Penentuan Jarak dalam Astronomi

Penentuan Jarak dalam Penentuan Jarak dalam AstronomiAstronomi

Kuliah Fisika Galaksi 1 Februari 2011

Urutan metoda penentuan jarak berdasarkan jarak maksimal yang dapat dicapai


Metoda langsung / primary method


Metoda yang dikalibrasi oleh

Pergeseran paralaks bintang

Earth

Sun

Earth 6 month later

Apparent changeof the direction

Parallactic angle Nearby star

Initial starposition

Star positi-on after 6month

More distant stars

Perlu diingat ! Perlu diingat ! Sebenarnya kita tidak mengukur paralaks absolut dari bintang tetapi paralaks relatif, yaitu perubahan posisi bintang relatif terhadap bintang-bintang latar belakang yang lebih redup dengan anggapan lebih jauh atau galaksi yang jauh

Paralaks tahunan bintang Paralaks tahunan bintang disebut juga paralaks disebut juga paralaks trigonometri karena gerak trigonometri karena gerak orbital Bumi. Bintang-orbital Bumi. Bintang-bintang dekat bergerak bintang dekat bergerak relatif terhadap bintang relatif terhadap bintang yang lebih redup dalam yang lebih redup dalam bentuk elips, bergantung bentuk elips, bergantung pada posisi bintang pada posisi bintang tersebut di langit.tersebut di langit.

Sudut paralaksnya berkurang Sudut paralaksnya berkurang

sebanding dengan sebanding dengan pertambahan jarak, dan bentuk pertambahan jarak, dan bentuk ellips-nya bergantung pada ellips-nya bergantung pada lintang ekliptikalintang ekliptika, , ββ: :

axes ratio axes ratio b/a = sin b/a = sin ββ

Degrade to line for β=0

The parallactic angle strongly depends on star’s distance

For (really) small angles, p ≈ 1/dp ≈ 1/d

ParsecParsec – new unit of interstellar – new unit of interstellar distancesdistances

Parsec = Parsec = PARPARallax +allax + SECSECond of arc:ond of arc:

The distance fromThe distance fromwhich 1 AU is seenwhich 1 AU is seenat the angle ofat the angle of1 arcsec1 arcsec

D(pc) = 1/p”D(pc) = 1/p”

1 pc ≈ 206265 AU ≈ 3.086·1016 m ≈ 3.26 light years

Large distances:1 kpc = 103 pc1 Mpc = 106 pc1 Gpc = 109 pc

Do we need 1 Tpc (TeraParsec) or not ?

Practical measurements of stellar parallaxes. I.Practical measurements of stellar parallaxes. I.Step 1: Step 1: Choose a large set Choose a large set (N>>1)(N>>1) of faint of faint

(presumably distant) reference stars.(presumably distant) reference stars. These stars are expected to set (fixed?) local These stars are expected to set (fixed?) local

coordinate frame used to reduce most of coordinate frame used to reduce most of optical aberrations and distortions and to optical aberrations and distortions and to measure shift of program stars relatively to measure shift of program stars relatively to this reference frame. this reference frame.

Step 2:Step 2: Measure Descart Measure Descart (x,y)(x,y) coordinates of coordinates of all program and reference stars on each frame all program and reference stars on each frame (time t(time tii, i=1,2,…,K; good if K >>1)., i=1,2,…,K; good if K >>1).

Step 3:Step 3: Choose one frame (near the middle of Choose one frame (near the middle of the set) as the set) as standard frame, standard frame, time labelled as ttime labelled as t00

If:If:◦All reference stars were fixed (say, very distant),◦All optical aberrations involved were negligible,◦Errors due to the differences in observational

conditions (temperature, focusing etc.) were absent,◦Measurement errors were also negligible,

……then all of then all of KK coordinate frames would be coordinate frames would be completely identicalcompletely identical

This is really not the case…The reduction procedure is necessary

Our goal: put all frames together and see Our goal: put all frames together and see how program star moves among how program star moves among reference starsreference stars

t1t2

tK-1 tK

t0

Reference stars

x

y

Proper Motion

Parallactic wobbleacross PM vector

Practical measurements of stellar parallaxes. Practical measurements of stellar parallaxes. I.I.

Step 4:Step 4: Find coordinate transformations from Find coordinate transformations from each ieach ithth frame to standard frame using frame to standard frame using (x,y)(x,y)ii and and (x,y)(x,y)00 coordinates of coordinates of NN reference stars, as reference stars, as

((RR for reference star) for reference star)

Here Here FFi-to-0i-to-0 is (weekly nonlinear) parametrized is (weekly nonlinear) parametrized function (matrice type) of the coordinates function (matrice type) of the coordinates (x,y)(x,y)

The unknown parameters can be estimated The unknown parameters can be estimated from the system of from the system of 2N2N conditional equations conditional equations by any suitable by any suitable nonlinear least-squares nonlinear least-squares algorithmalgorithm..

Now we are ready to match all coordinate Now we are ready to match all coordinate frames.frames.

iR

Rtoi

R

R

yx

Fyx

0

0

Transformation “matrix” Fi-to-0 may look as follows (taking into account frame scale differences, distortions, overall center’s shift etc.):

x0≈ a0x + ax·xi + bx·yi + cx·xi2 + dx·xi·yi + …

y0≈ a0y + ay·xi + by·yi + cy·yi2 + dy·xi·yi + …

Here index “i” used for ith frame, (x,y) are the Descart coordinates of reference stars

Here all a’s, b’s etc. are unknown constants that need to be calculated by the least squares for each of ith frame (i=1, 2, …, K)

Practical measurements of stellar Practical measurements of stellar parallaxes. I.parallaxes. I.

