Temperature Differences in the Cepheid Instability Strip Require Differences in the Period-Luminosity Relation in Slope and Zero Point


A graphical and an algebraic demonstration is made to show why the slope and zero point of the Cepheid period-luminosity (P-L) relation is rigidly coupled with the slope and zero point of the Cepheid instability strip in the HR diagram. The graphical demonstration uses an arbitrary (toy) ridge line in the instability strip, while the algebraic demonstration uses the pulsation equation into which the observed P-L relations for the Galaxy and the LMC are put to predict the temperature zero points and slopes of the instability strips. Agreement between the predicted and measured slopes in the instability strips argue that the observed P-L differences between the Galaxy and LMC are real. In another proof, the direct evidence for different P-L slopes in different galaxies is shown by comparing the Cepheid data in the Galaxy, the combined data in NGC 3351 and NGC 4321, in M31, LMC, SMC, IC 1613, NGC 3109, and in Sextans A+B. The P-L slopes for the Galaxy, NGC 3351, NGC 4321, and M31 are nearly identical and are the steepest in the sample. The P-L slopes decrease monotonically with metallicity in the order listed, showing that the P-L relation is not the same in different galaxies, complicating their use in calibrating the extragalactic distance scale.

stars: variables: Cepheids — P-L relations — distance scale

1 Introduction

There is evidence that the Cepheid period-luminosity relation is not universal but differs in slope and zero point from galaxy-to-galaxy at a level of up to as a function of period (cf. Tammann & Reindl 2002; Tammann et al. 2002; Tammann et al. 2003, hereafter TSR 03; Sandage et al. 2004, hereafter STR 04; Sandage & Tammann 2006 for a review). Drastic as this conclusion is for studies of the extragalactic distance scale, it has been strengthened in confirming studies by Ngeow et al. (2003), Kanbur & Ngeow (2004), Ngeow & Kanbur (2004, 2005, 2006), Ngeow et al. (2005), Koen et al. (2007), and from theoretical models as a function of chemical composition by many authors, starting perhaps with John Cox (1959, 1980), and including Christy et al. (1966, 1972), Iben & Tuggle (1975), Chiosi et al. (1992), and more recently Bono et al. (2000), Fiorentino et al. (2002), Marconi et al. (2005), and undoubtedly others.

These studies show that the position of the borders of the the , instability strip in the HR diagram depends on chemical composition. If the strip borders vary in position and slope, so must the slope and zero point of the P-L relation, as worked through the pulsation equation in the following sections.

Despite this evidence, the conclusion that different P-L relations apply in different galaxies has recently been challenged in the literature. In these papers it is said that the slopes of the Cepheid P-L relations in other galaxies satisfy the slope of the P-L relation in the LMC and therefore that no slope differences with LMC have been demonstrated conclusively (cf. Gieren et al. 2005a, b, 2006; Pietrzynski et al. 2006; Benedict et al. 2007; van Leeuwen et al. 2007 are examples).

However, this claim sets aside the parallel evidence that the slope and zero point of the ridge lines of the Cepheid instability strips of the Galaxy, LMC and SMC themselves differ in temperature at a given period (cf. Fig. 3 of STR 04), and hence, in luminosity.

The purpose of this paper is to again remind us that the slope of the P-L relation is rigidly coupled with the slope of the instability strip via the Ritter (1879) pulsation condition that constant. Hence, if the instability strip slope varies from galaxy-to-galaxy, so must the P-L slope.

Differences in the instability strips of the Galaxy and SMC were first set out by Gascoigne & Kron (1965). They were made secure as temperature differences by Laney & Stobie (1986), and have now been made definitive by the new CCD data by Udalski et al. (1999a, b) for LMC and SMC and by Berdnikov et al. (2000) for the Galaxy, as summarized for the Galaxy and LMC in Figure 20 of STR 04.

In the next section we show the pulsation equation graphically and demonstrate from it the stated premise; a slope difference in the ridge line of the instability strip leads to a slope difference in the P-L relation. The graphical solution here is parallel to the algebraic demonstration given elsewhere (TSR 03, § 7.3; STR 04, § 8), and made more explicit here in § 3.

2 A Graphical Solution Based on the Lines of Constant Period in the Hr Diagram

In an obvious way the Ritter pulsation condition can be put into the observable parameters of period, luminosity, mass, and temperature by also using the Stefan-Boltzmann black body radiation condition that . The Ritter plus black body condition is improved by model calculations for real stars by using details of the pulsating stellar atmosphere structure, leading to the more precise pulsation equation of .

