Transcript Слайд 1 - Tuorla Observatory

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
Moscow University
Content
• Chapter Five: Main-Sequence Fitting, or the
distance scale of star clusters
• Chapter Six: Statistical parallaxes
Chapter Five
Main-Sequence Fitting, or
the distance scale of star clusters
• Open clusters
• Globular clusters
• 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
• (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
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
• DataBase on open clusters: W.Dias, J.Lepine, B.Alessi
(Brasilia)
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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)
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
• 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)
• 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)
• 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”
Bolometric magnitudes and bolometric corrections
• Bolometric Magnitude, Mbol, specifies total
energy output of the star (to some constant):
M bol


 2.5 lg   j(  ) d   cb


• Bolometric Correction, BCV, is defined as the
correction to
V magnitude:
BCV  M bol  M V  2.5 lg
 j( ) d
 j( ) R
V
By definition, Mbol = MV + BCV
( )d
 const
BCV ≤ 0
Example:
BCV vs lg Teff: unique relation for all luminosities
From P.Flower (ApJ
V.469, P.355, 1996)
• Note: Maximum BCV ~0 at lgTeff~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 band
• For modern color-temperature and BCtemperature calibrations see papers by:
• P. Flower (ApJ V.469, P.355, 1996):
lgTeff - BCV – (B-V) from observations
• T. Lejeune et al. (A&AS V.130, P.65, 1998):
Multicolor synthetic-photometry approach;
lgTeff–BCV–(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])
• lgTeff – (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):
• Colors from UV to NIR vs Teff (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,
(m-M)0 and
error, mas its error,
magn.
[Fe/H]
Age,
Myr
E(B-V)
Praesepe
5.49±0.19
6.30±0.07
+0.11
~830
0.00
Coma 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
0.04
IC 2391
6.78±0.13
5.85±0.04
-0.040
30
0.01
IC 2602
6.64±0.09
5.89±0.03
-0.020
30
0.04
NGC 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
0.09
Hyades
21.51±0.14
3.34±0.02
+0.13
650
0.003
Pleiades problem:
HST gives smaller
parallax (by ~8%) •
ΔMHp ≈ -0.17m
MHp
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
(V-I)
• Bottom envelope (----)
can be treated as an
observed ZAMS
• (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)
MV
Solid line: theoretical isochrone with
Lejeune et al. (A&AS V.130, P.65, 1998)
color-temperature calibrations
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)
• Theoretical background:
– (a) Evolution tracks calculations for different
initial stellar masses (0.15-7MSun) and
metallicities (-2.5…+0.5) (also including αelement 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
Giants
• 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 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-CousinsGlass), WFPC2 (HST), …
• 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, Mbol - Teff
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 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 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 (Y=0.27-0.29) are used
• L.Girardi et al. (2002) database on
isochrones and evolutionary tracks is of
great value – it provides us with “readyto-use” multicolor isochrones for a large
variety of the parameters involved (age,
[Fe/H], [α/Fe], convection,…)
• 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
0.1
1
10 Gyr
• Theoretical isochrones (color - MV 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 !
• Theoretical 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)
Girardi et al. isochrones in modelling variables
Mbol – lg Teff (more detailed age grid)
Optics
NIR
• The same but
for “standard”
multicolor
system
How estimate age, extinction and the distance?
1st variant
• (a) Calculate color-excess CE for cluster
stars on two-color diagram like (U-B) – (BV). Statistically more accurate than for
single star. Highly desirable to use a set of
two-color diagrams as (U-B) – (B-V) and (BV) – (V-I) etc., to reduce statistical and
systematical errors
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 ΔM, to
fit proper ZAMS isochrone, i.e. cluster
turn-off point. Calculate true distance
modulus as (V-MV)0 = ΔV - RV∙E(B-V)
• (for V–(B-V) CMD)
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 (colormagnitude diagram) to ZAMS and isochrone
trying to fit cluster turn-off point
• (c) Calculate horizontal and vertical offsets:
H:
Δ (color) = CE (color excess)
V:
(m-M) = (m-M)0 + R· CE
(m-M)0 – true distance modulus
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 !
MS-fitting example: Pleiades, good case
Magnitudes offset
gives
ZAMS
ΔV=(V-MV)0+RV∙E(B-V)
↨
(m-M)0 = 5.60
E(B-V)=0.04
lg (age) = 8.00
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 et al.
(A&AS V.98, P.477,
1993)
Geneva isochrones
h Per cluster
(m-M)0=13.65
E(B-V)=0.56
lg (age)=7.15
RSG
(Red
SuperGiants)
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 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 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 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)
• 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
• 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
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]: δ(m-M)0 ≈ ±0.02m, 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 ≈ ±0.04-0.06m, 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≈±0.1m(random) ± 0.2m(systematic)
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
RR Lyrae
indicators, including
RR Lyrae variables
BHB
(EHB)
(with nearly constant
luminosity, see later)
TP
• D.An et al.
(arXiv:0808.0001v1)
• Isochrones (MS + giant
branch) for globular
clusters of different
[Fe/H] in (u g r i
photometric bands
(SDSS)
z)
u
g
r
i
3551Å
4686Å
6165Å
7481Å
z 8931Å
Å
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)
• 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 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
Gyr
Gyr
Gyr
“Vertical” method calibrations: magnitude difference vs
[Fe/H] for different ages
Gyr
• In 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
Multiple populations ?
He abundance
differences ?
NGC 1806
(LMC)
Chapter Six
Statistical parallaxes
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 mainsequence fitting
• Statistical parallax technique involves
space velocities of uniform sample of
objects – at first glance, it sounds as
strange and unusual…
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
Sun
VR
r
VT=k r μ
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
• 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 <MV>RR ≈ +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)
• 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)
• 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
• Detailed 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)
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:
- 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).
ΔMV
Refined
mean
True MV
Toy distribution
of accepted and
refined absolute
magnitudes
Expected
mean
Excpected and refined absolute magnitudes (distances)
differ due to systematic offset of the absolute
magnitude, ΔMV, 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, σM
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 star’s
random velocity vector η
- 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