Step 5:Step 5: With these F With these Fii,(K on total) ,(K on total) recalculate the coordinates of recalculate the coordinates of program program starsstars from i from ithth frame to standard one, as frame to standard one, as

((PP for program star) for program star) FFi-to-0i-to-0 are known now! are known now!Now we have Now we have KK calculatedcalculated coordinate coordinate

pairs (x,y) for each program star reduced pairs (x,y) for each program star reduced to standard frameto standard frame

Step 6:Step 6: Based on telescope properties Based on telescope properties (focal length) calculate equatorial (focal length) calculate equatorial coordinates coordinates ((αα,,δδ))ii of program star(s) on of program star(s) on each ieach ithth frame. Apparent coordinates frame. Apparent coordinates ((αα,,δδ))ii vary with vary with ii (with time!) (with time!)

iP

Ptoi

P

P

yx

Fyx

0

0

Now, it looks like we have transferred all positions of our (moving!) program star to single frame

Practical measurements of stellar Practical measurements of stellar parallaxes. II.parallaxes. II.

Apparent coordinates of program stars are Apparent coordinates of program stars are changing because ofchanging because of◦Parallactic shiftParallactic shift◦Stellar proper motionsStellar proper motions

Simplest modelSimplest model for time variations of for time variations of stellar coordinatesstellar coordinates::

HereHere pp is theis the parallaxparallax, , μμαα && μμδδ are two are two components of stellar proper motion,components of stellar proper motion, ξξαα,i,i && ξξδδ,i,i are the residuals from model adopted are the residuals from model adopted (considered as normally distributed random (considered as normally distributed random values)values)

iiii

iiii

tp

tp

,0

,0

(1)

Here Here ΔΔΦΦαα & & ΔΔΦΦδδ are the differences of the parallax are the differences of the parallax phase between two time points separated by phase between two time points separated by ΔΔttii=t=tii-t-t00

sinsincoscossinsin

,coscos

sinsincoscos

;sinsin)cos(,cos)sin(

0

0

where

… parallax phases

Φα Φδ

and auxiliaryangle φare calculatedfor each giventime point ti

|Φα , Φδ| < 1 λ – ecliptic longitudeλ0 – Sun ecliptic longitudeβ – ecliptic latitudeε – ecliptic inclination

2K2K conditional equations (1) can be resolved for unknown parametersunknown parameters pp μμαα μμδδ by the least-squares or maximum-likelihood least-squares or maximum-likelihood techniquetechnique (minimizing residuals ξα,i ξδ,i) for time moments (t1 t2 t3… tK),

2 K >> 3Important notes 1-2. Important notes 1-2. Trignometric parallax

and proper motion components are calculated simultaneouslysimultaneously! To calculate reliable parallax from (1), you should plane your observations so that some ΔΔΦΦαα && ΔΔΦΦδδ were of the order of 1 (unity)!were of the order of 1 (unity)!

Important note 3. Important note 3. To derive precise proper motions, we need long time intervals ΔΔtt , typically dozens of years…

Astronomer should live long enough to use his own observations!

Important note 4.Important note 4. Parallax and proper motions are calculated relatively to reference stars which are supposed to be very distant. If it is not the case, we calculate only relative relative parallax parallax & relative proper motion.relative proper motion.

We need to estimate the absolutization absolutization correctionscorrections as

Δp = pabs – prel & Δμ = μabs – μrel

The best way: to use objects with zero parallax and proper motions seen on the same frames: (1) compact distant galaxies or (2) quasars; (3) stars with known parallax & proper motions taken from the fundamental catalogs FK4- FK5, ICRS catalogs HIPPARCOS, TYCHO-2, …

Or (4) use appropriate statistical absolutization, based on Milky Way Galaxy dynamical model when (1-3) are not available…

Measuring parallaxesMeasuring parallaxesis very difficult and tedious jobis very difficult and tedious job……Extremely small angles…Before HIPPARCOS epoch, till 1992, ~16000

stellar parallaxes for ~8100~8100 stars have been measured with typical accuracy of ±16 mas with ground-based instruments.

Publications:Publications: “The General Catalogue of Trigonometric Stellar Parallaxes” (4th edition) by Van Altena W.F., Lee J.T., Hoffleit E.D. (Yale University Observatory, New Haven, 1995); Catalog I/238A in CDS

Absolutization: statistical

Mengukur paralaks dan proper motion adalah pekerjaan yang sulit dan membutuhkan waktu yang panjang !

HIPPARCOS satellite on Earth and above Earth: scanning the sky

Synchronous observationof two stellar fields separated by the basicbasicangleangle γ = 58º00'31.25"with scanning

HIPPARCOS actual orbitHIPPARCOS actual orbit(instead of geostationary)(instead of geostationary)

Perigee distance 500 km

Apogee distance 36000 km

HIPPARCOS optical layoutHIPPARCOS optical layout

29-cmhalf-mirror

Beam 2

Beam 1Basic angle

Beam combiner unit

Modulating grid

2 PMTs

Reduction to ICRF/ICRSReduction to ICRF/ICRS~240 quasars (from 608) were usedObservations of stellar radio-sources both

with VLBI and on-board, with HIPPARCOSMeasuring angles between HIPPARCOS stars

and radio-sources with HST…Results:Results:Final reduction accuracy ~0.6 mas~0.6 masFinal residual rotation of HIPPARCOS

coordinate system ~0.25 mas/year~0.25 mas/year

Why 240 not 608? Quasars proper Why 240 not 608? Quasars proper motions?motions? – No, complex jet structure! – No, complex jet structure!

