Spectral Analyses of DO White Dwarfs and PG1159 Stars from the Sloan Digital Sky Survey

a r X i v :a s t r o -p h /0012548v 1 29 D e c 2000A&A manuscript no.(will be inserted by hand later)1.Validity of General Relativity General Relativity (GR)is the correct description of gravity and space-time.The phe-nomena verified with three classic tests of GR are so well established that they are now used as tools in every-day astronomical practice and even in technological applications.The gravitational bending of light,famously detected in Eddington’s solar eclipse expedition is today used to determine the stellar content of our Galaxy and the Mag-ellanic Clouds (from stellar micro-lensing events detected by the OGLE,MACHO and EROS experiments).Lensing of distant galaxies by intervening galaxy clusters is used to determine the (dark)matter distribution in the latter.Gravitational redshift,first observed in spectra of the white dwarf Sirius B in 1925,has since been detected in the laboratory (Pound-Rebka experiment)and is now of necessity taken into account in surveying practice (the GPS system).The effect is also essential in timing radio pulsars—when compared to some millisecond pulsars,terrestrial clocks clearly run slower at full moon than at new moon.The magnitude of precession of the perihelion of Mercury is dwarfed by the same effect in the Hulse-Taylor pulsar,where the periastron shifts by 4.2◦per year.A similar system,Wolszczan’s binary pulsar,allows a confirmation of the Shapiro delay.Of course,GR also provides the framework for understanding the evolution of our expanding Universe.All these successes allow us to confidently use general relativity,even in domains where its validity has not yet been strictly proven.Observations of certain X-ray binaries (e.g.,Cygnus X-1and the so called X-ray novae),as well as of stellar motions in our Galaxy,and of velocities in the inner cores2Klu´z niak:Neutron stars and general relativityof other galaxies,strongly suggest the existence of black holes.However,the laws of GR have not yet been truly tested in the strongfield regime.1.1.Why neutron starsThe strength of gravity is conveniently parametrized by the mass to size ratio, (M/R)(G/c2).For black holes,of course,GM/(Rc2)∼1,as for the Schwarzschild radius R Sch=2MG/c2.For the Sun,GM⊙/c2≈1.5km,while the solar radius R⊙≈300000km,which yields M⊙/R⊙∼10−5(in units of c2/G).A similar value is obtained for mass/distance in the binary Hulse-Taylor pulsar,where relativistic effects in the or-bital motion are so clearly detected(because the pulsar period is so short≈0.06s,and known to10significantfigures).For white dwarfs,M/R∼10−3.But for neutron stars, M/R∼10−1,and GR effects just outside their surface are about as important as near the black hole surface.As a testbed for GR,neutron stars have one great advantage over black holes—they have a tangible surface which can support magneticfields and can emit X-rays and other radiation.A great deal can be learned about neutron stars without assuming the validity of GR.Hence,a great deal can be learned about GR by observing neutron stars.Today, about1000radio pulsars are known and about100X-ray binaries containing neutron stars,so also in sheer numbers neutron stars have an advantage over black holes.1.2.Basic referencesThe narrative presented in Sections1and2,to a large extent relies on well established observations and theories,which have made their way into excellent textbooks,where detailed references can be found to the literature.Among those,particularly useful in the context of these lectures are the ones by Shapiro and Teukolsky(1983),Lipunov(1992), M´e sz´a ros(1992),Glendenning(1997),and Frank,King and Raine(1985).2.A brief history of neutron stars.Before discussing in detail the properties of rapidly rotating,(at most)weakly mag-netized,compact stars—which are ideal astrophysical objects for testing strong-field pre-dictions of General Relativity—let us recount how they were identified.Klu´z niak:Neutron stars and general relativity3 2.1.Key datesThe basic chronology of the discovery of neutron stars can be found,together with the references,e.g.,in the text by Shapiro and Teukolsky.The following selection reflects my bias of what seems particularly important with the hindsight of today.1914:Adams discovered that the rather dim,L≈3×10−3L⊙,star Sirius B(orbiting Sirius),whose mass had been determined to be M≈0.85±0.10M⊙,has the spectrum of a“white”star—hence the name white dwarf.The unusual combination of low luminosity and high temperature implied a small radius,R≈2×104km.This conclusion was based on an application of the black-body formulaL=4πσB R2T4.(1)1925:Adams measures the redshift,z,of certain lines in Sirius B.Applying general relativity,one can infer the value of M/R from z,and from the known mass a value of the stellar radius,R∼104km.The agreement with the spectroscopically determined value was a great triumph of GR.1926:The Fermi-Dirac statistic is discovered.1926(December):Fowler identifies the agent holding up white dwarfs against gravity—it is the degeneracy pressure of electrons.1930:Chandrasekhar discovers theoretical models of white dwarfs,from which the maximum value for white dwarf mass follows,the famous1.4M⊙.Incidentally,M∼1M⊙and R∼few×103km imply a densityρ∼106g/cm3,which in turn implies a minimum period of possible rotation or vibration of a few seconds:(Gρ)−1/2∼3s.1932:Chadwick discovers the neutron.1932:Landau discusses cold,degenerate stars composed of neutrons.1934:Baade and Zwicky write:“With all reserve we advance the view that supernovae represent the transition from ordinary star to neutron stars.”This remains a remarkable contribution—two years after the discovery of neutrons,Baade and Zwicky correctly explain the mechanism of Supernovae(type II)explosions,find the correct value for the gravitational binding energy released in the creation of a neutron star,∼1053erg,and even identify a site where a neutron star is present(and was discovered35years later!): the Crab nebula.1938:Landau discusses the energy released inside ordinary stars with neutron-star cores(a theoretical precursor of what is now known as a Thorne-˙Zytkow object).At the time,the energy source of the Sun was not known.The great contribution here is the pointing out of the enormous energy released in accretion onto neutron stars.1939:Oppenheimer and Volkoffsolve the relativistic equations of stellar structure for a fermi gas of neutrons,and thus construct thefirst detailed model of a neutron star.4Klu´z niak:Neutron stars and general relativityTheyfind a maximum mass(≈0.7M⊙,lower than the one for modern equations of state), above which the star is unstable to collapse.Thus the road to the theoretical discovery of black holes is paved.1940’s are lost to the Second World War.1950’s:The basic physics of the interior of neutron stars is worked out by the Soviet school,including a detailed understanding of the superfluid phase.1962:Giacconi et al.discover thefirst extrasolar source of X-rays,Sco X-1.1967:Shklovsky derives a model for Sco X-1,in which the X-ray source is an accreting neutron star in a binary system.1967:Pacini points out that neutron stars should rotate with periods P<<1s,and may have magneticfields of surface value B∼1012G.The ensuing dipole radiation is not directly observable,as its frequency2π/P is below the plasma frequency of interstellar space.1967:Radio pulsars with P≤3s discovered by Hewish,Bell et al.1968:Gold gives the“lighthouse”model of radio pulsars.1968:Spin-down of radio pulsars is measured,˙P>0.From this moment,it is clear that pulsars are rotating,compact objects,ultimately powered by the kinetic energy of their rotation.1971:Giacconi et al.discover thefirst of accreting counterparts of radio pulsars,the X-ray pulsar Cen X-3,of period4.84s.Today,many are known,in the period range 0.7s≤P≤10000s.1978:Tr¨u mper et al.discover the∼40keV cyclotron line in the spectrum of the accreting X-ray pulsar Her X-1.From the formula hν=1keV×(B/108G),the inferred value of the magneticfield at the stellar surface is B p=few×1012G,in agreement with the estimates of the dipole strength of ordinary radio pulsars.1982:The discovery of millisecond pulsars by Backer,Kulkarni et al.1996:The discovery of kHz quasi-periodic oscillations(QPOs)in the X-rayflux of low-mass X-ray binaries(LMXBs).1998:The discovery of2.5ms pulsar in the transient LMXB SAX J1808.4-3658by Wijnands and van der Klis.2.2.The physics of identifying neutron starsIt should be apparent from the above review,that the basic physics behind identifying neutron stars is fairly simple.Of course,the discovery was possible only after decades of sustained technological development,particularly in thefield of radio and X-ray detectors, as well as much observational effort.Also,the existence of neutron stars would not have been so readily accepted without the solid theoretical foundations laid down over a periodKlu´z niak:Neutron stars and general relativity5 of many years.But the basic,incontrovertible,observational arguments are really based on two or three simple formulae.Let us accept the theoretical result,that a neutron star is a body of mass M∼1M⊙and radius R∼10km,hence of mean density¯ρ>1014g/cm3.How can we be certain that such bodies have been discovered?a)The mass can be determined directly in some binary systems by methods of classi-cal astronomy(as developed for spectroscopic binaries),essentially by an application of Kepler’s laws.For the binary X-ray pulsars,the errors are rather large,but it is clear that one or two solar masses is the right value.For the binary radio pulsars(the Hulse-Taylor and Wolszczan pulsars),where the pulse phase can be determined very precisely and relativistic effects give much redundancy,the mass has been measured very accurately (to0.01M⊙)and is close to1.4M⊙.For binary(millisecond)radio pulsars with white dwarf companions,the mass function is always consistent with these values.b)In bright,steady,X-ray sources,and especially in X-ray bursters(where the X-ray flux briefly saturates at a certain peak value),one can assume that the radiativeflux is limited,at the so called Eddington value,by a balance between radiation pressure on electrons and gravitational pull on protons.Since both forces are proportional to (distance)−2,there is a direct relation betweenflux and mass.Again,M∼1M⊙is obtained,for L X≈1038erg/s.c)The radius can be determined whenever a thermal spectrum is detected,by a combination of the black-body formula,eq.(1),and of Wien’s law giving the characteristic temperature of a body emitting the thermal spectrum.Thus,for X-ray pulsars,such as Her X-1,the spectrum gives a characteristic temperature of T∼10keV∼108K,which in combination with the luminosity L X∼1037erg/s gives an area of∼1010cm2,consistentwith the area of a“polar cap.”This is the area through which open magneticfield lines pass for a R∼10km star,rotating at P=1.24s,with a B∼1012Gfield.For the non-pulsating bright X-ray source Sco X-1,T∼1keV,L∼1038erg/s,i.e., R∼10km directly,as expected if the accreting material is spread over the whole surface.d)For pulsars,an upper limit to the stellar radius follows from causality,ωR<c, hence R<cP/(2π).For millisecond pulsars,this gives R<100km.e)The moment of inertia of certain pulsars(if they are powered by rotation)can be measured directly in“cosmic calorimeters.”If the luminosity of the Crab nebula (≈5×1038erg/s)is equated to Iω˙ω,for the known period(P=33ms)and its derivative of the Crab pulsar(or the known age of the nebula),the value I≈1045g·cm2is obtained.A similar,but less secure,argument can be given for the famous eclipsing pulsar PSR 1957+20(P=1.6ms,˙P≈10−19).It is thought that the power needed to ablate the6Klu´z niak:Neutron stars and general relativity0.02M⊙companion is∼1038erg/s(assuming isotropic emission from the pulsar).Again, I∼1045g·cm2is obtained.f)Finally,a lower limit to the density can at once be derived for rotating objects from Newton’s formula for keplerian orbital motion:ωK= 4π¯ρ/3.Since 2π/P=ω≤ωK,for any star rotating at a period P,the mean density satisfies¯ρ≥3πG−1P−2.With the known value of Newton’s constant,this gives directly¯ρ>2×1014g/cm3,for SAX J1808.4-3658(P=2.5ms)or the millisecond pulsars,such as PSR 1957+20(P=1.6ms).These basic results are subject to many consistency checks,which in all cases support the basic result that objects with a solid orfluid surface(i.e.,they are not black holes!) have been identified of dimensions M∼1M⊙and R∼10km:i)The gravitational energy released in accretion L∼GM˙M/R is consistent(for the discussed values M∼M⊙and R∼10km)with the mass accretion rate inferred from theoretical studies of binary evolution.ii)In some X-ray bursters,the photosphere clearly expands.Again,spectralfits for the temperature and for the radius of the photosphere(eq.