Download Cepheids

Astronomical Distances
or
Measuring the Universe
(Chapters 7, 8 & 9)
by Rastorguev Alexey,
professor of the Moscow State
University and Sternberg Astronomical
Institute, Russia
Sternberg Astronomical
Institute
Moscow University
Content
• Chapter Seven: BBW: Moving Photospheres
(Baade-Becker-Wesselink technique)
• Chapter Eight: Cepheid parallaxes and
Period – Luminosity relations
• Chapter Nine: RR Lyrae distance scale
Chapter Seven
BBW: Moving Photospheres (BaadeBecker-Wesselink technique)
Astronomical Background
• W.Baade-W.Becker-A.Wesselink-L.Balona
technique (BBWB) is applicable to pulsating
stars (Cepheids and RR Lyrae variables) or
stars with expanding envelopes
(SuperNovae)
• W.Baade (Mittel.Hamburg.Sternw. V.6, P.85,
1931); W.Becker (ZAph V.19, P.289, 1940);
A.Wesselink (Bull.Astr.Inst.Netherl. V.10,
P.468, 1946):
• photometry and radial velocities give rise to
radius measurement and independent
calculation of star’s luminosity
(a) For a pair of phases with the same
temperature and color (say, B-V), the
Cepheid’s apparent magnitudes V differ
due only to the ratio of stellar radii:
R
m  2.5 lg
R
R1
B-V
2
1
2
2
(b) Radii difference
Pulsation Phase
(R1-R2) can be calculated by integrating the
radial velocity curve, due to VR ~ dR/dt
R2
• A number of pairs
• (R1 – R2) ~∫VR·dt and R1/R2 ≈ 10-0.2m
give rise to the calculation of mean
radius, <R> = (Rmax + Rmin)/2
I. BBW Surface-brightness technique (SB)
Eλ
θLD
• θLD is “limb-darkened” angular diameter
• Flux incident, Eλ ~ Φλ·θLD2, where Φλ being
surface brightness (do not depending on the
star’s distance!)
• Apparent magnitude, mλ ~ -2.5 log Eλ , so
• 0.5∙log θLD ~ -0.1·mλ - Fλ + c ,
with Fλ as the “surface-brightness parameter”
• It can be easily shown that the “surfacebrightness parameter” is expressed as
• Fλ = 4.2207 – 0.1∙mλ0 - 0.5∙ log θLD =
=log Teff + 0.1 BC(λ),
• where BC(λ) is the bolometric correction to
the mλ magnitude, i.e.
• BC(mλ) = Mbol - Mλ
• The surface-brightness parameter is shown
to relate with the color CI (due to surface
brightness dependence on the effective
temperature Φλ ~ Teff4 and CI-Teff relations)
• Fλ ≈ a·CIλ + b
• Example relation is
shown; interferometric
calibrations (Nordgren
et al. AJ V.123, P.3380,
2002)
• The calibrations for surface brightness
parameter, Fλ, can be also derived from the
photometric data for normal stars (dwarfs,
giants and supergiants) with the distances
determined, say, by precise trigonometric
technique or other data
• See, for example, M.Groenewegen
“Improved Baade-Wesselink surface
brightness relations” (MNRAS V.353, P.903,
2004)
• From both relations,
• 0.5∙log θLD ≈ -0.1·mλ - Fλ + c
• Fλ ≈ a·CIλ + b
• log θLD ~ -0.2·mλ + 2a·CIλ + const
Light-curve
Color-curve
• Direct calculation of star’s angular
diameter vs time
• Radial velocity curve integration gives
absolute radius change, and being relate
to apparent angular diameter change,
distance to the pulsating star follows
II. Maximum Likelyhood technique by
L.Balona (MNRAS V.178, P.231-243,1977)
uses all light curve and precise radial
velocity curve
• Background:
• (a) Stefan-Boltzmann low: Lbol ~ σT4
• (b) Effective temperature and
bolometric correction are connected
with normal color (see examples)
• (c) Radius change comes from the
integration of the radial velocity curve
Luminosity classes:
Working interval
lg Teff – (B-V)0 relation by P.Flower (1996)
Working interval
ΔMbol
Luminosity
classes
ΔMbol – (B-V)0 relation by P.Flower (ApJ, V469, P.355, 1996)
Quadratic approximations can be written for
lgTeff, ΔMbol by (B-V)0
(and by other normal colors) in the range
0.2 < (B-V)0 < 1.6
lg Teff     ( B  V )0   ( B  V )
2
M bol     ( B  V )0   ( B  V )
( B  V )0  ( B  V )  E ( B  V )
0
2
0
Basic equations of the Balona approach
Lbol  4Teff R (Stefan - Boltzmann law)
4
2
Teff  R 
Lbol



4 
Sun
L bol TSun  RSun 
4
M bol  M
Sun
bol
2
Lbol
 2.5 lg Sun 
Lbol
R
 10 lg Teff  10 lg TSun  5 lg
RSun
Substituti ng M bol  M V  M bol
From the distance modulus expression,
d
M V  V  5 lg
 5  AV , after substituti on of all
(1 pc)
values we wright th e light curve MODEL as
R
V  A  ( B  V )  B  ( B  V )  5 lg
 C,
RSun
2
where
A, B, C
are the constants , V and ( B  V )
R  R  r (t )
is current radius with mean value  R 
are light and color curves , and
Here constants A, B and C:
A  2(  10 )  E ( B  V )  (   10 ),
B  (  10 ),
d
Sun
C  5  lg
 5  RV  E ( B  V )  M bol
 10 lg TSun 
(1 pc)
 (  10 )  (   10 )  E ( B  V )  (  10 )  E ( B  V ) 2
These constants enclose information on color
excess E(B-V) and the star’s distance d
How to calculate the radius change?