“Photocenter”movement reflects jetmotion that canbe masked to “proper motion”

HIPPARCOS observationsHIPPARCOS observationsHIPPARCOS satellite was launched on August,

1989 and lived on orbit till March, 1993Active observations lasted ~37 months

◦~118000 sample of stars complete to ~7.3m – included to HIPPARCOS catalog

◦~1 mln sample of stars complete to ~10.5m – included to TYCHO catalog

◦~2.5 mln stars later included to TYCHO-2 catalog (used also ~140 ground-based catalogs of positions and proper motions)

HIPPARCOS project stagesHIPPARCOS project stages

The HIPPARCOSInput Catalog(Catherine Turon et al.,ESA SP-1136, 1992)

HIPPARCOSworking catalog (after 37 month)

HIPPARCOS final catalog(ESA SP-1200, 1997 )

Data reduction by HIPPARCOS Data reduction by HIPPARCOS consortiumsconsortiumsFASTFAST (Fundamental Astronomy by Space

Techniques) – Jean Kovalevsky, 1992-1997NDACNDAC (Northern Data Analysis Consortium) –

Lennart Lindegren, Erik Hog, 1992-1997Approaches differ in details but not in general

conceptFinal HIPPARCOS catalog appeared on 1997----------------------------------------------Floor van Leeuwen “HIPPARCOS, the new Floor van Leeuwen “HIPPARCOS, the new

reduction of raw data”, 2007 – reduction of raw data”, 2007 – new versionnew version

HIPPARCOSHIPPARCOS TYCHOTYCHO TYCHO-2TYCHO-2Catalog system ICRS ICRS ICRSMean observationepoch & catalog epoch

J1991.25J2000.0

J1991.25J2000.0

J1991.5J2000.0

Number of stars 118 218 1 058 332 2 539 913Limiting magnitude

12.4m 11.5m 11.5m

Complete to: 7.3m 10.5m 11m

Mean accuracy of:Mean accuracy of:positions < 1 mas 7-25 mas 7-60 masproper motions < 1

mas/year- 2.5

mas/yearparallaxesparallaxes 1 mas1 mas -- --Mean photometric errors

0.002m 0.06-0.10m 0.013-0.10m

HIPPARCOS family of catalogsHIPPARCOS family of catalogs

HIPPARCOS: new revolution in HIPPARCOS: new revolution in astronomyastronomy

~20 000 stars with parallaxes better than 10%

~30 000 other stars with parallaxes better than 20%

~100 000 stars with broad-band magnitudes (close to standard B,V system) accurate within 0.002-0.003m

Nearly uniform covering of the sky with stars with

ICRS coordinates (~2-3 from HIPPARCOS and~40-60 from TYCHO-2 catalogs persquare degree) – first extension ofICRF/ICRS to optical range

HIPPARCOS parallaxes distributionHIPPARCOS parallaxes distribution<10%<10% negative parallaxesnegative parallaxesReasonsReasons:: (a) duplicity of stars (more than 70% are binary or

multiple); (b) position dependence on star’s brightness etc. (Some have been improved by F. van Leeuwen et al., 2007)

p < 0 p > 0 Mode of distribution ~1.5 mas

HIPPARCOS: new expectationsHIPPARCOS: new expectations

A:A: “Clean” Herzsprung-Russell diagram (good (good parallaxes + fine parallaxes + fine photometry)photometry) enables new luminosity calibration for many classes of stars (including some “standard candles”“standard candles”)

Main Sequence and Red Giantes stars

5lg5 pVMV

MVColors indicate spectral classes

HR diagrams for very accurate,HR diagrams for very accurate,moderate and “bad” parallaxesmoderate and “bad” parallaxes

p accurate to: 5% 10% 50%

MV

B-V

Broadening of the HR classes with increase of Broadening of the HR classes with increase of pp error: error:

Good forGood forcalibrationscalibrationsof standardof standard candles !candles !

HIPPARCOS: new expectationsHIPPARCOS: new expectationsB:B: Large

sample of stars to study the to study the galactic galactic structurestructure up to ~1 kpc from the Sun in more details than before: volume volume ~1000 times ~1000 times more!more!

CNS3sphere

RA

HIPPARCOS sphere

1

1

0

100

pc

HIPPARCOS: new expectationsHIPPARCOS: new expectationsC:C: Accurate and homogeneous proper motions

in ICRS (instead of compilations from large number of very inhomogeneous catalogs, like PPM, NPM & SPM) can be successfully used in kinematical researches of the Milky Way

(Less accurate but also homogeneous proper motions of ~2.5 mln stars from TYCHO-2 can also be added…)

Together with accurate distances, they give rise to an analysis of 2D velocity field2D velocity field

HIPPARCOS: new expectationsHIPPARCOS: new expectationsC:C: Adding radial velocities of tens of

thousands stars (measured during last two decades by CORAVEL, ELODIE/CORALIE, RAVE and SDSS projects), we derive 3D velocity 3D velocity fieldfield

It provides most detailed picture of the galactic dynamical state and serves as an excellent basic material to:◦derive galactic rotation curve◦estimate local fraction of dark matter◦search for “flows” of stars in solar vicinity◦search for traces of clusters disruption

Important example: Cepheid kinematicsImportant example: Cepheid kinematics“Double waves” of radial velocities and proper

motions vs galactic latitude reflect galactic differential rotation

Radial velocities and proper motions were used to calculate the rotation curve of young disk population and full set of local kinematical parameters: three components of velocity dispersion and solar

velocity components relative to local sample of Cepheids

l°

HIPPARCOS: new HIPPARCOS: new expectationsexpectations

D. D. 3D velocities of Cepheids and other “standard candles” (stars of a given color, RR Lyrae stars, subdwarfs, open star clusters etc.) open new way to improve their distance scales via statistical parallaxstatistical parallax method (will be considered later)

Similarly to trigonometric parallax, statistical parallax algorithm is considered to be almost free of many assumptions typical for most of”secondary””secondary” methods of distance determination

HIPPARCOS: new expectationsHIPPARCOS: new expectationsE:E: HIPPARCOS & TYCHO-2 extended

ICRF/ICRS reference coordinate frame to ~2.5 mln stars in optics. Because of dense and nearly uniform coverage of sky (approximately 40-60 stars per squared degree), new reductions of other “all-sky” catalogs (like GSC2, A2.0, UCAC2, 2MASS, SDSS,…) to ICRF/ICRS system became possible.

