pulsation models: cepheids and rr lyrae marcella marconi inaf astronomical observatory of...

Pulsation models: Cepheids and RR Lyrae

Marcella MarconiINAF Astronomical Observatory of Capodimonte,Naples,Italy

S. Agata sui due Golfi, May 2005

Outline


What are pulsating stars ?

Why to model pulsating stars ?

Why do Cepheids and RR Lyrae pulsate

and why the instability strip is a strip ?

What are the possible theoretical approaches

to build a pulsating model ?

What are the implications for the distance scale

and the main uncertainties ?

Pulsating stars

Pulsating stars are intrinsic variables showing cyclic or periodic variations on a time scale which is of the order of the free fall time. In the simplest case they pulsate radially

Period hours hundreds of days

Mag lim. oss. a few magnitudes

R/R up to ~ 0.3 for the classical ones (but even more for Miras)

Among the most studied pulsating stars:Classical Cepheids (intermediate mass He burning stars) with

1d P 100 d.RR Lyrae (old,low mass counterpart of Classical Cepheids) with

0.3 d P 1 d.S. Agata sui due Golfi, May 2005

The instability strip


General properties of CepheidsClassical Cepheids are intermediate mass stars (3M/M12) in the central Heliumburning evolutionary phase, crossing the pulsation instability strip during the so calledblue loop.


The theory of stellar evolution predicts that in this evolutionary phase the mass andluminosity are related:

log L/L = + logM/M + γ log Z +δ logY

Cepheids are observed to pulsate with periods 1d P 100 d. Since the discovery byMiss Leavitt (1912) of a Period-Magnitude (PL) relation for the Cepheids in the SMC,they have been considered powerful primary distance indicators (see HST KeyProject, lessons by B. Madore in this School).

Is the PL relation, in the form Mj=a+b logP universal ?

If the zero point and/or the slope of this relation depends on metal abundance the adoption of a universal PL relation (calibrated on LMC Cepheids) to infer the distance to external galaxies is affected by a systematic error. Many recent observational and theoretical investigations on this topic: Madore & Freedman 1990, Kochanek et al. 1997, Kennicutt et al. 1998, Sakai et al. 2004, Romaniello et al. 2005……, Bono et al. 1999, 2002, Caputo et al. 2000, Fiorentino et al. 2002………….

General properties of RR Lyrae


They are the most abundant class of pulsating stars in the Milky

Way and are found both in the field and in globular clusters.

pulsation periods range from ~ 0.3 d to ~ 1.0 d

pulsation amplitudes range from ~ 0.2 to ~ 1.8 mag.

RR Lyrae are important tracers of chemical and dynamical

properties of old stellar populations

RR Lyrae are low mass Helium burning stars, on the so called Horizontal Branch (HB) in the HR diagram

RRab and RRcRR Lyrae are traditionally divided into two classes:RRab with longer pulsation periods and asymmetric light

curves with amplitude decreasing toward the redder colours (Fundamental mode)

RRc with shorter pulsation periods and almost sinusoidal light curves (First overtone mode)


RR Lyrae are HB stars almost constant luminosity (apart from

evolutionary effects.............)

RR Lyrae as distance indicators

An accurate knowledge Mv(RR) provides fundamental

constraints to the distance to the galactic center, globular

clusters, nearby galaxies... see lessons by V. Ripepi, this school

The absolute magnitude Mv(RR) depends on metal abundance:

Mv(RR) = a + b [Fe/H]


The time scale of pulsation• As for an acoustic wave, the pulsation period should be of the

order of the time required to propagate through the diameter of the star:

~ 2R/vs where vs is the mean over the whole star in its equilibrium

state of the adiabatic sound velocity.

