MULTI-WAVELENGTH CHARACTERISTICS OF PERIOD–LUMINOSITY RELATIONS

The period–luminosity (PL) relation for classical Cepheids is a two-dimensional projection of the higher-order period–luminosity–color (PLC) relation. The PLC relation, in turn, is a finite portion of a plane, a so-called strip, embedded in the three-dimensional space of period, luminosity, and color. This strip is defined and delimited by physical limits (on temperature and luminosity) within which the atmospheres of these stars are found to be pulsationally unstable. A graphical representation of how these various projections arise and are inter-related was first given in Figure 3 of Madore & Freedman (1991). From this perspective it is clear that the width and orientation of the PLC strip in three-dimensional space will have direct and profound influence on the measured properties of the two-dimensional relations, projected and multiply transformed to observable (wavelength-dependent) quantities (magnitudes and colors).

It has long been known that the intrinsic width of the Cepheid PL relation (as commonly parameterized by its formal dispersion) decreases as a function of increasing wavelength (see, for example, Madore & Freedman 1991 for the optical; McGonegal et al. 1982 for the near-infrared; and more recently Madore et al. 2009 for applications in the mid-infrared). What is less commonly noted, although just as obvious, is the fact that the slope of the Cepheid PL relation does the opposite and increases systematically with increased wavelength. This anti-correlation of slope and dispersion is shown in Figure 1, which is an updated and expanded version of a plot first presented as Figure 6 in Madore & Freedman (1991). Scatter is the quantity of most interest when any such relation is being used for distance determinations. The slope, on the other hand, is of little or no consequence in most practical applications. In the following, however, we show that the behavior of these two quantities, slope and scatter, shares a common physical explanation and that they can be readily understood (i.e., predicted) from first principles. Moreover, there may indeed be some practical implications arising from this realization.

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The total luminosity of a radiating object is the product of its total (radiating) area times the mean surface brightness of that area. Generalizing the Stefan–Boltzmann law to a monochromatic (rather than bolometric) observation can be done by parameterizing the surface brightness by some power, a_λ, of the effective temperature, T_e. Thus, the entire (i.e., infinite) planar extent of the color–magnitude diagram is mapped by

$$\begin{equation} M_{\lambda } = -2.5 \log R^2 - 2.5 \times a_{\lambda } \log T_e + b_{\lambda }, \end{equation} \tag{ 1 }$$

where R is the radius of the star and λ is the wavelength of the observational bandpass. By imposing the following linear "constraints" on the above equation one can then define the (red/cool and blue/hot) boundaries of the instability strip: log R = clog P + d and log T_e = −elog P + f, where P is the period, and where it is to be noted that a_λ, c, and e are all numerically positive quantities. Applying these external constraints to the master Equation (1) above (where f_red and f_blue are, respectively, the red and blue intercepts of the blue and red boundaries of the instability strip) gives

$$\begin{equation} M_{\lambda }^{{\rm red}} = [-5 c + 2.5 e a_{\lambda }] \log P - 2.5 \times [a_{\lambda } \times f_{{\rm red}}] + b_{\lambda } - 5 d \end{equation} \tag{ 2a }$$

$$\begin{equation} M_{\lambda }^{{\rm blue}} = [-5 c + 2.5 e a_{\lambda }] \log P - 2.5 \times [a_{\lambda } \times f_{{\rm blue}}] + b_{\lambda } - 5 d. \end{equation} \tag{ 2b }$$

The full (wavelength-dependent) magnitude widths of the respective PL relations, |M_λ| (which are directly related and proportional to their intrinsic dispersions) are then found to be

$$\begin{equation} |M_{\lambda }| = 2.5 \times a_{\lambda } |f_{{\rm blue}} - f_{{\rm red}}|. \end{equation} \tag{ 3 }$$

Moreover, the (central) ridge-line equation of the mean PL relation becomes (M^red_λ + M_λ^blue)/2 or

$$\begin{equation} M_{\lambda }^{{\rm mean}} = [-5 c +2.5 e a_{\lambda }] \log P - 2.5 [a_{\lambda } \times f_{{\rm mean}}] + b_{\lambda } - 5 d ], \end{equation} \tag{ 4 }$$

where F_mean = (f_red + f_blue)/2.