[1]),assuming Eddington luminosity,constrain the M–R relationship,in a manner consistent with the values discussed above.iii)The surface magneticfield measured from the cyclotron line in X-ray pulsars agrees,to an order of magnitude(B p∼1012±1G),with the one inferred for radio pulsars, by applying the notion that the spin down in the latter sources is obtained through balancing the energy loss in the simple dipole formula˙E=−2|¨m|2/(3c2),where|m|= B p R3/2,with the kinetic energy loss of a body of moment of inertia I=1045g·cm2. Incidentally,for millisecond pulsars,the value inferred from spin-down,B p∼109±1G, is consistent with the absence of polar cap accretion(and of associated pulsations)in X-ray bursters and other LMXBs.Thus,as far as the magneticfield is concerned,two or three classes of neutron stars are known—ordinary radio pulsars and accreting X-ray pulsars(B∼1012±1G),millisecond radio pulsars(B∼109±1G),and low-mass X-ray binaries,where there is no evidence for such strong magneticfields(i.e.,B<109G).iv)The observed long-term spin-up and spin-down of accreting X-ray pulsars is also consistent with a moment of inertia I∼1045g·cm2,for torques which are expected at the mass-accretion rates derived from the observed X-rayflux,assumed to be L X∼GM˙M/R∼0.1˙Mc2,and the assumption that the lever arm corresponds to an Alfvenic radius,obtained by balancing the ram pressure with the dipole magnetic pressure,i.e., B2/(8π)∼ρv2r at r=r A,˙M=ǫ4πr2ρv r,B=B p R3/r3,whereǫ∼1is a geometric factor.Klu´z niak:Neutron stars and general relativity7 3.The maximum mass of compact stars3.1.Neutron stars or quark stars?It is clear that radio pulsars and some accreting X-ray sources contain compact objects of properties closely resembling those known from theoretical models of neutron stars. Specifically,there can be no doubt that rotating stars of M∼M⊙and R∼10km exist. However their internal constitution is not yet known.The expected mass and radius of “strange”(quark)stars is similar,the main difference being in that quark stars of small masses would have small radii—unlike neutron stars whose radius generally grows with decreasing mass—(Alcock et al.1986).The observed“neutron stars”could be made up mostly of neutrons,but some of them could also be composed partly,or even mostly,of quark matter.From the point of view of testing GR,the internal constitution of static(non-rotating) stars would matter little,as their external metric,directly accessible to observations, would be independent of their nature—the only parameter in the unique static,spher-ically symmetric,asymptoticallyflat solution(the Schwarzschild metric)is the gravita-tional mass,M,of the central body.However,for rapidly rotating stars,the metric does vary with properties of the body other than its mass,and it would be good to know the precise form of the equation of state(e.o.s.)of matter at supranuclear density.As we have seen,at least some low-mass X-ray binaries(LMXBs)contain stellar rem-nants of extremely high density,exceeding1014g cm−3,and many of them are not black holes because they exhibit X-ray bursts of the type thought to result from a thermonu-clearflash on the surface of an ultra-compact star.Further,in these long-lived accreting systems the mass of the compact star is thought to have increased over time by several tenths of a solar mass above its initial value,and in the process the stars should have been spun up to short rotational periods.The compact objects in the persistent LMXBs are expected to be the most massive stellar remnants other than black holes,hence the most stringent limits on the e.o.s.of dense matter is expected to be derived from the mass of the X-ray sources in low-mass X-ray binaries.Before we discuss how this can be done,let us turn to the maximum mass.3.2.The maximum mass of neutron starsOne quantity that depends sensitively on the e.o.s.is the maximum mass of afluid configuration in hydrostatic equilibrium.For neutron stars this maximum mass,and in general the mass–radius relationship,is known from integrating the TOV equations for a wide variety of e.o.s.(Arnett and Bowers1977).The mass of rotating configurations is also known(Cook et al.1994).Here,I will only briefly review the basic physics behind8Klu´z niak:Neutron stars and general relativitythe existence of the maximum mass and then give an example for strange stars,where the e.o.s.is so simple that the variation of mass with the parameter describing the interactions can be determined analytically.As we know from the work of Chandrasekhar and others,the maximum mass is reached when the adiabatic index reaches a sufficiently low value that the star becomes unstable to collapse.In the Newtonian case,this critical index is4/3,corresponding to the extreme relativistic limit for fermions supplying the degeneracy pressure,when the formula for kinetic energy of a particle E=d r =−Gmρd r=4πr2ρ,i.e.,if the pressure and density scale with somefiducial density,P∝ρ∝ρ0,thenm∝r∝ρ−1/2.Such scalings allow some general statements to be made about the maximum mass,such as the Rhoads-Ruffini limit:M<3M⊙,ifρ≥ρ0>2×1014g/cm3.3.3.Quark starsConversion of some up and down quarks into strange quarks is energetically favorable in bulk quark matter(because the Fermi energy is so high)and it has been suggested that at large atomic number,matter in its ground state is in the form of“collapsed nuclei”Klu´z niak:Neutron stars and general relativity9 with strangeness about equal to the baryon number(Bodmer1971).On this assumption, Witten(1984)discussed the possible transformation of neutron stars to stars made up of matter composed of up,down,and strange quarks in equal proportions,and found the maximum mass of such quark stars as a function of the density of(self-bound)quark matter at zero pressure isρ0≥4×1014g/cm3.Detailed models of these“strange”starshave been constructed(Alcock et al.1986,Haensel et al.1986).Here,I discuss only the maximum mass of such stars.Following Alcock(1991),take a gas of any relativistic particles—the e.o.s.is P g=ρg c2/3.If these are moving in a background of vacuum with uniform energy density ρv c2=B,i.e.,negative pressure p v=−B,then the e.o.s.connecting the total pressure p=p g+p v,with the total densityρ=ρg+ρv,isp=(ρ−ρ0)c2/3,(2) withρ0c2=4B.Witten(1984)showed that for this simple e.o.s.the maximum mass from the TOV equation is M=2M⊙10Klu´z niak:Neutron stars and general relativityequivalently,that quark matter composed of up and down quarks in1:2ratio is unstable to emission of neutrons through the reaction u+2d→n.This implies that the baryonic chemical potential at zero pressure of such quark matter satisfies(Haensel1996)µu,d(0)>939.57MeV.(3)As we neglect the masses of up and down quarks in our considerations,the baryonic chemical potential at pressure P is given by the expression(Chapline and Nauenberg 1976)µ(P)=(P+ρc2)/n=4(A/3)3/4(P+B)1/4,(4) where n is the baryon number density,andρc2=An4/3+B is the energy density.For matter(not in beta equilibrium)composed of deconfined up and down quarks in1:2 ratio,n=n u=n d/2and hence A=(1+24/3)(3¯h c/4)π2/3C−1/3,i.e.,µ(0)∝(B/C)1/4, where C≡1−2αc/πandαc is the QCD coupling constant.Inequality(1)then becomesBπ ρ0(0).Thus,through lowest order in the QCD interaction,thefiducial density is changed,but,this implies that the least upper not the e.o.s.Since the stellar mass scales asρ−1/2bound on the mass of the star as a function of the QCD coupling constant is given for non-rotating strange stars byM max(αc)= 1−2αc4.Measuring the mass of accreting neutron(or strange)starsFinally,we have to confront the question how the mass of the compact objects in LMXBs may be determined.Hopefully,a mass will be measured which will eliminate a class o equations of state of dense matter.Unfortunately,application to X-ray bursters of stan-dard methods for determining the mass function of the binary—and hence constraining the mass of the compact X-ray source—is exceedingly difficult,as the optical emission is usually dominated by that of the accretion disk(e.g.van Paradijs et al.,1996).However, reliable mass values obtained by this method may soon become available,particularly for transient sources,such as the accreting millisecond pulsar SAX J1808.4–3658.The mass of the compact object in an X-ray binary may also be determined by study-ing the time variability of the radiationflux formed in the accretionflow.Specifically,for sufficiently weakly magnetized stars,a maximum frequency is expected corresponding to the presence of the innermost(marginally)stable circular orbit allowed in general relativity(Klu´z niak,Michelson and Wagoner,1990).It has been reported that such a maximum frequency may have been observed,at least in one system where quasi periodic oscillations(QPOs)in the X-rayflux saturate at a particular value(Zhang et al.1998). In this manner,several e.o.s.were excluded(Klu´z niak1998)on the understanding that the maximum observed kHz QPO frequency implies a mass in excess of2M⊙;see also Kaaret et al.(1997).Similar considerations(Bulik et al.1999)exclude static(or slowly rotating)quark stars if the minimum density of quark matter isρ0>4.2×1014g/cm3, and the quark matter is taken to be described by the MIT bag model.The overall conclusion(Klu´z niak1998)is that neutron-star matter may be composed simply of neutrons with some protons,electrons and muons,as models of more exotic neutron-star matter(including hyperons or pion and kaon condensates)do not agree with the simplest interpretation of the kHz QPO data,namely that the maximum frequency observed in the low-mass X-ray binary4U1820-30,i.e.,1066Hz(Zhang et al.1998), is attained in the marginally stable orbit around a neutron star.If the compact stellar remnants in these systems are slowly rotating,the same conclusion would apply to ultra-dense matter in general,at densities greater than4.2×1014g/cm3,as matter composed of massless quarks would also be excluded for such densities(Bulik et al.1999).However, as we have seen,minimum densities smaller than4.2×1014g/cm3seem possible for more realistic models of self-bound quark matter,and this would change the conclusion.For rapidly rotating strange stars the conclusion may be drastically different,as the metric is greatly modified by a pronouncedflattening of the star(this effect is less im-portant for neutron stars).In general,the marginally stable orbit is pushed out by this effect,and a fairly low orbital frequency can be obtained for a low mass star.This is illus-trated in Fig.1(taken from Stergioulas et al.1999)which exhibits the frequency in theinnermost(marginally)stable circular orbit of general relativity(ISCO)as a function of stellar mass,M,for the Schwarzschild metric[the hyperbola f+=2.2kHz(M⊙/M)],as well as the ISCO frequency for strange stars rotating at Keplerian frequencies(i.e.,max-imally rotating,at the equatorial mass-shedding limit),for various values of the density at zero pressure,ρ0of eq.