For approximately one billion of starsone billion of stars ICRF/ICRS coordinates are now available

Improvement of the precision of astrometric Improvement of the precision of astrometric measurements with time: ~2000 years from Hipparchus measurements with time: ~2000 years from Hipparchus to HIPPARCOSto HIPPARCOS

Because of logarithmic scale on parallax and proper motion axis on the diagram, we see exponentially rapid growthexponentially rapid growth of the accuracy of astrometric measurements

We expect one or two order improvement in one-two decades (GAIA, SIM projects)

Post-HIPPARCOS epochPost-HIPPARCOS epochIn 2007, Floor van Leeuwen from HIPPARCOS community has completed new processing of HIPPARCOS original (raw) data (“Hipparcos, the New Reduction “Hipparcos, the New Reduction of the Raw Data”, of the Raw Data”, Astrophysics Astrophysics and Space Science Library, Vol. and Space Science Library, Vol. 350. Dordrecht: Springer, 2007350. Dordrecht: Springer, 2007; with DVD disc)

Radio-interferometers in Radio-interferometers in astrometryastrometryAngular resolution of the radio-interferometer

is given by the Rayleigh criterion

where θ is the minimal recognized angle, λ is an effective wavelength, D is basebase length For base length ~10000 km (intercontinental

VLBI) typical resolutions:OH (Hydroxyl) 2.3-18 cm 1.7-13 GHz 5-0.7 mas5-0.7 masH2O (Water) 1.4 cm 21 GHz 0.4 mas0.4 masNH3 (Ammonia) 1.3 cm 23 GHz 0.3 mas0.3 masCO (Carbon monoxide) 2.6 mm 115 GHz 65 65 μμasasSiO (Silicon monoxide) 2.3-7 mm 42-130 GHz 57 57 μμasas-0.2 -0.2

masmas

D 22.1sin

VVery ery LLarge arge BBaseline aseline IInterferometrynterferometry

Base length is the maximal distance between array elements

Submilliarcsecond angular resolution accessible with very large baseline arrays, opens very perspective way to investigate internal structure and dynamics of radio sources, molecular masers in star-formation regions, to measure reliable distances of masers up to few kpc

Main advantage of VLBI: direct operation in Main advantage of VLBI: direct operation in ICRF/ICRS systemICRF/ICRS system

The European VLBI Network (EVN) The European VLBI Network (EVN)

Europe – Europe – South AsiaSouth Asia~25 antennae~25 antennae

Examplesof VLBelements

MERLIN: MERLIN: MMulti-EElement RRadio LLinked IInterferometer NNetwork (Great Britain, Manchester University, 217km size)

Operates at frequenciesranging from 151 MHz to24 GHz (wavelength 2m – 1.2 cm) At 5 GHz (5.9 cm), theresolution of MERLIN isbetter than 50 mas(better then HST)

GBGB

US NRAO VLBA of 35-m antennaeUS NRAO VLBA of 35-m antennae

~10000 km~10000 km

Nice VLBANice VLBA picturespictures

TX Cam maser “spots” 50 mas × 50 mas

10 mas10 mas

Submilliarcsec radiointerferometrySubmilliarcsec radiointerferometryof quasars and X-ray sources withof quasars and X-ray sources withVLBI Space Observatory ProgramVLBI Space Observatory Program(VSOP) (8 m, 1.6-22.3 GHz, Japan)(VSOP) (8 m, 1.6-22.3 GHz, Japan)

PKS 0637-752

Chandra image

3C 345

VSOPimage

Radioastrometry – new Radioastrometry – new branchbranch

Several pictures follow that illustrate high precision of modern VLBI measurements of parallaxes and proper motions and their applicability for various astronomical investigations

Radioastrometry for galactic Radioastrometry for galactic astronomyastronomy

Loinard et al. arXiv:0708.2081v1, 2008Loinard et al. arXiv:0708.2081v1, 2008VLBI (8.42 GHz) observations of the

“supercomet” (hot gas clump) around T Tau protostar:

p = 6.78 mas p = 6.78 mas ± ± 30 30 μμasasD = 147.6 D = 147.6 ± ± 0.6 pc0.6 pc Accurate to ~0.4% ! Accurate to ~0.4% !

Paralaks Gerak Gugus / Paralaks Gerak Gugus / Metoda Titik KonvergenMetoda Titik Konvergen

What is “star cluster”?What is “star cluster”? – Group of stars of common origin, with similar ages and chemical composition

Star clusters:Star clusters:◦Globular clusters:Globular clusters: very massive (~105-106 M0) and old (>10Gyr)

◦Open clusters:Open clusters: loose (<104 M0, typically ~100M0) with age from 1Myr to ~10Gyr

Unlike stellar associationstellar association,star cluster is gravitationallygravitationallyboundbound, i.e. its Etotal < 0

The dynamics of the gravitationally bound stellar system in (quasi)equilibrium state, is governed by the virial theoremvirial theorem written in any of three forms:

2T + 2T + ΩΩ = 2E – = 2E – ΩΩ = T + E = 0, = T + E = 0,

where E is the total energy, T – total kinetic energy, Ω – total potential energy

Physical background of star clustersPhysical background of star clusters

For typical open cluster with 300 MO, ~2 pc radius we will have mean-square star’s velocity ~0.5 km/s~0.5 km/s

This is ~50-100 times less than mean velocity This is ~50-100 times less than mean velocity of cluster itself relative to the Sunof cluster itself relative to the Sun

ij

ij

N

ij ij

ji

N

iii

rGMV

rGM

rmGm

VMVmT

T

12

12

21

02

2

2

2

1

2

From stellar dynamics: virial theorem

where cluster kinetic energy

and cluster potential energy

Cluster velocity dispersion:

Main idea:Main idea: open star cluster moves in the space as a single bodysingle body, with nearly the same space velocities of stars, because relative velocities of cluster members are very low: ~0.1-1 km/s~0.1-1 km/s

Star cluster as viewedfrom infinite distance:velocity vectors seemto be parallel

If star cluster occupies wide enough area on wide enough area on the celestial spherethe celestial sphere (~10-30° in diameter, as for Hyades), and moves relative the Sun quickly enough (~20-40 km/s), the observer on the Earth will see cluster’s radiant (convergent point of space velocity vectors)

Star cluster as viewedfrom the Earth:velocity vectors seemto converge (perspectiveprojection)

~10-20°

Clusterradiant

Parallax of the Hyades “moving Parallax of the Hyades “moving cluster”cluster”

Radiant

λi – apparentangle between ith star and antiradiant(opposite to radiant point)

RadiantRadiant is the intersection ofgreat circles of the celestialsphere drawn through thevectors of stellar propermotions. Really, it is finite areafinite arearather than the point, becauseof the errors in proper motions

Simple geometric consideration:

λ

λ

VT

VRV

Sun

ri

i

iRi

iiiiT

iiR

ktgVr

rksinVV,cosVV

““Moving groups” offer unique wayMoving groups” offer unique wayof deriving of deriving the individual distancesthe individual distances to cluster stars !!!to cluster stars !!!

