Stellar radiation and stellar types. - State that fusion is the main energy source of stars - Explain that , in a stable star , there is an equilibrium between radiation pressure and gravitational pressure . -Define the luminosity of a star .
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-Statethatfusionisthemainenergysource of stars
-Explainthat, in a stablestar, thereisanequilibriumbetweenradiationpressure and gravitationalpressure.
-Define theluminosity of a star.
-Define apparentbrightness and statehowitismeasured.
-Applythe Stefan-Boltzmannlawto compare theluminosity of differentstars.
-StateWien’s (displacement) law and applyittoexplaintheconnectionbetweenthecolour and temperature of stars.
-Explainhowatomicspectramaybeusedto deduce chemical and physical data forstars.
-Describe theoverallclassificationsystem of spectralclasses.
-Describe thedifferenttypes of star.
-Discussthecharacteristics of spectroscopic and eclipsingbinarystars.
-Identifythe general regions of startypeson a Hertzsprung-Russell diagram
A starsuch as ourownSunradiatesanenormousamount of energyintospace—about 1026 J/s. Thesource of thisenergyis nuclear fusion in the interior of thestar, in whichnuclei of hydrogen fuse to produce helium and releaseenergy in theprocess.
Because of thehightemperatures in the interior of thestar, theelectrostaticrepulsionbetweenprotons can beovercome and hydrogennuclei can fuse.
Because of thehighpressure in stellarinteriors, thenuclei are sufficientlyclosetoeachothertogive a highprobability of collision and hencefusion.
Thesequence of nuclear fusionreactionsthattake place iscalledtheproton-protoncycle, and consists of:
Note thatthe net effect of thesereactionsistoturn 4 hidrogennucleiintoonehelium:
Theenergyreleased per reactionisaround 4 x 10-12 J.
Theenergyproducediscarriedawaybythephotons and neutrinos produced in thereactions. As theseparticlesmoveoutwardstheycollidewithsurroundingprotons and electrons and givethemsome of theenergy. Thus, gradually, most of theparticles in thestarwillreceivesome of thekineticenergyproduced. Themotion of theparticlesinsidethestar, as a result of theenergytheyreceive, can stabilizethestaragainstgravitationalcollapse.
Luminosity (L) istheamount of energyradiatedbythestar per unit of time; thatis, itisthepowerradiatedbythestar. Itsunitis W=J/s.
Luminositydependsonthesurfacetemperature and sufacearea of thestar.
Consider a star of luminosityL. Imagine a sphere of radius d centered at thelocation of thestar. Ifthestarisassumedtoradiateuniformly in alldirectios, thentheenergyradiated in a unit of time can bethoughttobedistributedoverthesurface of thisimaginarysphere. A detector of areaa placed somewhereonthisspherewillreceive a smallfraction of this total energy. Thefractionisequaltothe ratio of the detector area a tothe total surfacearea of thesphere; thatisthereceivedenergy per unit of time willbeaL/4πd2.
Thereceivedenergy per unit of time per unit of areaiscalledtheapparentbrightness and isgivenby
b = L/4πd2.
Theunits of apparentbrightness are W/m2
A black-body of surfaceareaA and absolutetemperatureTradiatesenergyaway in theform of electromagneticwaves at a rategivenbythe Stefan-Boltzmannlaw
L = σAT4,
σ = 5.67 x 10-8 W/m2K4
Black-bodyisany object that is a perfect emitter and a perfect absorber of radiation
b = σAT4/4πd2
Most of theenergyisemittedaroundthepeakwavelength. Callingthiswavelengthλ0, weseethatthecolour of thestarismainlydeterminedbythecolourcorrespondingtoλ0.
TheWiendisplacementlaw relates thewavelengthλ0tothesurfacetemperatureT:
λ0T = constant = 2.9 x 10-3 Km
1. Theradius of star A isthree times that of star B and itstemperatureisdoublethat of B. Findthe ratio of theluminosity of A tothat of B.
3. A star has halftheSun’ssurfacetemperature and 400 times itsluminosity. Howmany times biggerisit?
2. Thestars in question 1 havethesameapparentbrightnesswhenviewedformearth. Calculatethe ratio of theirdistances.
4. TheSun has anapproximateblack-bodyspectrumwithmost of theenergyradiated at a wavelength of 5 x 10-7 m. Findthesurfacetemperature of theSun.
The crucial pointforastronomerswasthatstarsemit light waves, and theyhopedthatthewavelengthscouldtellthemsomethingaboutthestars.
Forexample, once anobjectreaches 500°C it has justenoughenergytoemit visible red light, and isliterally red hot. As thetemperatureincreases, theobject has more energy and emits a greaterproportion of higher-energy, shorter, bluerwavelengths and ittransformsfrom red hottowhitehot, becauseitisnowemitting a variety of wavelengthfrom red toblue. Thefilament of a standard light bulboperates at approximately 3,000°C, whichcertainlymakesitwhitehot.
As well as measuringthetemperature of a star, astronomersworkedouthowtoanalysestarlight in ordertoidentify a star’singredients.
