Astronomical Distances or Measuring the Universe (Chapters 5 & 6) by Rastorguev Alexey, professor of the Moscow State University and Sternberg Astronomical.

Astronomical Distances or Measuring the Universe (Chapters 5 & 6) by Rastorguev Alexey, professor of the Moscow State University and Sternberg Astronomical Institute, Russia Sternberg Astronomical Institute Institute Moscow University

Content Chapter Five:Chapter Five: Main-Sequence Fitting, or the distance scale of star clusters Chapter Six:Chapter Six: Statistical parallaxes

Chapter Five Main-Sequence Fitting, or the distance scale of star clusters Open clustersOpen clusters Globular 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 the “standard candles” - 1 (a) Large statistics (N~100-1000 stars) reduce random errors as ~N -1/2. All derived parameters are more accurate than for single star(a) Large statistics (N~100-1000 stars) reduce random errors as ~N -1/2. All derived parameters are more accurate than for single star (b) All stars are of the same age. Star clusters are the only objects that enable direct age estimate, study of the galactic evolution and the star-formation history(b) All stars are of the same age. Star clusters are the only objects that enable direct age estimate, study of the galactic evolution and the star-formation history (c) All stars have nearly the same chemical composition, and the differences in the metallicity between the stars play no role(c) All stars have nearly the same chemical composition, and the differences in the metallicity between the stars play no role

Advantages of using star clusters as the “standard candles” - 2 (d) Simplify the identification of stellar populations seen on HRD(d) Simplify the identification of stellar populations seen on HRD (e) Large statistics also enables reliable extinction measurements(e) Large statistics also enables reliable extinction measurements (f) Can be distinguished and studied even at large (5-6 kpc, for open clusters) distances from the Sun(f) Can be distinguished and studied even at large (5-6 kpc, for open clusters) distances from the Sun (g) Enable secondary luminosity calibration of some stars populated star clusters – Cepheids, Novae and other variables(g) Enable secondary luminosity calibration of some stars populated star clusters – Cepheids, Novae and other variables

DataBase on open clusters: W.Dias, J.Lepine, B.Alessi (Brasilia)DataBase on open clusters: W.Dias, J.Lepine, B.Alessi (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 (V R ): 447 (25.17%) Proper motion and radial velocity: 432 (24.32%) Distance, age, PM and V R : 379 (21.34%) Chemical composition [Fe/H]: 158 ( 8.90%) “These incomplete results point out to the observers that a large effort is still needed to improve the data in the catalog” (W.Dias)“These incomplete results point out to the observers that a large effort is still needed to improve the data in the catalog” (W.Dias)

Astrophysical backgrounds of “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 of stellar evolution theory, i.e. on the evolutional interpretation of the CMDCommon solution can be found on the basis of stellar evolution theory, i.e. on the evolutional interpretation of the CMD

Difference with single-stars method:Difference with single-stars method: isochrones of different agesInstead 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 different ages (loci of stars of different initial masses but of the same age and metallicity)

Important note: lg T eff – M bolImportant note: Theoretical evolutionary tracks and theoretical isochrones are calculated in lg T eff – M bol variables Prior to compare directly evolution calculations with observations of open clusters, we have to transform T eff to observed colors, (B-V) etc., and bolometric luminosities lg L/L Sun and magnitudes M bol to absolute magnitudes M V etc. in UBV… broad-band photometric system (or others)Prior to compare directly evolution calculations with observations of open clusters, we have to transform T eff to observed colors, (B-V) etc., and bolometric luminosities lg L/L Sun and magnitudes M bol to absolute magnitudes M V etc. in UBV… broad-band photometric system (or others)

Important and necessary step: the empirical (or semi-empirical) calibration of “color-temperature” and “bolometric correction-temperature” relations from data of spectroscopically well-studied stars ofImportant and necessary step: the empirical (or semi-empirical) calibration of “color-temperature” and “bolometric correction-temperature” relations from data of spectroscopically well-studied stars of –(a) different colors –(b) different chemical compositions –(c) different luminosities with accurately measured spectral energy distributions (SED), with accurately measured spectral energy distributions (SED), or calibration based on the principles of the “synthetic photometry” or calibration based on the principles of the “synthetic photometry”

Bolometric magnitudes and bolometric corrections BM M bolBolometric Magnitude, M bol, specifies total energy output of the star (to some constant): Bolometric Correction BC V VBolometric Correction, BC V, is defined as the correction to V magnitude : BC V ≤ 0 M bol = M V + BC V By definition, M bol = M V + BC V >1

From P.Flower (ApJ V.469, P.355, 1996) Example: BC V vs lg T eff : unique relation for all luminosities

