1. 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

2. Content
Chapter Seven: BBWB: Moving Photospheres (Baade-Becker-Wesselink-Balone technique)
Chapter Eight: Cepheid parallaxes and Period – Luminosity relations
Chapter Nine: RR Lyrae distance scale

3. BBWB: Moving Photospheres (Baade-Becker-Wesselink-Balone technique)
Chapter Seven
BBWB: Moving Photospheres (Baade-Becker-Wesselink-Balone technique)

4. 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

5. (R1-R2) can be calculated by integrating the
(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:
(b) Radii difference
(R1-R2) can be calculated by integrating the
radial velocity curve, due to VR ~ dR/dt R1 R2
B-V
Pulsation Phase

6. A number of pairs
(R1 – R2) ~∫VR·dt and R1/R2 ≈ m
give rise to the calculation of mean radius, <R> = (Rmax + Rmin)/2

7. I. BBW Surface-brightness technique (SB)
θLD Eλ
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 lg Eλ , so
θLD ~ -0.2·mλ + Fλ + c ,
with Fλ as the “surface-brightness parameter”

8. 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)

9. 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)

10. From both relations,
θLD ~ -0.2·mλ + Fλ + c
Fλ ≈ a + a·CIλ + b
θLD ~ -0.2·mλ + a·CIλ + const
Direct calculation of star’s angular diameter vs time
Light-curve Color-curve

11. Radial velocity curve integration gives absolute radius change, and being relate to apparent angular diameter change, distance to the pulsating star follows

12. II. Maximum Likelyhood technique by L. Balona (MNRAS V. 178, P
II. Maximum Likelyhood technique by L.Balona (MNRAS V.178, P ,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

13. lg Teff – (B-V)0 relation by P.Flower (1996)
Luminosity classes:
Working interval
lg Teff – (B-V)0 relation by P.Flower (1996)

14. ΔMbol – (B-V)0 relation by P.Flower (ApJ, V469, P.355, 1996)
Working interval
Luminosity
classes
ΔMbol
ΔMbol – (B-V)0 relation by P.Flower (ApJ, V469, P.355, 1996)

15. Quadratic approximations can be written for lgTeff, ΔMbol by (B-V)0 (and by other normal colors) in the range < (B-V)0 < 1.6

16. Basic equations of the Balone approach

17. From the distance modulus expression,

18. Here constants A, B and C:
These constants enclose information on color
excess E(B-V) and the star’s distance d

19. How to calculate the radius change?
Moving photosphere
dS: ring area
-V0: photosphere velocity
Contribution of the
circular ring to the
observed radial velocity
(to the observer)

20. Calculating R(t)=<R>+r (t): integrating radial velocity curve Vr Ring contribution to the measured radial velocity:

21. Mean radial velocity weighted along star’s limb = observed Vr
p is the projection factor connecting
radial velocity and the photosphere speed

22. Projection factor values:
In the absence of limb darkening p=3/2
In most studies p=1.31 accepted
But: projection factor depend on
Limb darkening coefficient
Velocity of the photosphere
Instrumental line profile used to measure radial velocity

23. Theoretical spectral line will be distorted due to instrumental profile

24. 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

25. Some examples of the line profiles for different line broadening due to instrumental effects
Instrumental width
Theoretical profile
Solid: normal
Observed profile
Small
velocity
Good Gaussian fit

26. 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

27. 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)

28. Example: projection factor and line width vs photosphere velocity, V0
From Gauss tip
From line tip

29. 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 VR measurements have been performed by Moscow team for ~165 northern Cepheids with characteristic accuracy ~0.5 km/s and σ ~ 15 km/s

30. Radius change, r(t), can be calculated
by integrating the radial-velocity curve,
Vr (t), from some t=0 (where R(0)=<R>)

31. Example: TT Aql radial velocity curve (upper panel) and radius change (bottom panel)
Vr, km/s
ΔR, Solar units
Pulsation phase

32. Fitting the light curve for TT Aql Cepheid (+) with the model (solid line)
Pulsation phase

33. 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)

