Notes on Galactic coordinates: in the literature another systems is identified as: ( I I

Transcription

1 Galactic Astronomy Chapter... Astronomical measurements... Galactic coordinates:... Distances...4 Parallax and proper motion...4 Distances determined from velocities...6 Relativist Doppler Effect...7 Moving cluster method...8 Magnitude and colors of stars...0 Apparent magnitude... Standard photometric systems...3 Colors...6 Absolute magnitude...7 Absolute Energy distribution and bolometric magnitudes...8 Mass-to-light ratios...0 Surface brightness and isophotal radii...0

2 Chapter Astronomical measurements Galactic coordinates: Galactic equator (GEQ): great circle that most closely approximates the plane of the MW; ==> inclined by 6.87 from celestial equator; North Galactic Pole (NGP): J000 (, ) ( ,7.85 ) ( h m α δ = = 5, ' GNP GNP ) Galactic latitude b : angle from GEQ to the object along the great circle passing through the objects and the two Galactic Poles; Galactic longitude l : angle with respect to the direction of the center of the Galaxy; Coordinates of the Galactic Center: 0, 0, , h m l = b= α δ = = 45.6, 856.' ( ) ( GC GC ) ( ) ( ) Notes on Galactic coordinates: in the literature another systems is identified as: ( I I, ) l b. Abandoned in 958, longitude measured from a point where GEQ was intersecting the CEQ (arbitrary); New system II II l, b ; was known as ( ) * Notes on Center of the Galaxy: source Sgr A believed initially to be in the center of the MW. In fact this turned out to be located 5 from the real center; Transformation equations: if the longitude of the NCP is l = 3.93 sinb = sinδ sinδ + cosδ cos δ cos( α α ) ; GNP GNP GNP cos b sin( lcp l) = cos δ sin( α αgnp ) ; ( ) ( ) cos b cos l l = cosδ sinδ sinδ cos δ cos α α ; CP GNP GNP GNP sinδ = sin δ sinb+ cosδ cos b cos( l ); CP GNP GNP CP cos δ sin( α αgnp) = cos b sin( lcp l) ; ( ) b b ( l l) cos δ cos α α = cosδ sin sin δ cos cos ; GNP GNP GNP CP

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4 Distances Parallax and proper motion The lines ES and ES contain an angle at S that is defined as ϖ ; The angle ϖ is the parallax; If r is the radius of Earth orbit and d the distance to a star at S then, for d r: r = tan ϖ ϖ radian d In arcsec: ϖ " = 0665 ϖ radian Parsec = distance at which a star has a parallax of arsec: 3 pc= 0665 AU= km = 3.6 lyr ; Conversely, distance of star with observed parallax ϖ " ==> d = pc ; ϖ " HIPPARCHOS satellite mission, from Nov. 989 to March 993, measured the parallax for ~ 0000 stars with a precision of 0.5 mas down to magnitude V = 9 (with typical error of mas ==> for d = 00 pc ϖ = 5 mas d ~0% and 40% uncertainty in the determination of the intrinsic brightness); The largest parallaxes observed are for α Centauri, ϖ = 0.75" d =.3 pc and Barnard s star ϖ = 0.55" d =.8 pc ; Complete data compiled in TYCHO- catalog (Hog et al. 000, CDS-Vizier catalog number I/59) 4

