Astronomical Distances or Measuring the Universe Chapters 5

1
Astronomical Distancesor Measuring the
Universe(Chapters 5 6)by Rastorguev
Alexey,professor of the Moscow State University
and Sternberg Astronomical Institute, Russia
Sternberg Astronomical
Institute Moscow University
2
Content

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

3
Chapter Five

• Main-Sequence Fitting, or
• the distance scale of star clusters
• Open clusters
• Globular clusters

4

• 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

5
Advantages of using star clusters as the
standard candles - 1

• (a) Large statistics (N100-1000 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
• (c) All stars have nearly the same chemical
  composition, and the differences in the
  metallicity between the stars play no role

6
Advantages of using star clusters as the
standard candles - 2

• (d) Simplify the identification of stellar
  populations seen on HRD
• (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
• (g) Enable secondary luminosity calibration of
  some stars populated star clusters Cepheids,
  Novae and other variables

7

• 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 (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)
• These incomplete results point out to the
  observers that a large effort is still needed to
  improve the data in the catalog (W.Dias)

8
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 CMD

9

• 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 different ages
  (loci of stars of different initial masses but of
  the same age and metallicity)

10

• Important note Theoretical evolutionary tracks
  and theoretical isochrones are calculated in lg
  Teff Mbol variables
• Prior to compare directly evolution calculations
  with observations of open clusters, we have to
  transform Teff to observed colors, (B-V) etc.,
  and bolometric luminosities lg L/LSun and
  magnitudes Mbol to absolute magnitudes MV etc. in
  UBV broad-band photometric system (or others)

11

• 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 of
• (a) different colors
• (b) different chemical compositions
• (c) different luminosities
• with accurately measured spectral energy
  distributions (SED),
• or calibration based on the principles of the
  synthetic photometry

12
Bolometric magnitudes and bolometric corrections

• Bolometric Magnitude, Mbol, specifies total
  energy output of the star (to some constant)
• Bolometric Correction, BCV, is defined as the
  correction to V magnitude

gt1
BCV 0
By definition, Mbol MV BCV
13
Example BCV vs lg Teff unique relation for all
luminosities
From P.Flower (ApJ V.469, P.355, 1996)
14

• Note Maximum BCV 0 at lgTeff3.8-4.0 (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 band

15

• For modern color-temperature and BC-temperature
  calibrations see papers by
• P. Flower (ApJ V.469, P.355, 1996)
• lgTeff - BCV (B-V) from observations
• T. Lejeune et al. (AAS V.130, P.65, 1998)
• Multicolor synthetic-photometry approach
• lgTeffBCV(U-B)-(B-V)-(V-I)-(V-K)--(K-L),
• for dwarf and giants with Fe/H1-3
• (with step 0.5 in Fe/H)

16

• lgTeff (B-V)
• for different luminosities based on observations
• (from P.Flower, ApJ V.469, P.355, 1996)
• Shifted down by ? lgTeff 0.3 relative to next
  more luminous class for the sake of convenience

17

• T.Lejeune et al. (AAS V.130, P.65, 1998)
• Colors from UV to NIR vs Teff (theory and
  empirical corrections)

18

• 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

19
Revised HIPPARCOS parallaxes of nearby open
clusters (van Leeuwen, 2007)
20
Pleiades problem HST gives smaller parallax (by
8) ?MHp -0.17m

• Combined MHp (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 squares) is due to
• Larger Fe/H
• Larger age 650 Myr
• Bottom envelope (----) can be treated as an
  observed ZAMS

MHp
(V-I)
21

• (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

22
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. (AAS V.130, P.65, 1998) color-temperature
calibrations
MV
23
ZAMS and Hyades isochrones sensitivity to the
age for 650100 Myr (from Y.Lebreton, 2001)

• Fitting color of the turn-off point

ZAMS
24

• Best library of isochrones recommended to
  calculate cluster distances, ages and
  extinctions
• L.Girardi et al. Theoretical isochrones in
  several photometric systems I. (AA V.391, P.195,
  2002)
• Theoretical background
• (a) Evolution tracks calculations for different
  initial stellar masses (0.15-7MSun) and
  metallicities (-2.50.5) (also including
  a-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

25

• L.Girardi et al. Theoretical isochrones in
  several photometric systems I. (AA V.391, P.195,
  2002)
• Distribution of spectra in Padova library on lg
  Teff 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
26

• 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)

27
Metallicity effects on isochronesmodelling
variables, Mbol - Teff
Turn-off point
28
Metallicity effects on isochrones optics
Turn-off point
29
Metallicity effects on isochrones NIR
Turn-off point
30

• The corrections ?M and ?CI (CI Color Index) vs
  ?Fe/H or ?Z 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