As in previous papers we again use the van Albada-Baker (1973) pulsation equation. Although it was calculated by them to apply to the lower mass RR Lyrae stars, comparisons show that their predicted P-L relation is nearly identical with many other pulsation equations calculated for higher mass Cepheids. Examples are the equations by Iben & Tuggle (1975, their eq. 3), Chiosi et al. (1992, their eq. 5), Simon & Clement (1993, their eq. 2), and Saio & Gautschy (1998). The near identity among the equations is discussed in Sandage et al. (1999, hereafter ).

The pulsation equation by van Albada & Baker is


It can be made into an equation, , for the lines of constant period in the HR diagram once a mass-luminosity relation for Cepheids is used to eliminate mass from equation (1).

Observational determinations of many Cepheid masses are not available, and we must rely on theoretical mass values from calculated evolution tracks that pass through the instability strip. A summary of such tracks is given in Tables 1 to 5 of SBT 99 for tracks calculated from the Geneva models, in Table 11 for the Padua tracks, and Table 12 for the Saio-Gautschy tracks. Detailed references for these models are in SBT 99. The models of Marconi et al. (2005) for solar metallicity and by Bono et al. (2000) for lower metallicities were also studied.

From all the models, normalized at at , we have adopted


from the tracks. This is everywhere within of the Geneva and Padua tables in SBT 99 for all metallicities.

Putting equation (2) into equation (1) gives the equation of the lines of constant period to be


This produces a family of lines in the , HR diagram as is varied.

Figure 1 shows such a family for values of 0.4, 0.7, 1.0, 1.3, and 1.6. The ridge-line instability strip for the Galaxy is shown, using its equation of from STR 04, Figure 20. The blue and red strip borders are arbitrarily drawn parallel to the Galaxy ridge line using a temperature width of from the Galaxy ridge line. This is slightly wider than is observed (Fig. 20 of STR 04), but is drawn to accommodate the dashed strip line of a toy galaxy shown with the equation , similar to the instability strip of LMC (again Fig. 20 of STR 04), but drawn here without the break at 10 days. The toy galaxy strip (the dashed line) has been made to intersect the Galaxy strip at to insure that the separate P-L relations also cross at this period.

The ridge-line P-L relations are obtained in an obvious way by reading the (ordinate) values at the intersections of the instability strip with the constant period lines for both the Galaxy and the toy model. The fact that the resulting ridge-line P-L relation obtained for the Galaxy in this way differs from that of the toy model because of the different slopes of the instability strip ridge lines is obvious from this construction.

For , the instability strip has higher temperatures for the toy model than for the Galaxy at a given period. Hence, the intersection of the ridge-line strip with the constant period lines occurs at brigher lumimosities for the toy than for the Galaxy, giving a P-L relation for the toy model that is brighter than for the Galaxy for all periods smaller than . The opposite is true for . Hence the P-L relations will have different slopes, as was to be shown.

The discussion here in words could complete the promised demonstration that the slope of the P-L relation is rigidly coupled with the slope of the instability strip ridge line in the HR diagram. However, to make the point more explicit, even to the point that the discussion becomes unnecessarily more elementary, bordering on pedantry, Figure 2 displays the two different P-L relations obtained by reading Figure 1 in this way. The slope values for the Galaxy and the toy are marked in the Figure, based on the adopted instability equations adopted for Figure 1. These slopes are similar to the actual slope values measured for the Galaxy and the SMC from STR 04 (their eq. [17]) and Tammann et al. (2008), hereafter TSR 08 (their eq. [5]), and set out again in Table 1 here later.

3 The Algebraic Solution Using Data From the Galaxy and the Lmc

We can apply the pulsation equation directly to show the algebraic solution for the same problem using real data, both for the equations of the instability strips of the Galaxy and LMC and the observed P-L relations. The demonstration made here uses the equations for observed P-L relations from STR 04 in their equation (17) for the Galaxy and their equations (12) and (13) for the LMC. These are put into equation (1), which, together with the adopted mass-luminosity equation (2), gives a predicted , instability ridge-line relation. This predicted line is then compared with the observed instability strip equations shown in Figure 20 of STR 04.

