
Astronomy 141
Life in the Universe
Prof. Scott Gaudi

Lecture 2: Requirements for Habitability
Key Ideas
(period)2 is proportional to (semimajor axis)3
Equilibrium temperature is proportional to 1/squareroot of distance
Compositional gradients exist the solar system
Small bodies cool quicker
Colder and bigger bodies hold more atmosphere
What properties are conducive to life?
Heat
Keep water liquid
Supply energy for life?
Elements of Life
Water
Carbon, Oxygen, etc.
Atmosphere
Liquid Water
Thermal bath
Protection from UV, etc.?
Large Size
Keep heat
Plate tectonics
Hold atmosphere
Kepler's Laws Revisited
First Law
Orbits are ellipses with the Sun at one focus.
Second Law
Line from Sun to Planet sweeps out equal areas in equal times
Third Law
Period squared equals semimajor axis cubed.
Third Law of Orbital Motion
P = period
a = semimajor axis of the orbit
M1 = mass of the first body
M2 = mass of the second body
G = gravitational constant
For planets orbiting the Sun, one mass is much smaller than the
other, so the constant of proportionality is the same for every planet.
Equilibrium Temperature
Brightness of the Sun decreases as distance squared .
A perfectly absorbing surface illuminated by light with a given energy per area per time will ultimately reach an equilibrium temperature such that:
Energy per area per time = constant times temperature to the fourth power.
The radiation from the sun heats the planet to its equilibrium temperature:
For the Sun, assuming a perfect absorber and that the energy is absorbed
on one hemisphere but emitted over the entire surface, and assuming
that the distance from the Sun (d) is equal to its average distance (the semimjaor axis a),
This neglects the effect of albedo  higher albedo means lower temperature
(at fixed distance).
Composition Gradients
Because of the variation of temperature, the composition of the material that formed
the planet varies as a function of distance from the Sun.
Cooling Time
Cooling time = total thermal energy divded by rate at which energy is emitted.
The rate at which energy is emitted is the area of the body times the energy
emission rate per unit area, which is just a constant times temperature to the fourth power.
The total thermal energy of an object is proportional to its temperature and total number of particles (total mass).
The total mass depends on the volume, or radius cubed.
So the total energy of a body radius R and temperature T is proportional
The cooling time is then,
Hotter bodies cool faster
Larger bodies cool slower
Holding on to an Atmosphere
Escape Velocity and Temperature
The minimum velocity to just escape the gravity of an object of mass M and radius R is:
Atmospheric Retention
The higher the temperature, the faster the molecules are moving
For a given temperature, heavier gas molecules move slower, and so are less likely to escape
More massive planets hold on to gases better
See A Note about Graphics to learn
why the graphics shown in the lectures are generally not reproduced with
these notes.
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