Astronomy 340 Fall 2005

1. Astronomy 340Fall 2005 Class #4 15 September 2005

2. Announcements • HW #1 distributed at the end of class today; due on Tues Sep 27. • How’s the book search? • Schedule change: MIDTERM WILL BE ON OCT 20, NOT OCT 25

3. Review • Restricted 3-Body Problem • “pseudo-potential”: • U = (n2/2)(x2+y2)+(μ1/r1)+(μ2/r2) • Lagrangian points • Where is the orbit stationary? Where the forces balance • Most stable are L4,L5 • Jacobi integral • CJ = n2(x2+y2) + 2[(μ1/r1)+(μ2/r2)] – v2 • Where v=0, we have “zero-velocity” surfaces which bound the regions of allowable orbits • Tides • Tide raising force • V3 = -G(ms/a3)Rp2P2(cos Φ)

4. More on Tides • Tidal vs centrifugal deformation • Consider three axes  a,b,c • Tidal deformation  a > b=c, where a points towards satellite • Centrifugal deformation  a=b>c, where c is the polar axis • For a synchronously rotating satellite in hydrostatic equilibrium  (b-c)=1/4(a-c). • Mimas  [(b-c)/(a-c)]=0.27  suggesting non-uniformity of the interior (differentiated)

5. Transition Slide • We’ve covered enough dynamics for now  we’ll see more later in the semester • Kepler’s Law/Orbital Parameters • Lagrangian points/zero velocity surface • Tidal forces • Now onto radiation…

6. Solar Heating and Transport • Why? • Astrophysics is all about how energy gets from point A to point B • Sun responsible for most of energy in solar system • Surface temperature • Atmospheric temperature • Mass loss from comets • Temperature • Measure of kinetic energy; E=(3/2)nkT • n = # cm-3 • k = Boltzman’s constant • T = temp • Thermal  E = (1/2)mv2  so temp is related to velocity (consider simple case of escape velocity of an atmosphere from a planet a given distance from the Sun

7. Radiation • Bf(T) = (2hf3/c2)[1/(ehf/kT-1)] • Λmax = (0.29/T)  wavelength at the maximum of the BB curve • f = frequency • Units = erg s-1 cm-2 Hz-1 ster-1 • In limit hf << kT, then • Bf(T) ~ (2f2/c2)kT • True in the radio regime

8. Radiation • What do we measure? • F = ΩB(T) (erg s-1 cm-2 Hz-1) • Integrate over frequency and solid angle • F = 4π∫Bf(T)df = σT4 – this is a measure of the effective temperature – the flux emitted by any source can be described by a single temperature. Similarly, the Sun emits radiation as a function of its temperature

9. What happens when solar radiation meets a planetary surface? • Fin = (Lo/4πD2)πRp2 • This heats the surface and the surface radiates….how much? • In general • L = 4πR2σT4 • So if the planet’s luminosity arises solely from incoming solar flux, then • Teq = [(L0/4πD2)(1/4σ)]1/4 equilibrium temperature just balances radiation in with radiation out.

10. Complications • Albedo – the amount of radiation that is actually absorbed as opposed to being reflected or hitting at non-incident angles • Fin=(1-Ab)(L0/4πD2)πRP2 • But it’s even more complex… • Albedo • Rotation period  what do you think the effect is • angle of Sun

11. ∫0∞(1-Av)(L0/4πr2)cos(α(t)-α)cos(δ0(t)-δ)dv • Heating of planetary surfaces via conduction… • depends on the characteristics of the surface material • Depends on temperature gradient • Q = heat flux = -ζ (dT/dx)  this is empirical • X= distance • ζ = thermal conductivity (erg s-1 cm-1 K-1

12. Properties of surfaces – thermal heat capacity and specific heat • CP = (dQ/dT)P = thermal heat capacity = amount of heat needed to raise the temp of one mole of matter by 1 degree K at constant P (can also do the same for volume) • Specific heat = amount of energy needed to raise temp of 1 gram of material by 1 degree K at constant temperature and pressure. Usually shown as cP or cV. • Related via: cP = (CP/mm) where mm is that mass of a mole of the stuff  can substitute V for P.