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Asteroid’s Thermal Models AS3141 Benda Kecil dalam Tata Surya Prodi Astronomi 2007/2008 Budi Dermawan

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Thermal Infrared Radiation (1) Direct information about the asteroid’s size Ex. of thermal energy dist. Delbó 2004

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Thermal Infrared Radiation (2) Thermal energy dist. emission of a black body A p 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 ( A p & T eff ) 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)

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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): is the direction cosine of the normal to the surface with respect to to solar direction; S 0 is the solar constant; r is the heliocentric distance Absorption ( U a ) and emission ( U e ) energies:

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Asteroid Surface Temperature (2) Conservation of energy implies dU a = dU e For a surface element at the sub-solar point ( = 1 ): Delbó 2004

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Equilibrium Model (EM) Distribution of surface temperature (sphere: = cos ; is the solar colatitude)

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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 km 2 ) Direct relationship between the asteroid effective diameter and the measured infrared flux Function of p v

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Constraints on Diameter & Albedo From (visible) absolute magnitude H Delbó 2004

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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

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Implementation of STM Guess p v Given the H value, calculate D from [1] From [2] obtain A, and with = 0.756 calculate T SS [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 p v parameter and iterate the algorithm [1] [2] [3] [4] [5] [6] [7]

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STM-like Asteroid Model Surface temperature distribution Delbó & Harris 2002

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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

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FRM Formulas Consideration: an elementary surface strip around the equator (width d ) of the spherical asteroid (radius R ) Conservation of the energies: The sub-solar maximum temperature: The temperature dist. (a function of the latitude only):

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Implementation of FRM Guess p v Given the H value, calculate D from [1] From [2] obtain A, and calculate T SS [3] Calculate the temperature dist. on the surface of sphere [4] Calculate the model flux [5] Calculate the 2 [6] Change the value of p v parameter and iterate the algorithm FRM does not require any correction to the thermal flux for the phase angle [1] [2] [3] [4] [5] [6]

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FRM-like Asteroid Model Surface temperature distribution (depends on the latitude only) Delbó & Harris 2002

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Spectral Energy Distributions (SED) of STM & FRM At r = 1 AU, = 0.1 AU, = 0 , p v = 0.15, D STM = 1 km, D FRM = 5 km Delbó & Harris 2002 STM FRM

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Observed Thermal Flux of STM & FRM At r = , = 0 , = 0.9, p v = 0.1, G = 0.15, D = 100 km Harris & Lagerros 2002

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Model Constraint on D and p v D - p v dependencies for a 10 m flux measurement and H max = 10.47 of 433 Eros at lightcurve maximum Harris & Lagerros 2002

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Near-Earth Asteroid Thermal Model (NEATM) Assumptions: a spherical shape, STM surface temp. dist., is a free parameter Changing T ss 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

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Implementation of NEATM Guess p v Given the H value, calculate D from [1] From [2] obtain A, and provide initial guess of -value to calculate T SS [3] Calculate the temperature dist. on the surface of sphere [4] Calculate the model flux [5] Calculate the 2 [6] Change the value of p v parameter and iterate the algorithm [1] [2] [3] [4] [5] [6]

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Thermal Models on Sub-solar Temperature Delbó 2004 Solid line: = 1 ; dashed line: = 0.756 (STM), dotted- line: = 0.6 ; dashed- and dotted-line: = (FRM)

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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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Model Fits (2) Solid line: STM, dashed line: FRM, dotted-line: NEATM Delbó 2004

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Model Fits (3) Solid line: STM, dashed line: FRM, dotted-line: NEATM Delbó 2004

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Histogram Delbó 2004

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Radiometric Results (1) Delbó 2004

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Radiometric Results (2) Delbó 2004

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Radiometric Results (3) Delbó 2004

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