The slopes of the corresponding (wavelength-dependent) PL relations are then

$$\begin{equation} \Delta M_{\lambda }/\Delta \log P = A_{\lambda } = -5 c + 2.5 e a_{\lambda }. \end{equation} \tag{ 5 }$$

The explicit dependence¹ of both the slope (through Equation (5)) and the width/scatter (through Equation (3)) on the same wavelength-dependent term, a_λ, now becomes clear.

At fixed period, the total width of the PL relation at any given wavelength is simply the difference of Equations (2a) and (2b), with the formal dispersion being a fixed multiplicative fraction of that width. A quick check shows that for a temperature width of the Cepheid instability strip of 700 K (or Δlog T_e = 0.05) this corresponds to a color width of Δ(B − V) = 0.5 mag and a corresponding magnitude width of ΔM_V = 0.8 mag. If the stars are uniformly distributed within and across the strip this would correspond to a formal dispersion of $\sigma _V = 0.8/\sqrt{12} = \pm 0.23$ mag, which compares well with the observed dispersion of ±0.27 mag for the LMC V-band PL relation as given in Madore & Freedman (1991).

2.1. A Graphical Approach

Figure 2 graphically captures the substance of the above cascade of equations, and visually illustrates just how the slopes of the PL relations and their widths/dispersions are coupled. Three PL relations are shown. The lower relation (labeled 3.6 at its terminal point to the right of the diagram) represents a long-wavelength PL relation having the characteristic low scatter and a steep slope. The thick parallel lines (flanking each of the mean PL relations) represent the upper and lower boundaries of the instability strip as projected into the PL plane. The light, dotted lines trace the mean relations. The thick dashed lines running diagonally across each of the PL relations represent lines of constant color/temperature.

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It is important to note that the mapping of the PL relation at one wavelength into a PL relation as observed at another wavelength is solely dependent on the differential response of surface brightness to temperature across the respective wavelengths (as parameterized bf a_λ in Equation (1)). Hereafter color is taken to be synonymous with temperature. Given that the influence of temperature on surface brightness is known to be an increasingly sensitive function of decreasing wavelength, and given that temperature is itself a generally decreasing function of increasing period (longer-period Cepheids are generally cooler than their short-period counterparts), the mapping of the 3.6 μm PL relation into the B band is shown by the two thick arrows of decreasing amplitude in going from hot temperatures (the arrow above the circled A) to cooler temperatures (the shorter arrow above the circled letter B). This systematic decrease in the vector's magnitude with period (i.e., with mean temperature) is readily seen to be responsible for the systematic decrease in the slope of the PL relation in going from longer to shorter wavelengths (as explicitly given in Equation (5)).

The vectors originating from points in the long-wavelength PL relation having the same temperature (as tracked by the dashed lines) have (by definition) exactly the same magnitude/length. Those vectors are shown terminating in the B-band PL relation just below the circled letters C and D. In other words, the slope of the lines of constant temperature, Δmag/Δlog P, is independent of the wavelength of the PL relation.

There is a fixed temperature difference in crossing the instability strip at a given period. This manifests itself as an increasingly larger magnitude width of the PL relation when observed at progressively shorter and shorter wavelengths (as can be seen in Equation (3)).

There is also, as noted above, a general decrease in temperature (increase in color) along the PL relation as one increases the period. This results in the decreased magnitude of the correction for temperature as a function of period, which in turn manifests itself as a reduction of the overall slope of the PL relation as one goes to shorter and shorter wavelengths.

The significant point here is that both of the above effects are controlled by a single physical parameter, the temperature. Accordingly, an increased slope of the Cepheid PL relation must be accompanied by a causally connected decrease in the intrinsic dispersion of the PL relation. The longer-wavelength PL relations will be steeper and their intrinsic scatter must be smaller than their shorter-wavelength counterparts.

Looking at Equation (4) more closely it is clear that the slope of the PL relation is made up of two linearly additive, but physically independent parts: (1) a constant term "5c," which is essentially the slope of the period–radius (PR) relation converted into a period–area (PA) relation, and then cast into a magnitude (i.e., by multiplying log R by 2 × −2.5), and (2) the slope of the period–color relation (i.e., $-2.5 \times e \times (a_{\lambda _1} -a_{\lambda _2})$). It is this last term that contains the wavelength-dependent index, a_λ, characterizing the sensitivity of surface brightness to temperature variations for any given bandpass. If the (wavelength-independent) slope of the PA relation is simply subtracted from the (wavelength-dependent) PL slopes one would be left with the purely wavelength-dependent, period–surface-brightness relation (PSB) slope (for instance, transforming Equation (1) gives M_λ + 2.5log R² = −2.5 × a_λlog T_e + b_λ). Given that the constant of proportionality between slopes of the PSB relations and their respective dispersions must be the same number, independent of bandpass, we have then a means of determining the slope of the PA relation by demanding that the ratio of the PSB slope to PL dispersion must be constant and independent of wavelength.

