T. Lloyd Evans
Cepheid variable stars are supergiants with luminosities 500-30,000
times greater than that of the Sun, although their surface
temperatures are similar to the Sun's temperature. They undergo
regular radial pulsations (i.e., the star expands and contracts), with
periods mainly in the range 1-50 days, and can be distinguished at
great distances. More than 400 Cepheids are known in the Galaxy and
about 1000 Cepheids have been found in each of the two nearest
galaxies, the Magellanic Clouds, as well as substantial numbers in
other nearby galaxies. The close relationship between period and
luminosity which was found by Henrietta S. Leavitt in 1912 has given
Cepheids a unique role in establishing the distances of the nearer
galaxies and hence the distance scale of the universe.
The regularity of the light curve of a Cepheid variable star is
matched by that of the radial-velocity curve, which is almost a mirror
image of the light curve with minimum radial velocity (i.e., maximum
velocity of approach) at light maximum. The light amplitude is
typically between 0.5 and 2 magnitudes in visual light and the
velocity amplitude usually lies in the range 30-60 km s-1. The first
Cepheid velocity curves were measured toward the end of the nineteenth
century and were interpreted as the results of orbital motion. It was
only after orbits had been computed for a substantial number of
Cepheids that it was realized that these orbits were physically
implausible.
The pulsation hypothesis gained increasing acceptance after 1910,
especially because the surface temperature changes over the cycle. Sir
Arthur Eddington's theoretical work from 1917 onwards showed that
Cepheids are single stars that undergo radial pulsations because they
function as a heat engine. Later work by S.A. Zhevakin, J.P. Cox,
Robert F. Christy, and others has provided a deeper understanding of
the mechanism. Energy is stored in the form of the second ionization
of helium during the compression stage of the cycle and then released
as the helium recombines during the expansion stage. The restriction
of Cepheid pulsations to stars in a limited temperature range follows
from the requirement that the second helium ionization zone lies near
the transition from the nearly adiabatic interior, where any driving
is almost canceled by an equal amount of damping, to the nonadiabatic
exterior where the thin outer layers lack the heat capacity to
modulate the outward flow of radiation. The pulsation is a property of
the stellar envelope and is independent of the
nuclear-energy-generating core.
Classical Cepheids are comparatively young stars with masses of
several times the solar mass. This follows from their strong
concentration toward the plane of the Milky Way and their low space
velocities. Their presence in star clusters allows their ages to be
estimated as up to about 108 yr. Observations of the Cepheids in the
Magellanic Clouds show that the classical Cepheids are confined to a
narrow strip in the period-luminosity diagram, whereas the less common
Type II Cepheids are fainter than them at a given period. The presence
of Type II Cepheids in globular clusters and in the galactic halo
population allows their age to be estimated as up to 15 x 109
yr, so
that they must be much less massive than the classical Cepheids. The
Type II Cepheids can also be distinguished from the classical Cepheids
by the shape of the light curves and by spectroscopic peculiarities.
The light curves of most classical Cepheids are asymmetrical, with a
rapid rise to maximum light and a slower fall. The form of the light
curve changes with period in a systematic way known as the Hertzsprung
progression. A bump appears on the descending branch of the light
curve of stars with periods of about a week and is found at earlier
phases in stars of successively longer periods so that the bump is
near maximum light in stars of 10-day period which may show a double
maximum. The bump falls on the rising branch in stars of longer
period. Stars of the shortest or longest periods have smooth light
curves. The amplitude of the pulsation increases slowly with period up
to about 10 days, where there is a drop in amplitude; it then
increases more rapidly to longer periods. The bumps may represent an
echo of the surface pulsation from the deep interior; an alternative
explanation is that they result from a resonance when the second
overtone period is about one-half of the fundamental period.