Note: Maximum BC V ~0 at lgT eff ~3.8-4.0 (for F3-F5 stars), when maximum of SED coincides with the maximum of V-band sensitivity curveNote: Maximum BC V ~0 at lgT eff ~3.8-4.0 (for F3-F5 stars), when maximum of SED coincides with the maximum of V-band sensitivity curve Obviously, the bolometric corrections can be calculated to the absolute magnitude defined in each bandObviously, the bolometric corrections can be calculated to the absolute magnitude defined 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): lgT eff - BC V – (B-V) from observations lgT eff - BC V – (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; lgT eff –BC V –(U-B)-(B-V)-(V-I)-(V-K)-…-(K-L), for dwarf and giants with [Fe/H]=+1…-3 (with step 0.5 in [Fe/H])

lgT eff – (B-V)lgT eff – (B-V) for different luminosities; based on observations (from P.Flower, ApJ V.469, P.355, 1996) Shifted down by Δ lgT eff = 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 T eff (theory and empirical corrections)Colors from UV to NIR vs T eff (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 nearby open clusters (van Leeuwen, 2007) Cluster Parallax, error, mas (m-M) 0 and its error, magn. [Fe/H] Age, Myr E(B-V) Praesepe 5.49±0.196.30±0.07+0.11~8300.00 Coma Ber 11.53±0.124.69±0.02-0.065~4500.00 Pleiades 8.18±0.135.44±0.03+0.026 1000.04 IC 2391 6.78±0.135.85±0.04-0.040 300.01 IC 2602 6.64±0.095.89±0.03-0.020 300.04 NGC 2451 5.39±0.116.34±0.04-0.45 ~700.055 α Per 5.63±0.096.25±0.04+0.061 500.09 Hyades 21.51±0.143.34±0.02+0.13 6500.003

M Hp – (V-I)Combined M Hp – (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 red squaresHyades MS shift (red squares) is due to –Larger [Fe/H] –Larger age ~650 Myr ----Bottom envelope (----) can be treated as an observed ZAMS (V-I) M Hp Pleiades problem: HST gives smaller parallax (by ~8%) ΔM Hp ≈ -0.17 m

(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 & isochrone for different colors, by HIPPARCOS parallaxes (M.Pinsonneault et al. ApJ V.600, P.946, 2004) Solid line: theoretical isochrone with Lejeune et al. (A&AS V.130, P.65, 1998) color-temperature calibrations MVMV

ZAMS and Hyades isochrones: sensitivity to the age for 650±100 Myr (from Y.Lebreton, 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 several photometric systems I. (A&A V.391, P.195, 2002)L.Girardi et al. “Theoretical isochrones in several photometric systems I. (A&A V.391, P.195, 2002) Theoretical background:Theoretical background: –(a) Evolution tracks calculations for different initial stellar masses (0.15-7M Sun ) 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

L.Girardi et al. “Theoretical isochrones in several photometric systems I. (A&A V.391, P.195, 2002)L.Girardi et al. “Theoretical isochrones in several photometric systems I. (A&A V.391, P.195, 2002) Distribution of spectra in Padova library on lg T eff – lg g plane for [Fe/H] from -2.5 to +0.5 Wide variety of stellar models, from giants to dwarfs and from hot to cool stars, to compare with observations in a set of popular photometric bands: UBVRIJHK (Johnson-Cousins- Glass), WFPC2 (HST), … Giants

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: modelling variables, M bol - T eff Turn-off point

Metallicity effects on isochrones: optics Turn-off point

Metallicity effects on isochrones: NIR Turn-off point

The corrections ΔM and ΔCI (CI – Color Index) vs Δ[Fe/H] or ΔZ to isochrones, taken for solar abundance, can be found either – from theoretical calculations, – or empirically, by comparing multicolor photometric data for clusters with different abundances and with very accurate trigonometric distances

Metallicity differences can be taken into account byMetallicity differences can be taken into account 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

priorIdeally, we should estimate [Fe/H] (or Z) 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

L.GirardiL.Girardi et al. (2002) database on isochrones and evolutionary tracks is of great value – it provides us with “ready- to-use” multicolor isochrones for a large variety of the parameters involved (age, [Fe/H], [ α /Fe], convection,…)

Example: L.GirardiExample: Normalized transmission curves for two realizations of popular UBVRIJHK systems as compared to SED (spectral energy distributions) of some stars (from L.Girardi et al., 2002) See next slides for ZAMS and some isochrones

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

L.GirardiTheoretical isochrones (NIR color-magnitude diagrams) for solar composition (Z=0.019) and cluster ages 0.1 Gyr, 1 Gyr and 10 Gyr (L.Girardi et al., 2002, green solid lines) 0.1 1 1 Gyr What are fancy shapes !

