4. Simple physical considerations

The basic physics connecting the luminosity and color of a Cepheid to its period is well understood. Using Stephan's law

L = 4 R² T_e⁴

the bolometric luminosities L of all stars, (including Cepheids), can be derived. The radius R is a geometric term, parameterizing the total emitting surface area 4 R², and the effective temperature T_e is a thermal term, used to parameterize the areal surface brightness, given by T_e⁴. Expressed in magnitudes, Stephan's Law becomes

M_BOL = -5 log R - 10 log T_e + C

and it is schematically shown in Figure 2. It should be noted that the entire M_BOL - log T_e plane is mapped by Equation (2), and that within the context of this relation alone, for any value of log T_e, an unbounded range of radius R is independently possible. However, once R and T_e are each specified, M_BOL is uniquely defined. Additional constraints, outside of Stephan's Law itself, must be involved if bounds on permitted values of the independent geometric and thermal variables are to be considered. An important constraint is provided by stellar evolution. In terms of timescales and allowed equilibrium configurations, the core-hydrogen-burning main sequence is one of the most striking and well known examples of such a ``constraint'' on populating the HR diagram. Hydrostatic equilibrium can be achieved for long periods of time along the hydrogen-burning main sequence, and as a result we are constrained to observe most of the stars there most of the time. There are, of course, other constraints.

Figure 2. Stefan's Law expressed in graphical form projected onto the theoretical M_BOL - logT_e plane where loci of constant radius are indicated by upward sloping lines.

For mechanical systems it is well known that P ^1/2 = Q , where Q is a structural constant, and P is the natural free pulsation period, determined by gravity through , the mean density of the system, in turn defined by M = 4/3 R³ , where M is the total mass of the system. If it is assumed that mass is predominantly a function of R and T_eff, then the pulsation period can be used as the second observable parameter instead of requiring the radius to be observed directly.

Figure 3. The PLC relation expressed in graphical form as projected onto the observational M_V - (B-V) plane, where loci of constant period are downward sloping lines.

If we then linearly map log (T_e) into an observable intrinsic color (i.e., B-V), and map radius into an observable period, we thereby predict a new two-parameter description of the luminosity of (pulsating) stars. This is precisely the physical basis for the period-luminosity-color (PLC) relation for Cepheids, as was so elegantly introduced and explained over a quarter of a century ago (Sandage 1958, Sandage & Gratton 1963, Sandage & Tammann 1968). In its linearized form for pulsating variables, Stefan's law takes on the following form of the PLC: M_V = logP + (B-V) ₀ + . In analogy to plotting Stephan's Law in the theoretical M_BOL - log T_e above, Figure 4 shows the PLC mapped into the observational M₀ - (B-V) ₀ color-magnitude diagram. Again the entire plane is mapped. To see how the PLC relates to the more commonly referred to relations (the PL and PC relations), the reader is referred to a more detailed discussion in the sections ahead, the caption to Figure 4, and the Cepheid section in Jacoby et al. (1992) where an empirical analog of Figure 4 is given.

Figure 4. The Cepheid Manifold: Projections of the PLC plane (shown shaded) onto the three principal co-ordinate systems (luminosity [L], increasing up, period [log P], increasing to the right and color [(B-V)] becoming bluer to the lower left). The backward projection onto the L-P plane gives the period-luminosity relation. Projecting to the left gives the position of instability strip within the color-magnitude diagram. Projecting down gives the period-color relation.

Stars can be found evolving across many parts of the color-magnitude diagram. However, only in very narrowly defined regions do they become pulsationally unstable. Cepheid pulsation in particular occurs because of the changing atmospheric opacity with temperature in the helium ionization zone. This zone acts like a heat engine and valve mechanism, alternately trapping and then releasing energy, thereby periodically forcing the outer layers of the star into motion against the restoring force of gravity. Not all stars are unstable to this mechanism. The cool (red) edge of the Cepheid instability strip is thought to be controlled by the onset of convection, which then prevents the helium ionization zone from driving the pulsation (see Baker & Kippenhahn 1965; and Deupree 1977 for references). For hotter temperatures a blue edge is encountered when the helium ionization zone is found too far out in the atmosphere for significant pulsations to occur. Further details and extensive references can be found in the monograph on stellar pulsation by Cox (1980).

Figure 5. Magellanic Cloud Cepheid period-luminosity relations at seven wavelengths, from the blue to the infrared, constructed from a self-consistent data set (Madore & Freedman 1991). LMC Cepheids are shown as filled circles; SMC data, shifted to the LMC modulus, are shown as open circles. Note the decreased width and the increased slope of the relations as longer and longer wavelengths are considered.