NextPaper(s)• ModellingthebehaviourofaccretionflowsinX-ray
binariesEverythingyoualwayswantedtoknowaboutaccretionbutwereafraidtoaskDone,Gierlinski&Kubota2007A&ARv..15....1D
• Sec1and2ONLY-ORSec7onlythisisaverylongarticle!• ----------------------------------------------------------------------------------------• 2014MNRAS.437.1698X-rayemissionfromstar-forminggalaxies-III.
CalibrationoftheLX-SFRrelationuptoredshiftz≈1.3Mineo,S.;Gilfanov,M.;Lehmer,B.D
PeriodandMagneticField
A. Harding 2013
accreting sources
Population• Themost'common'
observationalpopulationarenon-accretingpulsars
• Periodsfrom..0.001-100secs
• 22ordersofmagnituderangeindP/dt
• dipolemagneticBs~1019(P/dP/dt)1/2wherePisinseconds,Bingauss
• ‘Millisecondpulsars’arerotation-powered,buthavedifferentevolutionaryhistories,involvinglong-livedbinarysystemsanda‘recyclingaccretionepisodewhichspun-uptheneutronstarandquencheditsmagneticfield
• Wewillnotdiscuss– Xray-DimIsolatedNSs(XDINSs),CentralCompactObject(CCOs)RotatingRadioTransients(RRATs),AXPSandMagnetars...
Longair13.5.3-13.5.5Open circles are in binaries
The diversity of non-accreting NSs
DegenerateCompactObjects-SeeLongairpg394-398• Thedeterminationoftheinternalstructuresofwhitedwarfsand
neutronstarsdependsupondetailedknowledgeoftheequationofstateofthedegenerateelectronandneutrongases
• Intheseobjectsdegeneracypressureismainforcebalancinggravity• pressureisindependentofthetemperaturefordegeneratestars,
onlyneedthefirsttwoequationsofstellarstructure(2.6)tocarryouttheanalysis,
• dp/dr��� �ρ/r2;dM/dr����ρ����• Ineqs13.16-13.24theeqsforawhitedwarfarederived
• M=5.836/μ2eM¤.μe=2forwhitedwarfanthusthe
ChandrasekarmassforawhitedwarfMCh=1.46M!(eq.13.24)• ForNSgeneralrelativityisimportant..
WhiteDwarfs…CourtesyofC.Reynolds
• Size,andpressure
Mass of particle producing degeneracy pressure
Number of nucleons per degenerate particle
• So,anapproximateexpressionforradiusofwhitedwarfis:
• Exactcalculationgives
DegneracyandAllThat-Longairpg395sec13.2.1-2.• Inwhitedwarfs,internalpressuresupportisprovidedbyelectron
degeneracypressureandtheirmassesareroughly<1M¤• thedensityatwhichdegeneracyoccursinthenon-relativisticlimitis
proportionaltoT3/2
• Thisisaquantumeffect:Heisenberguncertaintysaysthat δpδx>h/2π– Thuswhenthingsaresqueezedtogetherandδxgetssmallerthemomentum, p,increases,particlesmovefasterandthushavemorepressure
• Considerabox-withanumberdensity,n,ofparticlesarehittingthewall;thenumberofparticleshittingthewallperunittimeandareais1/2nv(visvelocity)– themomentumperunittimeandunitarea(Pressure)transferredtothewallis2nvp;P~nv p=(n/m) p2(mismassofparticle)
InotherwordsHeisenberguncertaintyprinciple,ΔxΔp≥�
whereΔxistheuncertaintyofthepositionmeasurements,Δpistheuncertaintyofthemomentummeasurements,andħish/2π.• Aspressureincreases,systemwillbemorecompactand,for
electronswithinit,theirdelocalization,Δx,willdecrease.• Thus,theuncertaintyinthemomentaoftheelectrons,Δp,willgrow.
nomatterhowlowthetemperaturedrops,theelectronsmustbetravellingatlargervelocities,contributingtothepressure
• Whenthistermexceedsthatofthepressurefromthethermalmotionsoftheelectrons,theelectronsarereferredtoasdegenerate,andthematerialistermeddegeneratematter.– Thisanalysisistrueevenatalmostzerotemperature.