• For a self gravitating structure in hydrostatic equilibrium, rotation, pulsation…) and no magnetic field and for which the pressure vanishes on the surface, the virial theorem can be written in the form:

- = 3∫PdV = 3∫P/ dm = 3∫ vs ²/ dm = 3 vs ²/ M vs (- / 3 M ) ½

( / - )½ (=r²dm)S. Agata sui due Golfi, May 2005

for given mass and radius, decreases as increases

By replacing M=4/3R³ we obtain

~2/ (4/3 G ) = costant Period-density law

For known stars 106 /o 10-9

white dwarfs red supergiants

3 s 1000 dS. Agata sui due Golfi, May 2005

Why to model pulsating stars

On the basis of a simple theoretical approach P = cost

but for Stephan-Boltzman law L = 4 R² Te4

P=P(L,M,Te) Period-Luminosity-Color-Mass (PLCM) relation


Observed P and colors constraints on L and/or M

For Classical Cepheids L=L(M,Y,Z) PLC (!) projecting on the PL plane PL

1) Connection to the distance scale problem ( cosmology) 2) Test of evolutionary theories ( ML relations, convection,

overshooting.........)

Driving mechanisms

In the classical instability strip the pulsation mechanisms areexpected to be connected with the position in the HR diagram phenomena related to L, Te (R)

The pulsation key mechanisms are related to the ionizationregions of the most abundant elements in a stellar envelope: H,He ed He +(Zhevakin 1953, 1954):

mechanism• k mechanism


mechanism

The adiabatic exponent 3-1 = (dlogT/dlog) decreases in the

ionization regions.

Initial contraction phase


In a ionization region most of the released energy during the phases of contraction goes into ionization:

increases

small L variation because L T4energy is trapped

Energy excess during

the subsequent expansion phasepulsation work

small increase of T

k mechanism

Opacity variations in the H, HeI, HeII ionization regions:

(k=1/λ) k n T-s


stellar interior positive n and s: opacity decreases during contraction of the stellar envelope producing heat loss

ionization regions s becomes large and negative: small temperature variations cause an increase of

during contraction further radiation trapping energy excess pulsation work

mechanism : effect of T variations on L variations

k mechanism: effect of k variations on L variations


Both the k and mechanisms are efficient in drivingthe pulsation but the phenomenon is started by a stochasticfluctuation of the external layer properties (e.g. contraction)

Why the instability strip has a BLUE boundary?

If the model effective temperature is too high the H and He/He+ ionization regions are very external low density, small mass takes part in the pulsation driving through the k and mechanisms damping prevails no pulsation


pulsationNo pulsation

Blue edge

Only when the ionization regions

are deep enough the mass involved

in the pulsation driving mechanisms

prevails pulsation

Why the instability strip has a RED boundary?

Moving toward lower effective temperature the depth of the driving ionization regions increases and the mass taking part in the phenomenon is larger increasing pulsation efficiency


But convection also becomes more efficient at lower effective temperatures k and gradients are reduced

quenching of pulsation (Baker & Kippenhahn 1965)

When the quenching effect due to convection prevails pulsation

is no more efficient red boundary of the instability strip

No pulsationpulsation

Red edge

The theory of stellar pulsation

The equations of stellar pulsations are the fundamental equations of hydrodynamics and heat transfer.


24

1

rm

r

m

H

t

P

t

E

2

22

22

4r

GM

m

Pr

t

r

Mass conservation equation

Momentum equation

Energy equation FrH 24

+ heat transfer equation

The lagrangian description is usually preferred better physical interpretation of equations

Radially pulsating envelope model

• Spherical simmetry: the star varies its volume and luminosity on the pulsation time scale but the shape remains spherical.

• No rotation, no magnetic fields.


• The core is excluded (nuclear reactions evolve on much longer time scales, pulsation mechanisms excited in the envelopes)

• The envelope is divided in150-250 mass zones.

Theoretical analysis by steps

• Linear adiabatic analysis (Eddington 1918) Periods


• Linear NONadiabatic analysis (Baker & Kippenhahn 1962, Cox 1963, Castor 1971, Iben 1971….) Periods, blue boundary of the instability strip (red boundary fixed with ad hoc assumptions)

• NONlinear convective analysis Periods, more accurate pulsation amplitudes (max-min) and lightcurve morphology, blue and red boundaries of the instability strip

• NONlinear radiative analysis (Christy 1966, Stellingwerf 1974…..) Periods, amplitudes (max-min), blue boundary of the instability strip

Linear adiabatic models


Adiabatic oscillating mass elements do not

gain/lose heat.