(2).It turns out that for these maximally rotating models,the ISCO is always at1.7to1.8km above the stellar surface,the increase of the ISCO orbital√frequency for these models can then be understood in terms of Kepler’s law:2πf∼Fig.1.The frequency of the co-rotating innermost stable circular orbit as a function of mass for static models(thin,continuous line)and for strange stars rotating at the equa-torial mass-shedding limit(thick lines,in the style of Fig.1).For the static models,this frequency is given by the keplerian value at r=6GM/c2,i.e.,by f+=2198Hz(M⊙/M), and the minimum ISCO frequency corresponds to the maximum mass,denoted by afilled circle,an empty circle,and a star,respectively forρ0/(1014g cm−3)=4.2,5.3,and6.5. Note that the ISCO frequencies for rapidly rotating strange stars can have much lower values,and f+<1kHz can be achieved for strange stars of fairly modest mass,e.g.1.4M⊙,if the star rotates close to the equatorial mass-shedding limit.Thisfigure is from Stergioulas et al.1999.to the marginally stable orbit(Kaaret1997,Zhang1998,Klu´z niak1998).But with the data gathered to date,it seems easier to constrain the e.o.s.of dense matter,on the assumption that the QPO frequency saturates in the ISCO,than to show that this as-sumption is indeed correct.One difficulty is that the physics of accretion disks is still very poorly understood.New data is being gathered daily and new experiments are planned which may lead to a break-through in thisfield.I thank the organizers of this School for their wonderful hospitality in Guanajuato. 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7The Spectral Analysis of Random Signals Summary.When one calculates the DFT of a sequence of measurements of a random signal,onefinds that the values of the elements of the DFT do not tend to“settle down”no matter how long a sequence one measures.In this chapter, we present a brief overview of the difficulties inherent in analyzing the spectra of random signals,and we give a quick survey of a solution to the problem—the method of averaged periodograms.Keywords.random signals,method of averaged periodograms,power spectral den-sity,spectral estimation.7.1The ProblemSuppose that one has N samples of a random signal1,X k,k=0,...,N−1,and suppose that the samples are independent and identically distributed(IID). Additionally,assume that the random signal is zero-mean—that E(X k)=0. The expected value of an element of the DFT of the sequence,a m,isE(a m)=EN−1k=0e−2πjkm/N X k=0.Because the signal is zero-mean,so are all of its Fourier coefficients.(All this really means is that the phases of the a m are random,and the statistical average of such a m is zero.)On the other hand,the power at a given frequency is(up to a constant of proportionality)|a m|2.The expected value of the power at a given frequency 1In this chapter,capital letters represent random variables,and lowercase letters represent elements of the DFT of a random variable.As usual,the index k is used for samples and the index m for the elements of the DFT.In order to minimize confusion,we do not use the same letter for the elements of the sequence and for the elements of its DFT.587The Spectral Analysis of Random Signalsis E(|a m|2)and is non-negative.If one measures the value of|a m|2for some set of measurements,one is measuring the value of a random variable whose expected value is equal to the item of interest.One would expect that the larger N was,the more certainly one would be able to say that the measured value of|a m|2is near the theoretical expected value.One would be mistaken.To see why,consider a0.We know thata0=X0+···+X N−1.Assuming that the X k are real,wefind that|a0|2=N−1n=0N−1k=0X n X k=N−1n=0X2k+N−1n=0N−1,k=nk=0X n X k.Because the X k are independent,zero-mean random variables,we know that if n=k,then E(X n X k)=0.Thus,we see that the expected value of|a0|2isE(|a0|2)=NE(X2k).(7.1) We would like to examine the variance of|a0|2.First,consider E(|a0|4). Wefind thatE(|a0|4)=NE(X4i)+3N(N−1)E2(X2i).(See Exercise5for a proof of this result.)Thus,the variance of the measure-ment isE(|a0|4)−E2(|a0|2)=NE(X4i)+2N2E2(X2i)−3NE2(X2i)=Nσ2X2+2(N2−N)E2(X2i).Clearly,the variance of|a0|2is O(N2),and the standard deviation of|a0|2is O(N).That is,the standard deviation is of the same order as the measure-ment.This shows that taking larger values of N—taking more measurements—does not do much to reduce the uncertainty in our measurement of|a0|2.In fact,this problem exists for all the a m,and it is also a problem when the measured values,X k,are not IID random variables.7.2The SolutionWe have seen that the standard deviation of our measurement is of the same order as the expected value of the measurement.Suppose that rather than taking one long measurement,one takes many smaller measurements.If the measurements are independent and one then averages the measurements,then the variance of the average will decrease with the number of measurements while the expected value will remain the same.Given a sequence of samples of a random signal,{X0,...,X N−1},define the periodograms,P m,associated with the sequence by7.3Warm-up Experiment59P m≡1NN−1k=0e−2πjkm/N X k2,m=0,...,N−1.The value of the periodogram is the square of the absolute value of the m th element of the DFT of the sequence divided by the number of elements in the sequence under consideration.The division by N removes the dependence that the size of the elements of the DFT would otherwise have on N—a dependence that is seen clearly in(7.1).The solution to the problem of the non-decreasing variance of the estimates is to average many estimates of the same variable.In our case,it is convenient to average measurements of P m,and this technique is known as the method of averaged periodograms.Consider the MATLAB r program of Figure7.1.In the program,MAT-LAB takes a set of212uncorrelated random numbers that are uniformly dis-tributed over(−1/2,1/2),and estimates the power spectral density of the “signal”by making use of the method of averaged periodograms.The output of the calculations is given in Figure7.2.Note that the more sets the data were split into,the less“noisy”the spectrum looks.Note too that the number of elements in the spectrum decreases as we break up our data into smaller sets.This happens because the number of points in the DFT decreases as the number of points in the individual datasets decreases.It is easy to see what value the measurements ought to be approaching.As the samples are uncorrelated,their spectrum ought to be uniform.From the fact that the MATLAB-generated measurements are uniformly distributed over(−1/2,1/2),it easy to see thatE(X2k)=1/2−1/2α2dα=α331/2−1/2=112=0.083.Considering(7.1)and the definition of the periodogram,it is clear that the value of the averages of the0th periodograms,P0,ought to be tending to1/12. Considering Figure7.2,we see that this is indeed what is happening—and the more sets the data are split into,the more clearly the value is visible.As the power should be uniformly distributed among the frequencies,all the averages should be tending to this value—and this too is seen in thefigure.7.3Warm-up ExperimentMATLAB has a command that calculates the average of many measurements of the square of the coefficients of the DFT.The command is called psd(for p ower s pectral d ensity).(See[7]for more information about the power spectral density.)The format of the psd command is psd(X,NFFT,Fs,WINDOW)(but note that in MATLAB7.4this command is considered obsolete).Here,X is the data whose PSD one would like tofind,NFFT is the number of points in each607The Spectral Analysis of Random Signals%A simple program for examining the PSD of a set of%uncorrelated numbers.N=2^12;%The next command generates N samples of an uncorrelated random %variable that is uniformly distributed on(0,1).x=rand([1N]);%The next command makes the‘‘random variable’’zero-mean.x=x-mean(x);%The next commands estimate the PSD by simply using the FFT.y0=fft(x);z0=abs(y0).^2/N;%The next commands break the data into two sets and averages the %periodograms.y11=fft(x(1:N/2));y12=fft(x(N/2+1:N));z1=((abs(y11).^2/(N/2))+(abs(y12).^2/(N/2)))/2;%The next commands break the data into four sets and averages the %periodograms.y21=fft(x(1:N/4));y22=fft(x(N/4+1:N/2));y23=fft(x(N/2+1:3*N/4));y24=fft(x(3*N/4+1:N));z2=(abs(y21).^2/(N/4))+(abs(y22).^2/(N/4));z2=z2+(abs(y23).^2/(N/4))+(abs(y24).^2/(N/4));z2=z2/4;%The next commands break the data into eight sets and averages the %periodograms.y31=fft(x(1:N/8));y32=fft(x(N/8+1:N/4));y33=fft(x(N/4+1:3*N/8));y34=fft(x(3*N/8+1:N/2));y35=fft(x(N/2+1:5*N/8));y36=fft(x(5*N/8+1:3*N/4));y37=fft(x(3*N/4+1:7*N/8));y38=fft(x(7*N/8+1:N));z3=(abs(y31).^2/(N/8))+(abs(y32).^2/(N/8));z3=z3+(abs(y33).^2/(N/8))+(abs(y34).^2/(N/8));z3=z3+(abs(y35).^2/(N/8))+(abs(y36).^2/(N/8));z3=z3+(abs(y37).^2/(N/8))+(abs(y38).^2/(N/8));z3=z3/8;Fig.7.1.The MATLAB program7.4The Experiment61%The next commands generate the program’s output.subplot(4,1,1)plot(z0)title(’One Set’)subplot(4,1,2)plot(z1)title(’Two Sets’)subplot(4,1,3)plot(z2)title(’Four Sets’)subplot(4,1,4)plot(z3)title(’Eight Sets’)print-deps avg_per.epsFig.7.1.The MATLAB program(continued)FFT,Fs is the sampling frequency(and is used to normalize the frequency axis of the plot that is drawn),and WINDOW is the type of window to use.If WINDOW is a number,then a Hanning window of that length is e the MATLAB help command for more details about the psd command.Use the MATLAB rand command to generate216random numbers.In order to remove the large DC component from the random numbers,subtract the average value of the numbers generated from each of the numbers gener-ated.Calculate the PSD of the sequence using various values of NFFT.What differences do you notice?What similarities are there?7.4The ExperimentNote that as two ADuC841boards are used in this experiment,it may be necessary to work in larger groups than usual.Write a program to upload samples from the ADuC841and calculate their PSD.You may make use of the MATLAB psd command and the program you wrote for the experiment in Chapter4.This takes care of half of the system.For the other half of the system,make use of the noise generator imple-mented in Chapter6.This generator will be your source of random noise and is most of the second half of the system.Connect the output of the signal generator to the input of the system that uploads values to MATLAB.Look at the PSD produced by MATLAB.Why does it have such a large DC component?Avoid the DC component by not plotting thefirst few frequencies of the PSD.Now what sort of graph do you get?Does this agree with what you expect to see from white noise?Finally,connect a simple RC low-passfilter from the DAC of the signal generator to ground,and connect thefilter’s output to the A/D of the board627The Spectral Analysis of Random SignalsFig.7.2.The output of the MATLAB program when examining several different estimates of the spectrumthat uploads data to MATLAB.Observe the PSD of the output of thefilter. Does it agree with what one expects?Please explain carefully.Note that you may need to upload more than512samples to MATLAB so as to be able to average more measurements and have less variability in the measured PSD.Estimate the PSD using32,64,and128elements per window. (That is,change the NFFT parameter of the pdf command.)What effect do these changes have on the PSD’s plot?7.5Exercises63 7.5Exercises1.What kind of noise does the MATLAB rand command produce?Howmight one go about producing true normally distributed noise?2.