Degeneracy-continued• Theaveragedistancebetweenparticlesisthecuberootofthe
numberdensityandifthemomentumiscalculatedfromtheUncertaintyprinciple p~h/(2πδx)~hn1/3
• andthusP=h2n5/3/m-ifwedefinematterdensityasρ=[n/m]then• P~ρ5/3independentoftemperatureρ ismassdensity• DimensionalanalysisgivesthecentralpressureasP~GM2/r4
• Ifweequatethesewegetr~M-1/3e.qadegeneratestargetssmallerasitgetsmoremassive
• Athigherdensitiesthematerialgets'relativistic'e.g.thevelocitiesfromtheuncertaintyrelationgetclosetothespeedoflight-thischangesthingsandP~ρ4/3;
• thisisimportantbecausewhenweuseP~GM2/r4wefindthatthepressuredoesnotdependonradiusandjustgetanexpressionthatdependsonmass-thisistheChandrasekarmass.(see13.2.2inLongair)
WhenDoesRelativityBecomeImportant• Particles are moving at relativistic speeds when density
satisfies:
MaximumMassofaCompactObject-Longair13.2.2• The Chandrasekar limit (maximum mass of a white dwarf) is when it costs
less energy for a electron to fuse with a proton to form a neutron then to climb higher in the Fermi sea.– Above this limit the compact object becomes all neutrons (a neutron star)
An alternative way of looking at this is to calculate the equation of state(EOS) of degenerate matter and use hydrostatic equilibrium.
– Pe=(1/20)(3/π)2/3(h2/me) (ρ/μemp)5/3 ----ρ is the total mass density and μemp is the mass per electron (composition of the material) - or more simply
– Pe~ρ5/3 - non relativistic• for relativistic matter Pe=(1/8)(3/π)1/3ch(ρ/μemp)4/3 - notice the appearance of the
speed of light– Pe~ρ4/3
• in hydrostatic equilibrium (remember dP(r)/dr=GM(r)ρ(r)/r2 ; P~GM2/R4
• Setting the 2 pressures equal produces the Chandrasekhar limit at which awhite dwarf collapses to a neutron star M~1.46M� (but depends on its composition μe, eg an iron core??))
ChandrasekarLimiteqs13.15-13.25• ElectrondegeneracyisresponsibleforbalancinggravityinWhite
Dwarfs
gravity
Pressure gradient = weight
NeutronStarSizeCourtesyofC.Reynolds
• So,wecantrytoestimateradiusofneutronstargivenwhatweknowaboutwhitedwarfs
– Weknowthat
– Soweexpect
• Byanalogytowhitedwarfs,neutronstarshave(toacrudeapproximation)…
• Where…
– I.e.,degenerateparticleshavemassmn,andµ=1
InsideaNeutronStar?
see fig 13.13 in Longair
Lots more going on... a lot is uncertain Lattimer and Prakash 2004
RadiusofNS• Usethe'known'densityofnuclearmatter(ρNeutron~1.2x1014g/cm3)andtheChandrasekarmassgivesaradius• RNS~(3MChandra/4πρNeutron)1/3~10kmconsistencybetweentheobservedspinperiods,andneutronstars
Cen X-3 Schreier et al 1971
EOSofNeutronStar-Size/MassRelation• RatherComplex
–HavetouseGeneralRelativisticformofhydrostaticequilibriumequation–Neutronsdonʼtbehavelikeanidealdegenerategas…• strongforceinteractions
arecrucial–Thereremainuncertaintiesaboutthe“equationofstate”ofneutronstars
FundamentalPhysics:TheNeutronStarEquationofState(EOS)
dP/dr = -ρ G M(r) / r2
• High mass limit sets highest possible density achievable in neutron stars (thus, in nature, �the MOST dense�).