Linear small amplitude approximation,

equations are linearized

Small amplitude linearized equationsLet’s assume

0r

r )1(0 rrR

rx 0 ↔ )1(0 fff

The linearized equations become:

xx

3

P

P

g

xR

dx

Pd

P

P

x

..

4ln

m

H

PtP

P

t

13

1

T

T

xxdTdT

T

H

H

R

lnln14

4

mass

momentum

energy

radiative transfer


The Linear Adiabatic Wave Equation (LAWE)

By combining the time derivative of the momentum equation with the other two conservation equations we obtain:

m

H

rrrrrP

dr

d

r

1

1Pr

143 3

.

4141

....

For adiabatic oscillation: 0

m

H

If solutions have the form: tier

21

414

431

Pr1

P

dr

d

rdr

d

dr

d

r

The solution =0 is discarded because it corresponds to the perfectly static case


An eigenvalue problem(Cox J. P. “Theory of Stellar Pulsation”)

The LAWE is an ordinary second order differential equation solution contains two integration constants.

21

414

431

Pr1

P

dr

d

rdr

d

dr

d

r


The LAWE is a linear homogeneus equation in (if is a solution, also multiplied by a constant A is a solution) the absolute value of the pulsation amplitude is one of the two constants

With the remaining integration constant we fix one of the two boundaryconditions (at th center or at the surface). The other boundary condition is satisfiedby varying the other available parameter, namely the angular frequency of pulsation σ

Only some eigenfrequencies 0, 1,….and eigenfunctions 0, 1……satisfy the

boundary conditions both at the center and at the surface.

Results of the adiabatic analysis

The fundamental mode eigenfunction 0 has no nodes in the range 0 r R, while k has k nodes in the range 0 r R.

For each k we have two possibilities : (a) k is real constant amplitude pulsation k(r,t)= k (r) e ±ikt; (b) k is imaginary dynamical instability on a free fall timescale

If k is real the pulsation period is Pk=2/ k

Linear adiabatic models provide periods for the various pulsation modes(fundamental, first overtone, second overtone…………………….)

Linear NONadiabatic modelsIn this case we cannot neglect heat exchanges

m

H

PtP

P

t

13

1


(- H/ m) 0

The linearized full energy equation can be written in the form

And for solutions of the form eiωt as

m

H

P

i

P

P

131

Solutions imply complex frequencies =+i increasing or decreasing

pulsation amplitudes pulsation stability can be investigated

Driving and damping regions in a stellar envelope

1P

PIn the adiabatic case


m

H

P

i

P

P

131

In the NONadiabatic case:

For a zone gaining heat at the phase of

minimum radius ∫PdV>0 driving

For a zone releasing heat at the phase of

minimum radius ∫PdV<0 damping

The growth rate and the blue edge If the sum of the closed cycle ∫PdV over all mass shells in

the star is positive the driving regions prevail over the damping ones the stellar envelope does pulsate


where W is the work done on the system by the restoring forces around a

complete closed cycle and Ψ is the total oscillation energy or, equivalently

i 4

W

The growth rate of pulsation is

dtdW /

2

1where <dW/dt>W/ is the average rate (over one period) at which the

restoring forces do work on the system and we have used σ=2π/Π

If η > 0 W>0, κ < 0 with a time dependence e±it.e-κt

oscillations are excited

If η < 0 W<0, κ > 0 oscillations are damped

Transition from η < 0 to η > 0 defines the BLUE EDGE

The red edge of the instability strip in the linear NONadiabatic approximation

Most linear models do not properly treat the coupling between convection and pulsation in the oscillating stellar envelope.

Convection is either neglected (e.g Castor 1971, Stellingwerf 1975,

Bono & Stellingwerf 1994) or included by means of the mixing

length theory (e.g. Baker & Kippenhahn 1965, van Albada & Baker

1971, Tuggle and Iben 1973, Chiosi & Wood 1990, Saio & Gautschy

1998, Alibert et al. 1999)

On this basis the red edge can be evaluated only by means of some ad

hoc assumptions (e.g. where the growth rate is maximum) or by

displacing it at fixed ΔTe from the blue edge.

