Today in Astronomy 142: observations of stars - Astro Pas Rochester

Today in Astronomy 142: observations of starsWhat do we know about individualstars?! Determination of stellarluminosity from measured fluxand distance• Magnitudes! Determination of stellar surfacetemperature from measurementof spectrum or color• Blackbody radiation• Bolometric correctionAt right: the old open cluster M29(2MASS/IPAC).Astronomy 142 1

JanuaryviewpJulyviewStarrTrigonometric parallaxRelatively nearby stars appear tomove back and forth with respectto much more distant stars as theEarth travels in its orbit.Since the size of the Earth’s orbit isknown very accurately from radarmeasurements, measurement ofthe parallax p determines thedistance to the star, precisely andaccurately.EarthAUp is an angle5

Parallax (continued)Since the Earth’s orbitis (nearly) circular,parallax can bemeasured no matterwhat the direction tothe star.6

Trigonometric parallax (continued)Mean distance between centers of Earth and Sun:Thussmall angleapproximationwhere the parallax p is measured in radians.! Parallax is usually expressed in fractions of an arcsecond(π radians = 180 degrees = 10800 arcminutes = 648000arcseconds); the distance to an object with a parallax of 1arcsecond, called a parsec, is! Good parallax measurements: about 0.02 arcsec from theground, ~ 0.001 arcsec from the Hipparcos satellite.7

Parallax examplesWhat is the largest distance that can be measured by parallaxwith ground-based telescopes? With the Hipparcos satellite?First note that we can write the distance-parallax relation asr = 1 parsec ⇥✓ ◆ arcsec! If the smallest parallax we can measure is 0.01 arcsec, thelargest distance measurable is 1/0.01 = 100 parsecs.! Similarly, Hipparcos got out to 1/0.001 = 1000 parsecs.! The ESA satellite Gaia, launched last year, will measureparallax with an accuracy of 0.00002 arcsec, thus perhapsstretching out to 50 000 parsecs = 50 kpc.• Compare: we live 8 kpc from the center of our Galaxy.p8

Parallax examples (continued)If we lived on Mars, what would be the numerical value ofthe parsec?Mars’s orbital semimajor axis is 1.524 AU, so the “parsec” is afactor of about 1.524 larger:But Mars’s orbit is twice as eccentric as Earth’s, so those lastdecimal places are affected by the orbit.9

ThethirtyneareststarsFigure:ChaissonandMcMillan,AstronomyToday10

Astronomer units: magnitudesAncient Greek system: first magnitude (and less) are thebrightest stars, sixth magnitude are the faintest the eye cansee.The eye is approximately logarithmic in its response:perceived brightness is proportional to the logarithm of flux.To match up with the Greek scale,or, for two stars with apparent magnitudes m 1 and m 2 ,11

Magnitudes (continued)magnitude = -2.5 log 10 (flux) + constantLogarithmic scale often used formeasurements based on human perception• Decibels in audio (not inverted10 log 10 power)• Richter scale for earthquakes(not inverted)12

Magnitudes continuedBrightnessMagnitude! Magnitudes are dimensionless: they are related to ratiosof fluxes or distances.! Another legacy from the Greeks: magnitudes runbackwards from the intuitive sense of brightness.• Brighter objects have smaller magnitudes. Fainterobjects have larger magnitudes. They’re like “rank.”! Fluxes from a combination of objects measured all atonce add up algebraically as usual. The magnitudes ofcombinations of objects do not.! Calibration is done through comparisons13

Magnitudes -the reference or zero pointA practical definition of zero apparent magnitude is Vega(α Lyrae), which has m = 0 at nearly all wavelengths (lessthan 20 microns).Absolute magnitude is the apparent magnitude a starwould have if were placed 10 parsecs away.An A0 star at 10 pc has magnitude zero in all bandsBolometric magnitude is magnitude calculated from theflux from all wavelengths, rather than from a small rangeof wavelengths.In this week’s workshop you’ll show that the absolute andapparent bolometric magnitudes, M and m, of a star arerelated by:14