dS: ring area
-V0: photosphere velocity
(to the observer)
Contribution of the
circular ring to the
observed radial velocity
Calculating R(t)=<R>+r (t): integrating radial
velocity curve Vr
Ring contribution to the measured radial
velocity:
1) V ( )   r cos   ring contributi on to Vr ;
2) 2 r sin  r d  2 r sin  d  ring area ;
3) cos   projection effect to light flux ;
4) (1     cos  )  limb darkening
Ring " weight " :
2
W ( )  2 r sin  cos   (1     cos  )
2
Mean radial velocity weighted along star’s
limb = observed Vr
 /2
Vr 
V ( ) W ( ) d
0
 /2
 W ( ) d
0
 /2
  r 
2
sin

cos
 (1     cos  ) d

0
 /2
 sin  cos  (1     cos  ) d
1
  r
p
0
p is the projection factor connecting
radial velocity and the photosphere speed
Projection factor values:
• In the absence of limb darkening p=3/2
• In most studies p=1.31 accepted
r   pVr
• But: projection factor depend on
– Limb darkening coefficient
– Velocity of the photosphere
– Instrumental line profile used to measure
radial velocity
Theoretical spectral line will be distorted
due to instrumental profile
Theoretica l line profile :

20 r
v
F (v) 
v  1     ,
2
V0 
V0

v
( here
 cos  , 0  v  V0 )
V0
2
Instrument al Gaussian p rofile :
I N ( v, V )  ( 2 )
1 / 2
 (v  V )
 exp  
2
2

1
2



Line profile calculation enables to derive
theoretical value of the projection factor, p
Observed line profile (with taking into
account line broadening due to instrumental
effects) can be calculated as the convolution
V0
f (V )   F ( v )  I N (v,V )  dv
0
Some examples of the line profiles for different line
broadening due to instrumental effects
Instrumental width
Solid: normal
Observed profile
Good Gaussian fit
Theoretical profile
Small
velocity
Some examples of the line profiles for different line
broadening due to instrumental effects
Instrumental width
Theoretical profile
Normal curve
Observed profile
Large
velocity
Normal curve
(blue line)
Bad Gaussian fit
• Modern calculations confirm the
variation of p by at least ~5-7% due to
different effects
• (Nardetto et al., 2004; Groenewegen,
2007; Rastorguev & Fokin, 2009)
Example: projection factor and line width
vs photosphere velocity, V0
From Gauss tip
From line tip
• Projection factor p and its variation
with the velocity V0 , limb darkening
coefficient ε and instrumental width σ
should be “adjusted” to proper
spectroscopic technique used for radial
velocity measurements
• Example: More than 20000 VR
measurements have been performed by
Moscow team for ~165 northern
Cepheids with characteristic accuracy
~0.5 km/s and σ ~ 15 km/s
Radius change, r(t), can be calculated
by integrating the radial-velocity curve,
Vr (t), from some t=0 (where R(0)=<R>)
t
t
dr
r(t ) ~  dt ~   p Vr  dt
dt
0
0
ΔR, Solar units
Vr, km/s
Example: TT Aql radial velocity curve (upper
panel) and radius change (bottom panel)
Pulsation phase
Fitting the light curve for TT Aql Cepheid (+)
with the model (solid line)
V
V
A (B V )  B  (B V )
2
 5  lg
( R  r)  C
RSun
A  1.69  0.20 B  0.18  0.08
C  14.09  0.15  R  (82  1.6) R
Sun
Pulsation phase
Steps to luminosity calibration
• <R>, A, B, C enable:
– To calculate mean bolometric and visual
luminosity of the star and its distance
– To refine transformations of Teff to normal
colors and bolometric corrections
(additionally)
• BBWB technique has been applied to
Cepheid and RR Lyrae variables and
turned out to be very effective and
independent tool for P-L calibrations (see
later)
• For Fundamental tone Cepheids:
• log R/RSun = 1.23 (±0.03) + (0.62 ±0.03) log P
by M.Sachkov (2005)
Cepheid radii can
be used to classify Cepheids by
FU
the pulsation
1st
mode (first-overtone pulsators
have shorter periods at equal radii)
• Well known possibility to measure the
distances to some SuperNova remnants
via their angular sizes and the envelope
expansion velocities is also can be
considered as the modification of the
moving photospheres technique
• Possible explosion asymmetry can induce
large errors to estimated parameters
t1
t2
SN
t2
D   ~ VR (t )  dt
t1
VR
Δφ
D
• Optical and NIR interferometric direct
observations of the radii changes of
nearest Cepheid are of great
importance and very promising tool to
measure their distances and correct the
distance scale
• B.Lane et al. (ApJ V.573, P.330, 2002):
• Palomar PTI
85-m base length
optical
interferometer
Distance within
10% error
Phase
• P.Kervella et al. (ApJ V.604, L113, 2004):
interferometric (ESO VLT) apparent
diameter measurements (filled circles) as
compared
to BW SB
technique
(crosses)
D ~ 566
±20 pc
Very good
agreement
Chapter Eight
Cepheid parallaxes and
Period – Luminosity relations
•
•
•
•
•
Astrophysical background
Overtone pulsators
Use of Wesenheit index
Luminosity calibrations
P-L or P-L-C ?