(This problem reviews material related to the PSD.)Suppose that onepasses white noise,N(t),whose PSD is S NN(f)=σ2N through afilter whose transfer function isH(f)=12πjfτ+1.Let the output of thefilter be denoted by Y(t).What is the PSD of the output,S Y Y(f)?What is the autocorrelation of the output,R Y Y(τ)? 3.(This problem reviews material related to the PSD.)Let H(f)be thefrequency response of a simple R-Lfilter in which the voltage input to thefilter,V in(t)=N(t),enters thefilter at one end of the resistor,the other end of the resistor is connected to an inductor,and the second side of the inductor is grounded.The output of thefilter,Y(t),is taken to be the voltage at the point at which the resistor and the inductor are joined.(See Figure7.3.)a)What is the frequency response of thefilter in terms of the resistor’sresistance,R,and the inductor’s inductance,L?b)What kind offilter is being implemented?c)What is the PSD of the output of thefilter,S Y Y(f),as a function ofthe PSD of the input to thefilter,S NN(f)?Fig.7.3.A simple R-Lfilter647The Spectral Analysis of Random Signalsing Simulink r ,simulate a system whose transfer function isH (s )=s s +s +10,000.Let the input to the system be band-limited white noise whose bandwidth is substantially larger than that of the fie a “To Workspace”block to send the output of the filter to e the PSD function to calcu-late the PSD of the output.Plot the PSD of the output against frequency.Show that the measured bandwidth of the output is in reasonable accord with what the theory predicts.(Remember that the PSD is proportional to the power at the given frequency,and not to the voltage.)5.Let the random variables X 0,...,X N −1be independent and zero-mean.Consider the product(X 0+···+X N −1)(X 0+···+X N −1)(X 0+···+X N −1)(X 0+···+X N −1).a)Show that the only terms in this product that are not zero-mean areof the form X 4k or X 2k X 2n ,n =k .b)Note that in expanding the product,each term of the form X 4k appears only once.c)Using combinatorial arguments,show that each term of the formX 2k X 2n appears 42times.d)Combine the above results to conclude that (as long as the samplesare real)E (|a 0|4)=NE (X 4k )+6N (N −1)2E 2(X 2k ).。

a rXiv:079.4286v1[astro-ph]26Se p27On the Radial Distribution of White Dwarfs in the Globular Cluster NGC 63971D.S.Davis 2,H.B.Richer 2,I.R.King 3,J.Anderson 4,J.Coffey 2,G.G.Fahlman 5,J.Hurley 6,J.S.Kalirai 7,8ABSTRACT We have examined the radial distribution of white dwarfs over a single HST/ACS field in the nearby globular cluster NGC 6397.In relaxed popula-tions,such as in a globular cluster,stellar velocity dispersion,and hence radial distribution,is directly dependent on stellar masses.The progenitors of very young cluster white dwarfs had a mass of ∼0.8M ⊙,while the white dwarfs themselves have a mass of ∼0.5M ⊙.We thus expect young white dwarfs to have a concentrated radial distribution (like that of their progenitors)that be-comes more extended over several relaxation times to mimic that of ∼0.5M ⊙main-sequence stars.However,we observe young white dwarfs to have a signif-icantly extended radial distribution compared to both the most massive main sequence stars in the cluster and also to old white dwarfs.Subject headings:globular clusters:individual (NGC 6397)—stars:Population II,white dwarfs —Stellar Dynamics1.IntroductionThe large proper motions of some pulsars indicate that they have space velocities of hundreds of kilometers per second,which were presumably imparted to them when they became neutron stars(for recent reviews see Podsiadlowski et al.2005;Romani2005).The question naturally arises,could something of this sort have happened to white dwarfs?They are,of course,not observed to have comparably high velocities,and their births do not involve anything as energetic as a supernova,but there is already a hint in the literature that white dwarfs may have higher velocities than their progenitors did.Weidemann(1977) and Williams(2002)have observed that the numbers of white dwarfs observed in open clusters are lower than expected—a natural explanation being that a fraction of the white dwarfs have escaped from the cluster.To investigate this question further,Fellhauer et al. (2003)performed N-body simulations and found that a kick of magnitude approximately twice that of the cluster velocity dispersion(several km s−1)would deplete a loosely bound open cluster of almost all its white dwarfs.The hypothesis that white dwarfs begin their lives with a small velocity excess leads to scenarios that are particularly conducive,for dynamical reasons,to testing in a globular cluster.The progenitors of the newest white dwarfs in a typical globular cluster had a mass of∼0.8M⊙,but they have become white dwarfs with a mass of∼0.5M⊙.Assuming a quiescent birth,they begin their lives with the spatial distribution that is appropriate for their progenitors,but over the course of a relaxation time—of the order of a few times 108years—they will acquire the more extended spatial distribution that goes with their new,lower mass.However,if white dwarfs are given natal kicks,they then start their lives with a velocity dispersion that is too large for their original spatial distribution,and in the course of a crossing time—only a million years or so—they acquire a more extended spatial distribution,perhaps even more extended than they will have after relaxation has made their spatial distribution correspond to their new masses.In a typical globular cluster,stars well down the white dwarf cooling sequence have been white dwarfs for dozens of relaxation times,while the youngest white dwarfs have entered that stage very recently.Moreover, these clusters have hundreds of white dwarfs,so that the test can be made in a statistically significant way.Finally,because globular clusters are old stellar populations,the mass difference between the progenitors of the youngest and oldest white dwarfs in our sample is <0.09M⊙.Though the white dwarf initial-final mass relation is uncertain(Ferrario et al. 2005;Kalirai et al.2005),the mass difference between the white dwarfs must certainly be much less than the mass difference between their progenitors.This allows us to treat all white dwarfs as essentially of equal mass.The question then is,do young white dwarfs in a globular cluster have a spatial dis-tribution that is more extended than that of their progenitors?Until recently,the problem with addressing this question has been that white dwarfs in a globular cluster,particularly the old,fainter ones,are at a magnitude where they cannot be easily observed.Today, however,deep imaging with HST is able to reach faint enough to image the entire white dwarf sequence in very nearby globular clusters(Richer et al.2006;Hansen et al.2007).We will show here that in NGC6397the radial distribution of its young white dwarfs is signifi-cantly more extended than that of their progenitors and also old white dwarfs.Davis et al. (2006)presented preliminary results on the white dwarf distribution in this cluster and(Heyl 2007a,b)then used Monte Carlo simulations to generate radial distributions of stars with kicks at birth and found that a kick of the order of1.8times the velocity dispersion of the cluster giants was required to account for the effect seen by Davis et al.(2006).2.ObservationsIn NGC6397we used HST’s ACS Wide Field Channel to image a singlefield centered 5′SE of the cluster center for126orbits(GO-10424,for details see Richer et al.2006). The exposure time was divided between the F814W and F606Wfilters.Thefield ranges from179′′to391′′from the cluster center,corresponding to1.3–2.8half-mass radii(Harris 1996).Proper motions,for the elimination offield stars,were measured on archival WFPC2 images.We were able to extend our magnitude limit considerably by blind measurement of the WFPC2images at every point where the ACS images told us that there was a star, but the shorter exposure time of the WFPC2images was still what set the limit of our completeness.Even so,artificial-star tests showed that at the magnitude that corresponds to a white dwarf age of5Gyr our completeness was still92%.Table1summarizes the major properties of NGC6397.The essence of this Letter is to compare the radial distributions of white dwarfs of various ages with those of main-sequence stars of various masses.Neither mass nor age is directly observable;in what follows we will need models of both main-sequence stars and cooling white dwarfs.To assign a mass to each main-sequence star,we use main-sequence models of Baraffe et al.(1997),with the metallicity appropriate for NGC6397.To assign an age to each white dwarf,we use a set of cooling models that were developed using interiors from Wood(1995)and atmospheres from Bergeron et al.(1995).Our cooling models are for0.5 M⊙stars with pure carbon cores and thick hydrogen-rich atmospheres.Shown in Figure1 is the colour-magnitude diagram(CMD)of NGC6397with the main-sequence masses and white dwarf cooling ages superimposed.3.Radial distributionsTo test whether two distributions differ significantly,we make use of two statistical tests: the familiar Kolmogorov-Smirnov(KS)test,which uses the maximum difference between the normalized cumulative distributions,and the Wilcoxon rank-sum(RS)test(equivalent to the Mann-Whitney U-test),which compares rankings within the merged,sorted distribution. Each of the tests gives the probability that two random samplings from the same parent population will show differences as large as the observed ones,though the RS test(because it can be applied as a one-tailed test)is generally more sensitive.Mass segregation is known to be present in NGC6397(King et al.1995;Andreuzzi et al. 2004).We test the sensitivity of our data set to mass segregation by comparing the radial dis-tributions of0.15–0.2M⊙main-sequence stars with0.5–0.6M⊙main-sequence stars(i.e.the presumed true mass of the white dwarfs).Figure2,left section,shows the cumulative radial distributions of these two groups,and the power with which the KS and RS tests can dis-tinguish between them.Our data are clearly able to detect mass segregation,even over the limited radial extent of ourfield.The white dwarfs that have had time to undergo dynamical relaxation should have a distribution similar to that of the∼0.5M⊙main-sequence stars,while the unrelaxed ones should have a more centrally concentrated distribution,like their more massive progenitors. However,NGC6397exists in the Galactic potential,and periodic changes in the lumpy potential could affect the relaxation process.In fact,the orbit of NGC6397has a disk-crossing time of∼100Myr(Milone et al.2006;Kalirai et al.2007),and one might expect disk shocking to play a role in the internal cluster dynamics.However,in relaxed clusters, such as NGC6397,we expect these effects to be most important close to the tidal radius, and insignificant at the half-mass radius(Gnedin et al.1999).To study the distributions of the white dwarfs,we need a“young”and an“old”sample. (When we refer to age here,we mean the time since the star became a white dwarf.)We want all the stars of the young sample to be young enough not to have experienced appre-ciable relaxation since they acquired the lower mass that goes with being a white dwarf. Therefore,we need to look at the relaxation time in our region—bearing in mind,however, that relaxation time is only an approximate time scale rather than an exact quantity.The Harris(1996)catalog gives a relaxation time of0.29Gyr at a half-mass radius of2.