5 Proper motion: change of position of a star due to intrinsic motion ==> angular rate of change produced by component of motion of a star across line of sight, after removing the Earth motion around the Sun; ==> measured: compare positions of stars at widely different epochs. Proper motion is directly proportional to star linear velocity perpendicular to the line of sight and inversely proportional to its distance from the Sun ==> selecting high proper motion stars is a good way to select nearby stars; Once the distance is known (using parallax) 3D motion of star can be deduce ==> unique information on dynamics of Galaxy; proper motion = vector, with magnitude, µ, and position angle, θ ; In equatorial system, proper motion is resolved into components: µ cosθ = µ δ perpendicular to CEQ - µ sinθ = µ α cosδ parallel to CEQ; where µ δ and µ α are the annual rate of changes in declination and right ascension; Star with larger proper motion = Barnard s star, with few hundreds of arcsecond per year. µ 0.3" yr. Typical values are only a More complete catalog to date: LSPM-north catalog (Lépine & Shara 004, astro-ph/04070) 6977 stars with proper motions larger than 0.5 arcsecond/yr complete at 99% for b > 5 and over 90% for lower Galactic latitudes, down to magnitudes V = 9 and with limiting magnitude V =.0. Astrometric systems: catalog objects with absolute positions and proper motions; ex. FK5 system (fifth Fundamental Katalog) accuracies of 0.03 arcsecond and ~0.8 mas/yr. More modern is the ICRS (International Celestial Reference System), which was established by observing extragalactic radio sources (Arias et al. 995, A&A, 303, 604). 5

6 Distances determined from velocities Distance + proper motion ==> transverse velocity v = t = t + t ; t α δ " " Transverse or tangential velocity: tα = Krµ α [ km/s] and tδ Kr µ δ [ km/s] ( pc km) ( arsec radian ) " s K = = and µ α = 5µ α cos δ ; ( yr s) = with To complete knowledge of the star s space velocity v ==> radial component, the radial velocity v r (positive if source moves away from us); 6

7 Relativist Doppler Effect Source of radiation, moving toward or away from an observer ==> shift in observed frequencies: ν β vr c = ( β) γν where 0 γ β ( ) Lowest order in vc, γ = and ν = ν ν0 = βν0; In terms of wavelengths λ = c ν ==> λ λ0 = ν ν0 λ = λ0; c For a receding source observed spectrum is redshifted relative to its rest wavelength. λ In general, redshift ==> z ; λ 0 v r In practice: o identify in spectra of stars known absorption lines produced by chemical elements and calculate the displacements from laboratory measurements; o After correcting for any components of Earth s orbital velocity ( 30 km s ) and rotation velocity ( 0.5 km s ) along line of sight = heliocentric radial velocity. 6 o For stars, measurements must be accurate to km s ± 3 0 c ; limited by turbulence in stellar photosphere to ± 0. km s ; o In some cases, radial velocities are variable ==> orbital motion of star companion or radial pulsation. 7

8 Moving cluster method A cluster of stars with physical diameter d receding from Earth will gradually shrink to a dot of light. At distance D its angular diameter is d given by: θ = ; D Taking natural logarithms of both side and θ differentiating with respect to time: D= vr θ ; Where vr = D the cluster radial velocity. If cluster not traveling directly away from us ==> it moves across the sky as it shrinks. Clearly, proper motion of individual stars will be directed towards the point on the sky that the whole cluster will occupy as a dot ==> convergent point of the cluster. By searching proper-motion catalogs, many moving clusters were discovered this way. For an approaching cluster, the convergent point is the point from which the cluster as expanded. Convergent point contains key information about space velocity of stars: o It defines the direction v v of this velocity; o Once it has been identified ==> distance to individual cluster stars. Consider plane that contains Sun + cluster star ==> velocity vector of cluster = v, pointing in the direction of convergent point. o Star radial velocity = v = vcosψ o Velocity transverse to line of sight = r 8 v = vsinψ. o Thus vt = vr tanψ ==> v t in terms of observable quantities; o Since vt = µ d* where µ = proper motion of star and d * its distance: vr tanψ vr tanψ µ = d* =. d µ * t ϖ 4.74 v o In terms of parallax: r µ = ; mas tanψ km s mas yr o We can obtain the distance of each cluster member (or parallax) from its observed proper motion, radial velocity and angular separation from the convergent point (Hanson 975 AJ, 80, 379).