Girardi et al. isochrones in modelling variables M bol – lg T eff (more detailed age grid) ZAMS

Optics NIR The same but for “standard” multicolor systemThe same but for “standard” multicolor system ZAMS

How estimate age, extinction and the distance? 1 st 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 distance? 1 st variant (b) If necessary, add corrections for [Fe/H] differences to ZAMS and isochrones family, constructed for solar abundance ΔM (V-M V ) 0 = ΔV - R V ∙E(B-V)(c) Shift observed CMD horizontally, the offset being equal to the color-excess found at (a) step, and then vertically, by ΔM, to fit proper ZAMS isochrone, i.e. cluster turn-off point. Calculate true distance modulus as (V-M V ) 0 = ΔV - R V ∙E(B-V) (for V–(B-V) CMD)

How estimate age, extinction and the distance? 2 nd 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 hv(c) Calculate horizontal and vertical offsets: H H: Δ (color) = CE (color excess) V V: (m-M) = (m-M) 0 + R· CE (m-M) 0 – true distance modulus

How estimate age, extinction and the distance? 2 nd 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 case Magnitudes offset gives ΔV=(V-M V ) 0 +R V ∙E(B-V) ↨ (m-M) 0 = 5.60 E(B-V)=0.04 lg (age) = 8.00 ZAMS G.Meynet G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

Young distant cluster, good case (m-M) 0 =12.55 E(B-V)=0.38 lg (age)=7.15 G.Meynet G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

h Per cluster RSG(RedSuper-Giants) (m-M) 0 =13.65 E(B-V)=0.56 lg (age)=7.15 G.Meynet G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

RSG (m-M) 0 =12.10 E(B-V)=0.32 lg (age)=8.22 G.Meynet G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

Older and older… (m-M) 0 =7.88 E(B-V)=0.02 lg (age)=9.25 G.Meynet G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

Very old open cluster, M67 (m-M) 0 =9.60 E(B-V)=0.03 lg (age)=9.60 G.Meynet G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

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

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

D.AnNew parameters of open clusters populated with Cepheid variables (from D.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 μ x - μ y vector-point diagram: (kinematic selection; reason – small velocity dispersion) Field stars Cluster stars

MS-fitting accuracy (best case, multicolor photometry) (D.An et al ApJ V.655, P.233, 2007) Random error of MS-fitting δ(m-M) 0 ≈ ±0.02 m –with spectroscopic [Fe/H]: δ(m-M) 0 ≈ ±0.02 m, i.e ~ 1% in the distance Systematic errors due to uncertainties of calibrations, [Fe/H] and α-elements, field contamination and contribution of unresolved binaries –δ(m-M) 0 ≈ ±0.04-0.06 m –δ(m-M) 0 ≈ ±0.04-0.06 m, i.e. 2-4% in the distance Uncertainties of Helium abundance may result in even larger systematic errors…

For distant clusters, with CMD contaminated by foreground/background stars, and uncertainties in [Fe/H], errors may increase to Δ(m-M) 0 ≈±0.1 m (random) ± 0.2 m (systematic) Δ(m-M) 0 ≈±0.1 m (random) ± 0.2 m (systematic) Typical distance accuracy of remote open clusters is ~10-15% Typical distance accuracy of remote open clusters is ~10-15%

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 variables (with nearly constant luminosity, see later) RR Lyrae BHB BHB (EHB) TP

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 3551Å g 4686Å r 6165Å i 7481Å z 8931Å more metal-deficient

Isochrones fitting example: M92 Age step 2 Gyr Theoretical background of this method is quite straightforward Galactic Globular Clusters are distant objects and very difficult to study, even with HST Reliable photometric data exist mostly for brightest stars: Horizontal Branch, Red Giant Branch and SubGiants

CMD for selected galactic globular clusters (HST observations of 74 GGC; G.Piotto et al., A&A V.391, P.945, 2002)CMD for selected galactic globular clusters (HST observations of 74 GGC; G.Piotto 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” methods of age estimates of globular clustersIllustration of the “vertical” and “horizontal” methods of age estimates of globular clusters M.Salaris & S.Cassisi, “Evolution of stars and stellar populations” (J.Wiley & Sons, 2005)

“Horizontal” method calibrations: Color offset vs [Fe/H] for different ages Gyr

magnitude difference vs [Fe/H] “Vertical” method calibrations: magnitude difference vs [Fe/H] for different ages Gyr

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

ω Cen NGC 1806 (LMC) Multiple populations ? He abundance differences ?

