Astronomy 340 Fall 2007

Astronomy 340Fall 2007 Lecture # 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)Mp1/3A5/3Z-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 Phobos Ida

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 - colors Jedicke et al. 2004  results indicate “space weathering”

Comparison with meteorite samples Points 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

Astronomy 340 Fall 2007