34. 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 classi-
fy Cepheids by
the pulsation
mode (first-over-
tone pulsators
have shorter peri-
ods at equal radii) FU
1st

35. 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

36. t1 SN t2 VR D Δφ

37. 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

38. B.Lane et al. (ApJ V.573, P.330, 2002):
Palomar PTI
85-m base length
optical
interferometer
Distance within
10% error
Phase

39. 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

40. Cepheid parallaxes and Period – Luminosity relations
Chapter Eight
Cepheid parallaxes and
Period – Luminosity relations
Astrophysical background
Overtone pulsators
Use of Wesenheit index
Luminosity calibrations
P-L or P-L-C ?

41. Cepheids “econiche” Cepheids as calibrators for these methods
100 pc … 50 Mpc pc

42. 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

43. 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

44. 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 , 2006)

45. 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)

46. 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

47. Relative fraction of Cepheids population depends on the evolution rate while crossing the instability strip:
slower evolution means more stars at the appropriate stage

48. The differences in the evolution rates at different crossings give
(a) Tracks are not parallel to the lines of constant periods
Evolutional period change happens
(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
The differences in the evolution
rates at different crossings give
rise to the questions on the
relative contribution of bright
and faint Cepeids for the same
period value
Cepheids evolution tracks,
the instability strip and
P=const lines (red)
lg L IS
An extra source
of P-L scatter

49. dP/dt is visible when observing on long time interval (~100 yr)

50. extragalactic Cepheids, Maybe, any spectroscopic
At equal periods: Cepheids with different masses but with different crossing number
lg (dP/dt) vs lg P: crossing number diagnostics (as compared to the theory)
Perspective tool to identify crossing number and to refine the P-L relation…
P increase +
… but it requires long
time monitoring
Impossible to use for
extragalactic Cepheids,
Maybe, any spectroscopic
features will be found
responsible for crossing
P decrease

51. The duration of the Cepheids stage is typically less than 0
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, ~ are expected to populate the Andromeda Galaxy (M31)

52. 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”

53. Hertzsprung Sequence of Cepheids (normalized) light curves
Hertzsprung Sequence of Cepheids (normalized) light curves. Bump position. (from P.Wils)

54. A. Udalski et al. “The Optical Gravitational Lensing Experiment
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 !

55. 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.

56. We can see Cepheids
even in distant galaxies

57. 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

58. 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

59. (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: ~ m fainter as compared to conventional P-L relations.
Lutz-Kelker
Bias: shifts
to top but
is too
uncerntain
for ~50%
errors

60. <MV > = (a + b) + b·(lg P – 1),
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

61. HIPPARCOS parallaxes alone disable to derive fine P-L relation!
<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
Blue: all Cepheids
HIPPARCOS
parallaxes alone
disable to derive
fine P-L relation!
(a + b)st
Partly,
overtone
pulsators?

62. “How to deal with the Lutz-Kelker bias
“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):

63. 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)

64. 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)

65. 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)

66. Example: Wesenheit index for (V-I) color
From true distance modulus we have:
Wesenheit index definition:
Constant!

67. Wesenheit index do not depend on the extincton but only
on the normal color
Normal color as well as <MV>
is linear on lg P
(see instability strip picture!),
so W(VI) is also linear on lg P
Substituting apparent color
we derive:
where const β = AV/E(V-I) ≈ 2.45±
follows from the extinction law

68. 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

69. Use of the Wesenheit index
M.Groenewegen & R.Oudmaijer, “Multi-colour 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)

70. F.van Leeuwen et al. “Cepheid parallaxes and the Hubble constant” (MNRAS V.379, P.723, 2007)
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)
W(VI) = α·lg P + γ

71. 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

72. MV ≈ (α + β·δ) ·lg P + (γ + β· ε)
To derive P-L relation: W MV:
MV ≈ W(VI) + β·(V-I)0
MV ≈ (α + β·δ) ·lg P + (γ + β· ε)

73. 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-M relationships in BV IJHK-bands for fundamental mode and 1st overtone Cepheids” (A&A V.371, P.592, 2001)

74. G.Tammann et al. (A&A V.404, P.423,2003):
P-C relations for Galaxy, LMC & SMC
L/SMC are
metal-defici-
ent relative
to the Galaxy
LMC/SMC
Cepheids are
bluer than in
the Milky
Way

75. Overall impression: [Fe/H] differences affect also P-L relations (slope & zero-point)
Key question: Systematic error induced to P-L based distances when we neglect metallicity differences

76. 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)
“…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”.