′3for NGC6397.Given that even the inner edge of ourfield is at a radius where the star density is lower than it is at2.′3,our stars should remain unrelaxed for∼1Gyr.For choice of the old-sample boundaries there are two considerations.The lower bound-ary in age should be high enough so that the stars have had enough time to have relaxed tothe distribution that is appropriate for their white dwarf mass.The upper age limit must be low enough to ensure that the completeness at the magnitude of the faintest white dwarf is not a function of crowding,and therefore radial position.We take the old group to be from 1.4–3.5Gyr(62stars).For the choice of the young-sample boundaries there is a different set of considerations. The lower boundary should not be less than a crossing time.However,this timescale is of the order of Myrs—shorter than the youngest inferred white dwarf age in our sample—so this boundary can be set to zero Gyr.In principle it would be advantageous to set the upper boundary to select only the very youngest(i.e.age much less than a relaxation time)white dwarfs for this sample,thereby ensuring that the young white dwarfs would have had very little time to undergo interactions with other stars.In practice,it is necessary to set the upper boundary to an age comparable to a relaxation time in order to increase the sample size,while maintaining a sample largely free from the effects of relaxation.We choose a young sample that goes from0.0–0.8Gyr(22stars).We note that we are in no danger of inducing a false signal by choosing an upper boundary that includes some relaxed white dwarfs,but we will rather dilute any signal that is truly there.While our definitions of young and old were guided by physical principles and observa-tional constraints,they remained somewhat arbitrary.In fact,there is some sensitivity to the upper boundary of the young group.If that boundary is placed anywhere from0.4to0.8Gyr the result remains strongly significant,increasing the boundary beyond this decreases the significance.Figure2,right section,compares the radial distributions of the young sample with the old one in NGC6397.Contrary to what one would expect supposing a quiescent birth scenario,the younger white dwarfs tend to be farther from the cluster center than old white dwarfs.This suggests that they have a higher velocity dispersion than old white dwarfs.However,this is not the key point.The most direct test would be to compare the radial distribution of young white dwarfs with that of their progenitors.Obviously,this test is impossible,as main-sequence stars of the mass of the progenitors no longer exist in the cluster.Instead we will test the white dwarf distributions against a series of main-sequence mass samples.Assuming that all main-sequence stars are relaxed,and that the trend is monotonic with mass,if we can show that the radial distribution of the young white dwarfs is inconsistent with the radial distribution of the highest mass main-sequence stars that we observe,we have shown that young white dwarfs are not drawn from the same population as their progenitors.We use the RS test to calculate the probability that the radial distributions of the young and old white dwarfs could come from samplings of the same parent distribution as main-sequence stars of various masses;the results are shown in Figure3.Setting the acceptancelevel at0.05(2σin a one-tailed normal distribution),we can reject the hypothesis that the young white dwarfs have a radial distribution consistent with that of main-sequence stars more massive than∼0.5M⊙in NGC6397.Furthermore,we note that we cannot reject the hypotheses that these same stars have radial distributions consistent with that of lower mass main-sequence stars.The rejection of the former hypothesis leads us to conclude that the young white dwarfs in NGC6397have a radial distribution more extended than that of their progenitors.Lending confidence to our method,Figure3also shows that the radial distribution of the old white dwarfs is not inconsistent with that of main-sequence stars of their supposed mass,but is inconsistent with that of low mass(M<0.25M⊙)main-sequence stars.4.Are white dwarfs born with a kick?We have shown that the radial distribution of young white dwarfs in NGC6397is more extended than would be expected given a quiescent birth.Other than a possible kick at birth,is there an alternate explanation for this observation?One might imagine a mechanism related to the white dwarfs being in binary systems,either through binary disruption or the inhibition of white dwarf formation through common envelope evolution or mergers.However,the binary fraction in NGC6397is low(Cool&Bolton2002),and any explanation invoking binaries is implausible.Alternatively,the newly born white dwarfs could undergo very close encounters during theirfirst passage through the inner regions of the cluster with their reduced masses,and have their velocities increased to greater-than-equilibrium values through stellar interactions alone.We have produced N-body simulations that accurately model both binary evolution and stellar encounters,but omit natal kicks; we do not observe the young white dwarfs to have extended radial distributions in these simulations.A full description of these N-body models can be found in Hurley et al.(2007). NGC6397has a collapsed core,and one may consider if interactions with the central density cusp could somehow create this effect.However,the white dwarf progenitors experienced the same potential as white dwarfs,so it is difficult to see how this could have an effect.Assuming the existence of a natal kick,we can estimate its size in the following way.At the radial distances that we are considering,the measured velocity dispersion of∼0.8M⊙giants in NGC6397is3.3km s−1(Pryor&Meylan1993).In accordance with equipartition of energy in a relaxed system,we expect the velocity dispersion of stars of mass M to be given byσvel=σgiantM⊙population are3km s−1and6km s−1,respectively.These values are comparable to the velocity dispersion of a typical globular cluster,but still well below their escape velocities.The fact that we are examining a relatively small range of radii,and that we have a small number of stars in the white dwarf sample,prevents us from putting tight constraints on the velocity dispersion of the young white dwarfs.However,we note that the young white dwarfs have an extended radial distribution,implying an extra source of velocity dispersion. The most natural explanation for this observation is the existence of a natal kick.We leave it to the theorists to search for a detailed mechanism,but note that asymmetric mass loss while the stars are on the asymptotic giant branch stages could generate a kick of the inferred magnitude(Fellhauer et al.2003).DSD would like to thank J.Wall for useful conversations and the UBC-UGF for funding. HBR is generally funded by NSERC,but support for this project was also provided through UBC by the Vice President for Research,the Dean of Science,the NSERC Emergency Fund, and the Department of Physics and Astronomy.HBR also thanks the Canada-US Fulbright Committee for the award of a Fulbright fellowship.JA,IRK,and JSK received support from NASA/HST through grant GO-10424.JSK is supported by NASA through Hubble Fellowship grant HF-01185.01-A,awarded by the Space Telescope Science Institute,which is operated by the Association of Universities for Research in Astronomy,Incorporated,under NASA contract NAS5-26555.REFERENCESAndreuzzi,G.,Testa,V.,Marconi,G.,Alcaino,G.,Alvarado,F.,&Buonanno,R.2004, A&A,425,509Baraffe,I.,Chabrier,G.,Allard,F.,&Hauschildt,P.H.1997,A&A,327,1054 Bergeron,P.,Wesemael,F.,&Beauchamp,A.1995,PASP,107,1047Cool,A.M.&Bolton,A.S.2002,in ASP Conf.Ser.263:Stellar Collisions,Mergers and their Consequences,ed.M.M.Shara,163Corradi,R.L.M.&Schwarz,H.E.1995,A&A,293,871Davis,S.,et al.2006,Bulletin of the American Astronomical Society,38,1215Drake,J.J.,Smith,V.V.,&Suntzeff,N.B.1994,ApJ,430,610Fellhauer,M.,Lin,D.N.C.,Bolte,M.,Aarseth,S.J.,&Williams,K.A.2003,ApJ,595, L53Ferrario,L.,Wickramasinghe,D.,Liebert,J.,&Williams,K.A.2005,MNRAS,361,1131 Gnedin,O.Y.,Lee,H.M.,&Ostriker,J.P.1999,ApJ,522,935Hansen,B.M.S.et al.2007,astro-ph/0701738Harris,W.E.1996,AJ,112,1487Heyl,J.2007,MNRAS,L91Heyl,J.S.2007,MNRAS,submitted,arXiv:0709.3118Hurley,J.R.,Aarseth,S.J.,&Shara,M.M.2007,ApJ,665,707Kalirai,J.S.et al.2007,astro-ph/0701781Kalirai,J.S.,Richer,H.B.,Reitzel,D.,Hansen,B.M.S.,Rich,R.M.,Fahlman,G.G., Gibson,B.K.,&von Hippel,T.2005,ApJ,618,L123King,I.R.,Sosin,C.,&Cool,A.M.1995,ApJ,452,L33Kraft,R.P.&Ivans,I.I.2003,PASP,115,143Milone,A.P.,Villanova,S.,Bedin,L.R.,Piotto,G.,Carraro,G.,Anderson,J.,King,I.R., &Zaggia,S.2006,A&A,456,517Peterson,R.C.,Rees,R.F.,&Cudworth,K.M.1995,ApJ,443,124Podsiadlowski,P.,Pfahl,E.,&Rappaport,S.2005,in ASP Conf.Ser.328:Binary Radio Pulsars,ed.F.A.Rasio&I.H.Stairs,327Pryor,C.&Meylan,G.1993,in ASP Conf.Ser.50:Structure and Dynamics of Globular Clusters,ed.S.G.Djorgovski&G.Meylan,357Richer,H.B.,et al.2006,Science,313,936Richer,H.B.,et al.2002,ApJ,574,L151Richer,H.B.,et al2004,AJ,127,2771Richer,H.B.,et al.1997,ApJ,484,741Romani,R.W.2005,in ASP Conf.Ser.328:Binary Radio Pulsars,ed.F.A.Rasio&I.H.Stairs,337Sirianni,M.,et al.2005,PASP,117,1049Weidemann,V.1977,A&A,59,411Williams,K.A.2002,PhD thesis,AA(UNIVERSITY OF CALIFORNIA,SANTA CRUZ) Wood,M.A.1995,in LNP Vol.443:White Dwarfs,ed.D.Koester&K.Werner,41Table1.NGC6397:Major PropertiesProperty Value References.—(1)Hansen et al.(2007);(2)Sirianni et al.(2005);(3)Kraft&Ivans (2003);(4)Harris(1996)Fig.1.—The CMD of NGC6397.The white dwarf ages,according to the Wood/Bergeron white dwarf cooling models,and the main-sequence masses,as determined from models of Baraffe,are indicated.The magnitude range of the old(lower)and young(upper)whitedwarfs are shown with arrows.Fig.2.—Left section:A comparison of the radial distributions of0.5–0.6M⊙and0.15–0.20 M⊙main-sequence stars for NGC6397.The upper panel shows the RS test and the lower panel the KS test.The vertical lines in the upper panel indicate the radial position of each star(0.5–0.6M⊙above and0.15–0.20M⊙below).Right section:A comparison of the radial distributions of the young and old white dwarfs.The panels are the same as in the leftfigure with the vertical bars denoting the locations of the old(upper)and young(lower)white dwarfs.Surprisingly,the young white dwarf population is less centrally concentrated thanthe old population.Fig.3.—The RS probability that the radial distribution of a given white dwarf population could be drawn from the same parent distribution as the radial distribution of a given mass of main-sequence star for NGC6397.The dashed lines indicate the equivalent of one-tailed 1σ,2σ,and3σresults.Note:the mass value of the points are the mean values of the mass bins.Adjacent points making up the lines are not independent,and are composed ofapproximately90%the same stars.。

