1. 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

2. 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.

3. Yarkovsky – O’Keefe – Radzievskii – Paddock Effect

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

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

6. 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

7. 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:

8. 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:

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

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

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

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

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

14. 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?

15. 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

16. 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:

17. Extra Slide 3: Long-Term Obliquity Evolution