1 / 2
 1 T
3 / 2
1
f ( V )  ( 2 )
L
exp  V  L  V 
 2

 

• where V  Vloc( re )  Vloc,mod ( re ) calculates for
expected distance, re

• and L is 3x3 covariance tensor for difference V
L = <ΔV·ΔV T>, T – transposition sign
Vloc (re) is what we measure !
• 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
Sun
Galactic disk
Covariance tensor L( re )  Lerr ( re )  Lresid ( re )  L( re )
After some advanced algebra:
 Vr 2
0

2
2
Observed errors Lerr ( re )   0
k 2 re  l

2 2
b
0
k
r

e
l  l b

 lb  correlation coefficient
~


2 2
b
k re  l  l b 
2
2

k 2 re  b

0
~
Ellipsoidal distribution Lresid ( re )  P  GS  L0  GS  PT
T
Systematic motion: (a) relative to the Sun and (b) rotation
~
~  T
T
L( re )  0.21 p    [ M  GS  L0  GS  M     ]
2
2
M
 ~
 

~
where   M  [ G0 V0  Vsys ( r )  r / p  P  Vsys ( r ) / r
~ ~
M , P , G0 , GS  knowncoordinate- dependentmatrixes,
L0  dispertsion matrix,Vsys  systematicmotion(rotation)
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,
N
 
 

F V1 ,V2 ,...,VN | A   f Vi


 
i 1
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 )




N

 
 
LF   ln F V1 ,...,VN | A   ln f Vi | A
i 1
For 3D-Gaussian distributions functions f,
LF can be written as a function of A


T

3
1 N
1
LF( A )  N  ln 2   ln Li  Vi  Li  Vi
2
2 i 1

Here |L| is matrix determinant, L-1 is inverse
matrix. By minimizing LF by A, we calculate all
important parameters
{A}, for example:
(u0 , v0 , w0 )  samplevelocity elative
r
to theSun
( U , V , W )  velocityellipsoid axes
(0 ,0 ,0,...)  rotationcurve parameters
p  re / r  1  M / 2.17  distancescale factor
Robust statistical parallax method:
applied 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
Local Oort’s approximation
Differential rotation contribution to space
velocity components in local approximation
r << R0 (or |RP –R0 | << R0 ):
2
r
r
r
2
2
2
2
RP  R0 ( 1  2 cos b  2 cosb cos l )  R0 ( 1  2 cosb cos l )
R0
R0
R0
RP  R0   r cosb cosl
To first order by the ratio r/R0
in the expansion for the angular velocity:
  0  r 0 cosb cosl
Differential rotation effect to radial velocity Vr :
From 1st Bottlinger equation (for radial velocity)
R0 (  0 ) sin l cosb
 Vr  

U 0 

 

   loc
T
 k r l    R0 cosl  r cosb  (  0 )  0 r cosb   G   V0   V pec
kr   

W 

R
(



)
sin
l
sin
b
b
0
0

 

 0
calculate contribution of the differential rotation to Vr :
R00
2
V  R0 (   0 ) sin l cos b  
sin 2l cos b
2
rot
2
Linearity on r,
Vr  A0 r sin 2l cos b ,
“double wave” on l
R00
A0  
 A Oort’s constant (definition)
2
rot
r
Differential rotation effect to tangential velocity Vl :
From 2nd Bottlinger equation (for velocity on l)
R0 (  0 ) sin l cosb
 Vr  

U 0 

 

   loc
T
 k r l    R0 cosl  r cosb  (  0 )  0 r cosb   G   V0   V pec
kr   

W 

R
(



)
sin
l
sin
b
b
0
0

 

 0
calculate contribution of the differential rotation to
Vl :
Vl rot  ( R0 cos l  r cos b )(   0 )  0r cos b 
  r R00 cos2 l cos b  r 20 cos l cos2 b  r0 cos b 
 2 A0 r cos2 l cos b  0r cos b
Vl rot  A0 r cos 2l cos b  ( 0  A0 ) r cos b
Linearity on r,
“double wave” on l
Oort constant A and the refinement of the distance scale
V
 A0 prtrue sin 2l cos b
rot
r
2
A0Vr depends on the distance scale: A0Vr ~ p -1
(decreases with increasing distances)
Vl
k
rot
rot
l
 kr
rot
l
 A0 r cos 2l cos b  ( 0  A0 ) r cos b
 A0 cos 2l cos b  ( 0  A0 ) cos b
A0μl do not depend on the distance scale, A0μl ≈ const
The requirement A0Vr
(p) ≈ A0μl – robust
adjusting the scale factor
p
method of the
Illustration of the robust technique
AVr
Optimal value of
the scale factor
Aμl
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
• 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):
• “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
• To eliminate the effects due to nonuniformity of the sample, bimodal
versions of the statistical parallax
method can be used (A.Rastorguev,
A.Dambis & M.Zabolotskikh “The ThreeDimensional Universe with GAIA”, ESA
SP-576, P.707, 2005)
• Example: RR Lyrae sample of halo and
thick disk stars
• 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