Table Of ContentACTAASTRONOMICA
Vol.55(2005)pp.1–16
Massandradius determination forthe neutron starinX-ray burst
source 4U/MXB1728-34
6
0 by
0
2 A.Majczyna and J.Madej
n
WarsawUniversityObservatory,Al. Ujazdowskie4,00-478Warsaw,Poland
a
J email: [email protected]
4
1
v Abstract
6
7 We analyzed archival X-ray spectra of MXB 1728-34obtained in 1996-99by
0 theProportionalCounterArrayonboardoftheRXTEsatellite. X-rayspectrawere
1 fitted to our extensive grids of model atmosphere spectra to determine the effec-
0
tivetemperature T ontheneutronstarsurface,logarithmofsurfacegravity logg,
eff
6
andthegravitationalredshift z simultaneously. Wehavechosenfittingbynumeri-
0
/ calmodelspectraplusbroadGaussianline,modifiedbyinterstellarabsorptionand
h
the absorption on dust. We arbitrarily assumed either hydrogen-heliumchemical
p
compositionofa modelatmosphere,orH-He-Femixtureinsolarproportion. The
-
o statistically bestvaluesof logg, and z subsequentlywereusedto determinemass
r andradiusoftheneutronstar. Weobtainedthebestvaluesoftheparametersforthe
t
as neutronstarinX-rayburstsourceMXB1728-34:masseither M=0.40 or 0.63M⊙
: (forH-HeorH-He-Femodels,respectively),radiusR=4.6 or5.3km, logg=14.6
v
or 14.6 andthegravitationalredshift z=0.14 or 0.22. Allthe aboveparameters
i
X haveverywide 1-s confidencelimits. Their valuesstronglysupportthe equation
r ofstateforstrangematterinMXB1728-34.
a
KeywordsStars: fundamentalparameters–stars: neutron–X-rays: bursts–
stars: individual(MXB1728-34)
1. Introduction
X-ray source MXB 1728-34 was discovered in 1976 by Forman, Tananbaum
&Jones(1976)inthesurveyobservationsoftheUhurusatellite. TypeIX-raybursts
fromthis sourcewere identifiedin the same year byLewin, Clark & Doty (1976)
inSAS-3satellitedata. Opticalcounterpartstillremainsunidentifiedprobablydue
to large extinction in the direction to this source. Recently Martí et al. (1998)
hasidentifiedaninfraredcounterparttoMXB1728-34,buttheconnectionwiththe
X-ray source has not yet been confirmed. MXB 1728-34was observedin a wide
energyrangefromradio(Martíetal. 1998)to g -rayenergies(cf.Claretetal.1994).
Thedeterminationofmassandradiusofaneutronstarsisaveryimportantand
interestingproblem,becauseitallowsonetodetermineortoconstraintheequation
2 A.A.
of state of superdense matter. Distance to the source is also necessary to localize
thesourcerelativelytotheGalacticcenter. ForMXB1728-34thefirstestimateof
its distance d and true radius R was given by van Paradijs (1978), d =4.2±0.2
kpc and R=6.5±0.4 km (see also Foster et al. 1986). Both parameters were
estimatedassumingthatthepeakfluxofagivenburstfromthissourceapproached
theEddingtonlimitingflux. Itbecameimmediatelyclearthatsucharadius R was
smallerthantheminimumradiusexpectedforaneutronstarofthecanonicalmass
of 1.4M⊙.
MXB 1728-34was frequentlyobservedin X-raysafter that year, and parame-
ters of the neutron star were estimated in several papers. Kaminker et al. (1989)
interpretedspectraofthissourceusingsimplifiedemissionmodels. Theyhavede-
rivedthefollowingvaluesoftheneutronstarparameters: mass M=1.4−2.0M⊙
and radius R=6.5−12 km assuming the distance d =6 kpc and pure helium
atmosphere.
DiSalvoetal. (2000)andGallowayetal. (2003)alsoassumed,thatluminous
bursts from this source are standard candles of the bolometric luminosity at the
Eddington limit. They concluded that the distance to this X-ray burster is in the
range4.4–5.1kpc.
Ontheotherhand,Shaposhnikovetal. (2003)obtainedothervalues: d=4.5−
5.0 kpc, M=1.2−1.6M⊙, and R=8.7−9.7 km. Theyalsoconcluded,thatthe
heliummassabundanceY>0.9inatmosphereoftheneutronstarinMXB1728-34.
TheseauthorsusedsemianalyticalmodelsofexpandingatmospheresbyTitarchuk
(1994),andTitarchuk&Shaposhnikov(2002).