• Radius is prop. to P1/4 at nuclear saturation density. Directly related to symmetry energy of nuclear interaction
• Other issues: have to use general relativistic eq for hydrostatic equil
• Maximum mass measurements, limits softening of EOS from hyperons, quarks, other �exotica�.
Li
FunFactfortheFamily• oneteaspoonofaneutronstarhasa
massof~5x1012kilograms.• http://videos.howstuffworks.com/
nasa/13498-chandra-neutron-stars-video.htm
C. Miller
AQuickTour• Somepropertiesofthevarious
'types'ofneutronstars....• IsolatedNeutronStars
– cooling– spinning(radiopulsars)– magnetars
• accretingneutronstars
IsolatedNeutronStars-NonAccreting• Theseobjectsarecooling
fromtheinitialhightemperatureofthesupernovaexplosion
• Recentresultsshowthattheyhaveanalmostpureblackbodyspectrum-
Burwitz et al 2001
• AfterNeutronstariscreatedinasupernova,ifitisisolateditcools
• Therateatwhichitcoolsdependsontheconductivityandheatcapacitywhichdependsonwhatitismadeofandadditionalphysicsnotwellunderstood.
(L.Cominsky)
NeutronStarContinuumSpectroscopyandCooling
Prakash and Lattimer
CoolingObservationalestimatesofneutronstartemperaturesandagestogetherwiththeoreticalcoolingsimulationsforM=1.4M�.• Lattimerand
Prakash2004
InterestingPhysics-WillNotDiscussFurther• Thephysicsof
howneutronstarscooldependcriticallyontheirexactcomposition
IsolatedNeutronStarsLongair13.5.1• Mostisolatedneutronstarsthatare
radioandγ-raypulsars�– rapidlyspinningneutronstarsthatemit
relativisticparticlesthatradiateinastrongmagneticfield
• Thepulsesoriginatefrombeamsofradioemissionemittedalongthemagneticaxis-thepulsarlosesenergybyelectromagneticradiationwhichisextractedfromtherotationalenergyoftheneutronstar.
• toproducepulsedradiationfromthemagneticpoles,themagneticdipolemustbeorientedatananglewithrespecttotherotationaxisandthenthemagneticdipoledisplaysavaryingdipolemoment
• EnergylossgoesasΩ4B2• Astheyradiatethestarspinsdown-
visiblefor~107yrshttp://www.jb.man.ac.uk/~pulsar/Education/Tutorial/tut/tut.html
Taylor 1991Proc. IEEE, 79, 1054
• Theshortestperiod(orangularvelocityΩ)whichastarofmassMandradiusRcanhavewithoutbeingtornapartbycentrifugalforcesis(approximately)Ω2R~GM/R2
• Withanaveragedensityoftheneutronstar ρ,Ω~(Gρ)1/2
– ArotationperiodofP=2π/Ω~1secrequiresdensityof108gm/cm3
• To'radiate'awaytherotationalenergyErot=1/2IΩ2~2x1046I45P-2ergs– τloss~Erot/L~60I45P-2L37-1yr(I=2/5MR2)– WherethemomentofinertiaIisinunitsof1045gmcm2
• IfthestarisspinningdownataratedΩ/dtitsrotationalenergyischangingatarateErot~IΩ(dΩ/dt)+1/2(dI/dt)Ω2~4x1032I45P-3dP/dtergs/sec
• Howeveronlyatinyfractionofthespindownenergygoesintoradiopulses-amajorrecentdiscoveryisthatmostofitgoesintoparticlesandγ-rays.
RadiationMechanism+MagneticField−dE/dt~µ0Ω4p2m0/6πc3.eq13.33Wherepisthemagneticmoment• Thismagneticdipoleradiationextractsrotationalenergyfromthe
neutronstar.– Iisthemomentofinertiaoftheneutronstar,
• -d[1/2IΩ2]/dt=-IΩdΩ/dt=Ω4p2m0/6πc3andsodΩ/dt�Ω3
• TheageofthepulsarcanbeestimatedifitisassumedthatitsdecelerationcanbedescribedbyalawdΩ/dt�Ωnifnisconstantthroughoutitslifetime
• Usingtherelationτ=P/(2dP/dt),thetypicallifetimefornormalpulsarsisabout105−108years.