V : cluster’svelocityvectorrelative tothe Sun

Projection of V to the line of sight and to the plane of the sky gives:

Radial velocity

Tangential velocity

To antiradiant

Mathematical Mathematical backgroundbackground

kk = 4738 (km = 4738 (km//s)s)/(/(pc·mas/yearpc·mas/year))

Application to Hyades=M45 open Application to Hyades=M45 open clustercluster

Large apparent size ~30°Large space velocity ~47 km/sEstimated internal velocity dispersion ~0.3 km/s

(Lindegren et al., 2000)Large proper motion ~50-200 mas/year (tangential

velocity ~20 km/s)

HyadesAge ~650 Myr

Pleiades

α Tau

Hyades: Comparing the results of distance Hyades: Comparing the results of distance determinations determinations

Bars:Bars: from new HIPPARCOS parallaxes of the stars

Histogram:Histogram: from the “convergent-point” method

Projected on the line of sight

Mean distance<D> ≈ 46.5 ± 0.3 pc

Hyades is the closest “moving” open cluster from ~10 suitable for “convergence-point” processing

Some other clusters are shown, with arrows indicating individual vectors of apparent proper motion of members (S.Madsen et al. A&A V.381, P.446, (S.Madsen et al. A&A V.381, P.446, 2002)2002)

OB-associations

““Convergent-point” method as applied to Hyades Convergent-point” method as applied to Hyades cluster, historically, played the critical role in the cluster, historically, played the critical role in the establishing of modern distance scale in our establishing of modern distance scale in our UniverseUniverse

In pre-HIPPARCOS epoch, this method became In pre-HIPPARCOS epoch, this method became the only real base for luminosity calibration of the only real base for luminosity calibration of main-sequences of open clusters, that allowed main-sequences of open clusters, that allowed routine usage of “main-sequence fitting” routine usage of “main-sequence fitting” technique applied by astronomers to open technique applied by astronomers to open cluster population not only of our Milky Way, but cluster population not only of our Milky Way, but also in other galaxiesalso in other galaxies

In turn, “MS fitting” was used for luminosity In turn, “MS fitting” was used for luminosity calibrations of other luminous “standard candles” calibrations of other luminous “standard candles” (such as Cepheids, red giants and supergiants)(such as Cepheids, red giants and supergiants)

ConclusionConclusionThe “convergent-point” method also rests upon

transparent idea (simple geometrical and physical considerations), and is considered as absolute absolute (primary)(primary) method

The “convergent-point” method is also of fundamental importance, specially for “MS-fitting”, and its “econiche” in the hierarchy of methods is next after the trigonometric parallax

Its disadvantages :Its disadvantages :◦ Needs measurements of reliable radial velocities of stars◦ Sensitive to systematic errors in proper motions◦ Applicable to very limited sample (~10) of star clusters:

Nearby Apparently large on the sky With large tangential velocities

Chapter FiveChapter Five

Main-Sequence Fitting, orMain-Sequence Fitting, orthe distance scale of star the distance scale of star

clustersclusters

Open clustersOpen clustersGlobular clustersGlobular clusters

Main idea:Main idea: to use the advantages of measuring photometric parallax of a whole stellar sample, i.e. close group of stars of common nature: of the same◦age,◦chemical composition,◦interstellar extinction,but of different initial masses

Advantages of using star clusters as Advantages of using star clusters as the “standard candles” - 1the “standard candles” - 1(a) Large statistics (N~100-1000 stars) reduce (a) Large statistics (N~100-1000 stars) reduce

random errors as ~Nrandom errors as ~N-1/2-1/2. All derived . All derived parameters are more accurate than for single parameters are more accurate than for single starstar

(b) All stars are of the same age. Star clusters (b) All stars are of the same age. Star clusters are the only objects that enable direct age are the only objects that enable direct age estimate, study of the galactic evolution and estimate, study of the galactic evolution and the star-formation historythe star-formation history

(c) All stars have nearly the same chemical (c) All stars have nearly the same chemical composition, and the differences in the composition, and the differences in the metallicity between the stars play no role metallicity between the stars play no role

Advantages of using star clusters as Advantages of using star clusters as the “standard candles” - 2the “standard candles” - 2(d) Simplify the identification of stellar (d) Simplify the identification of stellar

populations seen on HRDpopulations seen on HRD(e) Large statistics also enables reliable (e) Large statistics also enables reliable

extinction measurementsextinction measurements(f) Can be distinguished and studied even at (f) Can be distinguished and studied even at

large (5-6 kpc, for open clusters) distances large (5-6 kpc, for open clusters) distances from the Sun from the Sun

(g) Enable secondary luminosity calibration of (g) Enable secondary luminosity calibration of some stars populated star clusters – Cepheids, some stars populated star clusters – Cepheids, Novae and other variablesNovae and other variables

DataBase on open clusters: DataBase on open clusters: W.Dias, J.Lepine, B.Alessi W.Dias, J.Lepine, B.Alessi (Brasilia)(Brasilia)