Thomas Melvillsubjectedvarioussubstancesto a flame and noticedthateachoneproduced a characteristiccolour. Forexample, tablesaltgave off a brightorange flash of colour. Thedistinctivecolourassociatedwithsalt can betracedtoitsstructure at theatomiclevel. Theorange light isgeneratedbythesodiumatoms. Bypassingthe light fromsodiumthrough a prism, itispossibletoanalyseexactlywhichwavelengths are emitted, and thetwodominantemissions are both in theorangeregion of thespectrum.
Theexactwavelengthsemittedbyeachatomact as a fingerprint. So bystudyingthewavelengthsemittedby a heatedsubstance, itispossibletoidentifytheatoms in thesubstance.
Theprocessbywhich a substanceemits light iscalledspectroscopicemission. Theoppositeprocess, spectroscopicabsortion, alsoexists, and thisiswhenspecificwavelengths of light are absorbed byanatom. So, if a wholerange of wavelengths of light weredirected at vaporisedsalt, thenmost of the light wouldpassthroughunaffected, but a fewkeywavelengthswouldbe absorbed bythesodiumatoms in thesalt. The absorbed wavelengthsforsodium are exactlythesame as theemittedwavelengths, and thissymmetrybetweenabsorption and emissionis true forallatoms.
TheSunishotenoughtoemitwavelengthsovertheentirerange of visible light, butphysicists at thestart of the 19th centurynoticedthatspecificwavelengthsweremissing. Thesewavelengthsrevealedthemselves as fine blacklines in the solar spectrum. Themissingwavelengthshadbeen absorbed byatoms in theSun’satmosphere. Indeed, themissingwavelengthscouldbeusedtoidentifytheatomsthatmake up theSun’satmosphere.
Thistechnique of stellarspectroscopywas so powerfulthat in 1868 British Norman Lockyer and French Jules Janssenindependentlydiscoveredanelement in theSunbeforeitwasdiscovered in Earth. Theidentifiedanabsortion line in sun-light thatcouldnotbematchedwithanyknownatom. Itwasnamedhelium, after Helios, theGreeksun-god. Althoughheliumaccountsfor a quarter of theSun’smass, itisveryrareonEarth and itwouldbeovertwenty-fiveyearsbeforeitwasdetectedhere.
It has beenfound, however, thatmoststarshaveessentiallythesamechemicalcomposition, yet show differentabsorptionspectra. Thereasonforthisdifferenceisthatdifferentstarshavedifferenttemperatures. Considertwostarswiththesamecontent of hydrogen. Oneishot, about 25000 K, and theothercool, about 10000 K. Thehydrogen in thehotstarisionized , whichmeanstheelectronshaveleftthehydrogenatoms. Theseatomscannotabsorbthephotons and maketransitionstohigherenergystates. Thus, thehotstarwillnot show anyabsorptionlines at hydrogenwavelengths.
Following Galileo, astronomershadassumedthatthestarswerestationary. Althoughthestarsall moved acrosstheskyeverynight, astronomersrealisedthatthisapparentmotionwascausedbytheEarth’srotation.
Edmund Halley becameaware of subtlediscrepancies in therecorded positions of thestars Sirius, Arcturus and ProcyoncomparedwithmeasurementsmadebyPtolomymanycenturiesearlier. Halley realisedthatthesedifferenceswerenotdowntoinaccuratemeasurements, butweretheresult of genuineshifts in the positions of thesestarsover time.
In general, detectingpropermotion has requiredcarefulobservations of thecloseststarstakenacrossseveralyears.
Propermotionis a measure of motionacrosstheskyonly, and saysnothingaboutmotiontowardsorawayfromtheEarth, known as radial velocity. In ordertogetthisinformationweneedto combine spectroscopywiththeDopplereffect.
In 1842 Dopplerannouncedthatthemovement of anobjectwouldaffectanywavesitwasemitting. Whenanobjectemittingwavesmovestowardsanobserver, thentheobserverperceives a decrease in thewavelength, whereaswhentheemittermovesawayformtheobserver, thentheobserverperceivesanincrease in thewavelength. Alternatively, theemittermightbestationary and theobservermightbemoving, in which case thesameeffects are apparent.
Thethreespectra show howthe light emittedby a stardependsonits radial motion. Thespectrum in themiddle shows thewavelength of someabsorptionlinesfrom a starwhichisneithermovingclosertonorfartherfromtheEarth. Thespectrumbelow shows redshiftedabsorptionlinesfrom a starwhichismovingawayfromEarth-thelines are identical, excepttheyhaveallbeenshiftedtotheright. Thespectrumabove shows blueshiftedabsorptionlinesfrom a starwhichismovingtowardstheEarth-again, thelines are identical, exceptthis time theyhaveallbeenshiftedtotheleft. Measurement of theshiftallowsthedetermination of the radial velocity of thestar.
Astronomersrealizedthattherewas a correlationbetweentheluminosity of a star and itssurfacetemperature.
TheDanishastronomer, EjnarHertzsprungplottedluminosities versus surfacetemperature and the American, Henry Russell, plottedabsolutemagnitude versus spectralclass. Suchplots are nowcalled HR diagrams.
