SUJAN SENGUPTA
Indian Institute of Astrophysics, Bangalore
Chandrasekhar Centenary Conference
December 07-11, 2010
Multiple Scattering Polarization -
Application of Chandrasekhar’s Formalisms to the Atmosphere of
Brown Dwarfs and Extra-solar Planets
),(4
0
/ 0
pFe
1
1
),,(),(2
),,(),,(
dIpI
d
dI
d = dr, = a + s
),,(),,(),,(), ,(1
),,( 2
rIrrjrI
rr
rI
1
1
),,(),(2
),,(),,(
d
d
dIpI
I
V
U
Q
I
I
The state of polarized light is expressed by the Stokes parameters I, Q, U and V where I is the total intensity and Q, U, V are the polarized intensity. For linear polarization V=0.
jIds
dI
IVUQp /222
Scattering Polarization
Rayleigh Scattering : Scattering by atoms and molecules
Thomson Scattering : Scattering by free non-relativistic
electrons
Mie Scattering : Scattering by dust
)]'cos()'1()1('2
)'(cos)'1)(1('1[4
3)',';,(
2/122/12
22222
p
Phase function/matrix for Thomson and Rayleigh scattering
1'
')'1)(1(2
4
3)',(
2
22222
P )(μ,μ',gh
Polarization is the measure of anisotropy in a radiation field
Mass – less than 0.084 M ,
greater than 0.011 M
SUB-STELLAR MASS OBJECTS
BROWN DWARFS
EXTRA-SOLAR PLANETS
Mass – less than 0.011 M
Credit : NASA (1995)
Credit : C. Chauvin et al (2004) Teff 1400 K
L dwarfs :
1300 K < Teff < 2400 K
Signature of primordial lithium
T dwarfs :
600 K < Teff < 1300 K
Signature of methane
Credit : B. Openheimer et al. (1995)Credit : D. Kirkpatrick et al. (2000)Credit : M. C. Cushing et al. (2009)
Sengupta and Krishan, 2001, ApJ Letts, 561, L123
Menard et al. 2002, A & A Letts. 396, L35
Sengupta, 2003, ApJ Letts, 585, L155
Sengupta and Kwok, 2005, ApJ, 625, 996
Polarization of Dusty Brown DwarfsSingle Scattering
Credit : S. Sengupta and S. Kwok (2006)
Rotation Induced Oblatenesse
pe
R
RR
Darwin (1899) – Radau (1885) Relationship :
12
2
32
2
31
2
5
5
2
MR
I
GM
Re
Chandrasekhar, S. 1933, MNRAS, “The quilibrium of distorted
polytropes IV. The rotational and the tidal distortions as functions
of the density distribution”, Vol. 93, p-553, Table 1.
GM
RC e
32
3
2
12
22
31
2
5
5
2
2
3
MR
IC
DARWIN-RADAU-CHANDRASEKHAR RELATIONSHIP
Credit : S. Sengupta and M. S. Marley (2010)
Polarization of Dusty Brown DwarfsMultiple Scattering : Application of Chandrasekhar’s equations
The World Is Not Enough
8 planets within the solar system
51 Pegasi : The first confirmed exoplanet reported in 1995
PSR 1257+12 : a system of planets discovered in 1992
Radial Velocity
Transit
Imaging
Microlensing
Timing
506 planets (http://exoplanet.eu :Dec. 09, 2010)
Credit : F. Allard (2008)Credit : S. Berdyugina (2008)
Credit : Chauvin et al (2004)
2MASS 1207 b
Credit : Marois et al. (2008)
HR 8799
Credit : Marois et al. (2010)Credit : Lagrange et al. (2010)
b Pictoris b
Credit : S. Sengupta and M. S. Marley (2010)
Credit : NASA
Polarization by a Partially Cloudy Atmosphere ??
Credit : JPL, NASA (Spitzer)