31

• Metallicity differences can be taken into account
  by
• (a) Adding the corrections to absolute magnitudes
  ?M and to colors ?CI 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

32

• Ideally, 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 differences in Helium content (Y).
  Theoretically, can play important role. As a
  rule, evolutionary tracks and isochrones of solar
  Helium abundance (Y0.27-0.29) are used

33

• L.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, a/Fe, convection,)

34

• Example 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

35
0.1
1
10 Gyr

• Theoretical isochrones (color - MV magnitude
  diagrams) for solar composition (Z0.019) and
  cluster ages 0.1 Gyr, 1 Gyr and 10 Gyr (L.Girardi
  et al., 2002, green solid lines)

36
0.1
1
What are fancy shapes !
1 Gyr

• Theoretical isochrones (NIR color-magnitude
  diagrams) for solar composition (Z0.019) and
  cluster ages 0.1 Gyr, 1 Gyr and 10 Gyr (L.Girardi
  et al., 2002, green solid lines)

37
Girardi et al. isochrones in modelling
variablesMbol lg Teff (more detailed age grid)
ZAMS
38
Optics NIR
ZAMS

• The same but for standard multicolor system

ZAMS
39
How estimate age, extinction and the
distance?1st 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

40
How estimate age, extinction and the
distance?1st 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 ?M, to fit proper
  ZAMS isochrone, i.e. cluster turn-off point.
  Calculate true distance modulus as (V-MV)0 ?V -
  RVE(B-V)
• (for V(B-V) CMD)

41
How estimate age, extinction and the
distance?2nd 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 horizontal and vertical offsets
• H ? (color) CE (color excess)
• V (m-M) (m-M)0 R CE
• (m-M)0 true distance modulus

42
How estimate age, extinction and the
distance?2nd 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 !
43
MS-fitting example Pleiades, good case
Magnitudes offset gives ?V(V-MV)0RVE(B-V) ? (
m-M)0 5.60 E(B-V)0.04 lg (age) 8.00

ZAMS
G.Meynet et al. (AAS V.98, P.477, 1993) Geneva
isochrones
44
Young distant cluster, good case
(m-M)012.55 E(B-V)0.38 lg (age)7.15
G.Meynet et al. (AAS V.98, P.477, 1993) Geneva
isochrones
45
h Per cluster
(m-M)013.65 E(B-V)0.56 lg (age)7.15
RSG (Red Super- Giants)
G.Meynet et al. (AAS V.98, P.477, 1993) Geneva
isochrones
46
RSG
(m-M)012.10 E(B-V)0.32 lg (age)8.22
G.Meynet et al. (AAS V.98, P.477, 1993) Geneva
isochrones
47
Older and older
(m-M)07.88 E(B-V)0.02 lg (age)9.25
G.Meynet et al. (AAS V.98, P.477, 1993) Geneva
isochrones
48
Very old open cluster, M67
(m-M)09.60 E(B-V)0.03 lg (age)9.60
G.Meynet et al. (AAS V.98, P.477, 1993) Geneva
isochrones
49
Optical data D.An et al. (ApJ V.671, P.1640,
2007)(Some open clusters populated with Cepheid
variables)
50
The same, NIR data D.An et al. (ApJ V.671,
P.1640, 2007)
51

• New 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

52

• 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
53
MS-fitting accuracy(best case, multicolor
photometry)(D.An et al ApJ V.655, P.233, 2007)

• Random error of MS-fitting
• with spectroscopic Fe/H d(m-M)0 0.02m, i.e
  1 in the distance
• Systematic errors due to uncertainties of
  calibrations, Fe/H and a-elements, field
  contamination and contribution of unresolved
  binaries
• d(m-M)0 0.04-0.06m, i.e. 2-4 in the distance
• Uncertainties of Helium abundance may result in
  even larger systematic errors

54

• For distant clusters, with CMD contaminated by
  foreground/background stars, and uncertainties in
  Fe/H, errors may increase to
• ?(m-M)00.1m(random) 0.2m(systematic)
• Typical distance accuracy of remote open
  clusters is 10-15

55

• 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 (EHB)
TP
56

• D.An et al. (arXiv0808.0001v1)
• Isochrones (MS giant branch) for globular
  clusters of different Fe/H in (u g r i z)
  photometric bands (SDSS)

more metal-deficient
u 3551Å g 4686Å r 6165Å i 7481Å z 8931Å
Å
57
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
58

• CMD for selected galactic globular clusters (HST
  observations of 74 GGC G.Piotto et al., AA
  V.391, P.945, 2002)
• Bad cases for MS-fitting (except NGC 6397)

59

• 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)

60

• Illustration 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)

61

• Horizontal
• method
• calibrations
• Color offset
• vs Fe/H
• for different
• ages

