a sars,sincethesignalissmearedoutoveralargefrequency
From these two equations it is obvious that a line in
band due to Doppler’s effect (ref. Figure. 2).
the original space (“xy-space”) corresponds to a single
point in the parameter space (“ab-space”). On the other
hand a single point in xy-space is represented by a line
12000 in ab-space. Although sounding trivial, it actually solves
23.7 mHz 244 bins a complex problem: Identifying parameters of lines found
10000 in (pixel) images.
Figure2showswhathappenswhentransformingmany
8000 points froma single line. For eachof these points a line is
u.] drawn in ab-space and since each of these points belong
a.
er [ 6000 to the same xy-line, their lines are required to intersect
w
o in a single point in ab-space.Naturally, random points in
P
4000 xy-spaceconvertto randomlines inab-spaceonly adding
up to a random background.
2000 Trackingmillionsoflinesalongwiththeirintersections
is costly in terms of computing power and memory, thus
0 wechosetodiscretizethe ab-spaceinto rectangularpixels
476.02 476.03 476.04 476.05 476.06 476.07
andsimply counthowmanylinesaredrawnthrougheach
Frequency [Hz]
pixel. This approachallows us to discover the blue “line”
Fig.1. Power spectrum of the binary pulsar 47 Tuc J. inFigure3,whereonly95signalpointsarehiddenbeneath
Carsten Aulbert:Finding Binary Millisecond Pulsars with the Hough Transform 217
Fig.3.Recoveringahiddenline(top,blue)withaSNRof
about 20 (tiny white point (bottom)).
f , f : Called “external” variables, because they appear
p o
only as input parameters to the Hough transform and
are varied outside the scope of it. This separation al-
lows us to distribute the work onto a cluster of com-
puters.
So far, we have not said anything about our “input
image”. From the time domain data1 we use small pieces
ofthedata(typicallyafewminutes),createapowerspec-
trum for each piece and select peaks from the spectra.
Fig.2.Pointsfromalineinxy-spacetransformintolines Then, we drawthe registeredpeaks into a time-frequency
intersecting in a single ab-point. plane which serves as our input for the Hough transfor-
mation.
200,000 noise points while still be detectable at an SNR
of about 20! 47 Tucanae results
For binary pulsars,the frequency signaturecanbe de-
In this section we summarize some results obtained from
scribed by
47 Tucanae data. Figure 4 shows the outcome obtained
f(t)=fp(1+λocos(2πfo−Φo)), for the full parameter space (fp, Po =fo−1).
The maximum number counts (of how many lines are
where fp is the intrinsic frequency of the pulsar, fo the drawn through any pixel) found in each of the about 109
orbitalfrequency,λ the(dimensionless)frequencyampli-
o computed Hough planes are color coded in this image.
tude due toDoppler modulationandΦ the orbitalphase
o One canclearly see the signaturesofseveralnoisesources
at t=0. This equation can be recast into a line equation
(lines at and below 100 Hz as well as pulsars (173 Hz,
C = Ax+By, where 186 Hz) and their harmonics at twice, thrice, ... of the
A = sin(2πf −Φ ), fundamental frequency.
o o
A closer look at the Hough planes reveals that 47 Tu-
B = cos(2πf −Φ ),
o o canae C at 173.709 Hz is an isolated pulsar (Figure 5).
C = (f(t)−fp)/fp, Please recall that such a source does not feature any no-
x = f λ sinΦ , ticeable Doppler modulation in the solarsystem barycen-
p o o
ter and the parameter λ is therefore equal to zero. This
y = f λ cosΦ . o
p o o
in turn is shown by the maximum at the center/origin.