Some authors adopt one-dimensional turbolent convection

models into their linear nonadiabatic codes (e.g. Yecko, Kollath

& Buchler 1998, Xiong, Cheng & Deng 1998) red edge but a number of free parameters need to be

calibrated againts observations


Results of linear NONadiabatic RR Lyrae models

On the basis of linear nonadiabatic RR Lyrae fundamental and first overtone models, van Albada & Baker (1971) derive the following important relations:

eTL

L

M

MP

6500log48.3log84.0log68.0772.1log 0

125.06500

log09.0log014.0log032.0095.0log1

0

eTL

L

M

M

P

P

where P0 is the fundamental mode period and P1 is the first overtone one

These relations connect a pulsation parameter (the period) which

can be measured with high accuracy from observations, with

stellar intrinsic parameters, which in turn depend on age and

chemical composition

The relation for P0 implies that the parameter

A=logL/Lo -0.81logM/Mo

can be evaluated for RR Lyrae with measured period and Te

Mass-Luminosity relation

Helium content S. Agata sui due Golfi, May 2005

The Petersen diagram for double mode RR Lyrae

Some RR Lyrae pulsate both in the fundamental and in the first overtone mode (double mode pulsators or RRd).

• Comparison between pulsational and evolutionary mass

• Independent estimate of the stellar mass (pulsational mass)

• Mass discrepancy problem: pulsational masses significantly lower than the evolutionary ones (Cox, Hodson & Clancy 1983; Nemec 1985)

• Settled with updated input physics and nonlinear, nonlocal time-dependent convective models (Cox 1991, 1995; Bono et al. 1996)


van Albada & Baker’s relations suggest that the plot of log(P1/P0)

versus logP0 (Petersen 1973) is mainly a function of the stellar mass.

Results of linear NONadiabatic Cepheid models

Several authors predict the instability strip blue boundary as well as relations between the pulsation periods and the intrinsic stellar parameters (P(M,L,Te), P(M,R)) which are at the basis of the PL relation

theoretical constraint on the problem of PL universality

They do NOT find a signficant metallicity dependenc of

predicted PL relation (e.g. Saio & Gautschy 1998, Alibert et

al. 1999) )

or a differential dependence with the adopted bands, but

quite small when the V and I bands are adopted to correct

for reddening (see Chiosi, Wood & Capitanio al. 1999)

Nonlinear radiative models

In the nonlinear approach the fundamental equations are not linearized pulsation amplitudes can be obtained

The first applications of the nonlinear analysis were purely radiative

(e.g. Christy 1965, 1968, 1975; Buchler et al. 1990):

convection was neglected no indication on the red edge !

Also both luminosity and velocity amplitudes predicted by these

models turned out to be systematically larger than the observed ones

MAX→

←MIN

NONlinear convective pulsation models

In order to predict all the relevant observables of pulsation, namely periods, amplitudes, complete topology of the instability strip, a nonlinear convective code is required.


The crucial property of a nonlinear convective code is the

modeling of the coupling between convection and pulsation

in a stellar oscillating envelope.

The treatment of convection and the prediction of the red boundary of the

instability strip

As first suggested by Baker & Kippenhahn (1965) the increasing efficiency of convection toward lower effective temperatures has a damping effect on the and mechanisms, restoring stability.


Therefore, for an accurate evaluation of the red boundary we

need to take into account the coupling between dynamical and

convective velocity in the stellar envelope

nonlocal time-dependent theory

Several attempts in the literature (e.g. Stellingwerf 1982, Geymeyer 1993, Bono &

Stellingwerf 1994, Feuchtinger 1998)

Castor-Stellingwerf approach Define Φ(x,u´,t) as the number density of convective element in phase space,

where x is a space coordinate relative to the mean flow, u´ is the fluctuating part of the velocity (u=<u>+ u´)


Convective transport equation is the continuity equation in phase space:

Suu

uxt i

ii

i

''