Gyr
Gyr
Gyr
Gyr
62

• Vertical method calibrations magnitude
  difference vs Fe/H for different ages

Gyr
63

• In some cases isochrone fitting fails to give
  unique result because of multiple stellar
  populations found in most massive galactic and
  extragalactic globular clusters (?
  Cen L.Bedin et al., ApJ V.605, L125, 2004 NGC
  1806 NGC 1846 in LMC A.Milone et al.,
  arXiv0810.2558v1)

64
? Cen
Multiple populations ? He abundance differences ?
NGC 1806 (LMC)
65
Chapter Six

• Statistical parallaxes

66
Astronomical background

• Statistical 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
• Statistical parallax technique involves space
  velocities of uniform sample of objects at
  first glance, it sounds as strange and unusual

67
Main idea

• To match the tangential velocities (VT
  k r µ, proportional to distance scale of the
  sample of studied stars) and radial velocities VR
  (independent on the distance scale), under
  three-dimensional normal (ellipsoidal)
  distribution of the residual velocities

VTk r µ
VR
Sun
r
68
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
69
instead of being alike and pointed to the
galactic center
70

• One 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)
• Her estimate ltMVgtRR 0.6m was widely used and
  kept before early 1980th and even recently,
  differ only slightly on modern value for
  metal-deficient RR Lyrae (0.75m)

71

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

72

• 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, so
• only brief description follows

73

• Detailed description of the 3D statistical
  parallax technique can be found only in
  A.Rastorguevs (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)

74

• Photometric distances are calculated by stars
  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, ?M.
• Statistical parallax technique distinguishes
• expected distance re, calculated by accepted
  mean absolute magnitude of the sample (after
  luminosity calibration)
• refined distance r, calculated by refined mean
  absolute magnitude of the sample (after
  application of statistical parallax technique)
• true distance rt, appropriate to true absolute
  magnitude of the star (generally unknown).

75
Toy distribution of accepted and refined
absolute magnitudes
?MV
Refined mean
Expected mean
True MV

• Excpected and refined absolute magnitudes
  (distances) differ due to systematic offset of
  the absolute magnitude, ?MV, just what we have to
  found
• 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, sM

76
Kinematic model of the stellar sample

• Four components of 3D-velocity
• Local sample motion relative the Sun, V0
• Systematic motion, including differential
  rotation and noncircular motions, unified by the
  vector VSYS
• Ellipsoidal (3D-Gaussian) distribution of true
  residual velocities, manifested by stars random
  velocity vector ?
• Errors in radial velocity and proper motions

77

• Difference between observed space velocity and
  that predicted by the kinematical model is
  expected to have 3D-Gaussian distribution as
• where calculates for expected distance,
  re
• and L is 3x3 covariance tensor for difference

L lt?V?V Tgt, T transposition sign
Vloc (re) is what we measure !
78

• Vloc(re) is defined in the local astrocentric
  coordinate system (see picture) via
• VR radial velocity, independent on distance re
• Vl kre µl velocity on the galactic longitude
• Vb kre µb velocity on the galactic latitude

Vb
VR
Vl
re
Galactic disk
Sun
79
After some advanced algebra
Covariance tensor
Observed errors
Ellipsoidal distribution
Systematic motion (a) relative to the Sun and
(b) rotation
where
80
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,

• 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 )

81
For 3D-Gaussian distributions functions f, LF can
be written as a function of A

• Here L is matrix determinant, L-1 is
  inverse matrix. By minimizing LF by A, we
  calculate all important parameters A, for
  example

82
Robust statistical parallax methodapplied to
local disk populations

• Astronomical background
• 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
• As a result, scale factor can be estimated by
  requirement that both A values are equal to each
  other

83
Local Oorts approximation
Differential rotation contribution to space
velocity components in local approximation
r ltlt R0 (or RP R0 ltlt R0 )
To first order by the ratio r/R0 in the
expansion for the angular velocity
84
Differential rotation effect to radial velocity
Vr
From 1st Bottlinger equation (for radial velocity)
calculate contribution of the differential
rotation to Vr
Linearity on r, double wave on l
A Oorts constant (definition)
85
Differential rotation effect to tangential
velocity Vl
From 2nd Bottlinger equation (for velocity on l)
calculate contribution of the differential
rotation to Vl
Linearity on r, double wave on l
86
Oort constant A and the refinement of the
distance scale
A0Vr depends on the distance scale A0Vr p -1
(decreases with increasing distances)
A0µl do not depend on the distance scale, A0µl
const
The requirement A0Vr (p) A0µl robust method
of the adjusting the scale factor p
87
Illustration of the robust technique
AVr
Optimal value of the scale factor
Aµl
88
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

89

• 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

90
P.Popowski A.Gould (1998)

• 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

91

• To 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

92

• Statistical 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