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

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

3. 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)

4. 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:

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

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

7. 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

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

9. 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

10. 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]

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

12. 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

13. 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):

14. 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]

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

16. 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

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

18. 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

19. 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 

20. 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]

21. 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)

22. 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

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

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

25. Histogram Delbó 2004

26. Radiometric Results (1) Delbó 2004

27. Radiometric Results (2) Delbó 2004

28. Radiometric Results (3) Delbó 2004