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Asterosismologia Riccardo U. Claudi INAF Astronomical Observatory of Padova.
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Transcript of Asterosismologia Riccardo U. Claudi INAF Astronomical Observatory of Padova.
![Page 1: Asterosismologia Riccardo U. Claudi INAF Astronomical Observatory of Padova.](https://reader036.fdocuments.net/reader036/viewer/2022062701/5542eb4b497959361e8b819a/html5/thumbnails/1.jpg)
Asterosismologia
Riccardo U. Claudi
INAF Astronomical Observatory of Padova
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Asterosismologia
Sir Arthur Eddington (1882 – 1944)
„At first sight it would seem that
the deep interior of the sunand stars is less accessible
to scientific investigation thanany other region of the
universe.”
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Asterosismologia
Stelle Pulsanti nel diagramma HR
Un buon articolo di Review:
Gautschy & Saio 1996
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Asterosismologia
Cosa sono le oscillazioni di tipo solare?
•Dipendono fortemente dalle proprietà della stella
•Oscillazioni smorzate linearmente
•Eccitate in modo stocastico
Presenti anche in stelle non di tipo solare
( ) 1/ 2
3dyn
GMG
Rω ρ
−= =
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Asterosismologia
Le oscillazioni di tipo solare sono onde sonore stazionarie (modi - p)
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Asterosismologia
Proprietà delle oscillazioni
•ξnlm(r, , , t)= ξnl(r) Ylm(,)e-i nlmt
•Ylm(,)=(-1)m clmPl
m(cos ) cos(m - t)
•kh = 2 / h = [l(l+1)]1/2/r
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Asterosismologia
“Splitting” Rotazionale
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Asterosismologia
Identificazione dei Modi
n, n, , m, m
Per una determinata frequenza
nm
dobbiamo determinare tre numeri
"quantici”:
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Asterosismologia
n – ordine radiale, n=0,1,2,...
l - grado della armonica sferica, l=0,1,2, …
m – ordine azimutale, |m| l
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Asterosismologia
n
l
m
l-|m|
Numero dei nodi nella direzione radiale
Numero totale delle linee nodali sulla superficie
Numero delle linee nodali perpendicolari all’equatore
Numero delle linee nodali parallele all’equatore
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Asterosismologia
C. SchrijversC. Schrijvers
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Asterosismologia
= 1, m=0 = 1, m=1
Tim Bedding
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Asterosismologia
= 2, m=1 = 2, m=2
Tim Bedding
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Asterosismologia
= 3, m=0 = 3, m=1
= 3, m=2 = 3, m=3
Tim Bedding
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Asterosismologia
= 5, m=0 = 5, m=2
= 5, m=3
Tim Bedding
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Asterosismologia
= 8, m=1 = 8, m=2
= 8, m=3
Tim Bedding
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Asterosismologia
Dove è partito tutto…
Grec et al., 1980, Nature 288, 541
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Asterosismologia
Oscillazioni dei modi P Solari Osservate
Frequenze misurate da MDI su SOHO
Barra d’errore: 1000 σ
n=1
(Rodhes et al., 1997)
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Asterosismologia
Dal Sole alle stelle di tipo solare. I
Bassi valori del grado orizzontale: l3
MA
Alti valori del numero radiale: n
Descrizione asintotica dei modi p
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Asterosismologia
Relazione di dispersione delle onde acustiche
Quindi
Quando kr = 0 si ha il turning point rt:
Teoria asintotica: Frequenze
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Asterosismologia
Raggi
l=0
l=2
l=20
l=25
l=75
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Asterosismologia
2nl nl
lnυ υ α ε⎛ ⎞≈Δ + + +⎜ ⎟
⎝ ⎠
Grande separazione:
Teoria asintotica: modi p
1
, 1,
0
2R
n l n l
dr
cν ν ν
−
−
⎡ ⎤Δ = ≈ −⎢ ⎥
⎣ ⎦∫
( ), 2, 0
14
r
n ln l
dc drl l
dr r
νε
π ν
Δ≈ + ∫
Piccola separazione:
Tassoul, 1980
n-2,2
n-1,0
n,0
n-2,3
n-1,1
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Asterosismologia
Δn e n misurano rispettivamente la densità e la
composizione del core della stella.