Cepheids “econiche”
Cepheids as
calibrators
for these
methods
100 pc … 50 Mpc
pc
• Cepheids and RR
Lyrae variable stars
populate Instability
Strip on the HRD
(see blue strip)
where most stars
become unstable
with respect to
radial pulsations
• Instability Strip
crosses all
branches, from
supergiants to
white dwarfs
G.Tammann et al. (A&A
V.404, P.423, 2003)
• Instability strip
for galactic
Cepheids
(members of
open clusters
and with BBWB
radii) in more
details:
• MV-(B-V) and
MI-(V-I) CMDs
• Comprehensive review of the problem by:
Alan Sandage & Gustav Tammann
“Absolute magnitude calibrations of
Populations I and II Cepheids and Other
Pulsating Variables in the Instability
Strip of the Hertzsprung-Russell
Diagram” (ARAA V.44, P.93-140, 2006)
Cepheids: named after δ Cepheus star,
the first recognized star of this class
• Young (< 100 Myr) and massive (3-8 MSun)
bright (MV up to -7m) radially pulsating
variable stars with strongly regular
brightness change (light curves)
• Typical Periods of the pulsations: from ~1-3
to ~100 days (follow from P ~ (Gρ)-1/2
expression for free oscillations of the
gaseous sphere: mean mass density of huge
supergiants, ρ, is very small as compared to
the Sun)
• Luminosities: from
~100 to 30 000
that of the Sun
• Evolution status:
yellow and red
supergiants, fast
evolution to/from
red (super)giants
while crossing the
instability strip
(solid lines)
Evolution tracks for
1-25MSun stars on the HRD
• Relative fraction of Cepheids population
depends on the evolution rate while
crossing the instability strip:
• slower evolution means more stars at
the appropriate stage
• (a) Tracks are not
Cepheids
evolution tracks,
The
differences
in
the
evolution
parallel to the lines
the instability
rates
at different
give strip and
of constant
periods crossings
P=const
rise to the questions on
the lines (red)
lg bright
L
IS
relative contribution of
faint Cepeids
•and
Evolutional
period for the same
changevalue
happens
period
• (b) Evolution rate
during 2nd & 3rd
crossings is slower
than for 1st crossing,
defining the IS
population fraction
• (c) Luminosity
depend on the
crossing number
An extra source
of P-L scatter
• dP/dt is visible
when observing on
long time interval
(~100 yr)
P increase
+
P decrease
• At equal periods:
Cepheids
withlong
… but
it requires
different
masses
time monitoring
but withtodifferent
Impossible
use for
crossing number
extragalactic
Cepheids,
• lg (dP/dt) vs lg P:
crossing
number
Maybe,
any spectroscopic
diagnostics (as
features will be found
compared to the
responsible
for
crossing
theory)
• Perspective tool to
identify crossing
number and to refine
the P-L relation…
• The duration of the Cepheids stage is
typically less than 0.5 Myr and, taking also
in consideration the rarity of massive stars
at all, we guess that Cepheids form very
poor galactic population
• Statistics of Cepheids discovered:
~3000 proven and suspected in the Galaxy,
~2500 in the LMC,
~1500 in the SMC,
~Thousands are found and, ~50 000 are
expected to populate the Andromeda
Galaxy (M31)
• Easily recognized by its high brightness and
periodic magnitude change among other
stars, even in distant galaxies (up to 50
Mpc)
• Typical population of young clusters, spiral
and irregular galaxies
• Luminosity increases with the Period (P-L
relation)
• Cepheids are still among most important
“standard candles”
Hertzsprung Sequence of Cepheids
(normalized) light curves. Bump position.
(from P.Wils)
A.Udalski et al. “The Optical Gravitational Lensing
Experiment. Cepheids in the LMC. IV. Catalog…”
(Acta Astronomica V.49, P.223, 1999)
• Apparent mean
I magnitude
corrected for
differential
absorption
inside LMC vs
log P(days) for
fundamental
tone (FU) and
first overtone
(FO) cepheids:
• <I> ≈ a + b∙lg P
• PFO / PFU ≈ 0.71
• (Δ lg P ≈ 0.15)
W(I)
Brightest Cepheids
Distances are the same:
absolute magnitudes <MI>
are also linear on log P !
• Linearity of log P-log L relation retains also
for Cepheids absolute magnitudes
• Simple estimate:
• Brightest LMC Cepheids reach <MV> ≈ -7m
• These “standard candles” can be seen from
the distance ~50 Mpk (with ~27m limiting
magnitude accessible to HST)
• Brightest Cepheids can be widely used as
secondary sources of distance calibrations
to spiral galaxies hosted by SN Ia etc.
We can see Cepheids
even in distant galaxies
• Overtone Cepheids are clearly seen on the
“apparent magnitude – period” diagrams of
nearby galaxies as having smaller periods for
the same brightness
• Problem with Milky Way Cepheids: due to
distance differences, how to identify overtone
pulsators among Cepheids with different
distances ?
• Milky Way Cepheids sample is supposed to be
contaminated by unidentified first-overtone
pulsators
• This problem greatly complicates the extraction
of the P-L relation directly from observations
of Milky Way cepheids
Sources of calibrations of the
Cepheids absolute magnitudes
• (a) Trigonometric parallaxes
(HIPPARCOS and HST FGS)
• (b) Membership of Cepheids in open
clusters and associations
• (c) BBWB mean radii
• (d) Luminosity refinement by the
statistical parallax technique
(a) Cepheids trigonometric parallaxes
• P-L relation <MV> ≈ a + b∙lg P from van Leeuwen (2007)
data for 45 parallaxes (σp/p < 0.5). Total of 87 Cepheids
shown. Good slope, bad zero-point: ~0.7-0.8m fainter as
compared to conventional P-L relations.
Lutz-Kelker
Bias: shifts
to top but
is too
uncerntain
for ~50%
errors
• To separate the correction to P-L slope
from the correction to zero-point, we can
rewrite P-L relation in the equivalent form
• <MV > = (a + b) + b·(lg P – 1),
where (a + b) is <MV> (at P=10d)
with the “conventional” values for
•
(a + b) ≈ -3.9 … –4.2
• (a + b) can be considered as zero-point
“substitute”
• Common way to refine the constants
• <MV> (at P=10d) vs lg P (from new HIPPARCOS reduction
by van Leeuwen, 2007, and extinction data).
• Red: 45 Cepheids with σp/p < 0.5
HIPPARCOS
• Blue: all Cepheids
parallaxes alone
disable to derive
fine P-L relation!
(a + b)st
Partly,
overtone
pulsators?
• “How to deal with the Lutz-Kelker bias?” –
That’s the question not yet finally solved
(M.Groenewegen & R.Oudmaijer, A&A V.356,
P.849, 2000)
• Should the Lutz-Kelker correction be added
to each star of the sample (or the selection
by the parallax is always biased) ?
• Some authors use “reduced parallax”
approach instead of distances approach (see
C.Turon Lacarrieu & M.Creze “On the
statistical use of trigonometric parallaxes”,
A&A V.56, P.373, 1977):
0.2 MV
p ~ 10
• P-L relation: <M> = a + b·lg P
• (a) zero-point
• (b) slope
• Why not to estimate the slope of the
P-L relation directly from LMC
Cepheids ? –
• The slopes of the P-L relations in other
galaxies may differ due to systematic
differences in the metal abundances and
Cepheids ages (A.Sandage & G.Tammann,
2006)
• Before ~2003, most astronomers used
the slope of LMC Cepheids to refine
zero-point of Milky Way Cepheids
• If metallicity effects are really
important, application of single P-L
relation to other galaxies populated by
Cepheids can introduce additional
systematic errors (see detailed
discussion in A.Sandage et al., 2006)
Problems with the interstellar extinction
• Cepheids color excess are uncertain
because of:
– Finite width of the instability strip (~0.2m)
due to evolution effects
– Uncertainty of cepheids “normal colors”
•
• Wesenheit index (Wesenheit function)
is often used to reduce the effects of
the interstellar extinction (B.Madore in
“Reddening-independent formulation of
the P-L relation: Wesenheit function”;
ROB No.182, P.153, 1976)
Example: Wesenheit index for (V-I) color
• From true distance
modulus we have:
MV  V  10  5 lg p (mas)  AV
• Wesenheit index
definition:
Constant!