'

where derivatives are summed over relative coordinates and S is a source function thatcreates convective elements with small perturbed velocities (δu´≪ |u´|) and destroy themafter having traveled a mean free path “l”

By computing the moment of this equation with respect to (u´)², using theconservation equations and treating the flux of turbolent energy via a diffusionmodel with diffusivity |u´|l, one obtains the convective equation in the variable <(u´)²> = 2Et

urrrlr

lrrrDt

D 220

2/12/12

2

21

diffusive component

(convective overshoot)

phase lag term

containing the

driver 0

Interaction of the

convective and mean

velocity fields

(Castor 1968, Stellingwerf 1982)


By transforming the bolometric

lightcurves into the observational

bands UBVRIJK (static atmosphere

models) magnitude and color

curves mean (time-averaged over

the pulsation cycle) magnitudes and

colors.

Results of nonlinear convective models for Cepheids: I


Extended atlas of theoretical

light and radial velocity curves

are produced.Bono, Castellani, Marconi 2000, ApJ

There are two main ways to derive observed mean magnitudes:

Magnitude averaged values, e.g. (B), (V),….: mean over the magnitude curve

taken in magnitude units

Intensity averaged values , e.g. <B>, <V>,….: mean intensity over the

magnitude curve transformed

into magnitude

They differ from each other and from the static magnitudes, i.e. the values

the stars would have were they not pulsating.

For colors it is still more complicated: one has (B)-(V), <B-V>, <B>-<V>

Once mean magnitudes and colors have been obtained from the predicted

lightcurves, multiwavelength Period-Luminosity-Mass-ColorPeriod-Luminosity-Mass-Color relations can

be obtained for each assumed chemical composition

Constraints on the Mass-Luminosity relation if the period, the color and

the

distance are known

Constraints on the DISTANCE if periods and colors are known and a M-L

relation is assumed.

Light Radial velocity

Nonlinear convective models by Bono, Marconi & Stellingwerf (1999, ApJS) and Fiorentino et al. (2002) predict that the instability strip is a function both of metallicity and helium abundance.

The complete topology of the instability strip is obtained. This is a crucial point to evaluate the PL relation, which is obtained by evaluating the statistical magnitude of Cepheids at a given period.

Results of nonlinear convective models for Cepheids: II


As Y increases from

0.28 to 0.31 at Z=0.02,

the strip gets bluer

As metallicity increases the strip gets redder !

Implications for the universality of the PL


The strip in the Mbol -logP diagram moves toward longer periods as

metallicity increases.

The strip in the HR diagram gets redder as metallicity increase. But,

for each M and L, the effective temperature is anticorrelated with the

Period (← Period-Density relation + Black body)

At fixed period more metallic pulsators are fainter at variance with a number of empirical tests in the literature (Kennicutt et al. 1998, Sasselov et al. 1997, Sakai et al. 2004 and references therein)

Synthetic multiband PL relations


By populating with a mass law distribution the predicted instability strip at varying metallicity we find that :

optical synthetic PL depend on metallicity, appear to obey to a quadratic rather than to a linear relation and have a significant dispersion;

all these effect are reduced when we move toward NIR bands.

Caputo, Marconi, Musella 2000 A&A

Methods to derive Cepheid distances as based on nonlinear convective pulsation models

• PLC relations, if reddening is known: e.g. MV=α+βlogP+γ(B-V)0 for a given M-L relationIn the B, V bands you also need to take into account metallicity


=0.03

=0.04

• NIR PL relations (even if metallicity and reddening are poorly known) allow to drive accurate distances with an intrinsic uncertainty of the order of 0.05 mag.

Optical PL relations are quite dispersed due to the intrinsic width of the

instability strip and are not useful to derive direct distances to individual

Cepheids (σ0.25 mag)

If reddening is not known you need at leasttwo bands and two equations

For each metallicity and assumed M-L relation, you can use two

theoretical relations (PL and/or PLC) to evaluate 0 and E(B-V)


j,PLj =0 + Rj E(B-V) e.g. V,PLV =0 + 3.3 E(B-V)

j,PLC =0 + rj, E(B-V) with rj, function of Rj and of the color

term coefficient in the PLC

e.g. V,PLCB =0 + (3.10- ) E(B-V ) MV= + logP + (B-V)

True distance modulus and reddening for the measured metal abundance.