77. M.Groenewegen & R.Oudmaijer (A&A V.356, P.849, 2000):
ΔM = MGal – MLMC for VIK bands
(LMC is ~ 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

78. 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 …

79. “Obtaining accurate distances to stars is a non-trivial matter (Yes
“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).”

80. 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

81. (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 MS-fitting are good to about few per cent
Approximately galactic Cepheids are considered as possible cluster members

82. Metallicity differences have been taken into account empirically, by
L.Berdnikov et al. “The BVRIJHK period-luminosity relations for Galactic classical Cepheids” (AstL V.22, P.838, 1996) – 9 Cepheids in 5 open clusters:
Metallicity differences have been
taken into account empirically, by
adding the term proportional to
the difference of the galactocentric
distances (due to “mean” [Fe/H]
gradient across the galactic disk,
Δ[Fe/H] / ΔRG)
Very poor statistics!
Zero-points Slope
days
days
days
days
Multicolor (BVRCRICIJHK) P-L relations for galactic Cepheids

83. P-L slopes as compared to the LMC Cepheids
Optics | NIR
LMC m

84. 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)

85. 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:

86. Mean: <MV>≈-1.20m-2.84m·lg P
(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

87. BBWB P-L (circles) for 36 galactic Cepheids
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

88. LMC Cepheids: Instability Strip break on MV0-(B-V)0 CMD
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

89. 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

90. For
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

91. Close at P ~ 10d (after reducing LMC zero-points by ~0.15m)
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 zero-points by ~0.15m)
Systematic differences !
<MB>
<MV>
<MI>

92. 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.”

93. The use of Cepheids P-L-C (Period – Luminosity - Color) relations
P-L-C manifold is the “cousin”
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

94. βλ=ΔMλ / ΔCIλ taken for P=const
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 introducing a P-L-C relation of the form
Mλ = αλ·lg P - βλ·CIλ + γλ , CIλ is color index (λ)
where βλ is the slope of the constant-period lines on the appropriate CMD
βλ=ΔMλ / ΔCIλ taken for P=const

95. A.Sandage et al. (2004)
CMDs and the constant period lines (red) for LMC Cepheids
The slopes vary with P
P-L-C relation translates the magnitude
to the value an unreddened Cepheid
would have if it were lying on the ridge line
of the P-C and P-L relations
(except for observational errors)

96. Earlier, the use of P-L-C relation instead of P-L, gave unsatisfactory results because constant slope ΔMV/Δ(B-V) have been supposed for all P, L and colors

97. A.Sandage et al. (2004, 2006):
P-L-C constants α, β, γ depend on the period and vary from galaxy to galaxy

98. A.Sandage et al. (2004) LMC Cepheids
lg P<1
A.Sandage et al. (2004)
LMC Cepheids
Mλ-βλ(B-V) and Mλ-βλ(B-V) 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 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

99. 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:

100. 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-L-C relation to Milky Way Cepheids which have different P-L (BVI ) relations and (necessarily !) also different P-C relations (for B−V, B−I ). – Clear as day! (A.R.)

101. (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): extension by ~10% (ΔMV ≈ -0.2m)

102. 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
Relative magnitude

103. 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

104. Very promising tool to measure large extragalactic distances
W.Freedman et al. “The Cepheid P-L relation at mid-infrared 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

105. 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

106. 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 ≈ ± 0.10m

107. LMC distances from Cepheids: ΔM ≈ 0.6m

108. Most reliable conventional Cepheids distance scales derived by different approaches, differ from each other at the m level (10-15% in the distance)
Their systematic errors are unknown
Which one is true?