AV
W ( VI )  V  10  5 lg p (mas) 
(V  I )
E( V  I )
Wesenheit
Wesenheit index
do notindex
depend
on the extincton but only
the normal
color color
•on
Substituting
apparent
( V  I )  ( V  I )0  E( V  I )
Normal
color as well as <MV>
we derive:
is linear on lg P
AV
(see
instability
strip
picture!),
W ( VI )  V  10  5 lg p (mas)  AV 
 ( V  I )0 
E( VP  I )
so W(VI) is also linear on lg
AV
 MV 
 ( V  I )0  W ( VI )  MV    ( V  I )0
E( V  I )
where const β = AV/E(V-I) ≈ 2.45±
follows from the extinction law
• Wesenheit Index can be introduced for
any color: W(BV), W(VK) etc.
• W incorporates intrinsic color (and
Period – Color relation)
• The advantage of using the Wesenheit
index W instead of the absolute
magnitude M is that
– (a) Wesenheit index is almost free of any
assumptions on cepheids individual color
excess, particularly in our Milky Way, and
– (b) it reduces the scatter of the P-L
relations
Use of the Wesenheit index
• M.Groenewegen & R.Oudmaijer, “Multicolour PL-relations of Cepheids in the
HIPPARCOS catalogue and the distance to
the LMC” (A&A V.356, P.849, 2000)
• A.Sandage et al. “The Hubble constant: a
summary of the HST program for the
luminosity calibration of type Ia
SuperNovae by means of Cepheids” (ApJ
V.653, P.843, 2006)
• F.van Leeuwen et al. “Cepheid parallaxes and
the Hubble constant” (MNRAS V.379,
P.723, 2007)
• F.van Leeuwen et al. “Cepheid parallaxes and the
Hubble constant” (MNRAS V.379, P.723, 2007)
W(VI)
Wesenheit index
W(VI) for 14
Cepheids with most
reliable parallaxes
from HIPPARCOS
and HST FGS:
Route to P-L relation
W(VI)
W(VI) = α·lg P + γ
• To derive P-L relation:
• MV ≈ W(VI) + β·(V-I)0
• Normal colors are also linearly dependent
on the period: CI0 ≈ δ·lg P + ε
Example:
Period – Color (P-C)
relation for galactic
Cepheids from
G.Tammann et al.
(A&A V.404, P.423,
2003). Wide strip
(σCI≈0.07m)
(V-I)0 vs lg P
• To derive P-L relation: W
M V:
• MV ≈ W(VI) + β·(V-I)0
• MV ≈ (α + β·δ) ·lg P + (γ + β· ε)
• Serious problem: effects of [Fe/H]
• Theoretical approach:
• A.Sandage et al. “On the sensitivity of the
Cepheid P-L relation to variations of
metallicity” (ApJ V.522, P.250, 1999)
• I.Baraffe & Y.Alibert “P-L relationships in
BV IJHK-bands for fundamental mode and
1st overtone Cepheids” (A&A V.371, P.592,
2001)
• G.Tammann et al. (A&A V.404, P.423,2003):
• P-C relations for Galaxy, LMC & SMC
• L/SMC are
metal-deficient relative
to the Galaxy
LMC/SMC
Cepheids are
bluer than in
the Milky
Way
• Overall impression: [Fe/H] differences
affect also P-L relations (slope & zeropoint)
• Key question: Systematic error induced
to P-L based distances when we neglect
metallicity differences
• From A.Sandage et al. new stellar
atmospheres models and synthetic spectra
(ApJ V.522, P.250, 1999):
• At P = 10d:
bol
B
V
I
dM/d[Fe/H] (mag/dex) 0.00 +0.03 -0.08
-0.10
“…The agreement of the RR Lyrae
distance to the LMC, the SMC, and
IC 1613 with the Cepheid distance
determined on the basis of only a
mild (if any) metallicity dependence
of the P-L relation for classical
Cepheids is our principal conclusion”.
• M.Groenewegen & R.Oudmaijer (A&A V.356,
P.849, 2000):
• ΔM = MGal – MLMC for VIK bands
(LMC is ~0.3-0.4 dex metal-deficient as
compared to MW: W.Rolleston et al., A&A
V.396, P.53, 2002)
• Corrections disagree due to the differences
in the theoretical models !
• M.Groenewegen “Baade-Wesselink distances
and the effect of metallicity in classical
Cepheids” (A&A V.488, P.25, 2008)
• Problem was investigated by P-L relations based on
BBWB Cepheids radii
…
• “Obtaining accurate distances to stars is a
non-trivial matter (Yes! – A.R.)… The
metallicity dependence of the PL-relation
is investigated and no significant
dependence is found. A firm result is not
possible as the range in metallicity
spanned by the current sample of galactic
Cepheids is 0.3–0.4 dex, while previous
work suggested a small dependence on
metallicity only (typically −0.2mag/dex).”
