dynamics of asteroids classical theory the yarkovsky and yorp effects

Dynamics of asteroids

Classical theory

The Yarkovsky and YORP Effects

Classical model of asteroid dynamics

Collisions and gravitation perturbations dominate evolutionary processes

=> Should explain asteroid and meteorite delivery to the Earh – moon system (NEO) or Mars

How does the main belt resupply the NEO population ?

Injection into chaotic resonances

This graph was created in June 2007 using all asteroids with "well-determined" orbits (specifically, 156929 numbered asteroids)

Problems with the classical model

- Direction injection into chaotic resonances imply very short dynamical lifetime (a few millions of years).

Observations indicate CRE (Cosmic ray exposures) of 10 – 100 My for stony meteorites and even 0.1 – 1 Gy for iron bodies. Relatively few meteorites are found to have CRE ages shorter than a few millions years.

- There are roughly 5000 – 6000 kilometer-sized asteroids in the Mars-crossing and NEO populations. These bodies have a wide range of taxonomic types.=> Frequent disruption events are required among spectrally diverse asteroid to keep this population in a steady state.

Observations – Most potential parent asteroids for the km-sized inner solar system asteroids reside in dynamically stable regions far from resonant espace hatches.

Problems with the classical model

- Planet crossing asteroids are “fresh ejecta”, they should have a size-frequency distribution that has a steep power law.

Observations – the size – frequency distribution of km-sized NEO is shallow.

- Asteroids collisions should produce a wide range of asteroid spin rates.

Observations – The distribution of spin rates among observed asteroids (D < 10 km) contains an excess number of fast rotators and very slow rotators when this data is fit to Maxwellian distribution.

Histogram of f / f for 367 asteroids with D > 40 km. The dashed curve is ⟨ ⟩the corresponding Maxwellian distribution (Pravec et al., 2002)

Fast and Slow Rotation of Asteroids

Histogramof f/ f for225asteroidswith10km<D≤40km. The dashed curve is the ⟨ ⟩Maxwellian distribution for the same number of objects (Pravec et al., 2002)

Fast and Slow Rotation of Asteroids

Histogram of f / f for 156 asteroids with D ≤ 10 km. The dashed curve is the ⟨ ⟩Maxwellian distribution for the same number of objects. The short- dashed lines are the histogram for near-Earth asteroids only.(Pravec et al., 2002)

Fast and Slow Rotation of Asteroids

The Yarkovsky effectA little bit of history

Ivan Osipovich Yarkosky (1844 – 1902) – Civil engineer who worked on scientific problems in his spare time (!) first proposed the effect that now bears his name

In a pamphlet around 1900, Yarkovsky noted that the diurnal heating of a rotating object in space would cause it to experience a force that, while tiny, could lead to large secular effects in the orbits of small bodies.

Enst J. Öpik read Yarkovsky’s pamphlet around 1909 and discussed the importance of the Yarkovsky effect for moving meteoroids about the Solar System.

Öpik E.J. 1976 – Interplanetary encounters (Elsevier) – No mention of the original pamphlet !

The Yarkovsky effectDescription of the diurnal component

Axis of orientation perpendicular to the plane of rotation.

A fraction of the solar insolation is absorbed only to later be radiated away, yielding a net thermal force in the direction of the wide arrows. Because thermal reradiation in this example is concentrated at about 2 PM on the spinning asteroid, the radiation recoil force is always oriented at about 2 AM. Thus, the along-track component causes the object to spiral outward.

Retrograde rotation would cause the orbit to spiral inward.

The Yarkovsky effectDescription of the seasonal component

Axis of orientation parallel to the plane of rotation.

The seasonal Yarkovsky effect, with the asteroid’s spin axis in the orbital plane. Seasonal heating and cooling of the “northern” and “southern” hemispheres give rise to a thermal force, which lies along the spin axis.

The strength of the reradiation force varies along the orbit as a result of thermal inertia; even though the maximum sunlight on each hemisphere occurs as A and C, the maximum resultant radiative forces are applied to the body at B and D.

The net effect over one revolution always causes the object to spiral inward.

The Yarkovsky effectTheory and equations

The problem involves two fundamental parameters :

- The penetration wave of the thermal wave

Note – Equivalent expression with the thermal skin depth and thermal inertia :

- The thermal parameter

n : the frequency of the insulation function E (diurnal or seasonal)

T* : subsolar temperature defined by esT*4 = aE

A measure of the relaxation between the absorption and reemission at frequency ν

The Yarkovsky effectTheory and equations

Hear diffusion equation inside the body :

Boundary condition at the surface (radiative) :

A : Bond albedo

r : material densityK : thermal conductivityCp : specific heat at constant pressure

Photon’s momentum : p = E/c

The Yarkosvsky force is given by :

The Yarkovsky effectTheory and equations

Radiation pressure coefficient

R’ : radius of the body scaled to the thermal penetration wave

The Yarkovsky effectPredictions

- Obliquity and rotation dependence:F-functions are always negative for seasonal Yarkovsky effect => always produces a net decrease in a.Diurnal Yarkovsky effect becomes negligible in the limit of fast rotation and zero rotation

- Size dependence:The Yarkovsky effect vanishes for small and very large objectsLarge object – surface/mass is smallSmall object – thermal penetrates the entire body (for dust)Effect is maximum for objects of a few centimers

- Surface conductivity dependence:Maximum da/dt rates occur when the 2 fundamental parameters are close to 1

At higher conductivity => thermal wave penetrates through the entire body, and low conductivity (no thermal inertia) the Yarkovsky effect disappears.