9 Clusters where the method was applied:. Hyades ~ 00 stars at average distance of 45 pc;. Ursa Major Group ~60 stars at average distance of 4 pc; 3. Pleiades ~ 600 stars at average distance of 5 pc; 4. Scorpio-Centaurus ~00 stars at average distance of 70 pc; Until very recently (outdated by trigonometric parallaxes), distance to Hyades set the scale for essentially all Galactic and extragalactic distance measurements ==> Revisions: far-reaching repercussions; Other methods: Secular and Statistical parallaxes; Methods used in the past to estimate distances and absolute magnitude of rare but luminous stars. Until Hipparcos, such stars were playing important role as distance scale calibrator ==> can be used to determine distances as far as 500 pc; Methods may become important again in future when positions and parallaxes for second epoch will be available. 9

10 Magnitude and colors of stars Energy emitted by an object = flux per unit of frequency (monocrhomatic flux): [ ] f ν =W m Hz or [ ] f λ =W m nm Absolute energy distribution = Spectral Energy Distribution (SED) Figure shows SED for Vega; spectrophotometric magnitudes AB ν and ST λ refer to spectra of constant f ν and f λ ; Magnitude zero defined to be mean flux density of Vega in Johnson V passband ==> spectra produce same count rates. In practice = Astrometry: measure total energy in definite range of frequencies ==> integrate monochromatic flux W m - λ over λ ; 0

11 Apparent magnitude Response of the eye = logarithmic in intensity ==> equal steps of brightness = equal ratio of radiant energy; For two stars with apparent magnitudes and flux m f and m f: f m m = klog0 f Where the constant k is defined such that brighter objects have smaller magnitude; N. Pogson: magnitude system defined such that m= 5 f f = 00; m m =.5log0 or f f f f = ( m m ) Note: The magnitude difference when small is about equal to the fractional difference in f relative brightness. Since m m = 0.9log e ==> for nearly similar flux ratio: f f f ( ) = + f and f m m m.086 f In particular: m = f =.5 f and m =.5 f = 0 f. Due to lost in telescope system + atmosphere what we measure is not the flux but something proportional to the flux: f ftfrd where: 0 ν ν ν ν ν. T ν : transmission of the atmosphere : Since T ν a e where a column density of air along the line of sight. Airmass: defined as the ratio For plane parallel atmosphere: a a of air column density to the value at the zenith a 0. 0 airmass= sec z where z = zenith distance; With this expression one can take cares of dimming of light by atmosphere extinction: m z = k z+ const ; All it is needed is to observe star at different airmass (determining k) ( ) sec and extrapolate to secz = 0 (magnitude out of the atmosphere);

12 . F ν : transmission of the filters : The choice of F ν determines various photometric systems. Description of filter sets used by different magnitude system: describe by effective wavelength λ and Full Width Half Maximum (FWHM: wavelength difference λ between the points eff at which the F ν is half is peak value); 3. R ν : efficiency of the telescope system: This factor depends on the combination of the telescope and detector and difficult to measure. In practice must be calibrated by measuring the response of known source brightness. Until 950, international photographic system (bluer than eye response) was identified by while the photovisual system (similar to eye response) was identified by m pv ; m pg Inconvenients: a) Photographic plates are not linear ==> the relation between incident intensity and darkening must be empirically determine by calibrating each plates; b) Dynamic range is only a factor of 0 ==> several different plates must be used to cover larger ranges;