• There is no common agreement in the
reality of significant differences in the
slopes of the P-L relations in different
galaxies due to differences in [Fe/H]
• (F.van Leeuwen et al. (2007): “…The main
conclusion … is that within current uncertainties
<the P-L slope - A.R.> is the same in the Galaxy
as in the LMC…”)
• As a consequence, some ambiguity
remains in the calibrations of other
standard candles based on the Cepheids
distance scale
(b) Cepheids in open clusters and associations
• Another way to derive independently the
slope and zero-point of the P-L relations of
galactic Cepheids comes from open
clusters and associations (young stellar
groups hosted by the Cepheid variables)
• Their distances calculated by the MSfitting are good to about few per cent
• Approximately 30-50 galactic Cepheids are
considered as possible cluster members
L.Berdnikov et al. “The BVRIJHK period-luminosity
Metallicity
differences
have
been
relations for
Galactic classical
Cepheids” (AstL
V.22,
P.838,
1996)
Cepheids
in 5 open
clusters:
taken– 9into
account
empirically,
by
days
days
adding the term proportional
to
Very poor statistics!
the difference of the galactocentric
distances (due to “mean”
[Fe/H]
Zero-points
Slope
gradient across the galactic disk,
Δ[Fe/H] / ΔRG) ~ -0.05… -0.10 ±
days
days
Multicolor
(BVRCRICIJHK) P-L
relations for galactic
Cepheids
P-L slopes as compared to the LMC Cepheids
Optics |
NIR
LMC
m
• D.Turner & J.Burke
“The distance scale
for classical
cepheid variables”
(AJ V.124, P.2931,
2002)
• The list of 46
Cepheids as
possible members
of young stellar
groups (clusters
and associations)
• (Future prospects)
• D.An et al.
“The distances to open clusters from
main- sequence fitting. IV. Galactic Cepheids, the LMC, and
the local distance scale” (ApJ V.671, P.1640, 2007)
New
distances
New P-L
for the
galactic
Cepheids:
(c) Distance scale from BBWB radii
Comparing Cepheids P-L derived from
23 cluster Cepheids (red) with that
from Wesselink radii (blue)
(D.Turner & J.Burke, 2002)
Insignificant slop
difference ~0.19
Mean: <MV>≈-1.20m-2.84m·lg P
• A.Sandage et al.
(A&A V.424, P.43,
2004)
BBWB P-L
(circles) for 36
galactic Cepheids
P-L for 33 cluster
members (dots)
P-L are very close
within errors
• Rms scatter:
σ ~ 0.19…0.27m
MB0 Galaxy
• A.Sandage et al.
(A&A V.424,
P.43, 2004)
• LMC Cepheids:
Instability
Strip break on
MV0-(B-V)0 CMD
• Hint on P-L
break at P=10d ?
• Complicates the
use of P-L for
distance
measurements
Constant
lg P lines
• Nonlinear calculations of the Cepheids
models also seem to support an idea on
two Cepheids families divided by the
period value Plim ~ 9-10d
• Theory: large fraction of the Cepheids
with P < Plim are suspected to be 1st
overtone pulsators
For
•
•
•
0.5
1.0
1.5
2.0
lg P
“Broken” at P=10d P-L relations for LMC Cepheids (A.Sandage et al., 2004, 2006)
(a) Flatter P-L for P > 10d can introduce systematic distance errors to distant
galaxies where only brightest Cepheids are observed
(b) Poor statistics (~ 2 dozens) of brightest Cepheids introduces extra scatter
A.Sandage et al. (2004, 2006):
• Comparing “broken”
P-L (BVI bands) for
LMC (solid line) with
conventional P-L for
the Milky Way Galaxy
(BBWB & cluster
cepheids: dots &
circles)
• Close at P ~ 10d (after
reducing LMC zeropoints by ~0.15m)
Systematic differences !
<MB>
<MV>
<MI>
• A.Sandage et al. (2004) “closing speech”:
• “…The consequences of the differences in the slopes of the
P-L relations for the Galaxy, LMC, and SMC weakens the
hope of using Cepheids to obtain precision galaxy distances.
Until we understand the reasons for the differences in the
P-L relations and the shifts in the period−color relations,
(after applying blanketing corrections for metallicity
differences), we are presently at a loss to choose which of
the several P-L relations to use (Galaxy, LMC, and SMC) for
other galaxies.
• Although we can still hope that the differences may yet be
caused only by variations in metallicity, which can be
measured, this can only be decided by future research such
as survey programs to determine the properties of Cepheids
in galaxies such as M33 and M101 where metallicity
gradients exist across the image. But until we can prove or
disprove that metallicity difference is the key parameter,
we must provisionally assume that this is the case, and use
either the Galaxy P-L relations, or the LMC P-L relations, or
those in the SMC.”
• Detailed study of the Cepheids
populations in the Local Group galaxies
may seem “unfashionable” to most of
the astronomical community
• This is the reason why a number of key
questions still remain unexplained,
including the effects due to the
differences in metallicity
The use
of Cepheids
P-L-C
manifold
is theP-L-C
“cousin”
(Period – Luminosity - Color) relations
of the Fundamental Plane
for elliptical galaxies
• Cepheids P-L-C
manifold in
lg T – lg P – lg L
coordinates
• Projections give
P-L & P-C
relations and the
Instability Strip
• Constructing the P-L-C “plane”:
• The intrinsic scatter of the P-L relation
is due to the finite width of the
instability strip and the sloping of the
constant-period lines.
• The scatter can be reduced by
βλ=ΔMλ / ΔCI
forofP=const
introducing
a P-L-C
relation
the form
λ taken
• Mλ = αλ·lg P - βλ·CIλ + γλ , CIλ is color index (λ)
where βλ is the slope of the constantperiod lines on the appropriate CMD
• A.Sandage
P-L-C relation translates the magnitude
et
to al.
the value an unreddened Cepheid
(2004)
would have if it were lying on the ridge line
• CMDs
of theand
P-C and P-L relations
the
(except for observational errors)
constant
period
lines (red)
for LMC
Cepheids
• The slopes
vary with P
• Earlier, the use of P-L-C relation
instead of P-L, gave unsatisfactory
results because constant slope of the
constant period lines, ΔMV/Δ(B-V),
have been supposed for all P, L and
colors
• A.Sandage
et al. (2004,
2006):
• P-L-C
constants α,
β, γ depend
on the period
and vary
significantly
from galaxy to
galaxy
lg P<1
V
lg P>1
lg P<1
V
lg P>1
lg P<1
I
lg P>1
lg P<1
I
lg P>1
• A.Sandage et al. (2004)
• LMC Cepheids
MV - βV-(B-V)(B-V) and
MI –βI-(V-I)(V-I) vs lg P
relations: reduced
scatter
(σ ≈ 0.07…0.17m)
• Very attractive idea!