Magnitudes – zero pointmagnitude = -2.5 log 10 (flux) + constantmagnitude = -2.5 log 10 (flux/flux_comp)constant = 2.5 log 10 (flux_comp)flux_comp is the flux that gives youzero magnitudewe call the constant the zero point andis related to the flux that gives you zeromagnitude15

Magnitudes (continued)Example (IAA problem 11.2). Two stars, with apparentmagnitudes 3 and 4, are so close together that they appearthrough our telescope as a single star. What is the apparentmagnitude of the combination?16

Magnitudes (continued)Example (IAA problem 11.2). Two stars, with apparentmagnitudes 3 and 4, are so close together that they appearthrough our telescope as a single star. What is the apparentmagnitude of the combination?Call the stars A and B, and the combination C:First how much larger flux does A have than B?Add 1 to both sides:So17

Magnitudes and PhotometriccalibrationObserve an object at the same time as anotherobject with known brightness (two stars in thesame field of view).Find correction factor for known objectUse that correction factor to measure apparentmagnitude of your object.Apparent magnitude – how bright the object isobserved to be.18

Absolute magnitudeIf you know the distance to the object and youmeasure the apparent magnitude, then you candetermine the brightness for the object as if wereobserved at 10pc away.This is the absolute magnitude.Directly related to the stars intrinsic luminosityWays to know distances?19

Blackbody emissionBecause stars are opaque at essentially all wavelengths oflight, they emit light much like ideal blackbodies do.Blackbody radiation from a perfect absorber/emitter isdescribed by the Planck function:B (T )= 2hc251e hc/ kT 1Power per unit area,bandwidth and solid angle.B λ units: erg s -1 cm -2 wavelength -1 sterradian -1For a small bandwidth (wavelength interval) Δλ (

Blackbody emission (continued)Surface brightness, relative to the Sun’speak surface brightness30000 K10000 K3000 KWavelength (µm)Although stars likethe Sun are brightestat visible wavelengths,the Planckfunction for star-liketemperatures is muchwider than the visiblespectrum.So most of a star’sluminosity comes outat ultraviolet and/orinfrared wavelengths.21

Planck function30000 K10000 K3000 KB ⌫ (⌫) B ( )As a function ofphoton frequencyWavelength (µm)As a function of photonwavelength22

Blackbody emission (continued)B ⌫ (⌫)B ( )flux integrated between ν and ν + dνflux integrated between λ and λ+ dλ⌫ = c/B ⌫ (⌫)d⌫ = B ( )dAstronomy 142 23

Color and temperatureNote that a givenratio of flux withinthe B and Vwavelength ranges(see below) ischaracteristic of acertain temperatureof blackbody, andthat continuousstellar spectra areapproximatelyblackbodies.Figure: Chaisson andMcMillan, Astronomy Today24

Planck functionB (T )= 2hc251e hc/ kT 1h Planck’s constantk Boltzmann’s constantc speed of lightphoton energy distributionPhotons in a box derivationEnergy levels set with Planck’s constantEnergy levels populated proportional to exp(E/kT)Astronomy 142 25

Integrating the Planck functionB (T )= 2hc251e hc/ kT 1k Boltzmann constanth Planck constantZ 10B (T )d = 2k4 ⇡ 415h 3 c 2 T 4= B T 4⇡usingZ 10x =x 5e 1/xkThc⇡4dx =1 15σ B = 5.67 x 10 -5 erg cm -2 s -1 K -4is the Stefan-Boltzmannconstant.B = 2⇡5 k 415h 3 c 2Astronomy 142 26

Blackbody emission (continued)The total flux emitted into all directions at all wavelengthsfrom the surface of a blackbody isf =f =Z 10Z 10df = B T 4⇡ZZd B (T )2⇡2⇡d⌦ cos ✓B (T )Z ⇡/2cos ✓ sin ✓d✓ hemisphereL =4⇡R 2 BT 40cos ✓ sin ✓d✓ = B T 4Z ⇡/20σ B = 5.67 x 10 -5 erg cm -2 s -1 K -4 is the Stefan-Boltzmannconstant.Luminosity of a spherical blackbody with radius R:27