a r X i v :a s t r o -p h /0010299v 1 16 O c t 2000A&A manuscript no.(will be inserted by hand later)ASTRONOMYANDASTROPHY SICS1.IntroductionThe Hamburg/ESO Survey for bright quasars (HES,Wi-sotzki et al.1996)was initiated in 1990to perform aspectroscopic survey,based on digitized objective-prism photographs taken with the ESO Schmidt telescope,in the southern hemisphere.A nominal area of ∼9000deg 2in the southern sky is covered with an average limiting magnitude on the prism plates of B ∼17.5.It was con-ceived as a complement to the Hamburg Quasar Survey (HQS,Hagen et al.1995),which aims at a complete cov-erage of the northern sky except for the galactic plane.Similar surveys have been conducted in the past to searchDate Tel.Instr./CCDGrism Resol.Grat.[˚A ]for faint blue objects.Still in progress in the southern hemisphere are the Edinburgh-Cape survey (Stobie et al.1997),the Homogenous Bright Quasar Survey (Cristiani et al.1995),and the Montreal-Cambridge-Tololo survey (Lamontagne et al.2000).The main goal of both the HES and the HQS survey is the compilation of a large sample of bright QSOs.In a first step QSO candidates are selected by various broad-band color criteria and feature detection algorithms ap-plied to spectra derived from low resolution scans of the prism plates (Wisotzki et al.2000).Since among UV ex-cess objects (i.e.U −B <−0.4)less than 10%can be expected to be QSOs,the Hamburg Quasar Survey and the Hamburg/ESO Survey are also a rich source of white dwarfs of various types,and early-type subdwarfs.In a recent paper,Homeier et2S.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO SurveyTable2.Coordinates,magnitudes,cross-references and spectral types,if already existing,for the observed objects. Cross-references were taken from the Simbad database and McCook&Sion(1999).Most of the EC objects are listed in Kilkenny et al.(1997).Details of individual observing campaigns can be found in Table1.The surveyfields,given in column“field”,are ESO/SERC atlasfields.The B J magnitudes are accurate to better than0.m2.HE0004−1452PHL66598a Nov1.54m1800607000636.0−14361417.1HE0052−2511CT19595Oct1.52m900474005446.7−24552917.9HE0104−3056c94Sep3.6m300412010712.4−30401617.3HE0108−303692Sep3.6m300412011106.3−30203417.7HE0122−2244KUV01223-224597Oct1.52m300476012444.6−22290716.8HE0127−3110BPM47178,GD1363DAH a98a Nov1.54m900413012956.2−30550816.092Sep3.6m30098c Nov1.54m1800HE0149−2518b97Oct1.52m600477015159.5−25031517.1HE0200−555696Oct1.52m300153020157.9−55421617.5HE0203−5013JL281d96Oct1.52m300197020503.1−49590317.0HE0308−1313c98b Nov1.54m300616031107.2−13015317.1HE0309−2105EC03097−210598c Nov1.54m1000547031158.1−20540915.6HE0319−434495Oct1.52m300248032110.1−43340017.7HE0413−330692Feb2.2m300360041520.6−32591115.9HE0423−550295Oct1.52m900157042432.5−54553816.9HE0442−302796Oct1.52m1660421044429.4−30213616.2HE0446−253197Oct1.52m480485044901.4−25263616.9HE0449−255498a Nov1.54m1200485045153.8−25491516.594Nov1.52m180098a Nov1.54m1200HE0453−442395Oct1.52m900251045522.8−44181916.8HE0956−1121EC09565−1121DB93Mar3.6m300709095901.3−11352416.9HE1002−192993Mar3.6m300567100502.8−19442917.1HE1102−185093Mar3.6m300570110509.2−19070517.1HE1109−1953EC11091−1953DB93Mar3.6m300570111140.2−20093716.0HE1128−1421EC11285−1421DB96Mar1.52m300641113105.5−14380916.7HE1128−152496Mar1.52m300641113051.7−15412617.4HE1130−011196Mar1.52m300858113325.9−01283017.6HE1145+014596Mar1.52m300858114833.6+01285917.2HE1146−010996Mar1.52m300858114851.7−01261217.3HE1149−1320PG1149-133,EC11492DB292Feb2.2m300642115150.5−13371416.1HE1214−201796Mar1.52m300573121649.9−20335617.6HE1243−2134EC12436−213496Mar1.52m300574124618.5−21505716.3HE1308−145891Apr3.6m300646131112.9−15142817.2HE1350−161292Feb3.6m300649135334.9−16270517.3HE1352+0026PG1352+004DB496Mar1.52m300865135532.4+00112416.1HE1409−182193Mar3.6m300578141148.6−18350415.6HE1420−191496Mar1.52m300579142324.1−19275417.3HE1421−085693Mar3.6m300722142437.9−09095717.3HE1428−1235EC14289-123592Apr3.6m300650143139.6−12485615.9HE2147−0950c98Sep1.52m300744214955.7−09363116.6HE2237−0509c98Sep1.52m300819223940.7−04541714.0HE2237−3630LHS3841,LP984-7698Sep1.52m480406224005.2−36152017.6S.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO Survey3al.(1998)present an analysis of80DA white dwarfs detected through follow-up observations of the HQS. Reimers et al.(1994,1996,1998)reported on the detec-tion of several new magnetic white dwarfs in the HES.A summary of helium-rich subluminous stars can be found in Heber et al.(1996)and an analysis of47sdO stars from the HQS in Lemke et al.(1998).In this paper we present the analysis of40helium-rich white dwarfs using LTE model atmospheres.The objects have been found in the HES and have entered the quasar candidate sample because of U−B≤−0.18.Due to the high spectral resolution of the HES spectra(∼15˚A at Hγ),it is possible to identify most of the hot stars in the quasar candidate sample(see Reimers&Wisotzki1997for example spectra)by their hydrogen and/or helium absorp-tion lines.Featureless,very hot stars can be identified by their continuum shape.Therefore,the sample of stars pre-sented in this paper is restricted to a temperature regime of10000K<T eff≤20000K–as the analysis shows–in which neither strong absorption lines of helium nor a con-tinuum shape notably different from that of a quasar are present.Because it was not possible to make a definitive distinction between quasars and stars on the basis of the HES objective prism spectra alone,spectroscopic follow-up observations have been performed.2.Observational dataIn Table1and Table2we summarize the follow-up spec-troscopy and give coordinates,magnitudes,names from other surveys,and spectral types,if already existing,for the observed helium-rich white dwarfs.The instruments used for spectroscopy were the ESO Faint Object Spec-trograph and Camera(EFOSC),the Danish Faint Object Spectrograph and Camera(DFOSC),and the Boller& Chivens Spectrograph(B&C).A#in Table1refers to ESO numbering of grisms,for which details can be found in the according telescope manuals.In general a spectral range between about3800˚A and7000˚A up to9000˚A is covered with resolutions between about100to500˚A/mm. The coordinates have been derived from the Digitized Sky Survey I(DSS-I).Therefore,finding charts can easily be obtained from the online DSS-I.Magnitudes are given as B J magnitude,which is defined by the sensitivity curve of the hyper-sensitized Kodak IIIa-J emulsion combined with thefilter function of a Schott BG395filter.B J is roughly equal to B for objects around B−V=0and accurate to better than±0.m2for all sample stars.The spectra of all DB white dwarfs are depicted in Fig.1to3.The spectra of DBA and DBAZ white dwarfs can be found in Fig.4and Fig.5,respectively,and the spectra of the DZ,DQ,and DC white dwarfs are summa-rized in Fig.6.3.Spectral analysis 3.1.ModelsIn order to derive effective temperatures the observed spectra have beenfitted to model spectra of nearly pure helium atmospheres with a small admixture of hydrogen (10−6relative to helium by number)covering a temper-ature range from10000K to35000K in250K steps up to16000K,in500K steps between16000K and20000K, and1000K steps above20000K.log g was given between 7.75and8.5in0.25dex increments.Between16500K and20000K we extended the grid to log g=9.He i line profiles were calculated with our LTE atmosphere codes (see Finley et al.1997for a recent description)in which the improved He i line broadening theory of Beauchamp et al.(1997)was implemented.These Stark profiles have been calculated for temperatures ranging from10000K to 40000K and consider broadening by ions and electrons. Line broadening by neutral perturbers,which becomes im-portant at lower temperatures,was not included in our cal-culation of line profiles.This might lead to some uncertain-ties.However,a temperature determination from the con-tinuum slopes of HE0446−2531and HE0449−2554,for which we have absolutely calibratedflux spectra,showed the deviation in T effto be less than1000K at tempera-tures of about12000K.3.2.Determination of T effEffective temperature and log g were determined in aχ2fitting procedure based on a Levenberg-Marquard algo-rithm(cf.Press et al.1992)by comparison of He i line pro-files with synthetic spectra from the He/H atmospheres. The method is very similar to that described in detail in Homeier et al.(1998).We started thefitting procedure with both tempera-ture and log g as a free parameter.However,at the given data quality the dependence on log g turned out to be rather small,and it was thus not possible to determine it unambiguously.In most cases a higher log g could be com-pensated by a higher temperature,and vice versa,with roughly the sameχ2value.Furthermore,systematic ef-fects like small differences in the starting values for T effand log g might change the solution by much more than the statistical errors.We therefore determined T efffor log g fixed to8,parison of temperatures derived from fits with log g as free parameter andfixed to8only re-vealed small differences,which could often be regarded as equal within their statistical errors.This is reflected by the mean of allfitted log g values of8.15±0.24.In Table3 we have compiled the results of our analysis.The given errors for temperatures and log g are formal statistical er-rors from the covariance matrix of thefit,which do not reflect any systematic errors.As has been discussed by many authors using similar methods(see e.g.Homeier et al.1998;Napiwotzki et al.1999),external errors can be higher by a large factor.4S.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO SurveyFig.1.DB white dwarfsS.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO Survey5Fig.2.DB white dwarfs(cont.)6S.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESOSurveyFig.3.DB white dwarfs(cont.)3.3.Determination of hydrogen and metal abundancesAs for the DB stars,T effwas determined for DBA andDBAZ white dwarfs by comparing He i line profiles withthose from synthetic spectra of our He/H atmospheres.Wedid not perform selfconsistent calculations with hydrogenand metal lines already included in the atmospheres.Inthe next step model atmospheres for the derived effec-tive temperatures were calculated which contain calciumand/or hydrogen in estimated amounts for the respectivestar.The resulting temperature and pressure stratifica-tion was then used to compute detailed synthetic spec-tra with varying calcium and/or hydrogen abundances.For HE0446−2531also magnesium and iron were consid-ered.Abundances were then obtained by comparison ofobserved equivalent widths and line profiles to those fromthe synthetic spectra.Unless otherwise mentioned,equiv-alent widths of Hα,Hβ,and Hγwere determined between6543˚A and6583˚A,4841˚A and4881˚A,and4310˚A and4370˚A,respectively;equivalent widths of the Ca ii dou-blet were determined between3890˚A and3990˚A.When comparing HE0446-2531and HE0449-2554,thegreat difference in calcium abundances(approximately afactor of30)is surprising because temperatures are sim-ilar,and equivalent widths differ only by a factor of2.However,their hydrogen abundances also differ by abouta factor of20.This affects the atmospheric structure sincehydrogen contributes electrons,and changes the opacity.In turn the changed atmospheric structure does influencethe line spectrum.4.Discussion of individual objects4.1.DB white dwarfs4.1.1.HE0127−3110HE0127−3110was observed several times during thefollow-up observations performed for the HES:SpectraS.