We have used an explicit bolometric correction to change the values obtained from the observations into required in equations (1) and (2), and back to to compare the predictions from the pulsation equation with the observations. The bolometric corrections are interpolated from Table 6 of SBT 99 for the appropriate metallicities and surface gravities of the Cepheids. The turbulent velocity was assumed to be . The surface gravities vary with radius, mass, and luminosity and therefore with period as a surrogate as (eq. [49] of STR 04). The metallicities are assumed to be [A/H] for the Galaxy and for LMC. The mass is from equation (2). The obvious arithmetic is not shown.

The resulting predictions of the instability strip ridge-line equations are these:


for the Galaxy at all periods, and,


for days for the LMC, and


for days, also for the LMC.

Note that the break in the instability strip relation at days in equations (5) and (6) is mirrored in the break in the P-L LMC relations given in equations (12) and (13) of STR 04, and shown as Figure 4 there.

For comparison with the predictions in equations (4)–(6) here, the observed ridge lines of the strips in the Galaxy and the LMC, taken from the insert equations shown in Figure 20 of STR 04, are


for the Galaxy at all periods, and


for days, and


for days for the LMC.

The near agreement of the predicted slopes of the instability strips in equations (4)–(6) with the observed slopes in equations (7)–(9) is the demonstration we are seeking.

The agreement is good, but there is a disagreement in the temperature zero points between equations (4)–(6) and equations (7)–(9) by . The predicted temperatures are cooler than those observed. However, the difference is remarkably small, given the approximations we have made in the bolometric corrections, in the adopted temperature scale of SBT 99, their Table 6, and in the adopted van Albada-Baker theoretical zero point in equation (1).

The temperature offset could be made zero if the zero point of the mass in equation (2) would be made smaller by , but then the evolution mass would differ from the pulsation mass by this amount. This is the expression of the previous well known mass “problem” which is solved here by the temperature shift.

In this regard, it is useful to remark that many of the temperature scales in the current literature, for example as summarized by Sekiguchi & Fukugita (2000) or by Cacciari et al. (2005), and including the one in SBT 99 that we have used here, differ among themselves by as much as in at fixed . This, then, is the temperature uncertainty in the temperature zero point in Figure 20 of STR 04. Our shifting of the predicted temperature relative to the observed temperatures in Figure 3 by is not excessive.

The observed (solid lines) and the predicted (dashed lines shifted by in ) instability strips for the Galaxy and the LMC are shown in Figure 3. The agreement is satisfactory, showing again that differences in the instability strip loci causes differences in the slopes of the P-L relations. Hence the claims in the current literature, cited in the Introduction, that a universal slope exists for the Cepheid P-L relation are inconsistent with Figures 13 which show different positions of the instability strip in different galaxies.

4 Summary of Observed P-L Slope Differences in Selected Galaxies

The arguments given in the previous sections rely on knowledge of the temperatures of the instability strips. These can only be measured using reddening corrected colors, and these are reliable only if the reddening of the individual Cepheids can be determined by some method other than by using a fiducial period-color (P-C) relation. The reason is that if the temperatures of the instability strips differ from galaxy-to-galaxy, presumably because of chemical composition differences, the P-C relations will also differ. There will be no correct fiducial P-C template from which to determine the reddening if the chemical compositions vary greatly. The intrinsic P-C relations will differ from galaxy-to-galaxy depending on the chemical composition, and the reddenings are therefore indeterminate.

Presently, it is only the Galaxy, LMC, and SMC that can be subjected to the analysis given here because it is only for these galaxies that the reddening of their Cepheids have been determined by methods other than by comparing with some adopted fiducial P-C relation.

However, for some galaxies with enough Cepheids, and where the differential reddening between the Cepheids is small enough to be ignored, comparison of the P-L slopes can be made directly from the data. The result for the Galaxy, NGC 3351, NGC 4321, LMC, SMC, IC 1613, NGC 3109, and Sextans A and B is shown in Figure 4. The adopted data for the P-L relations are in Table 1. The equations for the apparent magnitude and absolute magnitude P-L relations are , and . Column (2) is the of the oxygen-to-hydrogen ratio from Table 4 of TSR 08. Column (3) shows the observed value of , which is the slope of the apparent magnitude P-L relation taken from the same sources (but changed slightly in a few entries here) as were used for Table 4 of TSR 08. Column (4) lists the apparent magnitude P-L intercept, , as observed. Column (5) lists the distance modulus adopted in TSR 08. The absolute magnitude P-L relation is in column (6), which is column (4) minus column (5). The literature source is in column (7). The resulting P-L relations, calculated from the and values in Table 1, are shown in Figure 4. The slopes for the NGC 3351/ NGC 4321 combination and the Galaxy are the steepest of those shown, and are similar. That of the LMC is next steepest.