The Yarkovsky effectSimulations

Mean drift rate of asteroids in the inner main belt over 1 My produced by the diurnal and seasonal Yarkovsky effects. Surface conductivity K values are 0.001, 0.01, 0.1, and 1 W m−1 K−1.

Rotation period P = 5 (D/2), where P is the rotation period in seconds and D the diameter in meters.

The low-K cases are dominated by the diurnal effect, whereas for high-K cases the seasonal effect is more important. Mobility decreases for small bodies with high K because the thermal wave penetrates throughout the body.

The Yarkovsky effectSimulations

Mean change in semimajor axis over the estimated collisional lifetimes of the bodies (Bottke et al. 2005b). Collisions are also assumed to reorient the spin vector of the bodies.

The YORP effectYarkovsky–O'Keefe–Radzievskii–Paddack effect

Term proposed by D. P. Rubincam in 2000

Affect the spin rate

Affect the obliquity

Indirect evidences on this effect until ~2007

Measurement of The YORP effectDirect measurement for 1862 Apollo (1400 m diameter)

Spin-up rate5.3*10-8 rad / day2

Kaasakainen, Nature, 2007

Measurement of The YORP effectDirect measurement on 2000 PH5 (54509)

Spin-up rate2*10-4 deg / day2

May increase the speed of rotation up to a period of 20 s !

Lowry et al., Science, 2007

Numerical simulations of the YORP effect

Assuming that the asteroid rotates around the shortest axis of the inertia tensor (with the moment of inertia C)

Body’s angular momentum is

Change of angular momentum is equal to the Torque

For C constant, the equation can be split into :

Numerical simulations of the YORP effect

Numerical simulations of the YORP effect

YORP-induced mean rate of change of the rotation rate ω and obliquity ε as a function of the obliquity for asteroid (6489) Golevka (assumed to be on a circular orbit at 2.5 AU). Eleven values of the surface thermal conductivity logK = −9, −8, . . . , −1, 0, 1 are shown. The lowest value—black—isidentical to the zero conductivity casea analyzed by Vokrouhlicky ́&Cˇapek (2002). The rotation effect shows a small dependence on K, whereas the obliquity effect has a significant dependence on K (and rotation period of 6 h).

The thermal history of asteroids

Insights into the differentiation of terrestrial planets

Le chainon manquant pour comprendre la formation

planétaire?

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Le chainon manquant pour comprendre la formation

planétaire?

Asteroid diversity

toward Mars toward Jupiter

High-temperature minerals Low-temperature mineralsMetals

Asteroid diversity

C-type : Dark carbonaceous objectsS-type : Stony objects

Evidences for differentiated (melted) objects

From meteorites….

Les Howardites-Eucrites-Diogenites (HED)

Lava flows ? Deep magmatic rocks Impact breccias

4-Vesta (from its south pole) With other asteroids

A complex geological history !

4-Vesta (centered on the equator)

Which formation process?

Crustal and mantle structure?

The thermal historyRadionucleides

Il faut chercher des traces « fossiles »

Short radioactive elements derived from stellar nucleosynthesis:

Al26

Half-life = 717.000 ans

U, Th, K are long-period radioactive elements…

Al26 Mg26 λ= 0,717 Ma

Study of refractory inclusions show the presence of Al26 (in more or less

constant abundance)

Is that enough to melt a planet?

The heat transfer equation with a sphere

Boundary condition

T = T0 at the surface

“Heated from inside, and cooled at the surface”

The heat transfer equation with a sphere

Find the non-dimensional expression of this equation

The heat transfer equation with a sphereExample of a body of 100 km in diameter

tACC = 0

tACC = 0,5 Ma

tACC = 1 MatACC = 2 Ma

3 Ma

Entire vaporations!

tACC = 0

tACC = 0,5 Ma

tACC = 1 Ma

In the center of a 100 km body

In the case of later accretion, lower Al26 concentration and lower maximum temperature

tACC = 1,5Ma

tACC = 2 MatACC = 2,5 Ma

tACC = 3 Ma

Complete melting

50% meltingIncipient melting

Combining thermal properties and accretion age/duration a wide range of thermal histories are possible

First moltenbody

(size ~100km)

4568 +2Ma

Age

millionsd’années

Source of heat(26Al: half-life 0,7 Ma)

Planet formation – Sequence of events

Very early

First solids(size - mm) End of planetary

accretion(size ~1000km)

+30 Ma

Impact melting