PHASE PLANE ANALYSIS OF THE PHOTOMETRICAL VARIATIONS OF LONG-PERIOD VARIABLES

Анотація

Using  the  phase  plane  diagrams,  the phase  light  curves  of  a  group  of  the  Mira-type  stars  and semi-regular variables are analyzed. As generalized coordinates  х  and  ẋ,  we  have  used  m  –  the  brightness  of  the  star  and  its  phase  derivative.  We  have  used  mean  phase light  curves  using  observations  of  various  authors.  The data typically span a large time interval (nearly a century). They  were  compiled  from  the  databases  of  AAVSO, AFOEV, VSOLJ, ASAS and approximated using a trigonometric  polynomial  of  statistically  optimal  degree.  As the  resulting  approximation  characterizes  the  autooscillation  process,  which  leads  to  a  photometrical  variability, the phase diagram corresponds to a limit cycle. For  all  stars  studied,  the  limit  cycles  were  computed. For a simple sine-like light curve, in e.g., L₂ Pup, the limit cycle  is  a  simple  ellipse.  In  a  case  of  more  complicated light  curve,  in  which  harmonics  are  statistically  significant, the limit cycle has deviations from the ellipse.   In an addition to a classical analysis, we use the error  estimates  of  the  smoothing  function  and  its  derivative  to  constrain an “error corridor” in the phase plane.