The slope of the Galaxy P-L relation in TSR 03 and STR 04 is based on averaging the results using the moving atmosphere method (the Baade-Becker-Wesselink procedure) and the independent main sequence fitting method. Nevertheless, the resulting slope of the P-L slope has been questioned as being too steep (cf. Gieren et al., 2005b; van Leeuwen et al., 2007). However, the slopes of the NGC 3351 and NGC 4321 combined P-L relation, and that of M31 by Vilardell et al. (2007) are equally steep as for the Galaxy. The M31 slope by Vilardell et al. has been redetermined by TSR 08. The original slope by Villardel and collaborators was based on values using the LMC P-C relation rather than the more correct higher metallicity P-C relation for the Galaxy. The resulting values turns out to depend on period as a further complication. But even discounting the M31 case, the steep slope for NGC 3351 and NGC 4321 from TSR 08 (their Fig. 2), supports the Galaxy slope that we derived in STR 04 and its difference from the P-L slope in LMC.

The strongest evidence for the difference as function of metallicity is the data for NGC 3109 (Pietrzynski et al., 2006) which has the well determined P-L slope of . This differs significantly from the slopes of either the Galaxy or NGC 3351/ NGC 4321 at , or for M31, and even for the LMC at . The six longest period Cepheids in NGC 3109 with are too faint by (Fig. 4 of Pietrzynski et al. 2006) compared with either the Galaxy or the LMC P-L relations.

Figure 4, similar in principle to Figure 5 in TSR 08, together with Figure 3 here, is our chief case for non-unique P-L relations between galaxies of different chemical compositions. The complications that this portends for determining the scale of extragalactic distances from Cepheids to within , unless special corrections for the difference are applied, is discussed elsewhere (cf. Saha et al. 2006; Sandage et al. 2006; TSR 08).

It is a pleasure to thank Bernd Reindl for his skill in the preparation of the diagrams and the text for publication. We also thank John Grula, Carnegie editorial chief, for his liaison with the press.
Name [O/H] Ref
(1) (2) (3) (4) (5) (6) (7)
Galaxy 1
NGC 3351/4321 mean mean 2
M31 3
IC 1613 6
NGC 3109 7
Sextans A/B 8

References. – (1) STR 04, eq. [17]; (2) TSR 08, Fig. 2; (3) Vilardell et al. 2007; (4) STR 04, eq. [8]; (5) TSR 08, eq. [5]; (6) Antonello et al. 2006; (7) Pietrzynski et al. 2006, Fig. 4; (8) Piotto et al. 1994.

Table 1: Observed (P-L) Relations for Ten Galaxies with Different Chemical Compositions
Figure 1: Schematic HR diagram in the vicinity of the Cepheid instability strip. The central line is the observed ridge line for the Galaxy taken from Figure 20 of STR 04 whose equation is . The dashed line is for a toy galaxy whose ridge-line equation is . The borders of the instability strip are put parallel to the Galaxy ridge line. Lines of constant period, calculated from equation (3), are marked with their values (in days).
Figure 2: The two P-L relations for the two ridge lines in Figure 1, determined from the intersections of the ridge lines of the Galaxy and the toy galaxy with the lines of constant period in Figure 1. The absolute magnitudes along the ordinate are transferred from Figure 1 by where the bolometric correction in is adopted to be zero.
Figure 3: The algebraic demonstration of the rigid coupling between the slopes of the instability strip and the slope of the P-L relation required by the pulsation equation. Predicted (dashed lines) slopes and zero points for these instability strip ridge lines in the Galaxy and the LMC are compared with the observed (solid) lines from Figure 20 of STR 04. The predictions are made by inserting the equations of the observed P-L relations for the Galaxy and the LMC into the pulsation equation (1). The predicted zero points are moved by 0.018 in , hotter.
Figure 4: The observed ridge lines of the P-L relations for eight galaxies listed in Table 1. The P-L relation for the Galaxy (not shown) is nearly identical with the combined NGC 3351 and NGC 4321 line, and has the steepest slope. The agreement between the Galaxy and the combined NGC 3351 and NGC 4321 slopes argues for the correctness of the steep slope for the Galaxy P-L relation.


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