In the HR diagram, the vertical axis representsluminosity in units of thesun’sluminosity (i.e., 1 correspondstothe solar luminosity of 3.9 x 1026 W). The horizontal axis shows thesurfacetemperature of thestar. Alsoshown at the top of thediagramisthespectralclassforeachstar, whichisanalternativewaytolabelthe horizontal axis. Thescaleontheaxesisnot linear.
Threeclearfeatures emerge fromthe HR diagram:
Moststarsfallon a stripextendingdiagonallyacrossthediagramfrom top lefttobottomright. Thisiscalledthemainsequence.
Somelargestars, reddish in colour, occupythe top right – these are the red giants (large, coolstars)
Thebottomleftis a region of smallstarsknown as whitedwarfs (small and hot).
In fact, about 90% of allstars are mainsequencestars, 9% are whitedwarfs and 1% are red giants. Anotherfeature of the HR diagramisthat, as wemovealongthemainsequencetowardhotterstars, themass of thestarincreases as well. Thus, therightend of themainsequenceisoccupiedby red dwarfs and theleftbybluegiants.
Mainsequencestars produce enoughenergy in theircore, from nuclear fusion of hydrogenintohelium, toexactlycounterbalancethetendency of thestartocollapseunderitsownweight.
OurSunis a typicalmember of themainsequence.
Red giants are verylarge, coolstarswith a reddishappearance. Theluminosity of red giantsisconsiderablygreaterthantheluminosity of mainsequencestars of thesametemperature; they can, in fact, be a millionoreven a billion times bigger. Themass of a red giant can be as much as 1000 times themass of theSun, buttheirhugesizealsoimpliessmalldensities. In fact, a red giantwillhave a central hotcoresurroundedbyanenormousenvelope of extremelyteneous gas.
White dwarfsformwhen a starcollapsingunderitsowngravitationstabilizes as a result of electrondegeneracypressure. Thismeansthattheelectrons of thestar are forcedintothesame quantum states. Toavoidthat, thePauliexclusionprincipleforcesthemtoacquirelargekineticenergies. Thelargeelectronsenergies can thenwithstandthegravitationalpressure of thestar.
Anexampleis Sirius B, a star of massroughlythat of theSunwith a size similar tothat of theEarth. Thismeansthatitsdensityisabout 106 times thedensity of theEarth.
A variable star is a star whose brightness as seen from Earth fluctuates.
This variation may be caused by a change in emitted light or by something partly blocking the light, so variable stars are classified as either:
Intrinsic variables, whose luminosity actually changes; for example, because the star periodically swells and shrinks.
Extrinsic variables, whose apparent changes in brightness are due to changes in the amount of their light that can reach Earth; for example, because the star has an orbiting companion that sometimes eclipses it.
This group consists of several kinds of pulsating stars that swell and shrink very regularly by the star's own mass resonance. Generally the Eddington valve mechanism for pulsating variables is believed to account for cepheid-like pulsations: a certain helium layer of the star has variable opacity depending on the ionization degree, greater opacity for the greater level of ionization. At minimum the star is contracted so that the layer has the higher ionization and opacity, and therefore absorbs fusion energy for the star to expand. When the star swells up to a certain size, the ionization suddenly switches from higher to lower, switching the opacity to lower too. The inner fusion energy now radiates more easily through this star layer, so the star shrinks to the original contracted state, and the cycle begins anew.
Cepheids are important because they are a type of standard candle. Their luminosity is directly related to their period of variation. The longer the pulsation period, the more luminous the star. Once this period-luminosity relationship is calibrated, the luminosity of a given Cepheid whose period is known can be established. Their distance is then easily found from their apparent brightness. Observations of Cepheid variables are very important for determining distances to galaxies within the Local Group and beyond. A relationship between the period and luminosity for classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt in an investigation of thousands of variable stars.
Theseappear as twoseparatestarswhenviewedthrough a telescope. They are in orbit, around a common centre, the centre of mass of thetwostars.
Thecommonperiod of rotationfor a binaryisgivenby
Theinnerstaristhemostmassive of thetwo.
Thetwostars are alwaysdiametricallyoppositeeachother.
Where d isthedistancebetweenthetwostars.
Iftheplane of thetwostarsissuitablyorientedrelativetothat of theEarth, the light fromone of thestars in thebinarymaybeblockedbytheother, resulting in an eclipse of thestar, whichmaybe total orpartial. If a brightstarisorbitedby a dimmercompanion, the light curve has a similar patterntotheoneshown in the figure.
Thissystemisdetectedbyanalysingthe light fromoneorboth of itsmembers and observingthatthereis a periodicDopplershifting of thelines in thespectrum. A blueshiftisexpected as thestarapproachestheEarth and a redshift as itmovesawayfromtheEarth in itsorbitarounditscompanion.
Ifλ0isthewavelength of a spectral line and λthewavelengthreceivedonEarth, theshift, z, of thestarisdefined as
z = | λ – λ0|/ λ0.
Ifthespeed of thesourceissmallcomparedwiththespeed of light it can beshownthat
z = v/c