Some Cepheids of short period have nearly sinusoidal light curves
with amplitudes of only about 0.5 magnitude. These stars are uncommon
in the Galaxy but account for about 10% of the Cepheids known in the
Magellanic Clouds. A monumental study by Cecilia Payne-Gaposchkin and
Sergei Gaposchkin showed that they are systematically brighter than
the period-luminosity relation defined by the stars with asymmetrical
light curves. The ratio in periods at a given luminosity is 0.6, which
in view of the observational uncertainties is probably identical to
the well-defined ratio P1 / P0 =
0.71 found for double-mode Cepheids in
the Galaxy. The latter are Cepheids, mostly in the period range
P0 =
2.0-4.3 days, whose light curves may be represented as the sum of
simultaneous pulsations in the fundamental and first overtone periods
of P0 and P1 days, respectively. An
analogous situation is found in the RR Lyrae stars in globular clusters.
Evolutionary tracks covering the mass range and evolutionary states in
which Cepheids occur have been calculated by Icko Iben, Jr. and
others. The mass range 3-9 M (solar masses) covers all but stars of
very long period. The most important point is that a star may cross
the Cepheid instability strip on the Hertzsprung-Russell (HR) diagram
more than once during its evolution (see
Fig. 1). The star leaves the
main sequence after the exhaustion of H in the core and then expands
to become a red giant while burning H in a shell surrounding the
temporarily inert He core. It crosses the instability strip rapidly on
a Kelvin-Helmholtz or thermal time scale. It climbs the red giant
branch to the red giant tip and after the ignition of He burning in
the core it may make a loop to higher temperature in the HR
diagram. This loop may extend to sufficiently high temperature (or
blue color) to intersect the instability strip, in which case two more
crossings will occur. Core helium burning is a relatively long-lived
evolutionary stage and the star may remain in the instability strip
for much longer, by perhaps a factor of 50, than it did in the first
crossing. The exact location of the blue loops is a function of mass
and of chemical composition, so that above and below the most
favorable mass, the number of Cepheids will decline quickly. The more
massive stars evolve more rapidly in any case, so that the peak
residence time will correspond to a relatively low mass with a decline
in the numbers of longer-period stars which is accentuated by the
relative rarity of massive stars. All the Cepheids at the extremes of
the mass (period) distribution must be on their first crossing of the
instability strip. It has been estimated that they account for about
10% of all Cepheids.
The distribution of periods has been found for Cepheids in the Galaxy,
the Magellanic Clouds, and several other nearby galaxies. There is
inevitably selection in favor of the brighter long-period Cepheids and
those of larger amplitude at a given period, but differences between
the period distributions in the Galaxy and the two Magellanic Clouds
are well established. The distributions in these three systems may be
characterized by the shortest periods, the periods that are most
common, and the longest periods for fundamental mode pulsators:
(The second, third, and fourth columns above list the shortest,
maximum frequency peak, and longest periods, respectively.) The
maximum frequency peak period in the SMC, about 2.9 days, is uncertain
because there is a wide double maximum in the period
distribution. There is some doubt as to whether the Cepheid-like stars
of the very longest periods are genuine Cepheids or are some other
type of variable star. Even so, the overall distributions are very
wide and are difficult to account for theoretically unless there is a
spread of metal content (abundances of elements heavier than helium)
within each galaxy.
The principal difference between the Cepheids in the three galaxies
is the successively smaller value of the shortest and maximum
frequency peak periods in the order Galaxy-LMC-SMC, which is in order
from high to low metal abundance. This is readily explained by the
differences in the evolutionary tracks resulting from the differences
in composition. The blue loops in the core helium burning stage
extend to higher temperature the lower the metal abundance, so that
the maximum frequency Cepheids occurs at lower mass and luminosity and
shorter period. Stars of lower mass are relatively more numerous, so
that a star system of lower metal abundance will contain more
Cepheids, other factors being equal. The more metal-deficient Small Magellanic Cloud contains roughly as many
Cepheids as the
Large Magellanic Cloud whose total mass is four
times greater.
