Thermal Observations of Small Bodies

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1 Thermal Observations of Small Bodies 3 Physics of Thermal Models and Interpretation of Data 19 th Advanced School in Astrophysics Observatório Nacional, Rio d Janeiro November 6, 2014

2 ! Simple models sizes! Radia/ve equilibrium! NEATM beaming parameter! Deriving albedo! Thermophysical Model (TPMs)! Theory! Finite Difference implementa/on! Surface Roughness! Non- spherical shape! Other complica/ons

3 Bennu at 45 µm from Spitzer! Want to derive size! If asteroid were uniform T and spherical: R radius Δ obs distance ε λ emissivity B Planck function! Know Δ, assume ε (silicates typically in thermal- IR Vary R and T to fit data

4 ! However, we know surfaces are not uniform T Energy balance at surface S o solar constant (1367 W m -2 ) A B Bond albedo r AU heliocentric distance [AU] θ i incidence angle σ Stefan-Boltzmann const

5 ! Now integrate over visible hemisphere for total flux! Or, for a sphere θ e emission angle! There are no free parameters in this temperature calc Equilibrium Model

6 ! Fluxes measured at low phase angles higher than EQM! ie, higher temperatures " Beaming parameter ar/ficial fudge factor to fit data Beaming parameter! Lower η = higher T! S/ll get flux from:! Vary R and η to find good fit to data! NEATM Near- Earth Asteroid Thermal model! Valid for any object, not just NEAs

7 ! Have size, what about albedo?! Visible flux: p v geometric albedo (light reflected at zero phase, relative to that of a perfectly diffuse (Lambert) sphere) Φ phase correction! Generally, don t have simultaneous visible measurement! Predict from tabulated absolute magnitude H v Absolute magnitude (brightness at r=1au, Δ=1AU, α=0; ie, observed at 1AU from the Sun)! Combining these two equa/ons, find:! We ve encountered 2 albedos: A B and p v! A B = p v q where q is the phase integral (q~039 is a good guess)

8 ! How realis/c is this model?! Good for sizes and albedos as long as asteroid is:! Observed at low phase angle (NEATM no nightside emission)! Somewhat spherical Γ = 0 J m -2 K -1 s -1/2 Γ = 150! However, if interested in finer precision, thermal iner/a, higher phase angles Need to consider explicitly! Heat conduc/on & storage! Surface roughness! Shape

9 ! Heat conducted away from surface when hot! Heat stored in subsurface! Heat conducted back to surface when cool Addi/onal source/sink in energy balance at surface! How do we compute temperature gradient at surface?! Fully consider heat diffusion! Compute temperatures at depth and as a func/on of /me Note thermal inertia:! 2 nd boundary condi/on

10 ! Cannot solve this heat diffusion problem analy/cally! Finite difference Becomes: j time n space Define and rearrange to group /me terms:! This defines a set of linear equa/ons! If number of equa/ons = number of unknowns, can solve

11 ! Example of linear equa/ons Depth step 1 Known ! Surface boundary condi/on Excess flux:! Iterate un/l average excess flux over rota/on is constant (zero)

12 ! Now, change Γ and R un/l find a good fit to data! Forward Modeling! Temperature dist Flux dist Match disk- integrated flux Γ = 100 J m -2 K -1 s -1/2 Γ = 1500 J m -2 K -1 s -1/2

13 ! r = 1 AU! P rot = 24 hrs! ρ, c p, k for typical silicates

14 ! r = 1 AU! P rot = 24 hrs! ρ, c p, k for typical silicates! Thermal skin depth ~15 cm

15 ! r = 1 AU! P rot = 24 hrs! ρ, c p, k for typical silicates! Thermal skin depth ~15 cm

16 ! Surfaces are not perfectly smooth! Shadowing, self- hea/ng, mul/ple scamering! Extra terms in surface energy balance equa/on:

17 ! Spherical- sec/on craters easiest & as accurate as others! Analy/c sol ns for E ref and E self! Increased emission at low phase angle beaming! Addi/onal parameter fit (roughness)! Need 2 phase angles to break degeneracy with thermal iner/a

18 ! Symmetry of spheres provides model efficiency! But many objects not really close to spherical! Shape models generally given as x, y, z coordinates of vectors that combine to make facets! Same heat diffusion solu/on as before! Large- scale shape effects! One /me mapping of facets rela/ve to each other

19 ! Example (101955) Bennu! Spherical shape Γ= 600 J m - 2 K - 1 s - 1/2! Radar shape Γ= 310 J m - 2 K - 1 s - 1/2! Oblateness affects regional /lts

20 ! In principle, follow same surface spot around rota/on! Difficult with flyby missions

21 ! Temperature and depth dependent k and c p! eg, radia/on between grains: k = a + bt 3! Similar finite difference approach! Track changes in parameters with temperature/depth Vesta! 3D heat transfer! Only important if looking at small scales (few cm)

22 18 km 535 m Itokawa: TI~750 (Müller et al 2005) YU55: TI~600 (Müller et al 2013) Bennu: TI~310 (Emery et al 2014) Eros: TI~150 (Müller et al 2007)

23 ! Interpret data through forward modeling! Vary relevant parameters to find best fit! NEATM simple radia/ve equilibrium! Reliable sizes and albedos at small phase angles! No physical interpreta/on of temperature distribu/on! Thermophysical model (TPM) Heat diffusion! Use to determine physical property (Γ) of surface! Can consider/constrain surface roughness! Also relevant for disk- resolved observa/ons! Either type of model can incorporate non- spherical shapes! T- dependent parameters and lateral heat flow! 2 nd order effects generally can ignore! Next lecture, we ll talk about current and future direc/ons in thermal observa/ons of small bodies

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