• Least squares solutions
for lg P>1 (lower) &
lg P<1 (upper) are
shown
• Note:
• Reduced scatter of the P-L-C relation
may seem attractive, but in the
absence of independent data on the
intrinsic color differences (say, due
to metallicity or extinction
differences), this technique can give
rise to artificial errors in the
absolute magnitude
• A.Sandage explains:
• A.Sandage et al. (2004): “The P-L-C
relation can only be used if it can be proven
that the Cepheids under consideration follow
the same P-L and P-C relations. Otherwise
any intrinsic color difference is multiplied by
the constant-period slope β and erroneously
forced upon the magnitudes with detrimental
effects on any derived distances.”
• It would be wrong idea to apply the LMC P-LC relation to Milky Way Cepheids which have
different P-L (BVI ) relations and
(necessarily !) also different P-C relations
(for B−V, V−I ). – Clear as day! (A.R.)
(d) Luminosity refinement by the statistical
parallax technique
• The applicability of the statistical parallax
technique to the Milky Way Cepheids requires
very accurate radial and tangential velocities.
Observational errors obviously should not exceed
the intrinsic scatter of space residual velocities.
HIPPARCOS proper motions joined with the
extensive set (~20000) of CORAVEL radial
velocity measurements made by the Moscow group
(N.Gorynya et al., VizieR Cat. III/229, 2002)
became real base to Cepheids statistical
parallaxes, and enabled to A.Rastorguev et al.
(AstL V.25, P.595, 1999) to derive the corrections
to the Cepheids distance scale (L.Berdnikov et al.,
1996), i: extension by ~7% (ΔMV ≈ -0.15m)
Relative magnitude
NIR/MIR prospects
• An example of the Cepheid
light curve (SU Cyg) in
UBVRIJK bands (top-down):
• The amplitude decreases
with wavelength, as well as
the scatter of multicolor P-L
relations
• Few brightness estimates in
NIR (>2.2 μm) suffice to
estimate mean magnitude
and the distance
• NIR/MIR data are of great
importance for distance
scale refinement and
luminosity calibrations
• W.Freedman et al.
“The Cepheid P-L relation at
mid-infrared wavelengths.
I.” (ApJ V.679, P.71, 2008)
• Single-epoch observations
for 70 LMC Cepheids from
SPITZER (1 point at each
light curve!)
• Slope steeper in MIR
(>3.3 mμ) than in NIR
and optics
• W.Freedman et al. “The Cepheid P-L relation at midinfrared wavelengths. I.” (ApJ V.679, P.71, 2008)
• P-L relations in MIR (>3.3 mμ) are nearly the same:
• RMS scatter of MIR P-L relations is ~0.07m,
so even single MIR observation can give the
distance precise to ~8%
• Very promising tool to measure large
extragalactic distances
Addendum to Cepheids discussion:
LMC as the touchstone of the distance scale
adjustment
• LMC is hosted by a wide variety of stellar
populations, from very young to old ones;
as a consequence, LMC with its ~50 kpc
distance, is very “suitable” stellar system
to apply different tools of distance
measurements used different “standard
candles”, such as Cepheids, RR Lyrae, RCG,
Tip Red Giants, MS-fitting, binaries, Miras
etc.
• Ideally, all distances to LMC should agree
within appropriate errors of methods used
• In the mid-1980’s, measurement of H0 with
the goal of 10% accuracy was designated as
one of three “Key Projects” of the HST
(M.Aaronson & J.Mould, ApJ, V.303, P.1,
1986), with LMC as the central object
• Real HST observations began in the 1991
• Results in: W.Freedman et al. “Final
Results from the Hubble Space Telescope
Key Project to Measure the Hubble
Constant” (ApJ V.553, P.47, 2001)
• LMC weighted “mean” distance:
•
(m-M)0 ≈ 18.50 ± 0.10m
LMC distances from Cepheids: ΔM ≈ 0.6m
HST KP
Systematic difference
between Cepheids and
RR Lyrae distances is
clearly seen: “short” vs
“long” distance scale ?
(m-M)0
• Most reliable conventional Cepheids
distance scales derived by different
approaches, differ from each other at
the 0.15-0.2m level (7-10% in the
distance)
• Their systematic errors are unknown
• Which one is true?
• Concluding remark of A.Sandage &
G.Tammann (2006):
• “Clearly, much work lies ahead. We are
only at the beginning of a new era in
distance determinations using Cepheids.
It can be expected that much will be
discovered and illuminated in the years
to come”.
Chapter Nine
RR Lyrae distance scale
RR Lyr star is an archetype of old halo population
(Horizontal Branch) short-period variable stars
RR Lyr is the brightest and nearest star of this
type with its <MV> ≈ 9.03 ± 0.02m, <B-V> ≈ 0.44 ±
0.04m, D ≈ 260 pc , (HIPPARCOS, 2007)
• M.Marcony & G.Clementini (ApJ V.129,
P.2257, 2005)
• Periods < 1d
• Examples of
LMC RR Lyrae
light curves in
BV bands
Observations vs
theory
Large amplitude,
Δm ~ 1m
• Examples of the
LMC RR Lyrae
light curves in I
band (OGLE
program,
I.Soszynski et al.,
Acta Astr. V.53,
P.93, 2003)
• RR are simply
discernible among
field stars in
stellar systems
• <MV> ≈ +1m
T.Brown et al. (AJ
V.127, P.2738,2004)
• HST ACS
• M31 RR Lyrae light
curves example
• <[Fe/H]> ≈ -1.6 from
periods distribution
• NIR light curve of RR Lyr star: look at small
amplitude. RR Lyr: nearest star of this type.
Not so bright
as Cepheids
but very
important for
distance scale
subject in the
galactic halos,
bulges and thick
disk populations
• RR Lyrae variable
stars in the
Instability Strip on
the HRD
• In contrast to
Cepheids, RR Lyrae
variables are among
oldest stars of our
Milky Way
• Evolution status:
horizontal branch
(HB) stars
• RR Lyrae variables populate galactic halos
(H) and thick disks (TD) (as single stars),
and globular clusters of different [Fe/H]
• Evolution stage: Helium core burning
• Age: ≥ 10 Gyr
RR Lyrae
M2
• LifeTime: ~100 Myr
V const ?