The approximately 400 known classical Cepheids in the Galaxy include
only 17 which are well-established members of star clusters and as
these are divided between 14 clusters, there are too few stars in any
one cluster to study the distribution of stars in the
Hertzsprung-Russell diagram. The Magellanic Clouds contain much richer
clusters and NGC 1866 in the Large Magellanic Cloud contains at least
seven Cepheids with periods in the range 2.6-3.5 days. Comparison of
the observed and theoretical HR diagrams leads to an estimated metal
(elements heavier than helium) abundance Z = 0.016, a Cepheid mass of
about 4.9 M, and an age of 86 x 106 yr. There are
still differences in
detail in the numbers and positions, especially of the red giant
stars, between the observed cluster and that which is calculated theoretically.
The main value of the Cepheids in the small clusters in the Galaxy
is that they may be used to establish the zero point of the
period-luminosity relation. The 17 stars noted previously and another 8
of longer period which belong to the loose stellar groups known as
associations have been used to establish the zero point to an accuracy
of ± 0.1 magnitude, excluding uncertainties in the distance scale for
star clusters.
Our knowledge of the masses of stars is obtained from binary systems.
Spectroscopic orbits need to be obtained for both components of a
binary and the inclination of the orbital plane to the line of sight
must be found or the mass of one component has to be established
independently. The orbital periods of Cepheid binaries are generally
long: The shortest known is the 507-day orbital period of the 9.7-day
variable star S Muscae. This means that the velocity amplitudes are
quite small and that eclipses that would establish the orbital
inclination are unlikely to occur. The mass of the companion must be
deduced from its spectrum. The light of the Cepheid is always dominant
at visible wavelengths so the spectrum of the companion, which is
usually a much hotter B star, is only readily distinguishable in the
far ultraviolet. This means that observations must be made from a
satellite. A few estimates of around 5-6 M for Cepheids of periods of
4-10 days are available so far, in broad agreement with the mass
estimates from evolutionary tracks. The latter have generally been
found to give larger masses than methods based on the pulsation
properties of the stars and more precise dynamical mass estimates are
needed to clarify the situation.
The limiting orbital period below which a Cepheid on its first
crossing of the instability strip would suffer disturbance to its
evolution is about 20 days but a Cepheid crossing the instability
strip for the second time on a blue loop has previously expanded to a
much larger radius at the red giant tip. The minimum period to avoid
overflowing the Roche lobe with consequent severe mass loss is several
hundred days in this case. The longer periods of all the Cepheid
binaries studied to date are in accordance with the theoretical
estimate that most are on their second or a subsequent crossing of the
instability strip.
STARS, CEPHEID VARIABLE
CLASSICAL AND TYPE II CEPHEIDS
CEPHEID LIGHT CURVES
THEORETICAL EVOLUTIONARY TRACKS OF CEPHEIDS
CEPHEID PERIOD DISTRIBUTIONS
Galaxy
2.3 days
5.4 days
67 days
LMC
1.6
3.4
135
SMC
1.1
2.9:
210
CEPHEIDS IN STAR CLUSTERS
CEPHEIDS IN BINARY SYSTEMS
Christy, R.F. (1966). Pulsation theory. Ann. Rev. Astron.
Ap. 4 353.
Cox, A.N. (1980). The masses of Cepheids. Ann. Rev. Astron.
Ap. 18 15.
Cox, J.P. (1980). Theory of Stellar Pulsation. Princeton University
Press, Princeton.
Fernie, J.D,. (1990). The structure of the Cepheid instability strip.
Astrophys. J. 354 295.
Payne-Gaposchkin, C. and Gaposchkin, S. (1966). Relation of light
curve to period for stars in the Small Magellanic Cloud. Vistas in
Astronomy 8 191.
Pel, J.W. (1985). Fundamental parameters of Cepheids. In
Cepheids: Theory and Observations, B.F. Madore, ed.
Cambridge University Press, Cambridge, p. 1.
Zhevakin, S.A. (1963). Physical basis of the pulsation theory of
variable stars. Ann. Rev. Astron. Ap. 1 367.
See also Star Clusters, Globular, Variable Start; Stars, BL
Herculis, W W Virginis, and RV Tauri Types; Stars, Cepheid Variable,
Dwarf; Stars, Cepheid Variable, Period-Luminosity Relation and Distance
Scale; Stars, Pulsating, Overview; Stars, Pulsating, Theory.