13 c) Old photographic magnitude catalogs are affected by relatively high uncertainties: ( ± 0. mag); but differential photometry much more accurate: ± 0.03 mag; Photometers : are linear and have large dynamical range ==> precision ± 0.0 mag in absolute calibration and ± 0.00 for differential photometry; Since 980s, CCD is used for photometry ==> detector is linear and have imaging capability ==> allows to observe stellar fields: crowded center of globular clusters; Standard photometric systems Johnson & Morgan (953): UBV + 7 bands in the near infrared; As long as different observers use same set of filters ==> ratio of brightness is the same; but the spectral response of the equipment is varying from standard system; In practice: possible to matches contemporary detector to standard filters + brightness measured with several different filters usually provide sufficient knowledge of deviations between program and standard stars ==> enable accurate magnitude measurements. For example: if ' V and ' B are the observed values ==> transformation equation: ( ) V = α V + β B V + γ ' ' ' V V V Constants determined such that transformed UBV magnitudes of standard stars agree as closely as possible with published UBV mag (see Leggett 99 ApJS, 8, 35); Johnson & Mitchell: Extension of standard systems toward the NIR RIJKLMN; Others sets used: Cousin: RI (most used); Glass (974): define H between J and K; With many adjustments, practically, there are no more standard sets; NIR : JHKL (see Elias et al. 98, AJ, 87, 09) Thuan & Gun: avoid wavelength at which spectrum of night sky shows prominent emission; Hipparcos: exceptionally broad, to maximize throughput; HST: approximate Johnson s bands; NICMOS: wavelength shifted by 0; ex. F75W =.75µm 3

14 Note: the zero point of each scale is chosen such that Vega has U = B = V = H = V = 0 ; On this scale, then the Sun has V = 6.74 ; Sirius has V =.45 and the faintest stars have V 7 ; p T 4

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16 Colors Color index: formed by taking the differences in magnitude measured in two different bands; If A and B denote two different filters and S λ is the combined telescope-receiver-filter sensitivity: ( ) CI.. = m m = const.5log AB A B 0 0 dλs λ dλs λ ( ) A f ( ) B f The color index is usually denoted as: A B, ex. B V and U B; ==> Since the C.I. measure the ratio of fluxes near λ eff of two different bands, it is independent of distances (not true for galaxies at high redshift). Utility: for stars, broad-band colors yield information on T eff determine the distribution of energy; o B V? T eff since cool stars are red and hot star blue; λ λ, g and z, because these parameters o U B? z because the U band contain emission line (which opacity is sensible to metallicity) while B is dominated by continuum; o Complication for U B is that is also sensible to Balmer jump (sudden onset of continuum absorption from n = level of H) at 370nm; Intermediate filters = Strömgren and DDO ==> devised with narrower bands 0nm FWHM 30nm; Strömgren filters: uvby + β n ( H β ) and β ω (continuum around H β ); The usual color indices in this system are:. b y, v b. c ( u v) ( b y) ; temperature indicator for hot stars and luminosity indicator for cooler stars; m u b b y ; for cooler stars, measure the level of line-blanketing (dimming of the blue part of the spectrum caused by millions of heavy element absorption lines); Also indicator of spectral peculiarity; β β ; luminosity indicator for hot stars and temperature indicator for cooler stars; 3. ( ) ( ) 4. n ω REFERENCES: Strömgren 966, ARA&A, 4, 433; Olsen 987, A&A, 89, 73; Edvarsson et al. 993, A&A, 75, 0 6

17 DDO filters: sets of five filters The usual color indices in this system are:. C ( 4 45) ; sensible to T eff. C ( 45 48) ; sensible to g 3. C ( 4 4) ; sensible to the strength of spectral line due to CN and Fe atoms ==> abundance of heavy elements; REFERENCE: Claria et al. 994, MNRAS, 68, 733 Note: interstellar extinction (due to scattering) is more efficient in the blue band ==> reddening + dimming effects ==> colors must be corrected before deducing physical characteristics of stars; Absolute magnitude If F is the fluxes detected of an object at distance D, then the flux f of the same object at distance d will be: f D = d F The absolute magnitude M is the apparent magnitude for an object at standard distance D : f d m M =.5log = 5log, where d is given in parsec; F D For D= 0 pc, the distance modulus : m M = 5logd 5 Utility: allows deducing d from m and M or vice versa M from m and D; Ex. M V, which measure the energy in the band V; Note the total energy is measured through the bolometric magnitude; ==> Since the distance to the Sun is A. U pc modulus of the Sun is m M = 3.5 mag; The most luminous stars are =, using M ( ) = time brighter and the least luminous V 4 0 times fainter;, the distance 7