A differentapproachto this problemwas used by Li et al. (1999), who inter-
pretedpowerspectraoftheX-raylightcurveandcomparedthemtothemodelby
Osherovich&Titarchuk(1999)andTitarchuk&Osherovich(1999). Astheresult
Lietal. (1999)constrainedtheareaonthe(M, R)plane,wheretheneutronstarof
MXB1728-34islocated.ConstrainedlocalizationofMXB1728-34wascharacter-
isticforastarbuiltupofstrangematter.
We present here for the first time the estimation of mass and radius of MXB
1728-34, obtainedby fitting of archivalRXTE X-ray spectra of this source to the
newextensivegridsofmodelatmospheresandtheoreticalspectraofneutronstars.
This technique also allowed us to obtain the upper limit of the distance to MXB
1728-34. Theoreticalmodel spectra were obtained with the ATM21 code, which
computes model atmospheres of hot neutron stars with the account of Compton
scatteringonfreeelectrons. Thecodetakesintoaccountangle-averagedCompton
scatteringofX-rayphotonswithinitialenergiesapproachingtheelectronrestmass.
Detailed description of the equations and numerical methods were given in a
longseriesofearlierpapers(Madej1991a,b;Joss&Madej2001;Majczynaetal.
2002;Madej,Joss&Róz˙an´ska2004;Majczynaetal. 2005).
Vol.55 3
2. Modelatmospheresandtheoretical spectra
Model atmospheres used in this paper are based on the equation of radiative
transferofthefollowingform
µ¶ Inr¶(z,zµ) =k ′n (1−e−hn /kT)(Bn −In ) (1)
¥
c2 dw ′ n
+ (cid:18)1+2hn 3In (cid:19)I 4p Z n ′s (n ′→n ,~n′·~n)In ′(z,~n′)dn ′
w ′ 0
¥
dw ′ c2
− In (z,µ)I 4p Z s (n →n ′,~n·~n′)(1+2hn ′3In ′)dn ′,
w ′ 0
which defines a very elaborate source function Sn of our models (Madej 1991a).
ThesignificanceofsymbolsinEq. 1iscommonlyknownandwillnotberepeated
here.
We stress here, that ourtheoreticalmodelsuse sophisticated Comptonscatter-
ingcross-sections s (n →n ′,~n·~n′), whichallowforalargephotonenergychange
at the time of a single scattering off electrons with a relativistic thermal velocity
distribution (Guilbert 1981). Frequently used Kompaneets approximationwas re-
jectedinourcalculationsinordertoobtainhighnumericalaccuracyoftheoretical
ComptonisedX-rayspectra.
The actual code is able to compute modelatmospheresand theoreticalX-rays
spectraofveryhotneutronstars,takingalsointoaccountnumerousbound-freeand
free-free monochromatic opacities of various elements, and the equation of state
of idealgas. The codeis “exact”in thatit solvesthe equationof transfercoupled
with the equation of radiative equilibrium using partial linearisation and variable
Eddingtonfactorstechnique(Mihalas1978).
Wecalculated3extensivegridsofmodelatmospheresofhotneutronstarscor-
respondingtovariousarbitrarilyassumedchemicalcompositions.
– Hydrogen-heliummixturewithsolarheliumnumberabundance, N /N =
He H
0.11 (223models),
– Hydrogen-helium-ironmixturewith N /N =0.11 and solar iron number
He H
abundance, N /N =3.7×10−5 (228models),
Fe H
– Hydrogen-helium-ironmixture with N /N =0.11 and 100 × solar iron
He H
numberabundance, N /N =3.7×10−3 (229models).
Fe H
Computedmodelscovertherangeof 1×107≤T ≤3×107 Kwithstepof 106
eff
K, and the rangeof surface gravity 15.0≥logg≥logg (cgs units)with step of
cr
0.1. Hereweintroducedthecriticalgravity g ,forwhichaccelerationexertedby
cr
theradiationpressuregradientanddirectedoutwardjustbalancesgravity. Finally,
we transformed numerical ASCII models to FITS format required by the XSPEC
package.
