1. A Quest for Double-Mode Variables
And Other Tidbits from the Zoo of Pulsating Stars
Genk, 13 May 2006
VVS Werkgroep Veranderlijke Sterren Patrick Wils

2. Based on Data from: CCD Visual 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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3. In Collaboration with:
Adam Sodor (HU)
Christopher Lloyd (UK)
Ennio Poretti (IT)
Klaus Bernhard (AT)
Sebastian Otero (AR)
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4. 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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5. Radial Pulsation versus Non-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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6. 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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7. 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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8. Double-Mode Strings Genk, 13 May 2006
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9. Fundamental-Mode Pulsation
Radial pulsation
All layers inside the star move in the same direction
No “nodes” inside the star
Example is exaggerated: the pulsations will never go deep into the core of the star
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10. 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:
Sine functions are an exact representation of the vibrations of a string, but not for the brightness of a pulsating star: a Fourier series is a just a mathematical representation for it, as there is no “pulsation sine” function.
To model the vibration of a “double-mode string” one needs only 2 sine functions: 2 Fourier terms with one period and its first harmonic; for “double-mode” stars one needs 2 series with an “infinite” number of Fourier terms (2 incommensurate periods).
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11. 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)
1st Overtone:
“Dwarf” Cepheids (DCEPS)
RRc
Mira/SR variables?
Fundamental mode:
Most Cepheids
(Pop. I + II)
HADS
RRab
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12. Fundamental Mode/1st Overtone Pulsation
RRab
1st overtone:
RRc
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13. 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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14. Double-Mode Pulsation
Two stable radial pulsation modes
Linear combinations of independent modes
Important objects for pulsation theories
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15. Double-Mode Pulsation: HADS
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16. Double-Mode Pulsation: RRd
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17. 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
Ratio for DCEP(B), RRd, HADS(B): ~
3 Triple-Mode variables:
AC And, V829 Aql, V823 Cas
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18. Petersen Diagram Period ratio versus fundamental period
“Spread” due to different metallicity or mass
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19. 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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20. Double-Mode Stars Discovered
Type
New
Total (mag<14)
DCEP(B) 4 22
HADS(B) 8 16
RRd
References:
IBVS 5501
A&A 440, 1097
IBVS 5698
And references therein
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21. Stars of Special Interest
Two modes:
High amplitude -> 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 (< 1st/2nd overtone)
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22. Beat Period Double-mode Cepheids = “Beat” Cepheids
1/Beat period = 1/Period1 – 1/Period2
Close periods => Long beat periods
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23. 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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24. Blazhko Effect: Examples
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25. Radial Mode + Non-Radial Mode Pulsation
HADS: GSC : 1 radial + 2 non-radial modes
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26. Hertzsprung Sequence Genk, 13 May 2006
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27. 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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28. Fourier Analysis: Hertzsprung Sequence
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29. Fourier Analysis: Generalized Phase Differences
Linear combination modes
Double-mode G11: HADS, RR, Cepheids
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30. RV Tau Stars Alternating faint/bright minima (like Beta Lyr!)
Best known example: R Sct
Also double-mode with ratio 1:2 ?
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31. RV Tau Stars: Sub-types
RVa
RVb (binaries + circumstellar dust?)
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32. RV Tau: Visual Data RVa RVb Genk, 13 May 2006
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33. DI Car RW Aur ? W Vir (Population II Cepheid) ? Like R Sct (RVa) ?
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34. DI Car Carbon star: link with R CrB ? Genk, 13 May 2006
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35. Open Questions Fixed G11? : more (accurate) data needed
Why aren’t there any W Vir DM-stars?
DM-stars among LPV (BS Lyr)?
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36. 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?
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