Modeling the YORP EffectModeling the YORP Effect““Radiative Spin-up and Spin-down of Small Asteroids” – D. RubincamRadiative Spin-up and Spin-down of Small Asteroids” – D. Rubincam

AndAnd

““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 RiethmillerPresented by David Riethmiller26 September 200726 September 2007

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

Asteroids rotate with some angular Asteroids rotate with some angular velocityvelocity

YORP is a radiation “recoil” effect that puts YORP is a radiation “recoil” effect that puts a net torque on the asteroida net torque on the asteroid

We are interested in how the asteroid’s We are interested in how the asteroid’s spin rate evolves with time, due to the spin rate evolves with time, due to the YORP effect.YORP effect.

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

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Directed RadiationDirected Radiation

Observed radiant Observed radiant intensity proportional intensity proportional to cos(to cos(θθ))

Consequence: area Consequence: area element viewed from element viewed from any angle measures any angle measures same radiancesame radiance

θθ

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Surface ModelingSurface Modeling

Surface of asteroid Surface of asteroid modeled as modeled as triangular facets, triangular facets, based on based on Gaussian Random Gaussian Random SphereSphere method method

Each facet Each facet absorbs sunlight absorbs sunlight and reradiates and reradiates immediatelyimmediately

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How to Use the ModelHow to Use the ModelPower deposited on surface element dA:Power deposited on surface element dA:

( )s sSdA F dA r N������������� �

rrss = unit vector from asteroid to sun = unit vector from asteroid to sun

NN = unit vector normal to surface element = unit vector normal to surface element

SS = insolation = insolation

FFSS = incident solar flux = incident solar flux

N

rs

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More How to Use ItMore How to Use It

Assume each surface element in radiative steady Assume each surface element in radiative steady state with respect to insolationstate with respect to insolation– (all energy absorbed gets re-radiated immediately(all energy absorbed gets re-radiated immediately– Surface elements act as blackbodiesSurface elements act as blackbodies

Lambert’s Law gives the “recoil” force per unit area on dA:Lambert’s Law gives the “recoil” force per unit area on dA:

2( )3S

sF

c

Nf r N

��������������������������� �

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The Important PartThe Important Part

If If rr is the vector from the asteroid center of is the vector from the asteroid center of mass to the area element, the torque on mass to the area element, the torque on each area element is given by:each area element is given by:

d dA τ r f

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ObliquityObliquity

Unlikely that the Unlikely that the net torque will net torque will maintain the maintain the asteroid’s spin asteroid’s spin axis orientationaxis orientation

How will torque How will torque components components change the change the orientation over orientation over time?time?

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Obliquity Evolution: Type IObliquity Evolution: Type I

Stable ε at 90° (Eros)

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Obliquity Evolution: Type IIObliquity Evolution: Type II

Stable ε at 0 and 180 (Deimos)

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Obliquity Evolution: Type IIIObliquity Evolution: Type III

Stable ε at ~35 and 150 (Geographos)

1313

Obliquity Evolution: Type IVObliquity Evolution: Type IV

Stable ε at 90 and 180 (Golevka)

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ConclusionsConclusions

YORP is a radiation YORP is a radiation “recoil” effect that puts a “recoil” effect that puts a net torque on an asteroid.net torque on an asteroid.

This net torque changes This net torque changes the spin axis orientation the spin axis orientation (obliquity) and rotation (obliquity) and rotation period of the asteroid period of the asteroid over time.over time.

Asteroid shape influences Asteroid shape influences its evolution type, its evolution type, characterized by points of characterized by points of stable obliquity.stable obliquity. Got ‘roids?

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Extra Slide 1: Spherical Harmonic Extra Slide 1: Spherical Harmonic Surface ModelingSurface Modeling

The models used are spherical harmonic The models used are spherical harmonic expansions on known asteroid shapes:expansions on known asteroid shapes:

max

1

( , )

( , ) (cos )cos

l

lmi lmilmi

lm lm

r A Y

Y P m

PPlmlm(cos (cos θθ) = Legendre polynomial of degree ) = Legendre polynomial of degree ll and order and order mm

θθ = co-latitude = co-latitude

λλ = east longitude = east longitude

AAlmilmi = shape coefficients derived from inner products of = shape coefficients derived from inner products of

spherical harmonics with numerical shapesspherical harmonics with numerical shapes

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Extra Slide 2: Gaussian Random Extra Slide 2: Gaussian Random SpheresSpheres

Original model called for spherical harmonic expansion on known Original model called for spherical harmonic expansion on known asteroid shapesasteroid shapesImproved model uses Gaussian Random SpheresImproved model uses Gaussian Random Spheres

The radii of a large sample of objects satisfy log-normal statistics The radii of a large sample of objects satisfy log-normal statistics with a with a variance variance σσ and a and a characteristic dimensional factor characteristic dimensional factor aa, and the , and the the the radius radius rr in the direction given by in the direction given by spherical angles spherical angles θθ and and φφ may may be expressed as:be expressed as:

2

( , ) exp ,1

ar s

0 0

( , ) (cos )( cos sin )l

ml lm lm

l m

s P a m b m

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Extra Slide 3: Long-Term Obliquity Extra Slide 3: Long-Term Obliquity EvolutionEvolution