Propertiesofneutronstartheoreticalspectrawereextensivelydiscussede.g. in
Majczyna et al. (2005), see also earlier theoretical papers of this series. As an
example,Fig.1presentsamodelspectrumwithparameterssimilartotheseofMXB
1728-34, see Section 6 (solid line). The relevant effective temperature equals to
4 A.A.
T eff = 2.3e+07 K
7 log g = 14.5
BB
6.5
6
H-He-Fe mixture
solar abundance of He and Fe
5.5
0 0.5 1
log energy (keV)
Figure 1: Comparison of the neutron star theoretical spectrum (solid line) and the blackbody
spectrum(dashedline). Sampleeffectivetemperature T ,surfacegravity logg andchemicalcom-
eff
positioncorrespond toone of thebest fits. TheBB spectrum inthisFigurewascomputed for the
temperature T .
eff
T =2.3×107 K,surfacegravity logg=14.5 (cgs),andchemicalcompositionis
eff
H-He-Fe of solar proportions. Fig.1 includesalso properlynormalizedblackbody
spectrum of the temperature T =T =2.3×107 K. Spectrumof blackbodywas
eff
routinelyusedinmanyearlierpapersfortheroughrepresentationofanX-rayburst
spectrum.
Fig. 1demonstratesbasicpropertiesofourmodelspectrum. First, thelatteris
shifted to higherenergiesas comparedwith the blackbody. Sucha shiftis typical
forscatteringatmospheres(Madej1974)andcausesthattheratioofcolortoeffec-
tive temperatures, T /T >1. Second, one can note that the shape of precisely
col eff
computedspectrumisdifferentthantheblackbodyandexhibitslowenergyexcess
andseveralspectrallines. InthisFigurespectrallinesarecausedbyhydrogeniciron
andappearinemission.
Boththespecificshapeofanexactspectrumanditsshifttowardhigherenergy
dependon T , logg and also the redshift z. Therefore, these parameterscan be
eff
determinedbycomparingoftheoreticalandobservedRXTEspectra. WhileRXTE
spectralresolutiondoesnotallowtoresolvemostofspectrallines,theycontribute
totheoreticalcountsinchannelsofRXTEproportionalcounters.
Vol.55 5
3. Observationsanddataanalysis
X-ray burster MXB 1728-34was observed by the RXTE satellite many times
at the time period 1996-1999. In our paper we have reanalyzed archival RXTE
spectralobservationstakenfromtheHEASARCdatabase,numbered10073-01-02-
00, 10073-01-03-000,20083-01-01-01,20083-01-02-01,20083-01-04-00,20083-
01-04-01. We haveselected spectraobtainedbythe PCA instrument, since itwas
sensitivein the energyband2-60keV, andhavechosendata fromthe toplayerof
detectors Pcu0, 1, 2. The above spectra correspond to the quiescent phase of the
burster.
Standard-2configurationwasusedtoanalyzethespectra. Thistypeofconfigu-
rationhasaverygoodenergyresolution,however,ithaspoortimeresolution. We
integratedrawspectraover96secondintervals,takingshorterspectrabinnedto16-
secatthetimeoftheirextraction.Weselectedspectraofthissourceoutsidebursts,
because wanted to avoid possible phases of radius expansion of the neutron star
(NS), andrapid changesof its luminosity. We did notincludecorrectionfor dead
time. Inourcase,countratewasnotlargeandneglectingthiscorrectiongenerated
errorlowerthan1%.
Duringpreparationof an observedspectrum forfitting we neglectedcountsin
channelsbelow3.0keV,sinceonecanexpectpoorenergyresolutioninlowenergy
channelsandtoostrongimpactofinterstellarabsorption. Dataabove20keVwere
ignoredduetopoorstatisticsofcounts. Weextensivelyusedthepubliclyavailable
softwareXSPECv. 11.1andtheresponsematrixv. 10.1. TheXSPECsoftwarewas
describedinArnaud(1996).
Claretet al. (1994)havediscovereda hardenergyexcessofthissourcein the
30–200keV energy band with the SIGMA telescope on board GRANAT satelite.
AuthorsfittedhardX-rayspectrumofMXB1728-34byathermalbremsstrahlung
modelwithaveryhighelectrontemperatureof38keV=4.4 ×108 K.Authorsdid
notexplainwhatistheoriginofthisextremelyhotmedium. Theexistenceofvery
hotelectronstherewasnotconfirmedatalatertime.
RXTEobservedthesourceMXB1728-34foraverylongaccumulatedtimepe-
riod, andthereforerecordedmanyhardX-rayphotons. As was written above,we
integratedour spectra over relatively short time period. For such integrationtime
hardpartofthespectrumisveryweak. Forexample,inourtemplatespectrumde-
scribedbythenumber24000themaximumoffluxexhibitsabout160counts/s/keV,
andabove20keVfluxdropsbelow2counts/s/keV.Forsuchalowfluxitisimpossi-
bletoobtainasignaltonoiseratiowhichisusefulforfittingofanyemissionmodels.