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO Survey7Fig.4.DBA white dwarfswere obtained in September1992(used here),September1994,November1994,and November1998.Spectra from1994show broad absorption troughs,varying in depth andprofile,whereas the September1992spectrum resembles aDB spectrum.After a careful inspection of individual tele-scope pointings to exclude the accidental observation of adifferent star,we conclude that HE0127−3110is variable.The only common feature of all spectra is an absorp-tion line at5876˚A which can be attributed to He i.Thebroad absorption features in the spectra of1994led tothe classification as a magnetic DA white dwarf with apolarfield strength of176MG(Reimers et al.1996).Inthis scenario the putative He i line at5876˚A is due toa hydrogen Hαcomponent(2s0→3p0),which becomesstationary only at afield strength of235MG with a min-imum wavelength of5830˚A.It is interesting to mentionthat HE2201−2250,whose spectrum looked almost iden-tical to the1994spectra of HE0127−3110(Reimers et al.1996),meanwhile also changed its spectroscopic appear-ance to a DB white dwarf.A detailed discussion will bepublished in a forthcoming paper.4.2.DBA white dwarfs4.2.1.HE1002−1929and HE1102−1850In the spectra of HE1002−1929and HE1102−1850the Hαline is clearly visible with equivalent widths of(3.6±1)˚A and(2.4±0.6)˚A,respectively.The Hβline(W(λ)=(1.1±0.6)˚A)can be detected only in the spectrumof HE1102−1850,despite the larger equivalent width ofHαin HE1002−1929and a similar temperature.Fromthe equivalent widths we calculated hydrogen abundancesof(5+5−2)·10−5and(2+2−1)·10−5for HE1002−1929andHE1102−1850,respectively.Metal lines could not be de-tected.4.2.2.HE2147−0950and HE2237−0509HE2147−0950and HE2237−0509show absorption fea-tures at the position of Hαwith equivalent widths of(3.5±0.5)˚A and(3.4±1.0)˚A,respectively.Because bothspectra are very noisy,this should be taken with caution.The measured equivalent widths would correspond to hy-drogen abundances of(4±2)·10−5and(6±4)·10−5,respectively.4.3.DBAZ white dwarfs4.3.1.HE0446−2531The most prominent lines in the November1998spec-trum of HE0446−2531are hydrogen lines of the Balmerseries up to Hγ,the He i line at4471˚A and a somewhatweaker He i line at5876˚A.The October1997spectrum istoo noisy to let the Balmer lines be discerned.Equivalentwidths for Hαand Hβwere determined between6530˚Aand6590˚A,and4820˚A and4885˚A,respectively.Togetherwith the equivalent width of Hγthey result in a hydrogenabundance of(7+3−2)·10−4.The equivalent width of the cal-8S.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO SurveyTable3.Results from the temperature and log g determination and derived spectral types for the analyzed stars. Equivalent widths are given in˚A ngstr¨o mHE0004−1452DB16800±1008.17±0.0616700±100HE0104−3056DB12600±3208.11±0.3012500±580HE0108−3036DB13600±2608.36±0.2113200±430HE0127−3110DB13800±1208.48±0.0913500±130variable(?),possibly magneticHE0149−2518DB16900±2808.21±0.1716500±310HE0200−5556DB18500±2508.26±0.1217900±230HE0203−5013DB16200±3108.37±0.2016000±290HE0308−1313DB18000±1708.90±0.2018000±350HE0319−4344DB18400±7107.82±0.3918600±800HE0413−3306DB16600±707.80±0.0516700±70HE0423−5502DB15600±2008.12±0.1415500±200HE0446−2531DBAZ12600±2507.77±0.2312600±280W(λ)=50.7+5−3/54.3±5(Ca ii,Oct97/Nov98),W(λ)=8.6±1(Hα),W(λ)=4.7±0.5(Hβ),W(λ)=2.9+0.5−1.0(Hγ)HE0449−2554DBAZ12500±2208.23±0.1512200±290W(λ)=25.0+2−3/22.8+2−3(Ca ii,Nov94/Nov98)W(λ)=2.8±0.5(Hα)HE0956−1121DB18800±908.02±0.0418400±80HE1002−1929DBA16500±1008.26±0.0616200±100W(λ)=3.6±1(Hα)HE1102−1850DBA17000±1008.25±0.0616700±120W(λ)=2.4±0.6(Hα),W(λ)=1.1±0.6(Hβ) HE1109−1953DB17100±707.85±0.0417000±60HE1128−1421DB16900±2408.04±0.1416600±240HE1128−1524DB26800±14208.33±0.1627800±1120HE1130−0111DB15000±3008.00±0.1915100±330HE1145+0145DB16900±2508.02±0.1516600±250HE1149−1320DB17300±1208.07±0.0717600±130HE1214−2017DB16700±2008.05±0.1215700±210HE1243−2134DB13800±2408.44±0.1513300±280HE1308−1458DB19500±1908.35±0.0718800±170HE1350−1612DBAZ15000±808.28±0.0614600±90W(λ)=8±1(Ca ii),W(λ)=2.2+0.3−0.5(Hα),W(λ)=1±0.5(Hβ)HE1352+0026DB15700±1508.09±0.1015400±170HE1409−1821DB18400±1107.88±0.0618300±130HE1420−1914DB17900±2208.24±0.1217200±370HE1421−0856DB17100±907.84±0.0517200±90HE1428−1235DB19500±507.87±0.0219500±60HE2147−0950DBA14600±2208.17±0.1314100±410W(λ)=3.5±0.5(Hα) HE2237−0509DBA13300±3608.34±0.2712900±380W(λ)=3.4±1(Hα)HE0052−2511———possibly DC or magneticHE0122−2244DZ——10000+2000−1000W(λ)=18.4+3−5/19.0±2(Ca ii,Oct97/Nov98)HE0309−2105DC———HE0442−3027———possibly DC or DQ HE0453−4423DC———HE1146−0109DQ———C2bandsHE2237−3630DC———binaryS.Friedrich ,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO Survey9Fig.5.DBAZ white dwarfs 4.3.2.HE 0449−2554Since the November 94spectrum is very noisy HE 0449-2554was reobserved in November 1998.Besides the dou-blet of calcium at 3934˚A and 3968˚A already seen in theNovember 94spectrum,the new spectrum clearly shows an H αline.At 4471˚A and 5876˚A also weak features can be spotted.They are consistent with line depths and line strengths of corresponding He i lines in a synthetic spec-trum with T eff=12200K.The equivalent width of the H αline ((2.8±0.5)˚A )corresponds to a hydrogen abun-dance of (3±1)·10−5.The unweighted mean equivalent width of the calcium doublet from both observing runs(23.9+2−3˚A )was taken to determine a calcium abundanceof (7±2)·10−8.4.3.3.HE 1350−1612In addition to He i lines also H α,H β,and the Ca ii dou-blet at 3934˚A and 3968˚A can be detected in the spectrum.Table 4.Element abundances (relative to helium)for the DBA,DBAZ,and DZ white dwarfsHE 0122−2244DZ 0.04+0.01−0.02HE 0446−2531DBAZ 70+30−202+3−1200±1002±1HE 0449−2554DBAZ 3±17±2HE 1002−1929DBA 5+5−2HE 1102−1850DBA 2+2−1HE 1350−1612DBAZ 2±120+10−15HE 2147−0950DBA 4±2HE 2237−0509DBA6±410S.Friedrich ,et al.:Cool helium-rich white dwarfs from the Hamburg/ESOSurveyFig.6.DZ,DQ,and DC white dwarfsU −B =−0.08±0.05)derived from UBV photometry they were not able to classify the object.However,our syn-thetic colours derived from prism plates of the HES and confirmed by spectroscopy yield B −V =−0.10±0.08and U −B =−0.88±0.08,respectively,which place HE 0309−2105well in the regime of white dwarfs in thetwo-colour diagram.Together with its featureless spec-trum it is therefore classified as DC.S.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO Survey114.5.3.HE0442−3027C2bands might be the reason for weakflux depressions at 4700˚A and5100˚A.Therefore,HE0442−302might rather be a DQ than a DC.4.5.4.HE1146−0109HE1146−0109displays strong C2bands at4670˚A and 5140˚A.Since no other features can be detected in the spectrum,the star is classified as DQ.4.5.5.HE2237−3630This star has a composite spectrum,which shows no clear features bluewards of5800˚A,and strong TiO bands red-wards of5800˚A.There are hints for absorption features at3730˚A and4026˚A,which could be attributed to he-lium,and one at5170˚A,which could possibly be due to magnesium and iron.We assume that HE2237−3630is a binary star with a helium-rich white dwarf component and a M-dwarf component.5.DiscussionWe have presented a model atmosphere analysis of cool helium-rich white dwarfs found in the Hamburg/ESO sur-vey for which follow-up spectroscopy has been performed. Since the selection criterium for these stars was confined to a continuum slope similar to QSOs,and no strong ab-sorption lines,this sample of helium-rich white dwarfs is biased towards T effbelow20000K.Among the33analyzed DB stars,we found4DBA, if we include HE2147−0950and HE2237−0509with a somewhat uncertain identification of Hα,and3DBAZ stars;hence,a fraction of21%in our sample shows traces of hydrogen.This is consistent with thefindings of Ship-man et al.(1987),who found19%DBA white dwarfs in their complete sample of cool DB stars from the Palomar-Green survey.Especially the DBAZ stars are important in the con-text of the ongoing discussion of accretion of hydrogen and metals from the interstellar medium.Since the diffusion times of metals in helium-rich atmospheres are short com-pared to evolutionary time scales(Paquette et al.1986),it is generally accepted that metals cannot be of primordial origin but must be accreted from the interstellar medium. Compared to the accretion-diffusion model of Dupuis et al.(1992,1993a,1993b),which predicts element abun-dances in cool white dwarf atmospheres in consideration of accretion from the interstellar medium,the derived Ca abundances for the DBAZ stars found and the magne-sium and iron abundance for HE0446−2531are consis-tent with accretion in solar element proportions at high accretion rates.For HE0446−2531it is also possible to determine metal-to-metal ratios.Whereas the Fe/Mg ra-tio falls within the predicted range,the Fe/Ca and Mg/Ca ratios are below their respective lower limits,but given the large errors they might still be compatible with accretion in solar element proportions.Contrary to the metals,which sink down in the at-mospheres and disappear very rapidly,hydrogen has the tendency tofloat on top of a helium-rich atmosphere.It is therefore accumulated with time in the surface layers. Thus one would expect a metal-to-hydrogen ratio,which is below the solar value.As already observed in other helium-rich metal-line white dwarfs(see Dupuis et al.1993b for a compilation)the Ca/H ratio is larger by a factor of103to 104than the solar value for the new DBAZ stars,support-ing the hypothesis that accretion of hydrogen is somehow reduced compared to accretion of metals.The mechanism most widely discussed for this“hydrogen screening”is the propeller mechanism of Illarionov&Sunyaev(1975),orig-inally developed for accretion of interstellar matter onto neutron stars.Wesemael&Truran(1982)first discussed it in the context of white dwarfs.They proposed that metals are accreted in the form of grains onto a slowly rotating, weakly magnetized(105G)white dwarf,whereas ionized hydrogen is repelled at the Alfv´e n radius.However,it re-mains to be demonstrated that the necessary conditions for rotation,magneticfield and hydrogen-ionizing radia-tion are fulfilled to make the propeller mechanism operate. Acknowledgements.The Hamburg/ESO Survey has been sup-ported by the DFG under grants Re353/33and Re353/40. Work on the Hamburg Quasar Survey in Kiel is supported by a grant from the DFG under Ko738/10-3.We want to thank Dr.E.Chavira(INAOE)for supplyingfinding charts of PHL objects.D.K.is grateful to Dr.Jay Holberg(Lunar and Plan-etary Laboratory)and Dr.Jim Liebert(Steward Observatory) for their hospitality during his sabbatical at the University of Arizona.ReferencesBeauchamp,A.,Wesemael,F.,Bergeron,P.,1997,ApJS108, 559Cristiani,S.,La Franca,F.,Andreani,P.,et al.,1995,A&AS 112,347Dupuis,J.,Fontaine,G.,Pelletier,C.,Wesemael,F.,1992, ApJS82,505Dupuis,J.,Fontaine,G.,Pelletier,C.,Wesemael,F.,1993a, ApJS84,73Dupuis,J.,Fontaine,G.,Wesemael,F.,1993b,ApJS87,345 Finley,D.S.,Koester,D.,Basri,G.,1997,ApJ488,375 Friedrich,S.,Koester,D.,Heber,U.,Reimers,D.:1999,Anal-ysis of UV and optical spectra of helium-rich white dwarfs with trace elements.In:Proceedings of the11th Euro-pean wokshop on white dwarfs,J.-E.Solheim&E.Meiˇs tas (eds.),ASP Conf.Ser.,vol.169,505Friedrich,S.,Koester,D.,Heber,U.,Jeffery,C.S.,Reimers,D., 1999,A&A350,865Greenstein,J.L.,Liebert,J.W.,1990,ApJ360,662 Hagen,H.-J.,Groote,D.,Engels,D.,Reimers,D.,1995,A&AS 111,195Heber,U.,Dreizler,S.,Hagen,H.-J.,1996,A&A311,L1712S.Friedrich,et al.:Cool helium-rich white dwarfs from the Hamburg/ESO Survey Homeier,D.,Koester,D.,Hagen,H.-J.,et al.,1998,A&A,338,563Illarionov,A.F.,Sunyaev,R.A.,1975,A&A39,185Kilkenny,D.,O’Donoghue,D.,Koen,C.,Stobie,R.S.,Chen,A.,MNRAS287,867Jaidee,S.,Lyng˚a,G.,1969,Arkiv f.Astronomi5,345Lamontagne,R.,Demers,S.,Wesemael,F.,Fontaine,G.,Ir-win,M.J.,2000,ApJ119,241Lemke,M.,Heber,U.,Dreizler,S.,Napiwotzki,R.,Engels,D.:1997,New results from the stellar component of the Ham-burg Schmidt survey:A sample of sdO stars.In:The thirdconference on Faint Blue Stars,A.G.D.Philip,J.Liebert,R.A.Saffer(eds.),L.Davis Press,Schenectady,N.Y.,375McCook,G.P.,Sion,E.M.,1999,ApJS121,1Napiwotzki,R.,Green,P.J.,Saffer,R.A.,1999,ApJ517,399Paquette,C.,Pelletier,C.,Fontaine,G.,Michaud,G.,1986,ApJS61,197Press,W.H.,Teukolsky,S.A.,Vetterling,W.T.,Flannery,B.P.,1992,Numerical Recipes in FORTRAN,2nd Edition,Cam-bridge University Press,CambridgeReimers,D.,Jordan,S.,K¨o hler,T.,Wisotzki,L.,1994,A&A285,995Reimers,D.,Jordan,S.,Koester,D.,Bade,N.,K¨o hler,T.,Wisotzki,L.,1996,A&A311,572Reimers, D.,Jordan,S.,Beckmann,V.,Christlieb,N.,Wisotzki,L.,1998,A&A337,L13Reimers,D.,Wisotzki,L.,1997,The Messenger88,14Sefako,R.R.,Glass,I.S.,Kilkenny,D.,et al.,1999,MNRAS309,1043Shipman,H.L.,Liebert,J.,Green,R.F.,1987,ApJ315,239Stobie,R.S.,Kilkenny,D.,O’Donoghue,D.,et al.,1997,MN-RAS287,848Wesemael,F.,Truran,J.W.,1982,ApJ260,807Wisotzki,L.,K¨o hler,T.,Groote,D.,Reimers,D.,1996,A&AS115,227Wisotzki,L.,Christlieb,N.,Bade,N.,Beckmann,V.,K¨o hler,T.,Vanelle,C.,Reimers,D.,2000,in press。

