Petersen diagram for pulsation variable stars
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Author: Dr. Jørgen Otzen Petersen, The Niels Bohr Institute, Copenhagen University, emeritus
Dr. Jørgen Otzen Petersen accepted the invitation on 14 August 2007 (self-imposed deadline: 14 February 2008).
Petersen diagrams are used to give information on physical properties of stars that pulsate with at least two periods simultaneously. Stars are physical systems in mechanical and thermodynamical equilibrium, and due to their spherical symmetry it is possible to discuss oscillations around the equilibrium state in detail. The simplest type of oscillation is radial pulsation with the stellar body simply expanding and contracting as observed for example in Ceheid variables. Historically, the easily observed periods of classical Cepheids have been the basis for the distance scale of the Universe, and pulsation periods of several types of variable stars have provided many valuable tests of stellar models and stellar evolution theory. Modern developments in this area are helio- and asteroseismology.
Stellar Pulsation Periods
Ever since Tycho Brahe discovered his Stella Nova in Cassiopia in 1572
and Fabricius noted the variability of the first known periodic variable star
Mira in 1596
variable stars have been fascinating and important for both amateur
observers and professional astronomers. The main reason why variable stars
are still discussed so intensively after more than four centuries is
that stellar pulsation periods can often be determined with an accuracy
far exceeding anything else directly characterizing stars, typically
with a relative error less than 
. And also important, observed periods can directly be compared with theoretical
periods calculated for a specified stellar model without any
difficult and/or uncertain tranformation — a unique situation in
astrophysics. Today sophisticated analyses based on detailed stellar
modelling can use observed oscillation periods to check subtle effects
such as rotation, magnetic fields and chemical composition of matter
inside stars.
Period Ratios, Double-Mode Cepheids and Stellar Opacities
The Petersen diagram (PD in the following) is a convenient visual
aid in astrophysical analyses of stars that have (at least) two well
determined pulsation periods. In particular PDs have been used to discuss
several groups of Cepheid-type variable stars, e.g. classical Cepheids,
RR Lyrae variables and
Scuti stars.
The designation Petersen diagram is due to the generosity of Arthur N. Cox, for many years head of the Theoretical Division of the Los Alamos National Laboratory and the leading authority of stellar pulsation theory. Art started to use Petersen diagram for period ratio diagrams at several conferences, and graduately it was used also in publications.
Figure 1 shows an early PD and Figure 2 illustrates several details
to be discussed in the following.
Petersen (1978) gives an overview of early applications of PDs.
Basically, all PDs plot the period ratio
of a shorter period 
and a longer one 
as function of
or log .
Since observed periods are so accurate error bars in PDs for observed points
are usually invisible in both coordinates. For calculated oscillation periods
errors/uncertainties in the build-in physics of the stellar models may of course
give large uncertainties. But when all physical properties of models are
| Invited by: | Dr. Søren Bertil F. Dorch, The Niels Bohr Institute and the Royal Library, Copenhagen, Denmark |