Chapter Six Statistical parallaxes

Astronomical background refineStatistical parallaxes provides very powerful tool used to refine luminosity calibrations of secondary “standard candles”, such as RR Lyrae variables, Cepheids, bright stars of constant luminosity, and isochrones applied for main- sequence fitting space velocitiesStatistical parallax technique involves space velocities of uniform sample of objects – at first glance, it sounds as strange and unusual…

Main idea V T = k r μ V RTo match the tangential velocities ( V T = k r μ, proportional to distance scale of the sample of studied stars) and radial velocities V R (independent on the distance scale), under three-dimensional normal (ellipsoidal) distribution of the residual velocities r V T =k r μ VRVR Sun

If all accepted distances are systematically larger (shorter) than true distances, then overestimated (underestimated) tangential velocities will generally distort the ellipsoidal distribution of residual velocities, and the velocity ellipsoids will look like

… instead of being alike and pointed to the galactic center

E.PavlovskayaOne of the first attempts to calculate statistical parallax of stars has been made by E.Pavlovskaya in the paper entitled “Mean absolute magnitude and the kinematics of RR Lyrae stars” (Variable Stars V.9, P.349, 1953) RR ≈ +0.6 mHer estimate RR ≈ +0.6 m was widely used and kept before early 1980 th and even recently, differ only slightly on modern value for metal-deficient RR Lyrae (~0.75 m )

First rigorous formulation of modern statistical parallax technique have been done by: S.Clube, J.DaweS.Clube, J.Dawe in “Statistical Parallaxes and the Fundamental Distance Scale-I & II” (MNRAS V.190, P.575; P.591, 1980) C.A.Murray C.A.Murray in his book “Vectorial Astrometry” (Bristol: Adam Hilger, 1983)

Modern (3D) formulation of the statistical parallax technique enables –(a) To refine the accepted distance scale and absolute magnitude calibration used –(b) To take into account all observational errors –(c) To calculate full set of kinematical parameters of a given uniform stellar sample (space velocity of the Sun, rotation curve or other systemic velocity field, velocity dispersion etc.) Advanced matrix algebra is required, soAdvanced matrix algebra is required, so only brief description follows only brief description follows

A.Rastorguev’sDetailed description of the 3D statistical parallax technique can be found only in A.Rastorguev’s (2002) electronic textbook in http://www.astronet.ru/db/msg/1172553 “The application of the maximum-likelyhood technique to the determination of the Milky Way rotation curve and the kinematical parameters and distance scale of the galactic populations” (in russian)

ΔM Photometric distances are calculated by star’s apparent and absolute magnitudes. Absolute magnitudes are affected by random and systematic errors. The last can be treated as systematic offset of distance scale used, ΔM. Statistical parallax technique distinguishes: r e accepted mean - expected distance r e, calculated by accepted mean absolute magnitude of the sample (after luminosity calibration); rrefined mean - refined distance r, calculated by refined mean absolute magnitude of the sample (after application of statistical parallax technique); r t true - true distance r t, appropriate to true absolute magnitude of the star (generally unknown).

Excpectedrefined systematic offset ΔM V Excpected and refined absolute magnitudes (distances) differ due to systematic offset of the absolute magnitude, ΔM V, just what we have to found Truerefined random factors σ M True and refined absolute magnitudes differ due to random factors (chemistry, stellar rotation, extinction, age etc.). Random scatter can be described in terms of absolute magnitude standard (rms) variance, σ M ΔMVΔMV Toy distribution of accepted and refined absolute magnitudes True M V Refined mean mean Expected

Kinematic model of the stellar sample: Four components of 3D-velocity: -Local sample motion -Local sample motion relative the Sun, V 0 -Systematic motion -Systematic motion, including differential rotation and noncircular motions, unified by the vector V SYS -Ellipsoidal (3D-Gaussian) distribution -Ellipsoidal (3D-Gaussian) distribution of true residual velocities, manifested by star’s random velocity vector η -Errors: -Errors: in radial velocity and proper motions

Difference between “observed” space velocity and that predicted by the kinematical model is expected to have 3D-Gaussian distribution as r ewhere calculates for expected distance, r e Land L is 3x3 covariance tensor for difference L =, T – L =, T – transposition sign V loc (r e ) is what we measure !