Latest Statistics - Version 2.9 (13/apr/2008):Number of clusters: 1776 Size: 1774 (99.89%) Distance: 1082 (60.92%) Extinction: 1061 (59.74%) Age: 949 (53.43%) Distance, extinction and age: 936

(52.70%)Proper motion (PM): 890 (50.11%) Radial velocity (VR): 447 (25.17%) Proper motion and radial velocity: 432

(24.32%) Distance, age, PM and VR: 379 (21.34%)Chemical composition [Fe/H]: 158

( 8.90%) ““TThese incomplete results point out to the observers hese incomplete results point out to the observers

that a large effort is still needed to improve the data in that a large effort is still needed to improve the data in the catalogthe catalog” (W.Dias)” (W.Dias)

Astrophysical backgrounds of Astrophysical backgrounds of “isochrone fitting” technique:“isochrone fitting” technique:

(a) Distance measurements: photometric parallax, or magnitude difference (m-M)

(b) Extinction measurements: color change, or “reddening”

(c) Age measurements: different evolution rate for different masses, declared itself by the turn-off point color and luminosity

-----------------------------------------------Common solution can be found on the basis Common solution can be found on the basis

of stellar evolution theory, i.e. on the of stellar evolution theory, i.e. on the evolutional interpretation of the CMDevolutional interpretation of the CMD

Difference with single-stars method:Difference with single-stars method:

Instead of luminosity calibrations of single stars, we have to make luminosity calibration of all Main Sequence as a whole: ZAMS (Zero-Age Main Sequence), and isochrones of isochrones of different agesdifferent ages (loci of stars of different initial masses but of the same age and metallicity)

Important note: Important note: Theoretical evolutionary tracks and theoretical isochrones are calculated in lg Tlg Teffeff – M – Mbolbol variables

Prior to compare directly evolution Prior to compare directly evolution calculations with observations of open calculations with observations of open clusters, we have to transform clusters, we have to transform TTeffeff to to observed colors, observed colors, (B-V)(B-V) etc., and bolometric etc., and bolometric luminosities luminosities lg L/Llg L/LSunSun and magnitudes and magnitudes MMbolbol to to absolute magnitudes absolute magnitudes MMVV etc. in UBV… broad- etc. in UBV… broad-band photometric system (or others)band photometric system (or others)

Important and necessary step:Important and necessary step: the empirical the empirical (or semi-empirical) calibration of “color-(or semi-empirical) calibration of “color-temperature” and “bolometric correction-temperature” and “bolometric correction-temperature” relations from data of temperature” relations from data of spectroscopically well-studied stars of spectroscopically well-studied stars of ◦(a) different colors(a) different colors◦(b) different chemical compositions(b) different chemical compositions◦(c) different luminosities (c) different luminosities

with accurately measured spectral energy with accurately measured spectral energy distributions (SED),distributions (SED),

or calibration based on the principles of the or calibration based on the principles of the “synthetic photometry”“synthetic photometry”

Bolometric magnitudes and bolometric Bolometric magnitudes and bolometric correctionscorrections

BBolometric MMagnitude, MMbolbol, specifies total energy output of the star (to some constant):

BBolometric olometric CCorrectionorrection, BCBCVV, is defined as the correction to VV magnitude:

bbol cd)(jlg.M

52

constdRj

djMMBC

VVbolV

)()(

)(lg5.2

BCBCVV ≤≤ 0 0By definition, MMbolbol = M = MVV + BC + BCVV

>1

From P.Flower (ApJV.469, P.355, 1996)

Example: BCV vs lg Teff: unique relation for all luminosities

Note: Note: Maximum BCMaximum BCVV ~0 at lgT ~0 at lgTeffeff~3.8-4.0 (for ~3.8-4.0 (for F3-F5 stars), when maximum of SED F3-F5 stars), when maximum of SED coincides with the maximum of V-band coincides with the maximum of V-band sensitivity curvesensitivity curve

Obviously, the bolometric corrections can be Obviously, the bolometric corrections can be calculated to the absolute magnitude calculated to the absolute magnitude defined in each banddefined in each band

For modern color-temperature and BC-temperature calibrations see papers by:

P. Flower (ApJ V.469, P.355, 1996):P. Flower (ApJ V.469, P.355, 1996): lgTlgTeff eff - BC- BCVV – (B-V) from observations – (B-V) from observations

T. Lejeune et al. (A&AS V.130, P.65, 1998):T. Lejeune et al. (A&AS V.130, P.65, 1998):Multicolor synthetic-photometry approach;Multicolor synthetic-photometry approach;lgTlgTeffeff–BC–BCVV–(U-B)-(B-V)-(V-I)-(V-K)-…-(K-L),–(U-B)-(B-V)-(V-I)-(V-K)-…-(K-L),for dwarf and giants with [Fe/H]=+1…-3for dwarf and giants with [Fe/H]=+1…-3(with step 0.5 in [Fe/H])(with step 0.5 in [Fe/H])

lgTlgTeffeff – (B-V) – (B-V)for different

luminosities; based on observations

(from P.Flower, ApJ V.469, P.355, 1996)

Shifted down by Δ lgTeff = 0.3 relative to next more luminous class for the sake of convenience

T.Lejeune et al. (A&AS V.130, P.65, 1998):T.Lejeune et al. (A&AS V.130, P.65, 1998): Colors from UV to NIR vs TColors from UV to NIR vs Teffeff (theory and empirical corrections)(theory and empirical corrections)

Before HIPPARCOS mission, astronomers used Hyades “convergent-point” distance as most reliable zero-point of the ZAMS calibration and the base of the distance scale of all open clusters

Recently, the situation has changed, but Hyades, along with other ~10 well-studied nearby open clusters, still play important role in the calibration of isochrones via their accurate distances

Revised HIPPARCOS parallaxes of Revised HIPPARCOS parallaxes of nearby open clusters (van Leeuwen, nearby open clusters (van Leeuwen, 2007)2007)

ClusterCluster Parallax, Parallax, error, error, masmas

(m-M)(m-M)00 and its and its error, error, magn.magn.

