*L* = 4
*R*^{2}
*T _{e}*

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 *R*^{2}, and the
effective temperature *T _{e}* is a thermal term, used to
parameterize the areal surface brightness, given by

*M _{BOL}* = -5

and it is schematically shown in Figure 2. It
should be
noted that *the entire* *M _{BOL}* -

**Figure 2.** Stefan's Law expressed in
graphical form projected onto the theoretical *M _{BOL}* -

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 *R*^{3} , 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}*
- (

If we then linearly map log (*T _{e}*) into an observable
intrinsic color (

**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.