BHB
(EHB)
TP
H
TD
Close luminosities for the
same [Fe/H] (rms ~ 0.15m)
• The duration of RR Lyrae stage (and HB
stage at all) is negligible as compared to
the age of stars, ≤100 Myr vs ~10-13 Gyr
(<1%), but comparable with lifetime of Red
Giants
• Therefore, HB population is comparable
with RG population in size, but RR Lyrae
form only a small fraction of all stars
above Turn-Off point on the CMD
• Thousands of RR Lyrae are found and
catalogued in the Milky Way halo and in its
globular clusters
• Post-ZAHB tracks for low-mass stars (by B.Dorman
(ApJS V.81, P.221, 1992) with [Fe/H]=-1.66 and
Blue & Red edges of the Instability Strip (IS)
• Masses
on ZAHB
are labelled
Stars
evolve from
ZAHB to
second RG
tip
IS
Zero-Age Horizontal
Branch (ZAHB)
Stellar evolution for low-mass stars
• Dots are separated by
10 Myr time interval
• HB position is almost
the same for clusters of
different age
HF
IS
Gyr age
• Universality of RR
Lyrae population
luminosity
• D.VandenBerg et
al. (ApJ V.532,
P.430, 2000)
theoretical ZAHB
levels for
different [Fe/H]
and [α/Fe] values
• [Fe/H] seems to
be the key
parameter
responsible for
RR Lyrae optical
luminosity
• The slopes of P - L and [Fe/H] - L
relations seems to be definitely found
from the theory as well as from
observations in globular clusters and
nearby galaxies differ by [Fe/H]
• Zero-point refinement is Main problem
in RR Lyrae distance scale studies
• From late 1980th, it became customary to
assume a linear relation between RR Lyrae
optical absolute magnitude and metallicity
of the form
•
<Mopt> = a + b·[Fe/H]
• The calibration problem reduced to finding
a and b by whatever calibration method was
used. The three most popular have been:
• (a) theory
• (b) the BBWB moving atmosphere method
• (c) distances of globular clusters; HST &
HIPPARCOS parallaxes
• (d) statistical parallaxe technique
(a) P.Demarque et al. (AJ V.119, P.1398, 2000)
• Is there universal slop of the <MV> - [Fe/H]
relation?
• Theoretical
slopes μ
(a) G.Bono “RR Lyrae distance scale: theory and
observations” (arXiv:astro-ph/0305102v1, 2003)
• RR Lyrae models in NIR do obey to a welldefined PLZK relation:
•
• <MK> ≈ -0.775 − 2.07·lg P + 0.167·[Fe/H]
with an intrinsic scatter of ~0.04m
(with small contribution from [Fe/H] term)
(a) M.Catelan et al. “The RR Lyrae P-L relation. I.
Theoretical calibration” (ApJSS V.154, P.633, 2004)
•
•
•
•
<MI> ≈
<MJ> ≈
<MH> ≈
<MK> ≈
+0.109 – 1.132·lg P + 0.205·[Fe/H]
-0.476 – 1.773·lg P + 0.190·[Fe/H]
-0.865 – 2.313·lg P + 0.178·[Fe/H]
-0.906 – 2.353·lg P + 0.175·[Fe/H]
• <MV> ≈ +1.258 + 0.578·[Fe/H] + 0.108·[Fe/H]2
• (this nonlinear function of [Fe/H] do not
depend on lg P:
•
<MV> ≈ +0.63m at [Fe/H]=-1.5 )
• Two important RR Lyrae features:
• (a) <MV> depend on [Fe/H] and practically
does not depend on period
• (b) <M> in NIR (JHK bands) depend on the
period and practically do not depend on
[Fe/H]
• [M/H] = [Fe/H] +
+ lg (0.638·10[α/Fe] + 0.362)
Takes into account
differences in [α/Fe]
(from theory: M.Catelan et
al., 2004)
• Usually, RR Lyrae distance scales are
characterized by the mean absolute
magnitude referred to [Fe/H] = -1.5, the
maximum of [Fe/H] distribution function
of RR Lyrae field stars and GGC (galactic
globular clusters)
• (b) BBWB technique rewiev: C.Cacciari &
G.Clementini in: Stellar Candles for the Extragalactic
Distance Scale ( Ed. D.Alloin & W.Gieren, Lecture Notes in
Physics, V.635, P.105-122, 2003):
• <MV>RR = (0.20±0.04)[Fe/H] + (0.98±0.05)
• C.Cacciari et al. (ApJ V.396, P.219, 1992):
<MV>RR vs [Fe/H]
<MV>RR ≈ 0.20∙[Fe/H] + 1.04
<MK>RR vs lg P
<MK>RR ≈ -2.79·lg P -1.08
• Notes to BBWB technique applied to RR
Lyrae variables:
• (a) Zero-points and slopes are in general
agreement with the theory
• (b) Interstellar absorption problems
seem to be not so important for RR
Lyrae (halo stars with small color
excess) as compared to the Cepheids
(c) RR Lyr – nearest RR-type variable - HST
trigonometric parallax
• G.Fritz Benedict et al. (AJ V.123, P.473,
2002) – HST FGS3 parallax for RR Lyr
•
pRR
•
For RR Lyr star itself:
•
MV(RR) ≈ +0.61 ± 0.1m
•
with [Fe/H] ≈ -1.39
•
<V> ≈ +9.0m
•
HIP: old HIPPARCOS
•
data (1997)
(c) RR Lyr HIPPARCOS trigonometric parallax
• F.van Leeuwen “HIPPARCOS, the new
reduction of the raw data” (Springer: 2007)
•
– new RR Lyr parallax:
pRR
•
•
•
pRR ≈ (3.88 ± 0.39) mas
means
MV(RR) ≈0.61±0.09m
•
•
in ideal accord with
HST FGS3 estimate
• Taking into account <MV>RR variation with
[Fe/H], we can extrapolate <MV>RR to RR
Lyrae with [Fe/H] ≈ -1.50 (at the maximum
of the metallicity distribution function) by
using the slope d<MV>RR/d[Fe/H] ≈ 0.2 as
•
<MV>RR ([Fe/H] = -1.50) ≈ +0.59m
~ 0.15m brighter than
its conventional value,
~0.72-0.75m
Archetype - RR Lyr star
is not the best
“standard candle” !
