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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/square-root 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?

--Keep water liquid
--Supply energy for life?

Elements of Life
--Carbon, Oxygen, etc.

--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 = semi-major 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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