A Quest for DoubleMode Variables

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A Quest for Double-Mode Variables

• And Other Tidbits from the Zoo of Pulsating Stars

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Based on Data from

• CCD
• All Sky Automated Survey ASAS3
• Northern Sky Variability Survey NSVS
• Individual observers
• C.W. Robertson (US)
• Joaquin Vidal (ES)
• Michael Koppelman (US)
• Neil Butterworth (AU)
• Paul Van Cauteren (BE)
• Russ Durkee (US)
• Stelios Klidis (GR)
• Visual
• AAVSO

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In Collaboration with

• Adam Sodor (HU)
• Christopher Lloyd (UK)
• Ennio Poretti (IT)
• Klaus Bernhard (AT)
• Sebastian Otero (AR)

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Pulsating Variable Stars

• Brightness variations because the star expands or
  contracts
• For normal stars, pulsations will die out
  rapidly
• Pulsations are stable only for certain periods in
  the life of a star the instability strip in
  the HR diagram

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Radial Pulsation versus Non-Radial Pulsation

• Radial pulsation
• Spherical shape is kept
• Internal layers expand or contract as a whole
• Generally large amplitude
• Non-radial pulsation
• Parts of the surface move outward while others
  move inward
• Nodes on the surface of the star
• Smaller amplitude because total surface area will
  change less

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Non-Radial Pulsation

• Delta Scuti stars
• Small amplitude
• Short periods
• Asteroseismology
• star quakes probe the interior of stars
• Source examples Delta Scuti Network

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Fundamental Mode versus Overtones String
Vibrations

• Guitar, piano, ...
• Fundamental mode 0 nodes,
• 1st overtone 1 node, 2nd overtone 2 nodes, ...
• Overtones are harmonics/higher octaves
• P(1O) P(F)/2
• P(2O) 2P(1O)/3 exact (low) integer fractions

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Double-Mode Strings
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Fundamental-Mode Pulsation

• Radial pulsation
• All layers inside the star move in the same
  direction
• No nodes inside the star

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Overtone Pulsation

• Radial pulsation
• Different layers may move in different directions
• One or more nodes
• inside the star
• (layers that do not move)
• Do not confuse with harmonics
• from Fourier analysis
• 1st overtone example

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Fundamental Mode/1st Overtone Pulsation

• Stability of the mode depends on the physical
  properties of the star (location in HR diagram)
• Determined by theoretical calculations and
• Phase difference amplitude ratio for different
  colours
• Pulsation constant Q (Mbol, Teff, g, P)

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Fundamental Mode/1st Overtone Pulsation

• Fundamental mode
• RRab
• 1st overtone
• RRc

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Higher Overtones

• Almost symmetric light-curve
• Short periods
• Some (very short period) Cepheids may be second
  overtone pulsators but no examples in the
  Galactic field
• Unclear whether RRe stars really exist

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Double-Mode Pulsation

• Two stable radial pulsation modes
• Linear combinations of independent modes
• Important objects for pulsation theories

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Double-Mode Pulsation HADS
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Double-Mode Pulsation RRd
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Double-Mode Pulsation

• Fixed ratio between 1st overtone and
  fundamental period
• No integer fractions (as in strings)
• Cepheids 0.705
• RR 0.745
• HADS 0.77
• 2 Galactic Cepheids are 1st/2nd overtone
  double-mode pulsators
• CO Aur, HD 304373
• Ratio for DCEP(B), RRd, HADS(B) 0.80-0.81
• 3 Triple-Mode variables
• AC And, V829 Aql, V823 Cas

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Petersen Diagram

• Period ratio versus fundamental period
• Spread due to different metallicity or mass

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How to Find Double-Mode Variables?

• Lots of data needed (wide-field surveys!)
• Period analysis (PDM, Period04, ...)
• Look for periodic stars with unusually high
  scatter

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Double-Mode Stars Discovered

• References
• IBVS 5501
• AA 440, 1097
• IBVS 5698
• And references therein

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Stars of Special Interest

• Two modes
• High amplitude -gt radial
• Not a standard ratio
• Binary variables? No linear combination modes!
• V371 Per
• Cepheid
• Periods 1.27 and 1.74 days
• Ratio 0.73
• V767 Sgr
• Cepheid?
• Periods 0.54 and 0.67 days
• Ratio 0.80 (lt 1st/2nd overtone)

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Beat Period

• Double-mode Cepheids Beat Cepheids
• 1/Beat period 1/Period1 1/Period2
• Close periods gt Long beat periods

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Blazhko Effect

• One or more frequencies very close to the main
  frequency
• Amplitude/phase modulation
• Beat periods 6-1000 days
• RR Lyrae stars, especially RRab
• Very few Galactic RRc Blazhko stars known
• Two possible causes, both based on rotation
• Resonance between radial and non-radial modes
• Oblique magnetic field

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Blazhko Effect Examples
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Radial Mode Non-Radial Mode Pulsation

• HADS GSC 4311-825 1 radial 2 non-radial modes

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Hertzsprung Sequence
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Fourier Analysis (Invariant) Phase Differences

• Describe brightness variations mathematically
  using sine functions (Fourier series)
• V V0 A1 sin(wt f1) A2 sin(2wt f2) ...
• Phase difference fundamental mode first
  harmonic
• Small amplitude harmonic
• (bump)
• Large amplitude harmonic
• (RV Tau?)

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Fourier Analysis Hertzsprung Sequence
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Fourier Analysis Generalized Phase Differences

• Linear combination modes
• Double-mode G11 HADS, RR, Cepheids

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RV Tau Stars

• Alternating faint/bright minima (like Beta Lyr!)
• Best known example R Sct
• Also double-mode with ratio 12 ?

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RV Tau Stars Sub-types

• RVa

• RVb (binaries circumstellar dust?)

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RV Tau Visual Data
RVa
RVb
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DI Car

• RW Aur ?
• W Vir (Population II Cepheid) ?
• Like R Sct (RVa) ?

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DI Car

• Carbon star link with R CrB ?

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Open Questions

• Fixed G11? more (accurate) data needed
• Why arent there any W Vir DM-stars?
• DM-stars among LPV (BS Lyr)?

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Conclusion

• A lot of variables have been under-observed,
  their real type not known
• Plenty of work that can be done by amateur
  astronomers
• Long time series
• Multi-colour data
• Data mining
• Visual observers
• more RV Tau stars!
• BS Lyr, other DM-Mira?