Modeling the YORP Effect “Radiative Spin-up and Spin-down of Small Asteroids” – D. Rubincam And “YORP-Induced Long-Term Evolution of the Spin State of Small Asteroids and Meteoroids: Rubincam’s Approximation” - D. Vokrouhlicky and D. Capek Presented by David Riethmiller 26 September 2007

Yarkovsky – O’Keefe – Radzievskii – Paddock Effect Asteroids rotate with some angular velocity YORP is a radiation “recoil” effect that puts a net torque on the asteroid We are interested in how the asteroid’s spin rate evolves with time, due to the YORP effect.

Yarkovsky – O’Keefe – Radzievskii – Paddock Effect

Directed Radiation Observed radiant intensity proportional to cos(θ) Consequence: area element viewed from any angle measures same radiance θ

Surface Modeling Surface of asteroid modeled as triangular facets, based on Gaussian Random Sphere method Each facet absorbs sunlight and reradiates immediately

How to Use the Model Power deposited on surface element dA: rs = unit vector from asteroid to sun N = unit vector normal to surface element S = insolation FS = incident solar flux N rs

More How to Use It Assume each surface element in radiative steady state with respect to insolation (all energy absorbed gets re-radiated immediately Surface elements act as blackbodies Lambert’s Law gives the “recoil” force per unit area on dA:

The Important Part If r is the vector from the asteroid center of mass to the area element, the torque on each area element is given by:

Obliquity Unlikely that the net torque will maintain the asteroid’s spin axis orientation How will torque components change the orientation over time?

Obliquity Evolution: Type I Stable ε at 90° (Eros)

Obliquity Evolution: Type II Stable ε at 0 and 180 (Deimos)

Obliquity Evolution: Type III Stable ε at ~35 and 150 (Geographos)

Obliquity Evolution: Type IV Stable ε at 90 and 180 (Golevka)

Conclusions YORP is a radiation “recoil” effect that puts a net torque on an asteroid. This net torque changes the spin axis orientation (obliquity) and rotation period of the asteroid over time. Asteroid shape influences its evolution type, characterized by points of stable obliquity. Got ‘roids?

Extra Slide 1: Spherical Harmonic Surface Modeling The models used are spherical harmonic expansions on known asteroid shapes: Plm(cos θ) = Legendre polynomial of degree l and order m θ = co-latitude λ = east longitude Almi = shape coefficients derived from inner products of spherical harmonics with numerical shapes

Extra Slide 2: Gaussian Random Spheres Original model called for spherical harmonic expansion on known asteroid shapes Improved model uses Gaussian Random Spheres The radii of a large sample of objects satisfy log-normal statistics with a variance σ and a characteristic dimensional factor a, and the the radius r in the direction given by spherical angles θ and φ may be expressed as:

Extra Slide 3: Long-Term Obliquity Evolution