[Fe/H][Fe/H] Age, Age, MyrMyr

E(B-V)E(B-V)

Praesepe 5.49±0.19 6.30±0.07 +0.11 ~830 0.00Coma Ber

11.53±0.12

4.69±0.02 -0.065 ~450 0.00

Pleiades 8.18±0.13 5.44±0.03 +0.026 100 0.04IC 2391 6.78±0.13 5.85±0.04 -0.040 30 0.01IC 2602 6.64±0.09 5.89±0.03 -0.020 30 0.04NGC 2451

5.39±0.11 6.34±0.04 -0.45 ~70 0.055

α Per 5.63±0.09 6.25±0.04 +0.061 50 0.09Hyades

21.51±0.143.34±0.02 +0.13 650 0.003

Combined MMHpHp – (V-I) – (V-I) HRD for 8 nearby open clusters constructed by revised HIPPARCOS parallaxes of individual stars (from van Leeuwen, 2007) and corrected for small light extinction

Hyades MS shift (red red squaressquares) is due to◦ Larger [Fe/H]◦ Larger age ~650 Myr

Bottom envelope (--------) can be treated as an observed ZAMS

(V-I)

MH

pPleiades problem:HST gives smallerparallax (by ~8%) ΔMHp ≈ -0.17m

(a) Observed ZAMS (in absolute magnitudes) can be derived as the bottom envelope of composite CMD, constructed for well-studied open clusters of different ages but similar chemical composition

(b) Isochrones of different ages are appended to ZAMS and “calibrated”

Primary empirical calibration of the Hyades MS & Primary empirical calibration of the Hyades MS & isochrone for different colors, by HIPPARCOS isochrone for different colors, by HIPPARCOS parallaxesparallaxes(M.Pinsonneault et al. ApJ V.600, P.946, 2004)(M.Pinsonneault et al. ApJ V.600, P.946, 2004)

Solid line: theoretical isochrone withLejeune et al. (A&AS V.130, P.65, 1998)color-temperature calibrations

MV

ZAMS and Hyades isochrones: sensitivity to ZAMS and Hyades isochrones: sensitivity to the age for 650±100 Myr (from Y.Lebreton, the age for 650±100 Myr (from Y.Lebreton, 2001)2001)Fitting

color of the turn-off point

ZAMS

Best library of isochrones recommended to calculate cluster distances, ages and extinctions:

L.Girardi et al. “Theoretical isochrones in L.Girardi et al. “Theoretical isochrones in several photometric systems I. (A&A V.391, several photometric systems I. (A&A V.391, P.195, 2002)P.195, 2002)

Theoretical background:Theoretical background:◦(a) Evolution tracks calculations for different initial

stellar masses (0.15-7MSun) and metallicities (-2.5…+0.5) (also including α-element enhanced models and overshooting)

◦(b) Synthetic spectra by Kurucz ATLAS9 ◦(c) Synthetic photometry (bolometric corrections

and color-temperature relations) calibrated by well-studied spectroscopic standards

Ages of open clusters vary from few Myr to ~8-10 Gyr, age of the disk

For most clusters, [Fe/H] varies approximately from -0.5 to +0.5

Necessary step in the distance and age determination – account for differences in metallicity ([Fe/H] or Z)

Metallicity effects on isochrones:Metallicity effects on isochrones:modelling variables, Mmodelling variables, Mbolbol - T - Teffeff

Turn-off point

Metallicity effects on isochrones: Metallicity effects on isochrones: opticsoptics

Turn-off point

Metallicity effects on isochrones: Metallicity effects on isochrones: NIRNIR

Turn-off point

Metallicity differences can be taken into Metallicity differences can be taken into account byaccount by◦(a) Adding the corrections to absolute

magnitudes ΔM and to colors ΔCI to ZAMS and isochrone of solar composition. These corrections can follow both from observations and theory.

◦(b) Direct fitting of observed CMD by ZAMS and isochrone of the appropriate Z – now most common used technique

These methods are completely equivalent

Ideally, we should estimate [Fe/H] (or Z) prior prior to fitting CMD by isochrones

If it is not the case, systematic errors in distances (again errors!) may result

Open question:Open question: differences in Helium content (Y). Theoretically, can play important role. As a rule, evolutionary tracks and isochrones of solar Helium abundance (Y=0.27-0.29) are used

Example:Example: Normalized transmission curves for two realizations of popular UBVRIJHK systems as compared to SED (spectral energy distributions) of some stars (from L.GirardiL.Girardi et al., 2002)

See next slides for ZAMS and some isochrones

Theoretical isochrones (color - MV magnitude diagrams) for solar composition (Z=0.019) and cluster ages 0.1 Gyr, 1 Gyr and 10 Gyr (L.GirardiL.Girardi et al., 2002, green solid lines)

0.1

1

10 Gyr

Theoretical isochrones (NIR color-magnitude diagrams) for solar composition (Z=0.019) and cluster ages 0.1 Gyr, 1 Gyr and 10 Gyr (L.GirardiL.Girardi et al., 2002, green solid lines)

0.1

110 Gyr

Girardi et al. isochrones in modelling Girardi et al. isochrones in modelling variablesvariablesMMbolbol – lg T – lg Teff eff (more detailed age grid)(more detailed age grid)

ZAMS

The same but The same but for “standard” for “standard” multicolor multicolor systemsystem

ZAMS

ZAMS

OpticsOptics

NIR NIR

How estimate age, extinction and the How estimate age, extinction and the distance?distance?11stst variant variant(a) Calculate color-excess CE for cluster stars

on two-color diagram like (U-B) – (B-V). Statistically more accurate than for single star. Highly desirable to use a set of two-color diagrams as (U-B) – (B-V) and (B-V) – (V-I) etc., to reduce statistical and systematical errors

