1. ASTRONOMY 340 FALL 2007 Lecture # 23 October 2007

2.  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

3. 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

4. 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)

5. 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

6. 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

7. Gravitational Energy E g = - κ(M p 2 /R p )

8. 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!

9. 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.

10. Asteroids Ida Phobos

11. 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

12. Distribution – SDSS results 200,000 asteroids – Ivezic et al. 2002

14. 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

15. Collisions  Collisions  numerical simulations  100-200 km diameter progenitors  Limits?  Surface ages  Vesta’s surface looks primordial, but it has a large impact crater

16. simulation

17. Asteroid Composition  How do you measure asteroid compositions?  Reflection spectroscopy Comparison with meteorites

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

19. Comparison with meteorite samples Points are real data, line is reflection spectrum of sample

20. 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