We also didnotattemptto fithardX-raypartof the spectrumsince ourmodelat-
mospheresaredarkabove20keV.Therefore,analyzingofhardX-rayspectrumfor
MXB1728-34withourmodelspectrawouldnotyieldanyinformationsregarding
T , logg andtheredshift z.
eff
Characteristic features of our spectra are: line-like absorption feature around
4.5 keV and flux excess around6.5 keV. We note the existence of such an excess
in each analyzed spectrum but sometimes this feature is notprominent. Line-like
featureappearsinsomeofspectra,butsometimesitdecreasestotheapparentnoise
level. Interpretationofthisfeatureisnotclear,andfittingamodelgaussianlineof
centralenergy4.5keVdonotimprovethefit. Thereforewedonotattempttomodel
thisline. Whatisimportant,thelineappearsonlywhenweusetheresponsematrix
6 A.A.
version1996,andinlaterversionsoftheresponsematrixthisproblemwassolved.
ThislinecanbeinterpretedastheinstrumentalxenonL–edgeline.
Situationis quitedifferentincase ofthe fluxexcessaround6.5keV.Approxi-
matingof thisexcessas a broadgaussianline seems necessary. PCA data did not
allowusforthephysicallymeaningfulinterpretationofthis“line”.Alsodetermina-
tionofthecentroidenergycorrespondingtothislineisimpossibleusingthefitting
procedure. In case of spectra recorded by other instruments like MECS on board
BeppoSAXsatellite such a flux excesswas interpretedas the fluorescentironline
producedincolddiskilluminatedbyhardX-rayphotons(seeDiSalvoetal. 2000).
OurmodelatmospheredonotincludeprocesseswhichproducedFeKa line,andwe
appendedthisline“byhand”usingtheXSPECmodelofagaussianline.
Figure 2: SamplefitoftheMXB1728-34spectrumof96-seclength, ObsID10073-01-03-000.
Wehavefittedwabs*plabs(ATM+gaussian)templatespectra.Fitisreasonable(c 2=0.812),andthe
lowerpanelexhibitsresidua.
IfwerestrictedouranalysisandusedthestandardtemplatespectraoftheXSPEC
software, than the best fits consist of a blackbody spectrum plus power law com-
ponentwith high energy cutoff modified by interstellar absorptionof cold matter.
Moreover,if we includedmulticolordisk blackbodymodelthenthe bestfityields
unrealistichightemperaturesinthe innerringofthediskequalto2.8keV. There-
fore, we replacedthe abovemodelbythe emission fromthecomptonisedneutron
star atmosphere, corrected for low energy absorption and a gaussian line. In our
solution emission from the accretion disk is assumed as negligibly small. X-ray
burster MXB 1728-34 exhibits frequentbursts, with average time intervals of 8.4
hours (Basinska et al. 1984). Therefore, its atmosphere does not cool down ef-
Vol.55 7
ficiently between bursts and then X-rays from the colder disk do not contribute
significantly.
Weperformedfittingof18spectrafromtheRXTEarchiveusingextensivegrids
of theoretical spectra computed by the code ATM21, The code was described in
detail e.g. in Madej (1991a); Madej, Joss & Róz˙an´ska (2004); Majczyna et al.
(2005),seealsothepreviousSection.
In order to determine the effective temperature, redshift and surface gravity
of the neutron star we fitted observed spectra of MXB 1728-34 with the model
wabs*plabs(ATM21+gaussian). Suchafittingformuladenotethesumofourthe-
oreticalspectrumplusgaussianlinemodifiedbyinterstellarabsorptionandabsorp-
tion caused by dust. Addition of absorption on dust is reasonable, because this
bursterislocatedinthedirectiontotheGalacticCenter,wheresuchtypeofextinc-
tionisveryhigh.
