ASTRONOMY 340FALL 2007Lecture #23 October 2007

Midterm: Thursday Oct 25 in class HW #3 due NOW HW #2, #3 solutions available HW #4 to be handed out on Tues Oct 30 Office Hours: Wed 1-4:30

Planetary Interiors/Size Apply the virial theorem 2Ek = -Ep What’s the kinetic energy?

Motion of electrons (degeneracy and electrostatic)

Protons don’t contribute much at all What’s the potential energy?

gravitational

Degeneracy Energy Mp has Np atoms of average mass number, A so

Np = Mp/Amp each atom has ZNp electrons Each electron occupies a volume with diameter, d,

so that d = (Amp/ZMp)1/3Rp

From quantum mechanics, Ek = p2/(2me) and pλ = h The de Broglie wavelength, λ, is the size of the

electron volume so λ=2πd (longest possible wavelength)

Degeneracy Energy cont’d Put that altogether and get:

Ek = (h2/2me)(4π2d2)-1 per electron volume

Substitute expression for d, multiply by ZNp to get total degenerate energy

EK = γMp5/3Z5/3A-5/3Rp

-2

Electrostatic Assume non-relativistic Ee ~ (1/4πεo)(Ze2/d) (per electron) Plug in d from previous page and multiply by NpZ

to get:

Ee ~ ξMp4/3Z7/3A-4/3Rp

-1

Gravitational Energy

Eg = -κ(Mp2/Rp)

Combine all the energies…. Use virial theorem so that 2Ek = Ee + Eg

Rearrange to get a relation between Rp and Mp

Rp-1 = (const)A1/3Z2/3Mp

-1/3 + (const)Mp

1/3A5/3Z-5/3

Peaks at log(M) ~ 27 (kg) and log(R) ~ 8 right around Jupiter!

Maximum radius Take dRp/dMp = 0, solve for MR(max) Get: MR(max) = (const) (Z7/3/A4/3)3/2

Insert this in for the mass in the long equation and get:

Rmax = (const) Z1/2/A

Rmax(H) ~ 1.2 x 108m The central pressure for a H body with maximum

radius is about the pressure needed to ionize H.

Asteroids

Ida

Phobos

Asteroid Distribution - orbit Note concentrations

in various regions of the plot

Each clump is an asteroid “family”

Major families Main belt (Mars-

Jupiter) Trojans Near-Earths

Size Distribtion Power law

N(R) = N0 (R/R0)-p

Theory says p = 3.5 based on collisionally dominated size distribution

Ivezic et al. 2000 p=2.3 +/- 0.05 for size distribution of 0.4-5.0km main belt asteroids Derived from SDSS data

Collisions Collisions numerical simulations

100-200 km diameter progenitors Limits?

Surface ages Vesta’s surface looks primordial, but it has

a large impact crater

simulation

Asteroid Composition How do you measure

asteroid compositions? Reflection

spectroscopy

Comparison with meteorites

Asteroid Composition - colorsJedicke et al. 2004 results indicate “space weathering”

Comparison with meteorite samplesPoints are real data, line is reflection spectrum of sample

Composition-results (note Table 9/4)

75% of asteroids are dark Look like “carbonaceous chondrites” Most of these are “hydrated” heated in

past so that minerals mixed with liquid water

12% are “stony irons” Fe silicates M-type albedos pure Ni/Fe, no silicate

absorption features