18 Two corrections must be applied to the magnitudes:. Dimming of star due to IS extinction (A mag): m M = 5logd 5+ A. K-correction (for galaxies only)- reddening due to redshift (K mag): m M = 5logd 5 + A+ K; For different photometric systems and galaxy types, Frei & Gunn (994) gives the value of a K = k z +.5log + z, where z is the redshift. parameter k( z ) related to K by the expression: ( ) ( ) NOTE: the K correction applies only to nearby galaxies, because spectra of galaxies evolve with time ==> different at higher redshift. Absolute Energy distribution and bolometric magnitudes The basic problem of absolute flux calibration = efficiency of telescope + receiver; Procedure:. Observing source from which we know theoretical SED: A) black body at known temperature B) synchrotron radiation;. Observing standard stars calibrated relatively to known sources; Stars calibrated in absolute energy: a Vir,? UMA, a Leo, a Lyr, Vega, a Tau optical data given in terms of relative absolute energy distribution, written in monochromatic magnitudes: f ν mλ.5log, with absolute flux f f ν at λ 0 0 = nm ; ν0 Ex. Vega f ν = 3560± 5 Jy (Mégessier 997); 0 Once the absolute magnitude is known for one single star it can be used as a standard. 8

19 Observation program:. Measure program star + standard star;. Correct for atmospheric extinction; 3. Use standard for calibration. The flux of a star averaged over one of the standard photometric bands X ( X U, BV,,...) f 0 X = dνs 0 X dνs ( ν) X f ν ( ν) Once f X has been determined for a standard star for other stars: ( ) = ( ) ( ) ( ) log f * log f std 0.4 X * X std X X = is: Apparent bolometric magnitude : the integral of the absolute magnitude over all the frequencies: bol.5log ( ν ν 0 ) Where m d f + C C bol is an arbitrary constant (determined by convention). bol Independent of C bol, however, the difference in bolometric magnitude of two stars gives the ratios of their luminosities. The difference between m bol and the magnitude in one of the standard photometric bands is the bolometric correction: BCX mbol X The bolometric correction is a function of spectral type ==> even with different luminosities, two stars with identical spectra have identical value of BC. Convention for C bol :. Chosen such that bol 0. MKS units ==> C bol = 8.90 V = ==> BC ( ) = 0 V 3. Chosen such that all bolometric correction are negative ==> BC ( ) = 0.9 X V 9

20 Mass-to-light ratios Brightness of an object, quantified by how many Sun placed at object distance would be LX 0.4( MX M X ) required to produce same flux: N = = 0 L X If one knows the mass (in unit of solar mass) of the object, NM, then the mass-to-light ratio: M MM ϒ X = = = NM L L 0.4( MX M X) 0 M M X X In general, mass-to-light ratio varies with X : Ex. for a giant E with B K = 3.3 and for the Sun 0.4( 3.3.) B K =., one gets: ϒ ϒ = 0 =.75 B K MM One can also define a bolometric mass-to-light ratio: ϒ bol = ; however, this Lbol L bol parameter depends on a lot of assumptions: Surface brightness and isophotal radii For extended objects like galaxy: surface brightness = radiative flux per unit solid angle; Because amount of light is and solid angle falls of as d d ==> surface brightness independent of distance d ; NOTE: redshift make the surface brightness to fall with distance; Suppose I = mag/arcsec = µ B at a given point in a galaxy ==> a square one arcsec on a side around that point emit as much light as a star of magnitude. Suppose a second galaxy with I = µ B lies behind the first one. What is I = I+ I? Answer = 0.5µ B, because amount of light coming from galaxies = 0 = 0 /.5 0.5/.5 Galaxies do not have sharp edges (we do not know where they end) ==> sizes are measured up to different levels of surface brightness ==> isophotal radius or isophotal diameter; Ex. D 5 is the diameter of a galaxy up to I = 5µ B ==> de Vaucouleurs Radius R5 = D5 Another well known radius is: Holmberg radius measure up to I = 6.5µ pg ; 0