To use the PL(V) and PL(I) relations is the same as using the so called Wesenheit relations:

e.g. MV-R (B-V)0 =α+βlogP(=V-R(B-V)- 0) for a given metallicity and M-L relation

AV=R E(B-V) (extinction law)

Application to the extragalactic distance scale

Applying the theoretical metal-dependent PL(I) and PLC(V,I)

relations to the VI dataset of the Key Project HST galaxies

Caputo et al. (2000) find a theoretical metallicity correction.

The HST distances based on universal PL need to be

corrected:

/ logZ ~ -0.27mag/dex

where logZ is the metallicity difference between the

observed galaxy and the LMC

h0/[O/H] ~ +0.124 dex-1

where h0 = H0/H0

(Ho(KP) increases by about 5 %)

The extension to sovrasolar chemical compositions and to varying helium

contents at fixed metallicity, suggests that the metallicity correction is more

complicated. It is negligible for period shorter than 10 d, whereas for longer

periods there is a change of sign around solar metallicity and a clear dependence

on the ΔY/ΔZ ratio

Fiorentino et al. 2002This behaviour is in agreement with empirical spectroscopic

results by Romaniello et al. (2005) and is confirmed by an

extended updated model grid (Marconi et al. 2005 submitted)

The predicted correction becomes in agreement with the

empirical one by Kennicutt et al. If Y/ Z 3.5

If reddening and chemical composition are not known you need three bands

PL, PLC relations…. depend on metallicity and helium unknownquantities in the evaluation of Cepheid extragalactic distances are: thedistance, the reddening and the chemical composition

Observations in at least three filters + 3 theoretical relations discriminate the metallicity and reddening effects on distance

determinations, in order to derive self consistent estimates of mean

and individual distances, reddenings and metal abundances

(Caputo, Marconi, Ripepi 1999 ApJ; Caputo, Marconi, Musella,Pont 2001 A&A)

The above method holds for each assumed ΔY/ΔZ ratio. Independent

constraints on this parameter are needed.

Comparison between predicted and observed light curves

• Wood Arnold & Sebo (1997) on the basis of a nonlinear convective code (Wood 1974), simulate the light curve of the LMC bump Cepheid HV 905

Tight constraints on the input parameters M, L, Te (M-L relation) 0(LMC)=18.51±0.05 mag.

• Bono, Castellani & Marconi 2002 reproduce the light curve of two Cepheids in the LMC from the OGLE (Udalski et al. 1999) database, with different period and light curve morphology good fit for the same 0 in the range 18.48-18.58 mag

• Keller & Wood (2002) applied the same technique to 20 bump Cepheids in the LMC (Macho project) 0(LMC)=18.55±0.02 (intrinsic error of the average)

Results of nonlinear convective models for RR Lyrae

(da Bono et al. 1997 A&AS, Bono et al. 1997 ApJ)


By taking into account the nonlinearity of the problem and thecoupling between dynamical and convective velocities, we areable to predict:

• the complete topology of the instability strip as a function of model parameters

• the detailed morphology of light and radial velocity curves and in turn to obtain accurate Bailey (Period-Amplitude) diagrams

Theoretical atlas of RR Lyrae light curves

Bono et al. 1997 ApJS

RRabRRc

The Bailey diagramTheory Observations

Bono et al. 1997 ApJS

The Instability strip

Bono, Caputo, Marconi 1995 AJ

Effect of metal abundance on the predicted RR Lyrae instability strip

Bono et al. 1997 ApJ

Methods to derive RR Lyrae distances asbased on nonlinear convective pulsation models


The theoretical Period-Wesnheit relations (Marconi et al. 2003, Di Criscienzo

et al. 2004)

The direct comparison between observed and modeled light and radial velocity

curves (see Bono, Castellani, Marconi 2000, Di Fabrizio et al. 2002, Marconi &

Clementini 2005)