How estimate age, extinction and the How estimate age, extinction and the distance?distance?11stst variant variant(b) If necessary, add corrections for [Fe/H]

differences to ZAMS and isochrones family, constructed for solar abundance

(c) Shift observed CMD horizontally, the offset being equal to the color-excess found at (a) step, and then vertically, by ΔΔMM, to fit proper ZAMS isochrone, i.e. cluster turn-off point. Calculate true distance modulus as (V-M(V-MVV))00 = = ΔΔV - RV - RVV∙E(B-V)∙E(B-V)

(for V–(B-V) CMD)

How estimate age, extinction and the How estimate age, extinction and the distance?distance?22ndnd variant variant(a) If necessary, add corrections for [Fe/H]

differences to ZAMS and isochrones family, constructed for solar metallicity

(b) Match observed cluster CMD (color-magnitude diagram) to ZAMS and isochrone trying to fit cluster turn-off point

(c) Calculate hhorizontal and vvertical offsets: HH: Δ (color) = CE (color excess)VV: (m-M) = (m-M)0 + R· CE

(m-M)(m-M)00 – true distance – true distance modulusmodulus

How estimate age, extinction and the How estimate age, extinction and the distance?distance?22ndnd variant variant(d) Make the same procedure for all available

observations in other photometric bands(e) Compare all (m-M)0 and CE ratios. For MS

fitting performed properly,◦distances will be in general agreement,◦CE ratios will be in agreement with accepted

“standard” extinction law

You can start writing paper !

MS-fitting example: Pleiades, good MS-fitting example: Pleiades, good casecase

Magnitudes offsetgives

ΔΔV=(V-MV=(V-MVV))00+R+RVV∙E(B-V)∙E(B-V)

↨(m-M)0 = 5.60E(B-V)=0.04lg (age) = 8.00

ZAMS

G.MeynetG.Meynet et al.(A&AS V.98, P.477,1993)Geneva isochrones

Young distant cluster, good caseYoung distant cluster, good case

(m-M)0=12.55E(B-V)=0.38lg (age)=7.15


h Per clusterh Per cluster

RSGRSG(Red(RedSuper-Super-Giants)Giants)

(m-M)0=13.65E(B-V)=0.56lg (age)=7.15


RSGRSG

(m-M)0=12.10E(B-V)=0.32lg (age)=8.22


Older and older…Older and older…

(m-M)0=7.88E(B-V)=0.02lg (age)=9.25


Very old open cluster, M67Very old open cluster, M67

(m-M)0=9.60E(B-V)=0.03lg (age)=9.60


Optical data: D.An et al. (ApJ V.671, P.1640, 2007)Optical data: D.An et al. (ApJ V.671, P.1640, 2007)(Some open clusters populated with Cepheid (Some open clusters populated with Cepheid variables)variables)

The same, NIR data: D.An et al. (ApJ The same, NIR data: D.An et al. (ApJ V.671, P.1640, 2007)V.671, P.1640, 2007)

New parameters of open clusters populated with Cepheid variables (from D.AnD.An et al., 2007)

The consequences for calibration of the Cepheids luminosities will be considered later

Important note:Important note: Open cluster field is often contaminated by large amount of foreground and background stars, nearby as well as more distant non-members

Prior to “MS-fitting” it is urgently recommended to “clean” CMD for field stars contribution, say, by selecting stars with similar proper motions on μx - μy vector-point diagram:

(kinematic selection; reason – small velocity dispersion)

Field stars

Cluster stars

Isochrones fitting is equally applicable to globular clusters, but this is not the only method of the distance estimates

Good idea to use additional horizontal branch luminosity

indicators, including RR Lyrae variablesRR Lyrae variables (with nearly constant luminosity, see later)

RR LyraeRR Lyrae

BHBBHB((EHB)EHB)

TPTP

D.AnD.An et al. (arXiv:0808.0001v1)

Isochrones (MS + giant branch) for globular clusters of different [Fe/H] in (u g r i z) photometric bands (SDSS)

Å

u 3551Åg 4686År 6165Åi 7481Åz 8931Å

more metal-deficient

Isochrones fittingIsochrones fittingexample: M92example: M92

Age stepAge step 2 2 GyrGyr

Theoretical background ofTheoretical background ofthis method is quitethis method is quitestraightforwardstraightforward

Galactic Globular ClustersGalactic Globular Clustersare distant objects andare distant objects andvery difficult to study,very difficult to study,even with HSTeven with HST

Reliable photometric dataReliable photometric dataexist mostly for brightestexist mostly for brighteststars: Horizontal Branch,stars: Horizontal Branch,Red Giant Branch andRed Giant Branch andSubGiantsSubGiants

CMD for selected galactic globular clusters (HST observations CMD for selected galactic globular clusters (HST observations of 74 GGC; of 74 GGC; G.PiottoG.Piotto et al., A&A V.391, P.945, 2002) et al., A&A V.391, P.945, 2002)

Bad cases for MS-fitting (except NGC 6397)Bad cases for MS-fitting (except NGC 6397)

For CMDs of globular clusters, without pronounced Main Sequence, there are other methods of age estimates, based on◦magnitude difference between Horizontal Branch

and Turn-Off Point (“vertical method”)◦color difference between Turn-Off Point and Giant

Granch (“horizontal” method)

Illustration of the “vertical” and “horizontal” Illustration of the “vertical” and “horizontal” methods of age estimates of globular clustersmethods of age estimates of globular clusters

M.Salaris & M.Salaris & S.CassisiS.Cassisi,,““Evolution of starsEvolution of starsand stellarand stellarpopulations” populations” (J.Wiley &(J.Wiley &Sons, 2005)Sons, 2005)

In some cases isochrone fitting fails to give unique result because of multiple stellar populationsmultiple stellar populations found in most massive galactic and extragalactic globular clusters (ωω Cen: Cen: L.Bedin L.Bedin et al., ApJ V.605, L125, 2004; NGC 1806 & NGC 1806 & NGC 1846 in LMC:NGC 1846 in LMC: A.MiloneA.Milone et al., arXiv:0810.2558v1)

ω Cen

NGC 1806 (LMC)

Multiple populations ?He abundancedifferences ?

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