4. Calculationsofmassandradius
Massandradiusoftheneutronstarwasdeterminedfromthevaluesofsurface
gravity logg and gravitationalredshift z. The effective temperature T was not
eff
usefulatthisstep. Gravitationalredshiftisgivenby:
2GM −1/2
1+z= 1− (2)
(cid:18) Rc2 (cid:19)
where G is thegravitationalconstant, M is theneutronstar mass, R isthe radius
measuredontheNSsurface, c denotesthespeedoflight.Gravitationalacceleration
ontheNSsurfaceequalsto:
GM 2GM −1/2
g= 1− (3)
R2 (cid:18) Rc2 (cid:19)
WesolveEqs. 1-2formass M andradius R,andobtaintheexplicitexpressions:
zc2 (2+z)
R= (4)
2g (1+z)
z2c4 (2+z)2
M= (5)
4gG (1+z)3
Bothmassandradiusofaneutronstararefunctionsonlyofthesurfacegravityg
andthegravitationalredshift z. Theeffectivetemperature T (whichisequivalent
eff
to the bolometric luminosity of an unit area on the NS surface) does not directly
influenceneither M nor R.
Moreover,ourmethodof M and R determinationforaneutronstarexhibitstwo
interestingproperties:
1. Valuesofbothmassandradiusareindependentonthedistancetothesource.
We also do not need to estimate and compare both the bolometric and apparent
X-rayluminositiesofaneutronstartoobtain M and R.
2.BothmassandradiusofaNSdonotdependontheestimateofthedimension-
lessparameter x ,whichisdefinedastherelativeareaoftheNSstarsurfaceactually
emittingX-rays. Ourmethodallowsonetomeasure M and R alsoincases,when
onlypartofthesurface(ofunknownvalue x )isvisibleinX-rays.
8 A.A.
4.1. Theminimummassofaneutron star
For the proper determination of surface gravities and gravitational redshifts it
is necessary to estimate physically reliable minimum mass for neutron stars (see
Section 6). We turn attention of the reader to the paper by Haensel et al. (2002),
who investigatedthe equationof state for dense matter and the minimummass of
neutronstars.
Haenseletal. (2002)predictedthatminimummassesofcoldnonrotatingneu-
tron stars are of the order Mmin ≈0.09M⊙, and this value depends very weakly
on the equation of state. If a neutron star rotates, than the minimum mass M
min
increases. Haensel et al. (2002) present the example that in the case of the most
rapidly rotating radio pulsar (P =1.56 ms or frequency 641 Hz) the minimum
rot
massofuniformlyrotatingcoldneutronstarisintherange 0.54−0.61M⊙ depend-
ingontheexactequationofstate.
ConstrainingofaNSmasswasnecessarytoeliminatemanyofthefitswhichwe
deemasphysicallyunrealistic, sincetheycouldimplylowNSmassesevenbelow
M . Ifthisconstraintwasnotintroduced,thentheaveragedNSmassesinMXB
min
1728-34wouldlowerthantheseclaimedinSection7ofourpaper.
5. Fittingprocedure
Wehavechosentotalof18spectrafrom6differentexposuresofRXTE,which
correspondeithertotheislandorbananastates. Weproceededinthefollowingway.
ForeachmodelatmosphereofagiveneffectivetemperatureT (ontheNSsurface),
eff
logarithm of surface gravity logg, and some trial chemical composition (see the
listingin Sec. 2), we havefixedparametersoftheline: centralenergy E =6.2
line
keV, and the width s =1.2 keV. Redshift z was also fixed at some value in the
range0.00–0.60. Atthetimeoffittingweiteratedthefollowingfreeparameters:
hydrogencolumndensity,twoparametersoftheplabsmodel,normalizationofthe
ATM21model,andthenormalizationoftheline(total5freeparameters).
We iterated the above five free parametersuntil the minimum of c 2 has been
achieved. Sucha procedurewas repeatedfor all othercombinationsof T , logg
eff
and z,andforallthreeavailablechemicalcomposition. Trialvaluesofthegravita-
tionalredshift z werechangedfrom0.00to0.60withstepsof0.01.
The actual fitting was restricted to the effective temperatures T in the range
eff
from 1×107 Kto 3×107 K,changingwithstepof 106 K.Insuchawaywehave
obtainedthetableofmorethan21000valuesof c 2,eachofthemcorrespondingto
theuniquesetofthethreefixedparameters: T , logg,and z,foragivenchemical
eff
composition. Then, we searchedforthe minimumof c 2 in the three-dimensional
space.
Inspectionofthetableof c 2 showed,thatthebestfittedmodelswithironabun-
dance 100 times greater than the solar value produced significantly higher c 2 at
numerous minima, than those obtained for hydrogen-heliummodels. We believe
thatironrichmodelsdonotrepresentrelevantchemicalcompositionoftheneutron
starsurfaceinquiescentstateofMXB1728-34. Therefore,fitstoironrichmodels
willneitherbepresentednordiscussedinsubsequentSections.