Blackbody emission (continued)Wavelength of the Planck function’s maximum value changeswith temperature according toReminders about solid angle:! Solid angle for cone with smallapex angle θ is! Differential element of solid angle in sphericalcoordinates:✓ 2 [0, ⇡]28

Solid angles29

S λ Ω ⊙ or B λ Ω ⊙Stars are blackbodies to (pretty good)approximation.Solar flux perunit band-widthas seen fromEarth’s orbit,com-pared to theflux per unitbandwidth of a5777 K blackbodywith the sametotal flux.(WikimediaCommons)30

Effective TemperatureL =4⇡R 2 BT 4for a black bodyT e is the temperature of the blackbody of the samesize as a star (not exactly a black body), that givesthe same luminosity:T e =✓L4⇡R 2B◆ 14The Sun’s luminosity and radius gives aneffective temperatureBlackbody: surface brightness and spectrumdescribed by 1 parameter, the temperature31

Color and temperature (continued)ρ Ophiuchi. a molecular cloud at T =60 K.A young stellar object. The star'ssurroundings have T = 600 K.The Sun's surface, T = 6000 K.Evolved stars in ω Centauri, at T =60,000 K.Figure: Chaisson and McMillan,Astronomy Today.32

Stellar photometryTo facilitate comparison of measurements by differentworkers, astronomers have defined standard bands, eachdefined by a center wavelength and a bandwidth, forobserving stars.! At visible wavelengths the bands in most common use(Johnson 1966) are as follows:BandWavelength (nm) Bandwidth (nm)U 360 70B 430 100V 540 90R 700 220! Measurement of starlight flux (or magnitudes) withinsuch bands is called photometry.33

Color and effective temperatureColor is the difference between apparent magnitude atdifferent wavelengths, or 2.5 times the logarithm of the ratioof fluxes in the two bands.Color index: the color between the B and V bands:In the absence of extinction, color is an index of the effectivetemperature T e of stars.For T e < 12000 K, and taking B = V = 0 at T e =10000 K (likeVega),relating color totemperature34

Color and effective temperature (continued)Surface brightness, relative to the Sun’speak surface brightness30000 K10000 K3000 KBVExample: for thesespectra, the visiblecolors aretop to bottom.Wavelength (µm)35

Color and effective temperature (continued)Surface brightness, relative to the Sun’speak surface brightness30000 K10000 K3000 KBVAt 30000 K most ofthe radiation isemitted bluer thanvisible wavelengthsAt 3000K most of theradiation is emittedredder than visiblewavelengthsWavelength (µm)36

Color index and effective temperature(Johnson 1966)37

Color, magnitude and bolometric correctionOnce the shape of a star’s spectrum (i.e. its temperature) isdetermined from its colors, the total flux is also determined.! In magnitude terms: the ratio of total flux to flux withinone of the photometric bands is expressed as a bolometriccorrection to the star’s magnitude in that band, usually V:! In blackbody terms: once the temperature of the body isknown, so is the total flux . If stellar spectrawere really blackbodies, the bolometric correction wouldbe38

Bolometric correction, BCBolometric correctionfor main sequencestars (Johnson 1966)Bolometric correction(continued)Shape of spectrum, and thiscorrection factor, can bedetermined if one knows thecolor of the star (differencebetween apparentmagnitudes at twowavelengths).Scaling factor for flux is thesame as an offset inmagnitude. This offset iscalled the bolometriccorrection, BC, and isdetermined from observed ortheoretical stellar spectra.39

Bolometric correction (continued)Example. Two stars are observed to have the same apparentmagnitude, 2, in the V band. One of them has color index B-V= 0, and the other has B-V = 1. What are their apparentbolometric magnitudes?From the graph, BC = -0.4 and –0.38 in these two cases, sotheir bolometric magnitudes are practically the same, too, 1.6and 1.62.! That these magnitudes are both less than the apparent Vmagnitude is a sign that these stars produce substantialpower at wavelengths far outside the V band. The bluerstar (with B-V = 0) turns out to be brightest at ultravioletwavelengths; the other one is brightest at red and infraredwavelengths.40