Published values of the slope of the PR relation have been recently tabulated by Molinaro et al. (2010). Their own work suggests values of c ranging from 0.71 to 0.75, with historical values, drawn both from theory and observations, ranging more widely from 0.67 to 0.77. We now explore the run of the aforementioned proportionality constant with bandpass, parameterized by the input slope of the PR relation. The abscissa in Figure 3, which is the ratio of the slope of the PL relation (Equation (5)) to the dispersion in the PL relation (effectively Equation (3)), will become independent of wavelength when the correct value of the scaled slope of the PR relation (i.e., −5c) is subtracted from the numerator, as given in Column 3 of Table 1. The intent is to find a slope of the PR relation that minimizes any residual trend of the proportionality constant with bandpass/wavelength. The circled dots in Figure 3 show that solution: no trend with bandpass (a formal slope of 0.000 ± 0.015) for an input value of c = 0.721 and a resulting scatter (±0.009) that is exceedingly small. In changing the slope of the PR relation to correspond to the Molinaro et al. limits, as given above, the two flanking solutions are realized. In both cases the dispersion is much enhanced, and there is a strong (and unphysical) trend seen with wavelength. This already small range in the PR slope (of only ±3% peak-to-peak) is now much more narrowly defined by the present study. Our final solution for the slope of the Cepheid PR relation and its error is c = 0.724 ± 0.006.

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The results of the above fitting, now adopting a PR slope of 0.724, are given in Table 1. The first column gives the bandpass, ranging from the blue to the mid-infrared, at which each fit was made. In the second column we list, in order of increasing wavelength, the published slopes, A_λ (equal to −5c + 2.5ea_λ; see Equations (4) and (5)) of PL relations taken from a merger of Madore & Freedman (1991) for the optical, Persson et al. (2009) for the near-infrared, and Madore et al. (2009) for the mid-infrared. Column 3 gives the slope of the PSB relation obtained by subtracting the constant and wavelength-independent slope of the PA relation (i.e., −5 × 0.724 = −3.62) from the respective PL slopes (as given in Column 2) leaving A_λ + 3.62 = 2.5ea_λ. As already derived above, the multi-wavelength slopes of the PSB relations and the dispersion/widths of the PL relations must each be proportional to the same factor, a_λ. As can be seen in Figure 3 the incorrect choice of the slope of the PR relation very quickly leads to a divergence with wavelength in the prediction of the scale factor (that must be independent of wavelength) required to convert the slope to a scatter. This diagnostic diagram can therefore be used to find the wavelength-independent slope of the PR relation that best predicts the wavelength-dependent scatter in the observed PL relations from the optical BVRI to the near-infrared JHK bands.

As can be seen in Figure 3 for a value of c = 0.72 the dispersion-to-slope ratio is independent of wavelength and has a value of 0.316. Using this proportionality factor we convert each of the wavelength-dependent PSB slopes to their corresponding wavelength-dependent PL relation dispersions. These predicted dispersions are given in Column 4 of Table 1 (e.g., for a B-band surface-brightness slope of 1.19 the predicted dispersion in the B-band PL relation is 1.19 × 0.316 = 0.376 mag). Column 5 of Table 1 contains the observed PL relation dispersions, self-consistently taken from the same references cited above for the slopes of the PL relations. The correspondence of the predicted and observed dispersions is impressive. A slight divergence occurs in the mid-infrared where the observed dispersions are upper limits, due to the fact that these magnitudes were derived from only two random-phase observations and that they contain additional scatter imposed by the back-to-front depth and tilt of the LMC, whose Cepheids were used for these demonstrations.

