14. A reddening-free formulation of the PL relation

The absolute calibration of colors and/or magnitudes of Cepheids is obviously wrought with many traps and uncertainties in methodology and procedure. One very appealing way to proceed is in an implicit formulation and calibration of the problem. Rather than explicitly solving for the reddening/extinction star by star, one can begin quite differently: First adopt a standard reddening line for the color system being employed, then form linear combinations of the magnitudes and colors so that the numerical quantity formed cancels the reddening (star by star) without ever explicitly determining it. Then this formulation can be used both for the calibration objects and for the target Cepheids. Forming this reddening-free magnitude directly parallels the more common reddening-free application to colors, such as Q = (U-B) - X (B-V) as defined by Johnson (1963). An example follows.

Suppose that the observed magnitude of the Cepheid is V and its apparent color is (B-V). As in the standard procedure, where individual reddening corrections are made, we assume that the ratio of total-to-selective absorption is known from independent determinations, and that its value is R = A_V / E (B-V). Then it naturally follows that a reddening free quantity W (called the Wesenheit function by Madore 1982) can be formed where W V - R (B-V). A simple expansion of the relevant terms shows that for small amounts of reddening the numerical value of W is independent of extinction, and equal to the value that would be calculated if the intrinsic magnitudes and colors were known (which they now need not be). That is, by definition,

W = V₀ + A_V - R [ (B-V)₀ + E (B-V) ]

W = V₀ + R E (B-V) - R (B-V) ₀ - R E (B-V)

W = V₀ - R (B-V) ₀ W₀

There exists some confusion in the literature as to the real motivation and physical justification for creating W for Cepheids. W in fact is a reddening-free quantity for stars, not exclusively for Cepheids. It is strictly defined by the properties of the interstellar medium, not by any properties of the stars to which it is to be applied. One is not at liberty, for example, to change the value of R from its normal value of 3.3, unless that correction derives from an independent study of the extinction law itself. Accordingly, with the exception of the small known dependence of R on (B-V) and E (B-V) itself, (manifest as non-linear curvature terms at large values of these quantities; true for all applications of reddening corrections, not just W) one cannot manipulate the form of W at all.

The reason for some of the confusion dates back to a similar realization of the PLC introduced by van den Bergh (1975), also called W. In that first appearance of a linear combination of V and (B-V) the complete reddening independence of the function was not fully implemented, as will become clear now. In the following discussion we will use W_BFM and W_vdB = V - (B-V) to distinguish between the differing definitions used by Madore (1982) and van den Bergh (1975), respectively.

For Cepheids one can begin to see what W_BFM is, both in terms of its parent relationship, the period-luminosity-color relation, and in terms of its own definition. Adopting a linearized form of the period-luminosity-color relation we have,

V = V₀ + A_V

V = M_V + mod₀ + A_V

V = log P + (B-V)₀ + + mod₀ + A_V

where mod₀ is the true distance modulus. By definition,

W_BFM V - R (B-V)

so, by substituting the period-luminosity-color relation into the definition of W_BFM we get

W_BFM = log P + (B-V) ₀ + + mod₀ + A_V - R (B-V)

or, expanding the reddening terms, regrouping and cancelling, we get

W_BFM = log P + [ - R] (B-V) ₀ + + mod₀

since, by definition, A_V equals R E (B-V). For Cepheids, the W_BFM can be reformulated as a period-luminosity-color relation. Here it must be emphasized that the zero point and the slope of the period dependence in the W_BFM formulation of the period-luminosity-color relation are identical to their counterparts in the V formulation; only the coefficient multiplying the intrinsic color changes from to [ - R].