是星星英语作文模板英文回答：What is a Star?A star is a massive, luminous sphere of plasma that emits electromagnetic radiation. Stars are formed whenlarge amounts of gas and dust collapse under their own gravity. The gravity causes the gas to heat up, and theheat causes the gas to glow.The size, color, and brightness of a star aredetermined by its mass, temperature, and age. The mass of a star is the amount of matter it contains. The temperatureof a star is determined by the amount of energy it produces. The age of a star is determined by how long it has been shining.Stars are classified into different types based ontheir spectral type. The spectral type of a star isdetermined by the color of its light. The color of a star's light is determined by the temperature of its surface.The most common type of star is a main-sequence star. Main-sequence stars are stars that are fusing hydrogen into helium in their cores. Main-sequence stars range in size from small, red dwarfs to large, blue giants.Other types of stars include white dwarfs, neutron stars, and black holes. White dwarfs are stars that have exhausted their hydrogen fuel and have collapsed undertheir own gravity. Neutron stars are stars that have collapsed even further than white dwarfs. Black holes are regions of spacetime where gravity is so strong that nothing, not even light, can escape.Stars are an important part of the universe. They provide light, heat, and energy to the planets that orbit them. Stars also play a role in the formation of new stars and planets.中文回答：什么是星星。

小学上册英语第3单元期末试卷(含答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The capital of Comoros is ________ (莫罗尼).2.My best friend has a __________ (美好的) smile.3.The _______ (章鱼) has eight arms.4.I enjoy watching ______ (纪录片) about animals. It helps me learn more about nature and ______ (生态).5.The ____ is known for its impressive speed and agility.6.My cousin is a talented __________ (运动员) in basketball.7.How many strings does a standard guitar have?A. 4B. 5C. 6D. 7答案: C8.He has a pet ___ . (fish)9.The __________ (历史的价值观) influence identities.10. A ______ has a long life cycle.11.Inertia is the tendency of an object to stay at ______ (rest).12.An indoor garden can improve your ______ (空气质量).13. A _______ can be used to change the direction of a force.14.What do you call the study of the universe?A. BiologyB. AstronomyC. GeologyD. Physics答案:B15.What is the capital of Comoros?A. MoroniB. MoutsamoudouC. DomoniD. Mitsamiouli答案: A16. A _______ (小猴子) swings from tree to tree in the jungle.17.What is the name of the process by which plants release oxygen?A. RespirationB. PhotosynthesisC. TranspirationD. Germination答案:b18.In chemistry, a solvent is the substance that does the _____.19. A __________ is a small, winged insect known for its beauty.20.The __________ (历史的影响) can create change.21.I like to eat ______ on my sandwich.22.The __________ is the capital city of Brazil. (巴西利亚)23.The house is ___. (colorful)24.I love to visit ______ (自然保护区) to learn about wildlife and conservation efforts. It’s important to protect our planet.25.Maple trees change color in the _____ (秋天).26.My aunt loves to explore new ____.27.__________ is the process of combining substances to create new ones.28.The __________ is a famous historical site in Egypt. (金字塔)29.The ______ is a large ball of gas that gives us light.30.What is the capital of the United States?A. New YorkB. Washington D.C. C. Los AngelesD. Chicago答案:B31. A _____ (植物) can tell us about the environment it grows in.32.The bicycle is ___ (new/old).33.The chemical symbol for niobium is _____.34.I like to _____ (read) adventure stories.35.What is the capital city of China?A. BeijingB. ShanghaiC. Hong KongD. Taipei答案: A36.The Rocky Mountains are located in __________.37.Which season is typically the warmest?A. WinterB. SpringC. SummerD. Fall答案: C38.The spider spins a ______ (网) to catch flies.39.The ancient Greeks held festivals to honor their _______.40.The _____ (野兔) is very fast and hard to catch.41.We have a ______ (快乐的) family dinner every Sunday.42.My favorite _____ is a stuffed rabbit.43. A solution is a homogeneous mixture of a ______ and a solvent.44.What do you call a place where animals are kept?A. ZooB. FarmC. ParkD. School答案: A45.The visible spectrum of light can be separated using a ______.46.__________ (温度) affects the rate of chemical reactions.47.What is the name of the popular show about a detective solving mysteries?A. Sherlock HolmesB. Law and OrderC. CSID. Monk答案: A48.What do we call the boundary between two countries?A. BorderB. LineC. EdgeD. Frontier答案: A49.The _____ (rain) is pouring.50.My cousin is a ______. She loves to act in plays.51.The _______ (The Trail of Tears) was the forced relocation of Native Americans.52.What do we call the distance around a circle?A. AreaB. DiameterC. CircumferenceD. Radius答案:C53.My cousin is a ______. She enjoys baking.54.Rocks come in three main types: igneous, __________, and metamorphic.55.Which animal is known for its ability to swim?A. CatB. DogC. FishD. Horse答案:C56.What color is the ocean?A. BlueB. GreenC. BrownD. Yellow答案: A57. A _______ is produced when acids and bases react together.58. A _______ is a reaction that occurs in the laboratory.59.The __________ (英法百年战争) lasted from 1337 to 1453.60.What is the term for a baby crocodile?A. HatchlingB. KitC. PupD. Chick答案:a61.The __________ (历史的资料) are invaluable for understanding.62.The ________ is a gentle giant in the jungle.63.The _____ (土壤类型) can affect plant growth.64.I like to build tall structures with my ________ (玩具名称).65.I have a toy _______ that zooms around the house quickly.66.The giraffe has a long ______ (脖子).67.The zebra has distinct _______ (条纹).68.I love to ______ (与朋友聊天) about books.69.The garden is alive with _______ and buzzing bees.70.My friend is a ______. He enjoys building things.71.My grandpa tells great ______. (我爷爷讲的故事很精彩。

烟台2024年08版小学三年级下册英语第2单元期末试卷考试时间：80分钟（总分:140）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题共100分)1. 听力题：A ______ is an animal that can swim and fly.2. 填空题：I can use my ________ (玩具名称) to create art.3. 填空题：I love to decorate my _________ (玩具房间) with colorful items.4. 选择题：What is the largest animal in the ocean?A. SharkB. WhaleC. DolphinD. Octopus答案: B5. 听力题：Astronomers use spectroscopes to analyze the ______ of light from stars.6. 听力题：She is ___ (laughing/sobbing) at the movie.7. 听力题：The ______ moves slowly and has a shell.8. 填空题：The _______ (The Age of Exploration) opened new lands for colonization and trade.9. 选择题：What is the capital city of India?A. MumbaiB. New DelhiC. KolkataD. Bangalore10. 选择题：What is the capital of Panama?a. Panama Cityb. Colónc. Davidd. Santiago答案:a11. 填空题：The ______ (猪) likes to roll in the mud.12. 选择题：What is the opposite of "clean"?A. DirtyB. NeatC. TidyD. Spotless13. 填空题：The _______ (Mount Rushmore) features the faces of four US presidents.14. 填空题：My dad is a _____ (工程师) involved in technical projects.15. 填空题：We visit the ______ (农场) to learn about agriculture.16. 填空题：A ____(desalinization) plant turns seawater into freshwater.17. 选择题：Which animal is known as man's best friend?A. CatB. DogC. BirdD. Fish答案:B18. 听力题：A base feels slippery and can turn __________ paper blue.19. 选择题：What do you call the main character in a play?A. ActorB. ProtagonistC. DirectorD. Supporting character答案:B20. 听力填空题：I believe in the power of friendship. Friends support each other and share good times. I feel lucky to have a friend like __________ who always makes me smile.21. 填空题：I enjoy riding my ______ around the neighborhood.22. 填空题：The __________ (历史的探讨) encourages inquiry.23. 选择题：What is the capital of Nigeria?A. LagosB. AbujaC. KanoD. Port Harcourt答案:B24. 填空题：Birds can ______ (飞) in the sky.25. 选择题：What do we call a group of stars that form a pattern?A. GalaxyB. ConstellationC. NebulaD. Asteroid26. 听力题：The ancient Greeks held the first Olympic Games in ________.27. 填空题：The __________ (历史的共享) enriches culture.28. 选择题：What is the color of grass?A. BlueB. YellowC. GreenD. Brown答案:C29. 选择题：What is the opposite of old?A. NewB. YoungC. RecentD. Fresh答案:A30. 填空题：I love to ______ (分享) my toys.31. 选择题：What is the name of the main character in "Peter Pan"?A. TinkerbellB. Captain HookC. WendyD. Peter答案:D32. 听力题：I have a new ___. (computer)33. 填空题：The best time of the year is ______ (假期). I can travel and visit my ______ (亲戚). We have so much fun!34. 选择题：What is the term for the distance light travels in one year?A. Light-YearB. Astronomical UnitC. ParsecsD. Cosmic Yard35. 选择题：What do you call a plant that grows in water?A. CactusB. FernC. Aquatic plantD. Tree答案:C36. 选择题：What do you call the process of making cheese?A. FermentationB. CurdlingC. PasteurizationD. Distillation答案:B37. 选择题：How many hours are there in a day?A. 24B. 12C. 36D. 2038. 听力题：The blanket is very ___ (soft/hard).39. 填空题：The _____ (乌龟) basks in the sun on a rock.40. 听力题：A ______ is a large area of land that rises above sea level.41. 听力题：I see a _____ (cat/dog) in the garden.42. 填空题：I enjoy ________ (听故事) before bed.43. 填空题：She is an _____ (科学家) who researches space.44. 选择题：What instrument do you play to make music?A. PencilB. GuitarC. BookD. Chair答案:B45. 选择题：How many points is a touchdown worth in American football?A. 3B. 5C. 6D. 7答案: C46. 选择题：What do you call the outer layer of the Earth?A. CrustB. MantleC. CoreD. Lithosphere答案:A47. 听力题：Chlorine is commonly used as a _____ to disinfect water.48. 选择题：What do we call a piece of fruit that is often orange?A. AppleB. BananaC. OrangeD. Grape49. 填空题：The _____ (蛇) can be found in many different habitats.50. 填空题：I can ______ (做) healthy choices.51. 选择题：What do we call a person who studies the effects of pollution on ecosystems?A. EcologistB. Environmental ScientistC. BiologistD. Chemist答案: A52. 选择题：What is the name of the largest rainforest in the world?A. Amazon RainforestB. Congo RainforestC. TaigaD. Temperate Rainforest53. 听力题：A __________ is a famous site for outdoor festivals.She is _______ (climbing) the stairs.55. 选择题：What is the name of the longest river in South America?A. AmazonB. NileC. MississippiD. Yangtze答案：A56. 填空题：I have a classmate who is really __________ (聪明).57. 听力题：The ability to conduct electricity varies among _____.58. 选择题：What is the capital of Estonia?A. TallinnB. TartuC. NarvaD. Pärnu答案:A59. 填空题：The chef is renowned for his _____ (创新菜肴).60. 填空题：My cousin is a __________ (小提琴演奏者).61. 填空题：The __________ (世界的变化) reflects the impact of human actions.62. 填空题：The _____ (pinecone) falls from pine trees.63. 选择题：What do you call a written work that tells a story?A. PoemB. BookC. ArticleD. Essay答案: BThe giraffe has a very long _________ (脖子).65. 听力题：My friend is a ______. He enjoys building models.66. 填空题:My favorite fruit is ________ because it is sweet.67. 听力题：An acid-base reaction produces ______.68. 选择题：What is the name of the famous clock tower in London?A. Big BenB. Eiffel TowerC. ColosseumD. Statue of Liberty答案:A69. 听力题：I see a _____ bird in the tree. (red/quick/small)70. 听力题：My brother is learning to play the ____ (guitar).71. 听力题：There are many stars in the _____ (sky/sea).72. 选择题：What is 8 x 2?A. 14B. 16C. 18D. 20答案:B73. 听力题：Many species of birds migrate to warmer __________ in winter.74. 听力题：My friend plays on the school ____ (football) team.75. 填空题：My favorite dish is _______ (披萨).The _____ (小马) enjoys running through the fields. 小马喜欢在田野中奔跑。