The comparison between the predicted and observed distribution of the firstovertone blue edge (Caputo et al. 2000)

The theoretical metal dependent PL(K) relation (Bono et al. 2003, MNRAS)

The theoretical PL(K) of RR Lyrae

There are empirical evidences for the existence of a PL relation in the K band

(Longmore 1986, 1990). On the basis of nonlinear convective models we derive

refined formulations of van Albada & Baker relations:

e.g.

log P(F) = 11.124(0.002)+0.844 logL-0.684 logM-3.378 logTe+0.009 logZ

log P(FO) = 11.816(0.002)+0.818 logL-0.680 logM-3.340 logTe+0.004 logZ

(Di Criscienzo, Marconi, Caputo 2004 ApJ)

Period is anticorrelated with the effective temperature for the HB Mass and

Luminosity level period is not correlated with the absolute magnitude in the

optical bands

no optical PL relation !


What are theoretical predictions for the PL(K) ?

The V-K color is a tight function of the effective temperature

Mk is strongly dependent on the effective temperature so that, moving from the blue (highest temperature) to the red (lowest temperature) edge of the instability strip, the period increases and Mk gets brighter


dMv/dlogP ~-0.14 whereas dMk/dlogP ~-2.1 with a luminosity dispersion logL=0.1 the accuracy on the K magnitude is

better than 0.1 mag., whereas Mv ~ 0.25 mag (Bono et al. 2001, 2003, MNRAS)

← with the dependence

on Z and L

MK=0.565-2.101 logP+0.125 logZ-0.734 logL/Lo =0.031 mag.

for F pulsators

MK=-0.016-2.265 logP+0.096 logZ-0.635 logL/Lo =0.025 mag.

for FO pulsators

logL = 0.1 dex MK ~ 0.07 mag

logZ = 0.3 dex MK ~ 0.03 mag

Bono et al. 2003 MNRAS

• V magnitudes heavily depend on logL • K magnitudes mildly depend on logL

(V-K) is very sensitive to logL and provides further constraints on

this unknown parameter

Accurate V,K observations of F and FO pulsators with known

period and metal content combined with theoretical metal dependent

PLC(V,K) relations can supply information on the intrinsic

luminosity.

Once known the luminosity and the metal content, the theoretical

MK -logP relation provide MK and in turn the distance.


The case of RR Lyr itself

For RR Lyr:

logP=-0.2466 (Kazarovets et al. 2001)

K=6.54 mag. (Fernley et al. 1993)

V=7.784 mag. (Hardie 1955)

[Fe/H]=-1.390.10 (Fernley et al. 1998)

theoretical PLC

logL = 1.642 0.024 + 0.535AV

For each adopted extinction correction one derives a luminosity

level to be inserted in the PLK relation

MK=0.565-2.101 logP+0.125 logZ-0.734 logL/Lo

in order to derive MK and a pulsation distance or parallax (=1/d)


Current parallax estimates remain

within the errors of the HST

absolute parallax (Benedict et al.

2002, dashed lines) and present a

better formal accuracy (3%

versus 5%) pulsation

parallaxes resemble direct

astrometric measurements

The use of this pulsational

approach for a large number of

well studied RR Lyrae, covering a

wide metallicity range, provide

relevant information on both

the slope and the zero point of the

Mv(RR)-[Fe/H] relation (see Ripepi, this School)


The theoretical PLC and Period-Wesenheitrelations for RR Lyrae

In the optical bands, at fixed mass and metallicity, RR Lyrae

models obey to a PLC relation which can be used to infer

accurate distances, provided that the reddening is known.

