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

Constraints on Diameter & Albedo

From (visible) absolute magnitude H

5

v

101329 H

effp

D

Delbó 2004

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

STM-like Asteroid Model

Surface temperature distribution

Delbó & Harris 2002

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

Model Fits (2)

Solid line: STM, dashed line: FRM, dotted-line: NEATM

Delbó 2004

Model Fits (3)

Solid line: STM, dashed line: FRM, dotted-line: NEATM

Delbó 2004

HistogramDelbó 2004

Radiometric Results (1)Delbó 2004

Radiometric Results (2)Delbó 2004

Radiometric Results (3)Delbó 2004