LLIGHT AND RADIAL VELOCITY IGHT AND RADIAL VELOCITY VARIATIONSVARIATIONS

DUE TO LOW FREQUENCY DUE TO LOW FREQUENCY OSCILLATIONS OSCILLATIONS

IN ROTATING STARSIN ROTATING STARS Jadwiga Daszyńska-Daszkiewicz Instytut Astronomiczny, Uniwersytet Wrocławski, Poland

Collaborators: Wojtek Dziembowski , Alosha Pamyatnykh

2222 NovemberNovember 2006, 2006, Porto Workshop Porto Workshop

INSTABILITY DOMAINS IN THE MAIN SEQUENCEINSTABILITY DOMAINS IN THE MAIN SEQUENCE

4.6 4.4 4.2 4.0 3.8 3.6

0

1

2

3

4

5

6

solar-likeoscillation

Dor

TAMS

ZAMS

1

2

4

12

30

60

Cep, field

Cep, NGC 3293

Cep, NGC 4755

Cep, NGC 6231

SPB Sct

OPAl 1996

X=0.70, Z=0.02, =1.0

logL

/Lo

logTeff A. A. PamyatnykhA. A. Pamyatnykh

for SPB pulsators often ~

Slow modes in the traditional approximationSlow modes in the traditional approximation

not too fast rotation: ( / crit)2 << 1

Cowling approximation

~ ~ << N(r)

Separation of the angular and radial dependences in eigenfunctions

s= 2/ (+1) (s) Ym (cos)eim

(cos) - the Hough functions

Modes with >0 propagate in the radiative zone (N>0).The radial wave number

Definition of mode degree, , for g-modes

s = 2/ 0 then (+1)

Retrograde r-mode with g-modes properties at s>|m|+1(Savonije 2005, Townsend 2005)

the Hough function

the Hough function

(, /2) – the normalized driving rate

For instability:

2/ - should match the thermal time scale in the driving zone

/2 – determines the r-dependence of eigenfunctions The pressure eigenfunction should be large in the driving zone

like (+1)/2 for high order g-modes in non-rotating stars

Radial displacementRadial displacement

Z = exp [i (m - t)]

in co-rotating system

m>0 - prograde modes

m<0 - retrograde modes

Oscillating atmospheric parametersOscillating atmospheric parameters

f (, /2)

Fx (Teff, log g)

hx (ns,Teff, log g)

Light variations in the x passband Light variations in the x passband

Pulsation velocity field Pulsation velocity field

Disc-averaged radial velocityDisc-averaged radial velocity

pulsational partpulsational part rotational partrotational part

the rotational contribution tothe rotational contribution to

arises from arises from rr, , n, n, FFbolbol, , gg

An example:An example:

M=6 MM=6 M, MS star, MS star

loglogTTeffeff= 4.205= 4.205

logL/LlogL/L= 3.204= 3.204

VVrotrot=0, 50, 150, 250 km/s=0, 50, 150, 250 km/s

Selected modes:Selected modes:

g-modes with g-modes with =1,2, most unstable =1,2, most unstable at each (at each (,m) ,m)

r-modes, most unstable with m= -1,-2r-modes, most unstable with m= -1,-2(only for V(only for Vrotrot ≥ 150 km/s) ≥ 150 km/s)

Hough functions for Hough functions for =1 and r mode with m= -1=1 and r mode with m= -1

Amplitudes of light and radial velocity variationsAmplitudes of light and radial velocity variations g-mode g-mode =1,m=0 and r-mode,=1,m=0 and r-mode, m=-1m=-1

Amplitudes of light and radial velocity variationsAmplitudes of light and radial velocity variations g-modes: g-modes: =1,=1, m= m= ±±11

Hough functions for Hough functions for =2 and r-mode with m= -2=2 and r-mode with m= -2

ProspectsProspects forfor modemode identificationidentification

Diagnostic diagrams ADiagnostic diagrams AVrad Vrad /A/AVV vs. A vs. AU U /A/AVV

fast rotation have a small effect on modefast rotation have a small effect on modestability stability butbut a large effect on visibility a large effect on visibility

there are large differences between modes there are large differences between modes in the light to radial velocity amplitude ratiosin the light to radial velocity amplitude ratios

rotation impairs mode visibility in the light rotation impairs mode visibility in the light butbut not in the mean radial velocity variations not in the mean radial velocity variations

Good prospects for mode identificationGood prospects for mode identification

g-modes with the same g-modes with the same and different m and different mddo not form regular multiplets and they haveo not form regular multiplets and they havedifferent visibility and instability propertiesdifferent visibility and instability properties