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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 lawN(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