Are perfect Black Body’s ever observed?Not really: In stars there are absorption linesand emission lines, absorption edges, morethan one temperature is seen from differentplaces. Many objects emit thermally as wellas non-thermally.Cosmic Microwave Background spectrum isa counter example. It’s really very close to aBlack Body.The frequency spectrum of the cosmicmicrowave background records interactionsbetween the evolving matter and radiationfields. Observations from ground-based,balloon-borne, and satellite instrumentsshow the CMB to agree with a blackbodyspectrum (dotted line) across 3 decades offrequency and 4 orders of magnitude inintensity. This agreement with a blackbodyspectrum indicates that the early universewas once in thermodynamic equilibrium,and limits energetic events since the BigBang: the cosmic microwave backgroundaccounts for more than 99.993% of theradiant energy in the universe.Credit Cobe team41

Wavelength vs TemperatureFor Black Body’s the wavelength of the emission peak is setby the temperature.There is little emission at vastly different other wavelengths.So you can often estimate the temperature of a thermal objectfrom what wavelength (or frequency) you observe it at.• 10 6 K stuff (plasma) emits in X-rays. Detected byCHANDRA• 10 4 K stuff (plasma) emits in UV. Detected by HST, IUE,FUSE• 10 3 K objects emit in optical. By eye.• 10 2 K stuff (cold gas and dust) emits in mi-far infrared.Detected by SIRTF/SST (previously ISO and IRAS).Note: there is also non-thermal emission -- then this ruledoesn’t apply (e.g. synchrotron emission)42

Near -Infrared Broad Band FiltersAtmospheric absorption along with filter transmissionJHKLMFrom Simmons and Tokunaga et al 200143

Some practical Astronomy! RA (right ascension) tells you when your object is up.• 12 hours is up highest Mar 21• 0 hours is up highest Sept 21! DEC (declination) tells you how far away from the north pole.Polaris is at +90. Anything below 0 degrees is hard to observe from thenorthern hemisphere.! Spectral types and luminosity class: A5 VA5 is the spectral typeV is the luminosity class.• V is called a dwarf, also is on the main sequence. Late type stars arefainter than the Sun, early type are much brighter than the Sun. Range inluminosities is about 10 5 , even though the range in mass is only 0.1 to100M⊙.• III’s are giants, typically 10 3 times more luminous than the Sun• I’s are supergiants, typically 10 5 times more luminous than the Sun.44

Practical astronomy (continued)• Using parallax to estimate distance:The distance in parsecs is 1/(parallax in arcsecs).• Estimating the absolute magnitude. The distance modulus is• X-ray emission from stars is often not from the stellar photosphere,(of order 5000K) but from the corona (of order a million K)• Far infrared emission from stars is often not from the stellarphotosphere but from circum-planetary dust that absorbs light fromthe star and reradiates it at longer wavelengths (of order 100K).45

Practical astronomy (continued)Spectral types: O,B,A,F,G,K,MClassified based on what their spectrum looks like.Is also a temperature scale.Is also a mass scale --- on the main sequence.M-stars are cool and low mass ~0.1 M ⊙ and faint (0.01 L ⊙ ).O stars are very very hot and more massive ~10M ⊙ andbright (10 4 L ⊙ ).Beta Pictorus (A5V) is more massive and luminous than thesun (a G star)Epsilon Eridani (K2V) is less massive and less luminous thanthe sun.46

Practical Astronomy (continued)Figuring out how bright things are in solar luminosities.• Figure out how bright the object is intrinsically.• Divide this luminosity by the luminosity of the Sun.Equivalently:• Take the absolute magnitude of the object, subtract theabsolute magnitude of the sun at the same band.• Convert magnitudes into luminosity.Example: A star has an absolute magnitude in V band of 7.0The sun has an absolute magnitude in V band of 4.8The star is about 1/10 th as bright as the sun. To do better than this usebolometric corrections.47

Summary! Magnitudes and how to calculate with them! Black body function! The relation between measured color and estimatedtemperature.! Bolometric correction! Broad band photometric measurements.48