asteroid’s thermal models as3141 benda kecil dalam tata surya prodi astronomi 2007/2008 budi...
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Asteroid’s Thermal Models
AS3141 Benda Kecil dalam Tata SuryaProdi Astronomi 2007/2008
Budi Dermawan
Thermal Infrared Radiation (1)
Direct information about the asteroid’s size
Ex. of thermal energy dist.
Delbó 2004
Thermal Infrared Radiation (2)
Thermal energy dist. emission of a black body
),()(Δ
)(2 effp TBA
F
Ap is the emitting area projected along the line-of-sight is the distance of the observer() is the material emissivity (common practice = 0.9 for = 5 – 20 m)
Sampling at several infrared wavelengths i , i = [1…N]
A solution (Ap & Teff) can be found by a non-linear least square fit(e.g. Levenberg-Marquardt algorithm: accuracies of ~10% in the effective diameter and 20 K in surface temperature)
Asteroid Surface Temperature (1)
• Temperature of a surface element: distance from the Sun, albedo, emissivity, angle of inclination to the solar direction
• Total incoming energy (incident):dS
r
SdU i
20
is the direction cosine of the normal to the surface with respect to to solar direction; S0 is the solar constant; r is the heliocentric distance
• Absorption (Ua) and emission (Ue) energies:
)1( AdUdU ia dSTdU e4
Asteroid Surface Temperature (2) Conservation of energy implies dUa = dUe For a surface element at the sub-solar point ( = 1):
2
04
)1(
r
ASTSS
Delbó 2004Delbó 2004
Equilibrium Model (EM)
Distribution of surface temperature(sphere: = cos ; is the solar colatitude)
otherwise 0)(2for cos)(
)( cos)1(
)( )1(
41
42
0
42
0
T
TTT
r
AS
Tr
AS
SS
Emitted Thermal Infrared Flux
Numerically integrating the contribution of each surface element visible to the observer
Evaluating on a “reference” asteroid (emitting projected area = /4 km2)
reference
dTBFreferenceΠ
2),(
Δ
)()(
Direct relationship between the asteroid effective diameter and the measured infrared flux
)()(2
reference
measuredeff F
FD
Function of pv
Standard Thermal Model (STM)
Assumptions: a spherical shape, instantaneous equilibrium between insolation and thermal emission at each point on the surface
Refined (Lebofsky et al. 1986; Lebofsky & Spencer 1989):
Introducing a beaming parameter (= 0.756) the tendency of the radiation to be “beamed”
towards the Sun Asteroids have infrared phase curves which could be
approximated by a linear function up to phase angles () of about 30
mean phase coefficient E = 0.01 mag/deg
Implementation of STM Guess pv
Given the H value, calculate D from [1]
From [2] obtain A, and with = 0.756 calculate TSS [3]
Calculate the temperature dist. on the surface of sphere [4]
Calculate the model flux [5] Scale the observed flux to zero
degree of [6] Calculate the 2 [7] Change the value of pv
parameter and iterate the algorithm
5
v
101329 H
effp
D[1]
2
04
)1(
r
ASTSS
[2]
204
)1(
r
ASTSS
[3]
41
cos)( SSTT [4]
2
02
2
sincos))(,(2Δ
)(
dTBD
F ii
[5]
5.210)()(
E
iobservedi ff
[6]
[7]2
1
2 )()(
N
i i
ii fF
Fast Rotating Model (FRM)
Also called iso-latitude thermal model For objects which: rotate rapidly, have high surface
thermal inertias (half of the thermal emission originates from the night side)
Assumptions: a perfect sphere, its spin axis is perpendicular to the plane of asteroid-observer-the Sun, a temperature distribution depending only on latitude
FRM Formulas
Consideration: an elementary surface strip around the equator (width d) of the spherical asteroid (radius R)
) 2( 2)1( 242
20 dRTdRr
AS
Conservation of the energies:
The sub-solar maximum temperature:
2
04
)1(
r
ASTSS
The temperature dist. (a function of the latitude only):
41
cos)( SSTT
Implementation of FRM Guess pv
Given the H value, calculate D from [1]
From [2] obtain A, and calculate TSS [3]
Calculate the temperature dist. on the surface of sphere [4]
Calculate the model flux [5] Calculate the 2 [6] Change the value of pv
parameter and iterate the algorithm
FRM does not require any correction to the thermal flux for the phase angle
5
v
101329 H
effp
D[1]
2
04
)1(
r
ASTSS
[2]
2
04
)1(
r
ASTSS
[3]
41
cos)( SSTT [4]
2
0
22
2
cos))(,(Δ
)(
dTBD
F ii
[5]
[6]2
1
2 )()(
N
i i
ii fF
FRM-like Asteroid Model
Surface temperature distribution (depends on the latitude only)
Delbó & Harris 2002
Spectral Energy Distributions (SED) of STM & FRM
At r = 1 AU, = 0.1 AU, = 0, pv = 0.15, DSTM = 1 km, DFRM = 5 km
Delbó & Harris 2002
STM
FRM
Observed Thermal Flux of STM & FRMAt r = , = 0, = 0.9, pv = 0.1, G = 0.15, D = 100 km
Harris & Lagerros 2002
Model Constraint on D and pv
D - pv dependencies for a 10 m flux measurement and Hmax = 10.47 of 433 Eros at lightcurve maximum
Harris & Lagerros 2002
Near-Earth Asteroid Thermal Model (NEATM)
Assumptions: a spherical shape, STM surface temp. dist., is a free parameter
Changing Tss the whole surface temp. dist. is scaled by -1/4
is not set to 0.01 mag/deg. NEAs are often observed at much higher (up to 90)
Require good wavelength sampling. If it is limited, use the default value = 1.2 (Harris 1998). Recently, Delbó et al. (2003) suggest = 1 for < 45 and = 1.5 for > 45
Implementation of NEATM Guess pv
Given the H value, calculate D from [1]
From [2] obtain A, and provide initial guess of -value to calculate TSS [3]
Calculate the temperature dist. on the surface of sphere [4]
Calculate the model flux [5] Calculate the 2 [6] Change the value of pv
parameter and iterate the algorithm
5
v
101329 H
effp
D
[1]
2
04
)1(
r
ASTSS
[2]
204
)1(
r
ASTSS
[3]
2/2/
coscos),( 41
41
π-π
TT SS
[4]
2
0
2
2
22
2
)cos(cos)),(,(Δ
)(
ddTBD
F ii
[5]
[6]2
1
2 )()(
N
i i
ii fF
Thermal Models onSub-solar Temperature
Delbó 2004
Solid line: = 1; dashed line: = 0.756 (STM), dotted-line: = 0.6; dashed- and dotted-line: = (FRM)
Model Fits (1)Solid line: STM, dashed line: FRM, dotted-line: NEATM ( = 1.22); r = 2.696 AU, = 1.873 AU, = 14.3
Harris & Lagerros 2002
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