109. 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”.

110. RR Lyrae distance scale
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)

111. 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

112. Examples of the LMC RR Lyrae light curves in I band (OGLE program, I
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

113. 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

114. NIR light curve of RR Lyr star: look at small amplitude
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

115. 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

116. Evolution stage: Helium core burning Age: ≥ 10 Gyr LifeTime: ~100 Myr
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
LifeTime: ~100 Myr
RR Lyrae M2
V const ?
BHB
(EHB) TP
Close luminosities for the
same [Fe/H] (rms ~ 0.15m) H TD

117. 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

118. Post-ZAHB tracks for low-mass stars (by B. Dorman (ApJS V. 81, P
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)

119. 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
Universality of RR Lyrae population luminosity HF IS
Gyr age

120. D. VandenBerg et al. (ApJ V. 532, P
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

121. 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

122. <Mopt> = a + b·[Fe/H]
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

123. (a) P.Demarque et al. (AJ V.119, P.1398, 2000)
Is there universal slop of the <MV> - [Fe/H] relation?
Theoretical
slopes μ

124. RR Lyrae models in NIR do obey to a well-defined PLZK relation:
(a) G.Bono “RR Lyrae distance scale: theory and observations” (arXiv:astro-ph/ v1, 2003)
RR Lyrae models in NIR do obey to a well-defined PLZK relation:
<MK> ≈ − 2.07·lg P ·[Fe/H]
with an intrinsic scatter of ~0.04m
(with small contribution from [Fe/H] term)

125. <MI> ≈ +0.109 – 1.132·lg P + 0.205·[Fe/H]
(a) M.Catelan et al. “The RR Lyrae P-L relation. I. Theoretical calibration” (ApJSS V.154, P.633, 2004)
<MI> ≈ – 1.132·lg P ·[Fe/H]
<MJ> ≈ – 1.773·lg P ·[Fe/H]
<MH> ≈ – 2.313·lg P ·[Fe/H]
<MK> ≈ – 2.353·lg P ·[Fe/H]
<MV> ≈ ·[Fe/H] ·[Fe/H]2
(this nonlinear function of [Fe/H] do not depend on lg P:
<MV> ≈ +0.63m at [Fe/H]=-1.5 )

126. 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] )
Takes into account
differences in [α/Fe]
(from theory: M.Catelan et
al., 2004)

127. 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)

128. <MV>RR = (0.20±0.04)[Fe/H] + (0.98±0.05)
(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 , 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 vs lg P
<MV>RR ≈ 0.20·[Fe/H]
<MV>RR ≈ -2.79·lg P -1.08

129. 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

130. (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
For RR Lyr star itself:
MV(RR) ≈ ± 0.1m
with [Fe/H] ≈ -1.39
<V> ≈ +9.0m
HIP: old HIPPARCOS
data (1997)
pRR

131. (c) RR Lyr HIPPARCOS trigonometric parallax
F.van Leeuwen “HIPPARCOS, the new reduction of the raw data” (Springer: 2007)
– new RR Lyr parallax:
pRR ≈ (3.88 ± 0.39) mas
means
MV(RR) ≈0.61±0.09m
in ideal accord with
HST FGS3 estimate
pRR

132. Taking into account <MV>RR variation with [Fe/H], we can extrapolate <MV>RR to RR Lyrae with [Fe/H] ≈ (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,
~ m
Archetype - RR Lyr star
is not the best
“standard candle” !

133. 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…

134. (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> ≈ (±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

135. (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

136. (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

137. (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 ≈ ·[Fe/H] from RR Lyrae
subdivided into 5 independent
groups by [Fe/H]
<MV>RR ≈ 0.79m ([Fe/H]=–1.5)

138. 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

139. 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”

140. 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

141. 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)

142. LMC distances from RR Lyrae: ΔM ≈ 0.5m

143. G. Fritz Benedict et al. (AJ V. 123, P
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

144. B.Schaefer (2008) test of 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

145. 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 much
too presise
estimates

146. 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

147. (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

148. 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.)

149. 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