Notes:
• (a) Estimated <MV> for individual stars can
differ from “mean” absolute magnitude by
~0.15m, intrinsic scatter for the HB stars
• (b) All RR Lyrae calibrations used the
trigonometric parallaxes (MS-fitting of
globular clusters populated by RR Lyrae
based on subdwarf stars, direct RR Lyrae
distances) suffer from Lutz-Kelker bias
• There is no common consensus in the
statement that the Lutz-Kelker correction
really should be applied to individual stars…
(c) M.Frolov & N.Samus (AstL V.24, P.174, 1998)
• K-band P-L RR Lyrae relation for 173 RR in 9 globular
clusters of different [Fe/H]
• <MK> ≈ -2.338 (±0.067) lg P – 0.88 (±0.06) (with rms
scatter ~0.06m)
• Synthetic P-L diagram
• P-L slope is very reliable,
but zero-point can differ
from this because it depends
on clusters distances adopted
(c) F.Fusi-Pecci et al. (AJ V.112, P.1461, 1996)
• <MV>HB calibration (<MV>HB ≈ >MV>RR)
• HST data for 9 M31 Globular clusters
• <MV>HB ≈ (0.13 ± 0.07) [Fe/H] +(0.95 ± 0.09)
• Zero-point depends on M31 distance adopted
(d) RR Lyrae statistical parallaxes
• Comprehensive rewiev of the problem in:
A.Gould & P.Popovsky “Systematics in RR Lyrae
statistical parallax. III” (ApJ V.508, P.844, 1998)
• <MV>RR ≈ 0.77±0.13m at [Fe/H]=–1.6 from 147 RR
Lyrae 3D velocity field and
• <MV>RR ≈ 0.80±0.11m at [Fe/H]=–1.71 from 865 RR
Lyrae 2D velocity field
• Fainter than from other data sources
(d) RR Lyrae statistical parallaxes
• A.Dambis & A.Rastorguev
“Absolute Magnitudes
and Kinematic Parameters of the Subsystem of RR Lyrae
Variables” (AstL V.27, P.108, 2001)
• <MV>RR ≈ 0.76±0.12m for “halo” RR population at
[Fe/H]=–1.6
• <MV>RR ≈ 1.01+0.15·[Fe/H] from RR Lyrae
subdivided into 5 independent
groups by [Fe/H]
<MV>RR ≈ 0.79m ([Fe/H]=–1.5)
Notes to (d):
• Statistical parallax technique seem to
underestimate RR Lyrae luminosity (and the
distance scale)
• C.Cacciari et al. (2003) note two main
sources of the systematical errors:
– (a) Contamination of halo RR Lyrae sample by
thick disk stars (with differential rotation in
the Milky Way)
– (b) Inhomogeneity of the galactic halo that can
include considerable fraction of stars came
from “accreting” Milky Way satellites identified
in SDSS in recent years
• Two halo subsystems would have different
dynamical characteristics and origins, the
slowly rotating subsystem being associated
to the Galactic thick disk, and the fast
rotating (possibly with retrograde motion)
subsystem belonging to the accreted outer
halo
• From new paper of E.Bell et al. (ApJ V.680,
P.295, 2008): “… a dominant fraction of
the stellar halo of the Milky Way is
composed of the accumulated debris from
the disruption of dwarf galaxies”
• Any kinematical inhomogeneity of the
sample used can introduce
unpredictable systematical errors to
the distance scale
• The statistical parallax technique,
very powerful in itself, need more
adequate kinematical models for halo
populations and more extended RR
Lyrae samples, with good radial
velocities and proper motions
• Comparing RR Lyrae <MV> vs [Fe/H]
theoretical relation (solid line) with the
statistical and trigonometric parallaxes
calibrations (A.Sandage et al., 2006):
systematical differences
Dambis & Rastorguev (2001)
LMC distances from RR Lyrae: ΔM ≈ 0.5m
(m-M)0
• G.Fritz Benedict
et al. (AJ V.123,
P.472, 2002) gave
a summary of all
the distance
measurements to
LMC galaxy,
performed by 21
different
methods, with
“mean” value close
to (m-M)0≈18.50m
• Ranking by value
B.Schaefer (2008) test of
Guess
what
happened
in
~2001
?
LMS distance measurements
• Main goal was to check if:
– (a) All measurements are really independent and
unbiased
– (b) All random errors are correctly reported
Consider the
relative deviation
parameter:
D = (μ-<μ>) / σ
• K-S (Kolmogorov-Smirnov) statistical test of
cumulative |D| distribution: theory (smooth)
vs published measurements (step function)
• An excess of small |D| is seen after 2002:
~68% are within 0.5 σ instead of 68% within
1 σ as it
follows from
the theory,
or too many
too presise
estimates
• An excess of large |D| is seen among the
results before 2001 (year of publication the
results of HST Key Project – HST KP), or
too much too inaccurate estimates
• (I) Before the year 2001, the many measures
spanned a wide range (from 18.1m to 18.8m), with
the quoted error bars being substantially smaller
than the spread, and hence the consensus
conclusion being that many of the measures had
their uncertainties being dominated by
unrecognized systematic problems
• (II) After 2001 (HST KP results), community has
generally accepted and widely popularized value
μ≈18.50±0.10m, and “independent” measures
clustered tightly around it, indeed, too tightly.
This concentration is a symptom of a worrisome
problem: (a) correlations between papers,
(b) widespread overestimation of error bars,
(c) band-wagon effect
Summary to Cepheids and RR Lyrae
• Cepheids as very bright and uniquely
identified stars are among most “popular”
standard candles in the distant galaxies, but
their distance scale is still uncertain by
~10% (systematic + random error)
• RR Lyrae are good standard candles used to
refine the distances to the “beacon” galaxies
(such as LMC/SMC, M31/33 etc.) in the Local
Volume (up to ~10 Mpc), and to calibrate
other secondary standard candles (SN Ia,
Tulli-Fisher & Faber-Jackson relations etc.)
• Much work has to be done with Cepheids
and RR Lyrae variables in recent years
and in the future, in the context of
GAIA and SIM observatories