In altre parole la massa e l’età della stella.
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Asterosismologia
/ Osc obs
obs
A NS N
σ=
1) Basso SNR:
Principali difficoltà nella determinazione dei modi p
2) “Aliasing”: Splitting of frequencies in side bands
- Day/night alternation
- Single Observing Site
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Asterosismologia
Dal Sole alle stelle di tipo solare. II
Piccola ampiezza dei modi p
Aph=4 ppm
Avr=0.23 ms-1
( )( )( ) ( )2
/ 4.7 0.3
/ 550 / 5777 /
SUN
eff SUN
L L ppmL
L nm T K M Mλ
δ
λ
±⎛ ⎞ =⎜ ⎟⎝ ⎠
( )( ) 1/23.4 1.4
/SUN
oscSUN
L Lcm s
M Mυ −= ±
Sole Stelle di tipo solare
Kjeldsen & Bedding, 1995
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Asterosismologia
Dal Sole alle stelle di tipo solare. IIIK
jeld
sen &
Beddin
g,
199
5
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Asterosismologia
Come misurare le pulsazioni stellari?
Variazioni radiali
Variazioni VR Variazioni L*
Serie temporali
Analisi di Fourier
FREQUENZE !
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Descrizione tecnica:La velocità radiale è la componente del moto di una porzione di superficie della stella nelladirezione dell’osservatore. La sua misura avviene osservando l’effetto Doppler sullo spettro della stella.
Difficoltà:A ~ 1000 m/s BinarieA ~ 10 m/s per pianeti con massa simile a GioveA ~ 0.30 m/s per pulsazioni di tipo solareA ~ 0.1 m/s per pianeti con massa simile alla Terra
Velocità Radiali
€
V
c=
λ − λ 0( )λ 0
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Le velocità vengono misurate confrontando la posizione delle righe spettrali della stella rispetto a quella misurata in laboratorioProblema:
Piccoli spostamenti dell’immagine della stella sulla fenditura di ingresso dello spettrografo possono causare errori importanti nelle misureSoluzioni:a)“Scrambling dell’immagine” usando fibre ottiche: metodo usato dal gruppo svizzero di Mayor (ELODIE, HARPS): precisione circa 10 m/s (1995 – 2002) fino a 1 m/s (2003)b) Sovrapposizione di righe dovute ad un gas a riposo rispetto all’osservatore (cella assorbente, in genere allo iodio); metodo usato da altri gruppi (Marcy & Butler, Texas, ESO, SARG):precisione circa 2-3 m/s, in funzione del software usato
Misure di Velocità Radiali ad alta precisione
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Asterosismologia
Misure di velocità radiale con la cella assorbente
La cella allo iodio del SARG
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Asterosismologia
Gratton et al. (2000)
SARG@TNG
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Asterosismologia
Spettri del SARG con la cella assorbente allo I2
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Asterosismologia
Riduzione dei DATI I2 FTS SPECTRUM
PSF
DECONVOLUTION
DOPPLER
Iobs()=K[TI2() IS(+Δ)]*PSF
STAR + IODINE 2
Vr
B STAR + IODINE
STAR
STELLAR TEMPLATE
AUSTRAL code by Endl M. et al., 2001
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Dove:VRMS = errore nella variazione di velocità radialeQ = Fattore di qualità dello spettro Ne- = Numero totale dei fotoni rilevati nell’intervallo spettrale
Ne- = F* Stel εtot texp /2.512V
Dove:F*=photons/cm2s per una stella V=0Stel= Area del telescopio (cm2)εtot = Efficienza totaletexp = Tempo di esposizioneV = Magnitudine visuale
Limite “Photon noise” per la misura delle velocità radiali
from Bouchy et al. 2001, A&A, 374, 733
VRMS = c /(Q Ne-)
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AsterosismologiaBouchy et al. (2001)
Errore sulla Velocità radiale
exp
2.512 vm
RMS
STAR Tel Tot
cV
tQ F A Eff =
⋅ ⋅
Fattore di qualità
Risoluzione
Lunghezza d’onda
Rotazione
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Accuratezza della Velocita’ Radiale (HARPS)
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Asterosismologia Claudi et al. (2005)
SARG and Solar like Stars
SARG Resolution:144,000