Note,thatinthecaseofablackbodyspectrumonecandetermineonlythequan-
tity (1+z)T ,andthevalueof logg remainsunknown. Separationof T and z,
eff eff
Vol.55 9
andthedeterminationof logg ispossibleonlybecauseshapesofrealstellarspec-
tra deviate from a blackbody. This is sometimes a marginal effect, therefore, the
redshiftandthesurfacegravitydeterminationscanbeuncertain.
6. Redshiftandsurfacegravitydetermination
Asiswellknown,fittingoftheobservedX-rayspectrawiththeXSPECsoftware
usuallydonotproduceuniqueresults.Inthisresearchwedidnotobtainauniqueset
of T , logg andz,whichwouldhaveavalueof c 2 distinctlylowerthanremaining
eff
fits. Instead,wealwaysobtainednumeroussetsoftheabovefittingparameterswith
c 2 veryclosetotheminimumvalue.
Foragivenchemicalcompositionwehaveselectedthesetof T , logg and z
eff
from thousands of fits which correspondsto the minimum value, c 2 . We have
min
appendedothersetswith c 2 intherange [c 2 ,c 2 +D ],where D representsthe
min min 1 1
increaseof c 2 smallenoughtobeinthe1-sigmaconfidencerange.
Sizes of various confidence ranges were estimated by Avni (1976), Lampton,
Margon&Bowyer(1976)andPressetal. (1996). Theoreticalmodelsfittedinthis
research have 5 free parameters and, typically, 38 degrees of freedom (38 d.o.f.).
We applied here values for Dc 2 taken from Press et al. (1996), page 692, which
givein ourcase D =5.89/38d.o.f.=0.16. Such a value of D was used in the
1 1
followingstagesofourresearch.
Foreachoftheanalyzed96secX-rayspectrumforMXB1728-34weobtained
usuallyfewhundredsetsof T , logg and z whichbelongtothe [c 2 ,c 2 +D ]
eff min min 1
range.
Shapeoftheconfidencelevelsindicatethattheseparametersarenotcorrelated,
seeFig. 3.
Weautomaticallycomputedmass M andradius R fromEqs. 4–5andrejected
allsetsofparameterswhere M<0.1M⊙. Insuchawaythenumberofacceptable
fits was significantly reduced in order to secure physical reliability of fits used in
furtherconsiderations.
Table1: Thebestvaluesand1-s confidence rangesfor logg and z
H-He H-He-Fe
z 0.14 0.22
best
0.06-0.21 0.06-0.41
logg 14.6 14.6
best
14.2-14.9 14.2-14.9
Our determinationsof the surface gravitationalredshiftcan be comparedwith
the value of z=0.35 given by Cottam et al. (2002) for other neutron star, EXO
0748-676.
Ourdeterminationsofthesurfacegravity logg fortheneutronstarMXB1728-
34areinagreementwiththeoreticalconsiderationsbyBejgerandHaensel(2004).
Both authors determined, that the maximum value of the surface gravity logg=
10 A.A.
Figure3: 1,2,and3-sigmaconfidence levelforthefitofthemodelwithsolariron
abundance. Hereweassumethat MNS≥0.1M⊙.
14.87 foraneutronstarbuildupofnormal(hadronic)matter,and logg=14.78 for
astrangestar.
7. Massandradiusdetermination
ThisSectionpresentsthefinalresultsofouranalysisforbothassumedchemical
compositions.
Ourresultsare explainedin detailin Tables2 and3. Eachrow ofbothTables
presentsdetailed analysis for each of the 96 sec spectra taken into accountin this
paper. ColumnsofbothTablesgive: identificationoftheobservation,nameofthe
spectrum,therangeof logg and z whicharelocatedinthe1-s confidencerange,
andtheminimum c 2. Thelasttwocolumnspresentthefinalresults: rangesofthe
NSmassesandradiicorrespondingtothe1-s confidencerange.
Inordertoproperlyweighttheaveraged1-s limitsofinvestigatedparameters,
wehavecomputedthearithmeticaveragesofthelowerandupperlimitsfor logg,
z, M and R inTables2and3.ResultsalsoaredisplayedinTables1and4andthey
representthebestvaluesofparametersforthecompactstarinMXB1728-34.
Theaveragedvaluesof M, R andtheir1-s confidencerangesarepresentedin
Table4.
As is evident, both mass and radius of the neutron star in MXB 1728-34 are
verylowascomparedwithcanonicalvalues. ThisistrueforbothH-HeandH-He-
Fe chemicalcompositionsof a neutronstar atmosphere. In anyevent, our M and