Here now is the point of confusion. van den Bergh (1969) noted the numerical similarity of R = 3.2 to the then espoused value of = 2.7, and at that point he chose his definition of W_vdB V - (B-V) which is quite a different approach. W_BFM has the advantage of being a strictly-defined, reddening-free magnitude adopting a well-determined quantity R, which does not attempt to anticipate the a priori unknown value of . Furthermore, residual scatter in the W_BFM - log P relation is also thereby distinctly of interest in its own right. Unlike scatter in the observed PL or PC relations, scatter in the W_BFM - log P relation cannot be due to reddening effects, be they differential or total. One is not at liberty to adjust the coefficient R for various samples of Cepheids (to minimize the scatter in the residual, for example) unless there is independent evidence that the ratio of total-to-selective absorption is different for that particular galaxy. The remaining scatter in a W_BFM - log P plot will be due to a combination of intrinsic (color?) correlations and photometric errors. As is so often the case it will be an understanding of the photometry and its quality that will ultimately limit our understanding of the intrinsic interrelations.

In order to illuminate the differences between the W_BFM and W_vdB , Figure 11 portrays the mapping of the PLC from its projection onto the traditional color-magnitude diagram (as an instability strip) and then into the three ``period-luminosity'' planes.

Figure 11. Projecting the observed instability strip, in the presence of reddening. (1) In the upper left panel the instability strip superimposed on the PLC is shown projected into the color-magnitude plane. Lines of constant period (P₁ and P₂) are shown as thick solid lines slanting down to the right. The blue and red edges of the instability strip are shown as thinner solid lines sloping down to the left. Arrows indicated the magnitude and direction of reddening which acts to increase the apparent width of the distribution by systematically scattering points to redder colors and to fainter magnitudes. A dashed line parallel to but fainter than the red edge of the instability strip illustrates the bounds of this effect. (2) The upper right panel is constructed from the first panel by projecting the instability strip and the reddening vectors into the Period-Luminosity plane. There is a systematic increase in the width of the apparent PL relation due to extinction, as in the CM diagram. (3) If the slopes of the lines of constant period are known a priori, then the projection of the instability strip into the W_vdB - log P plane can be performed. By definition, the intrinsic width goes to zero in this projection, but because the slope of the lines of constant period are not exactly parallel to the reddening trajectory, extinction does project into this plane and will widen the relation, and systematically shift the ridge line. (4) Since reddening trajectories are, in general, already determined a priori, it is possible to project the instability strip into the W_BFM - log P plane where (by definition) extinction effects are eliminated. In this case, however the intrinsic width does still contribute to the dispersion, but the disposition of the individual stars within this strip is not effected by reddening.

The upper left-hand panel shows a portion of the Cepheid instability strip in a color-magnitude diagram. The upwardly slanting solid lines give the red and blue edges of the strip, while the heavy downward-slanting lines labeled P₁ and P₂ are representative lines of constant period. Arrows indicate the slope of a reddening/extinction line. The upwardly sloping broken line indicates the apparent red edge of the instability strip as defined by stars reddened away from the intrinsic line.

The horizontal broken lines leading from Panel 1 to Panel 2 show the first mapping of the instability strip into the V-log P plane. The sloping lines of constant period in Panel 1 now become vertical, with downward extensions due to reddening increasing the apparent dispersion at fixed period.

Panel 3 to the lower right shows the effect of forming W_vdB given a priori knowledge of the value of the slope of the lines of constant period in the PLC. As can be seen, the projected width in the W_vdB - log P plane is non-zero in the presence of reddening. While the intrinsic width of the instability strip does project to zero (by definition, once is known from independent sources) there is residual widening due to reddening

Panel 4 shows the effect of forming W_BFM. Since the slope of a reddening line is well known from independent studies, differential reddening has no effect on the width at constant period. The only factor contributing to the width in W_BFM is the intrinsic width of the PLC (and as mentioned earlier photometric errors in the magnitudes and colors, which broaden all of the above relationships and projections).

Given Figure 11, it is perhaps worth noting that in the presence of differential reddening it is clearly inappropriate to determine the value of by minimization of residuals, as would be the case in an unrestricted application of a multi-linear regression fit to PLC data, for instance. For the example at hand, one can see immediately by inspection that a slope somewhere between the true value of and the reddening slope R would give the minimum residuals plotted as a function of period. The numerical value of that parameter for any given set of Cepheids is however of no universal significance.