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Asterosismologia
Programmi sulle velocità radiali di alta precisioneFIBRE
- Coralie ed Euler Telescopes (Svizzera, numerosi pianeti)
- Elodie (Svizzera-Francia, numerosi pianeti) - Advanced Fibre-Optic Echelle (USA)-Spectrashift (USA, astrofili, 1 pianeta)-HARPS (ESO dal 2003)
CELLA- Lick e Anglo-Australian Planet Search Programs (USA e Australia, numerosi pianeti)- Extrasolar Planets Discovery (San Francisco, numerosi pianeti)- ESO Coudè Echelle Spectrometer (ESO, 1 pianeta)- McDonald Observatory (USA, numerosi pianeti)
- SARG (Italia, un candidato pianeta)
ALTRO- Fringing Spectrometers for Planet Search (USA, test in laboratorio)- Absolute Astronomical Accelerometry (Francia, in costruzione)
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•Fourier Transforms•Wavelet Analysis•Autocorrelation analysis•Other methods
Metodi Numerici per l’analisi Metodi Numerici per l’analisi delle Serie Temporalidelle Serie Temporali
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L’analisi di Fourier tenta di fare il fit della serie temporale con una serie di funzioni seno ciascuna con un differente periodo, ampiezza e fase.Gli algoritmi che fanno questo eseguono Una trasformazione matematica dal dominio temporale al dominio dei periodi (o delle frequenze.
f (time) F (period)
Analisi di FourierAnalisi di Fourier
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Per una data frequenza ν (=1/period)La trasformata di Fourier é data da:
F (ν) = ∫ f(t) exp(i2νt) dt
Si ricordi la formula di Eulero:exp(ix) = cos(x) + isin(x)
La Trasformata di FourierLa Trasformata di Fourier
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Fourier AlgorithmsFourier Algorithms
Discrete Fourier Transform: the classic algorithm (DFT)Fast Fourier Transform: very good for lots of evenly-spaced data (FFT)Date-Compensated DFT: unevenly sampled data with lots of gaps (TS)Periodogram (Lomb-Scargle): similar to DFT
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Asterosismologia
Bedding & Kjeldsen (2003)
Alcuni pulsatori di tipo solare
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Asterosismologia
Stelle con molte frequenze individuate:
• Il Sole (G2 V) – continuous • α Cen A (G2 V) – dual-site • α Cen B (K1 V) – dual-site • Hyi (G2 IV) – dual-site • Boo (G0 IV) – several single-site• Ara (G3 V) – single-site (HARPS)• Vir (F9 V) – single-site (CORALIE)
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Asterosismologia
Solo la grande separazione:
υ Ind (G0 IV, metal-poor) – dual-site (UCLES & CORALIE)
• Procyon (F5 IV) – many, mostly single-site• HD 49933 (F5 V, COROT target) – single-site (HARPS)
• Hya (G giant) – single-site (CORALIE)
Solo eccesso di potenza:• 70 Oph A (K0 V) – single-site (CORALIE)
• e Oph , h Ser (G giants) – dual-site (CORALIE & ELODIE)
• d Pav, g Ser, b Aql – short segments (HARPS, UVES)
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Asterosismologia
Procyon Aα CMi; HR 2943; HD61421
F5 IV-V
Mv=0.363
d= 3.53 pc
M=(1.42 0.06) MSUN
R=(2.071 0.02) RSUN
Prediction (Kjeldsen &Bedding 1995):
Vosc= 1.11 m s-1
(L/L)V= 18 ppm
νMAX=1.0 mHz
Δν=54 Hz
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Asterosismologia
Serie Temporale di Velocità Radiale
int. err. =1.38 m/s
r.m.s. =4.48 m/s
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Asterosismologia
Grande Separazione
CR(Δν)=PS(νmax-1/2 Δν)PS(νmax+1/2 Δν)PS (νmax-Δν)
PS (νmax+Δν)[PS (νmax-3/2 Δν) PS(νmax+3/2 Δν)
PS (νmax-2 Δν)PS(νmax+2 Δν)]0.5
Δν= 56 ± 1 Hz <Δν>= 55.7 ± 1.4 Hz
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Asterosismologia
Ara: G3V planet-hosting star
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Asterosismologia
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Asterosismologia
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Asterosismologia
Ara (Bouchy et al. 2005)