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ASTRONOMY 340 FALL 2007 Lecture # 23 October 2007
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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
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Planetary Interiors/Size Apply the virial theorem 2E k = -E p What’s the kinetic energy? Motion of electrons (degeneracy and electrostatic) Protons don’t contribute much at all What’s the potential energy? gravitational
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Degeneracy Energy M p has N p atoms of average mass number, A so N p = M p /Am p each atom has ZN p electrons Each electron occupies a volume with diameter, d, so that d = (Am p /ZM p ) 1/3 R p From quantum mechanics, E k = p 2 /(2m e ) and pλ = h The de Broglie wavelength, λ, is the size of the electron volume so λ=2πd (longest possible wavelength)
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Degeneracy Energy cont’d Put that altogether and get: E k = (h 2 /2m e )(4 π 2 d 2 ) -1 per electron volume Substitute expression for d, multiply by ZN p to get total degenerate energy E K = γM p 5/3 Z 5/3 A -5/3 R p -2
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Electrostatic Assume non-relativistic E e ~ (1/4πε o )(Ze 2 /d) (per electron) Plug in d from previous page and multiply by N p Z to get: E e ~ ξM p 4/3 Z 7/3 A -4/3 R p -1
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Gravitational Energy E g = - κ(M p 2 /R p )
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Combine all the energies…. Use virial theorem so that 2E k = E e + E g Rearrange to get a relation between R p and M p R p -1 = (const)A 1/3 Z 2/3 M p -1/3 + (const)M p 1/3 A 5/3 Z -5/3 Peaks at log(M) ~ 27 (kg) and log(R) ~ 8 right around Jupiter!
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Maximum radius Take dR p /dM p = 0, solve for M R(max) Get: M R(max) = (const) (Z 7/3 /A 4/3 ) 3/2 Insert this in for the mass in the long equation and get: R max = (const) Z 1/2 /A R max (H) ~ 1.2 x 10 8 m The central pressure for a H body with maximum radius is about the pressure needed to ionize H.
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Asteroids Ida Phobos
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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
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Distribution – SDSS results 200,000 asteroids – Ivezic et al. 2002
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Size Distribtion Power law N(R) = N 0 (R/R 0 ) -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
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Collisions Collisions numerical simulations 100-200 km diameter progenitors Limits? Surface ages Vesta’s surface looks primordial, but it has a large impact crater
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simulation
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Asteroid Composition How do you measure asteroid compositions? Reflection spectroscopy Comparison with meteorites
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Asteroid Composition - colors Jedicke et al. 2004 results indicate “space weathering”
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Comparison with meteorite samples Points are real data, line is reflection spectrum of sample
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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
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