1. Astronomy 340 Fall 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.