Here, we split the variables into two types: For the binary pulsar 47 Tucanae J the situation is
different. It orbits its companion in less than 3 hours and
λ , Φ : These variables are considered in the Hough
o o
transform, i.e. the resulting Hough planes (formerly 1 KindlymadeavailableforusbythepulsargroupatJodrell
ab-space) are polar diagrams in these parameters. Bank, Manchester, UK
218 Carsten Aulbert:Finding Binary Millisecond Pulsars with the Hough Transform
−100
900
Hz] −50 800
m
hi) [
p 700
n( 0
si
*
da 600
b
m
50
a
l
500
100
400
−100 −50 0 50 100
lambda * cos(phi) [mHz]
Fig.4. 47 Tucanae global result map
Fig.6. 47 Tucanae J: The maximum in this Hough plane
isnotattheoriginanymore.TheoffsetindicatesaDoppler
173.709Hz, 101052 s, max at 0.26 mHz and 7.72 deg
850 modulationofabout12mHz agreeingwellwith literature
−100
values.
800
−80
−60 750
The most important advantageover “classical”search
−40 700 algorithms is the semi-coherent approach, allowing to
−20 650 combine data taken on several occasions into a single,
[mHz] 0 600 more sensitive search. However, it should be noted that
the Hough transform requires a large amount of CPU
20
550
power, a typical search over a frequency bandwidth of a
40
500 few hundredHz caneasilytake several10,000CPU-hours
60
— the current focus is to lower this number.
450
80
Finally, the author would like to thank the pulsar
100 400 groupatthe JodrellBank observatory,especially Michael
−100 −80 −60 −40 −20 [m0Hz] 20 40 60 80 100 Kramer and Dunc Lorimer, and the Albert-Einstein-
Institute for the opportunity to work on this project over
Fig.5.47TucanaeC:The maximuminthis Houghplane thepastyears.AtAEIespeciallythemembersofthedata
is at the origin, suggesting no Doppler modulation of the analysis group2 proved to be invaluable sources of knowl-
signal (isolated pulsar). edge. Special thanks go to Maria Alessandra Papa, Curt
Cutler, BernardSchutz, Alicia Sintes, Steffen Grunewald,
BerndMachenschalk,BadriKrishnan,ReinhardPrixand
the maximal measurable Doppler shift for this system is many more!
about 12 mHz. Figure 6 shows the Houghplane featuring
the maximum at a distance to the origin corresponding References
exactly to this Doppler shift.
John Middleditch and Jerome Kristian. A search for young,
More already known pulsars have been found in the
luminous optical pulsars in extragalactic supernova rem-
data set, but are not displayed here, please refer to 3 for
nants. Astrophysical Journal, 279:157–161, April1984.
more information.
S. M. Ransom, J. M. Cordes, and S. S. Eikenberry. A New
SearchTechniqueforShortOrbitalPeriod BinaryPulsars.
Astrophysical Journal, 589:911–920, June 2003.
Summary
CarstenAulbert.FindingMillisecondBinaryPulsarsin47Tu-
Although this new method has not revealedany new pul- canae by Applying the Hough Transformation to Radio
sarsinthewellobserverglobularcluster47Tucanae(yet), Data.PhDthesis,AlbertEinsteinInstitute,Potsdam,Ger-
many,2006.
many already known pulsars have been “rediscovered”in
a blind search. Since the obtained parameters for these 2 This group also uses the Hough transformation to search
sources agree remarkably well with the already published for continuous waves in data from gravitational waves detec-
values, the method has proven to be principally able to tors like GEO600 and LIGO. Please refer to recent papers for
discover new pulsars. further reading, e.g. 4, 5, 6.
Carsten Aulbert:Finding Binary Millisecond Pulsars with the Hough Transform 219
Alicia M. Sintes and Badri Krishnan. Improved hough search
for gravitational wave pulsars. Journal of Physics Confer-
ence Series, 32:206, 2006.
BadriKrishnan.Wideparametersearchforisolatedpulsarsus-
ing the hough transform. Classical and Quantum Gravity,
22:S1265, 2005.
Badri Krishnan, Alicia M. Sintes, M. A. Papa, Bernard F.
Schutz,Sergio Frasca, and Cristiano Palomba. The hough
transformsearchforcontinuousgravitationalwaves. Phys-
ical Review D, 70:082001, 2004.