Alternatively, you can use theoretical Wesenheit functions, as

defined and extensively adopted for Cepheids:

W(BV)=V-3.1(B-V) W(VI)=V-2.54(V-I) ……etc…..

to provide reddening free Period-Wesenheit relations

Di Criscienzo, Marconi, Caputo 2004


Comparison between the predicted and observed first overtone blue edge (FOBE)

The theoretical FOBE can be used

to match the observed distribution

of RRc samples belonging to

globular clusters at different metal

content (Caputo et al. 2000, MNRAS)

apparent distance modulus

absolute mean visual mag. Mv(RR)=V(RR)-

Mv(RR) vs [Fe/H] relation

By applying the method of the theoretical fitting of light curves to a sample of LMC RR Lyrae stars by using a refined version of Stellingwerf’s nonlinear convective code we obtain

0(LMC)=18.55±0.05 Marconi & Clementini (2005, AJ)

A comparison between the Cepheid and RR Lyrae distance scales: the case of LMC


by applying the same method with the same code to 2 LMC Cepheids

from the OGLE database (Udalski et al. 1999):

0(LMC)18.53±0.05Bono, Castellani, Marconi (2002)

These results are all consistent and in agreement with recent independent determinations of the LMC distance modulus (see lessons by B. Madore, this school)

By applying the same method but with a different code to a sample of 20 LMC Cepheids, Keller & Wood (2002) obtain

0(LMC)=18.55±0.02

Problems and uncertainties of current nonlinear convective pulsation models

The treatment of convection


The Cepheid Mass-Luminosity relation

The adopted model atmospheres

The adopted input physics

Uncertainties on the treatment of convection

The turbolent-convective model included in current pulsation codes is based on a number of free parameters.


Comparison with observations seems to suggest that α=1.5 is able to reproduce thepulsation properties in the blue part of the pulsation region (FO models), whereasas the effective temperature decreases a better agreement is obtained for α in therange 1.8-2.0 (see Di Criscienzo, Marconi, Caputo 2004, ApJ, Bono, Castellani,Marconi 2002 ApJL, Marconi & Clementini 2005).

In Stellingwerf’s approach the mixing-lenght approximation is used to close the

system of nonlinear differential dynamical and convective equations. Variations of

α =l/Hp affect both the pulsation amplitudes and the width and topology of the

instability strip:

amplitudes decreases and the strip narrows as α increases

The adopted static model atmopheres


Static model atmospheres (e.g. Castelli, Gratton, Kurucz1997a,b;

Bessell, Castelli Plez 2000) are used to convert luminosities and

effective temperatures into magnitudes (UBVRIJK) and colors, as a

function of mass and chemical composition.

L, Te, M, Z, (Y) BC, (B-V), (U-B), (V-I), ecc...

Current uncertainties on the Bolometric Correction scale and on the

various colors affect the comparison between predicted and observed

pulsation observables.

The Cepheid Mass-Luminosity relation

• Canonical ML: no core overshooting during the H burning phase.• Non canonical ML: mild or full core overshooting is assumed.

canonical logLcan=a + b logM + clogY+ dlogZ

mild overshooting logLmild=a + b logM + clogY+ dlogZ + 0.25

full overshooting logLfull=a + b logM + clogY+ dlogZ + 0.5 (Chiosi, Wood & Capitanio 1993)


Overshooting smaller mass at fixed L than the

canonical one

The ML relation significantly affects the coefficients of PLC and Wesenheit

relations, as well as the morphology of light and radial velocity curves.

Mass loss produces the same effect !

The effect of input physics

The assumed input physics, namely opacity and equation of state,

in both evolutionary and pulsational computations, affects the

model predictions.

e.g.

Key role of updated OP and OPAL opacity tables in the mass

discrepancy problem of RR Lyrae and in the modeling of metal

intermediate and metal-rich pulsators.

The evolutionary predictions, and in particular the ML relation

for Cepheids, depend on the adopted input physics.

Summary

Pulsation models of Classical Cepheids and RR Lyrae can provide fundamental constraint both to the distance scale problem and to stellar evolutionary models


On the basis of nonlinear convective pulsation models severalconsistent methods can be adopted to infer the distance toCepheids and RR Lyrae.

The pulsation period and the blue edge of the instability strip canalso be derived in the linear approximation but a nonlinear codeis required to model pulsation amplitudes and the inclusion of anonlocal time-dependent treatment of convection is needed inorder to properly model the red edge of the instability strip.

pulsation models: cepheids and rr lyrae marcella marconi inaf astronomical observatory of...

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