# User:Søren Bertil F. Dorch/Proposed/Petersen diagram for pulsation variable stars

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 Cepheid 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

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 $$10^{-5}\ .$$ And also important, observed periods can directly be compared with theoretical periods calculated for a specified stellar model without any difficult and/or uncertain transformation — 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 $$\delta$$ 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 gradually 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 $$P_s/P_l$$ of a shorter period $$P_s$$ and a longer one $$P_l$$ as function of $$P_l$$ or log $$P_l\ .$$ 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