Galactic disk Sun rere VbVb VlVl VRVR V loc (r e )V loc (r e ) is defined in the local “astrocentric” coordinate system (see picture) via: V R r eV R radial velocity, independent on distance r e V l = kr e μ lV l = kr e μ l velocity on the galactic longitude V b = kr e μ bV b = kr e μ b velocity on the galactic latitude

After some advanced algebra: Covariance tensor Observed errors Ellipsoidal distribution Systematic motion: (a) relative to the Sun and (b) rotation where

product f Individual velocities of all stars are independent on each other; in this case full (N-body) distribution function is the product of N individual functions f, A Maximum Likelihood principleobserved is most probable of all A F maximum-likelihood functionLF where N is the number of stars, A is the “vector” of unknown parameters to be found. Maximum Likelihood principle states that observed set of velocity differences is most probable of all possible sets. The set of parameters, A, is calculated under assumption that F reaches its maximum (or minimum, for maximum-likelihood function LF )

f LFA For 3D-Gaussian distributions functions f, LF can be written as a function of A |L|L -1 LFA {A} Here |L| is matrix determinant, L -1 is inverse matrix. By minimizing LF by A, we calculate all important parameters {A}, for example:

Robust statistical parallax method: applied to local disk populations Astronomical background: A A A, Oort constant, derived from proper motions alone, depends on the distance scale used, whereas A, derived only from radial velocities, do not depend on the distances A As a result, scale factor can be estimated by requirement that both A values are equal to each other

Local Oort’s approximation Differential rotation contribution to space velocity components in local approximation r << R 0 (|R P –R 0 | << R 0 ): r << R 0 ( or |R P –R 0 | << R 0 ): To first order by the ratio r/R 0 in the expansion for the angular velocity:

Differential rotation effect to radial velocity Vr : From 1 st Bottlinger equation (for radial velocity) calculate contribution of the differential rotation to V r calculate contribution of the differential rotation to V r : A A Oort’s constant (definition) Linearity on r, “double wave” on l

Differential rotation effect to tangential velocity V l : From 2 nd Bottlinger equation (for velocity on l) calculate contribution of the differential rotation to V l calculate contribution of the differential rotation to V l : Linearity on r, “double wave” on l

Oort constant A and the refinement of the distance scale A 0 Vr A 0 Vr ~ p -1 A 0 Vr depends on the distance scale: A 0 Vr ~ p -1 (decreases with increasing distances) A 0 μl A 0 μl ≈ const A 0 μl do not depend on the distance scale, A 0 μl ≈ const A 0 Vr (p) ≈ A 0 μl The requirement A 0 Vr (p) ≈ A 0 μl – robust method of the p adjusting the scale factor p

Illustration of the robust technique Optimal value of the scale factor A Vr AμlAμlAμlAμl

F.A.Q. How the corrections to absolute magnitudes are affected by the: (a) Shape of the velocity distribution (deviation from expected 3D-Gaussian form) (b) Vertex deviation of the velocity ellipsoid (velocity-position correlations) (c) Misestimates of the observation errors (d) Non-uniform space distribution of stars (e) Sample size (f) Malmquist bias (excess of intrinsically bright stars in the magnitude-limited stellar sample) (g) Interstellar extinction (h) Misidentification of stellar populations

systematic offsets P.Popowski & A.Gould (a) analytically (b) by Monte-Carlo simulations,Possible factors of systematic offsets have been analyzed by P.Popowski & A.Gould in the papers “Systematics of RR Lyrae statistical parallax. I-III” (ApJ V.506, P.259, P.271, 1998; ApJ V.508, P.844, 1998) (a) analytically and (b) by Monte-Carlo simulations, and applied to the sample of RR Lyrae variables

P.Popowski & A.Gould (1998): extremely robust and insensitive“Statistical parallax method … is extremely robust and insensitive to several different categories of systematic effects” “… statistical errors are dominated by the size of the stellar sample” … sensitive to systematic errors in the observed data … Malmquist bias should be taken into account prior to calculations

bimodal versions A.Rastorguev, A.Dambis & M.ZabolotskikhTo eliminate the effects due to non- uniformity of the sample, bimodal versions of the statistical parallax method can be used (A.Rastorguev, A.Dambis & M.Zabolotskikh “The Three- Dimensional Universe with GAIA”, ESA SP-576, P.707, 2005) Example: RR Lyrae sample of halo and thick disk stars

the absolute methodStatistical parallax technique is considered as the absolute method of the distance scale calibration, though it exploits prior information on the adequate kinematic model of the sample studied After HIPPARCOS, luminosities and distance scales of RR Lyrae stars, Cepheids and young open clusters have been analyzed by the statistical parallax technique

Astronomical Distances or Measuring the Universe (Chapters 5 & 6) by Rastorguev Alexey, professor of the Moscow State University and Sternberg Astronomical.

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Astronomical Distances or Measuring the Universe (Chapters 5 & 6) by Rastorguev Alexey, professor of the Moscow State University and Sternberg Astronomical.

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