An independent test using Cepheids in the Small Magellanic Cloud (SMC) is compromised by the extremely large geometric, extrinsically correlated scatter due to the extension of the SMC along our line of sight; however, the somewhat more distant galaxy IC 1613 can be used for this purpose. In the paper by Antonello et al. (2006) there are 48 Cepheids with complete BVRI photometry having periods ranging from 3 to 42 days. We have used these observations to derive the slopes and dispersions (the latter not having been published by these authors) for the four PL relations. The results are shown graphically in the lower left portion of Figure 2. The IC 1613 data are shown as filled dots enclosed by squares, each of which has an error bar derived from our calculated uncertainty on the slope at each wavelength. The points are plotted for an input value of the PR relation taken from the LMC solution. To within the uncertainties there is no obvious trend of the dispersion-to-slope ratio for the IC 1613 Cepheids using the LMC PR relation. There is, however, a systematic offset in the ratio which can be directly attributed to a narrower sampling of the Cepheid instability strip by the IC 1613 Cepheids. The conclusion remains that the slope of the PR relation is identical for these two galaxies. Extending the test to other systems with a wider range of metallicities will be of interest. However, until Gaia produces parallaxes for a sufficiently large sample of Milky Way Cepheids calibrating this relation in the galaxy will have to be postponed until secure reddenings and accurate distances are available for more than a handful of galactic Cepheids.

3.1. The Effects of Differential Reddening

While it could have been predicted, it is only now that we know that the decreased width and the increased slope of the Cepheid PL relation, as a function of increased observing wavelength, is inevitable. Both effects are driven by the wavelength-dependent temperature sensitivity of stellar surface brightness. At the longest wavelengths the PL relation asymptotically parallels the (wavelength-independent) PR relation; at shorter wavelengths the PL relation is increasingly dominated by the (highly wavelength-dependent) surface-brightness sensitivity to temperature.

It is of interest to use aspects of this formalism to put constraints on the slope of the PR relation. Since the dispersion must be non-negative the slope of the PA relation cannot be shallower than the steepest (longest-wavelength) slope of the PL relation. The steepest slope in Table 1 is −3.49 seen at 8.0 μm. This would require that the slope of the PR relation be steeper than c = −0.70, since there must still be at least one power of T_e flattening the slope of the PL relation, even at the longest-wavelength limit. Examination of the minimum dispersion solution of the generalized Wesenheit function gives a value of c = −0.70. And, more comprehensively, by requiring that the run of dispersion-to-slope be independent of wavelength gives for LMC Cepheids c = −0.724 ± 0.006, a value that also fits the Cepheids in IC 1613.

The implications of this discussion are not confined to classical Cepheids. For example, the somewhat surprising (at the time) discovery by Longmore et al. (1986) that RR Lyrae stars obey a fairly steep and tightly defined PL relation in the near-infrared, as compared to the same stars whose optical magnitudes show almost no dependence on period, is directly explained by the same argument given here in the main text. Inspection of the plots given by Longmore et al. (1986) would suggest that the scatter in the derived absolute magnitudes for RR Lyrae stars decreases by at least a factor of two in going from V to K as the slope of the PL relation rises from being essentially flat at V to having a value of about −2.5 at K. This observed trend is re-enforced and confirmed by multi-wavelength modeling of RR Lyrae stars by Catelan et al. (2004) whose Figure 2 dramatically tracks the systematic change of increasing slope and decreasing scatter occurring progressively from the ultraviolet to the near-infrared.

One note of caution is necessary. As the referee of this paper has suggested additional (magnitude) scatter may compromise these correlations in the case of RR Lyrae stars "caused by off-ZAHB (zero-age horizontal branch) evolution (which depends in turn on HB type) rather than intrinsic effects." The same caveat may apply to the Cepheids themselves given that stars of a given mass may make several crossings of the instability strip giving rise to first-, second-, and third-crossing scatter in period at a given luminosity and temperature.

Finally, we note that changes in the mapping of temperature to surface brightness might be modulated by additional physics, such as line blanketing changes in response to atmospheric metallicity variations, or surface gravity, effects especially in the near-ultraviolet, etc. As such, both the intrinsic width and the slope of the Cepheid PL relation will each be physically (and mathematically) forced to respond in a well-determined fashion. If the slope is thought to be increased, say, at a given wavelength because of a metallicity difference, then the prediction is that the intrinsic dispersion will be decreased and vice versa. However, uncertainties in differential reddening within any given Cepheid sample will always make these higher-order tests challenging.

We thank Eric Persson, Vicky Scowcroft and Mark Seibert for providing much valued comments on early drafts of this paper.