• Ifthelossofrotationalenergyisduetomagneticbreaking(seederivationinLongair13.40-13.42)B~���x������/d�/dt ��������
RadiationMechanism• Itisconventionaltosetn=3toderivetheageofpulsarsandsoτ=P/(2dP/dt)(seederivationinLongair13.35-13.37).• Usingthisrelationthetypicallifetimefornormalpulsarsisabout
105−108years.• extractingrotationalenergy-dErot/dt=-IΩ(dΩ/dt)=-4πI(dP/dt)P-3
Longair13.39• InmoreuseableunitsarotatingdipolehasaPoyntingfluxof
(Harding2013
in units of 1012 Gauss
• WhereradiopulsarslieintheP,dP/dtplot.– thelinescorrespondtoconstantmagneticfieldandconstantage.
• Ifmagneticbrakingmechanismslows-downtheneutronstarthen(seeeqs13.40-13.42)
• Bs≈3x1015(P/{dP/dt})1/2TBisinteslas
• Theradiopulsationsmakeup~10−4orlessofthespin-downpower
• Thepulsedradiation<10%ofthetotalspin-downpower.
• Mostofthepowerinpulsedemissionisin γraysaroundaGeV
• γ-raypeaksarenotinphasewiththeradiopulses,buttypicallyarrivelaterinphase,
Comparison of Spin Down Energy and γ-ray Luminosity of Pulsars
L γ-ray= spindown energy
L γ-rayα√ (spindown energy)
Caraveo 2010
Magnetars13.5.5TheirdefiningpropertiesoccasionalhugeoutburstsofX-raysandsoft-gammarays,aswellasluminositiesinquiescencethataregenerallyordersofmagnitudegreaterthantheirspin-downluminosities.• Theiraretwoclasses:the‘anomalousX-raypulsars’(AXPs)andthe
‘softgammarepeaters’(SGRs)Magnetarsarethoughttobeyoung,isolatedneutronstarspoweredbythedecayofaverylargemagneticfield.Theirintensemagneticfield,inferredviaspin-downtobe• intherange1014-1015Gisclosetothe‘quantumcriticalfield’
BQED≡m2ec3/αhq=4.4×1013G.,q=charge,αisthefinestructure
• wheretheLandaulevelseparationconstantexceedstherestmassenergyofanelectron,mec2=511keV.
Intheirmostluminousoutburstmagnetarscanbrieflyout-shineallothercosmicsoft-gamma-raysourcescombined[Kaspi2010]
S.Mereghetti-Madrid4-6June2007
Properties?1) �Persistent�X-rayemissionLx~1035erg/s;kT~0.5keV+hardpowerlawP~5-12shighspin-down10-11–10-13s/s;dEROT/dt<<Lx2)Short(<1s),super-EddingtonburstskT~30keV3)GiantFlares-rareevents!L~1044–>1046ergs
9 AXPs + 4 SGRs in our Galaxy and in Magellic. Clouds
What is a Magnetar ? Isolated neutron star powered by magnetic energy B~1014-1015 Gauss-the origin of strong magnetism in neutron stars is not well understood
SGR 1806-20 - INTEGRAL – Dec. 27, 2004 Giant Flare
E > 80 keV
Mereghetti et al. 2005, ApJ 624, L105
S.Mereghetti-Madrid4-6June2007
Mereghetti et al. 2005, ApJ 624, L105 2.8 light seconds
Initial giant pulse backscattered by the Moon
Peak affected by instrument saturation
SGR 1806-20 Giant Flare 2004 Dec 2004
AccretingNeutronStars:Longair13.5.2-AlsoCh14• Thesearethebrightest
x-raysourcesintheskyandwerethefirstx-raysourcesdiscovered
• Theyhaveawiderangeofproperties(spectralandtemporal)andshowanalmostbewilderingarrayofbehaviors
• Theirluminositiesrangeover6ordersofmagnitudeandarehighlyvariable