Habitable zone


Blackbody radiation




Blackbody radiation spectrum


The color of each curve reflects what your eye perceives.

An ideal blackbody radiates power according to the Stefan-Boltzmann law:

Radiation Power  =  Temperature^4  *  SurfaceArea^2  *  5.7*10^-8  Watts/K^4/m^2

The radiation is centered around a characteristic wavelength given by Wein's law:

Wavelength  *  Temperature  =  2.90*10^-3  meter Kelvins

      Temperature   Radiation  Relative     Wavelength
          (K)       (meters)   intensity
Earth     287       1.0*10-5        1       Infrared
Sun      5777       5.0*10-7   164000       Visible

The "Blackbody" simulation at phet.colorado.edu shows the blackbody radiation
spectrum as a function of temperature.


Blackbody radiation

f  =  Photon frequency
h  =  Planck constant
   =  6.62e-34 Joule seconds
E  =  Photon energy
   =  h f
C  =  Speed of light
I  =  Blackbody photon intensity as a function of frequency
k  =  Boltzmann constant
   =  1.38e-23 Joules/Kelvin
T  =  Temperature

The Planck law for the blackbody spectrum is

I  =  2 h f^3 C^-2  /  [exp(hf/kT) - 1]



How cold would the Earth be without CO2?

The Earth gains energy from the sun and loses it to blackbody
radiation. The equilibrium temperature occurs when these
are in balance. To estimate this temperature, assume that:

The Earth absorbs all the solar radiation falling onto it.

The Earth is at a constant temperature at all points on its surface.

Intensity of sunlight at the Earth's orbit = 1360 Watts/m^2

The Earth radiates energy as an ideal blackbody according to the
Stefan-Boltzmann law.

Plugging these values in for the Earth, what temperature do you get?
How about Venus and Mars?


Stellar lifetime

The luminosity of a star scales with mass as

Luminosity  ~  Mass^3.5

The lifetime scales as

Lifetime  ~  Mass  /  Luminosity
          ~  Mass^-2.5

The sun burns for ~ 10 billion years.

The minimum mass for hydrogen fusion is 0.08 solar masses.
Given the above scaling, how long does such a star last?
How about a 10 solar mass star? This is the minimum mass for a
supernova.


Goldilocks Zone

The luminosity of a star scales with mass as

Luminosity ~ Mass^3.5

The heating power absorbed by a planet from its host star scales as

Heating_power ~ Luminosity * Distance_to_star^(-2)

Define a "Goldilocks radius" as the ideal distance for a planet to be
from its host star to be at an ideal temperature for life.
If we say that the Goldilocks radius for a 1 solar mass star is
1 A.U., what is the Goldilocks radius for a stars of mass
{1/2, 1/4, 1/8, 0.08} solar masses?

http://en.wikipedia.org/wiki/Habitable_zone


Ultraviolet radiation

Assume the sun shines as an ideal blackbody with a temperature of 5777 Kelvin.
Using the "blackbody" simulation at phet.colorado.edu,
what fraction of the sun's energy is in the ultraviolet, visible, and infrared?

What temperature would the sun have to be for the ultraviolet fraction to
be 1/10th its value at 5777 Kelvin? How about 1/100th?




Absorption spectrum of water


Before the Earth had an oxygen atmosphere and ozone, UV radiation was a hazard
and the only safe place to be was underground or underwater.
Given the above spectrum, how far underwater do you have to go to escape
UV but still have visible light for photosynthesis?


Photosynthesis




Spectrum of photosynthesis


Using the blackbody spectrum tool and the above data,
can you produce an order-of-magnitude estimate for:

Rate of photosynthesis by planets for a 4000 K star divided by rate of
photosynthesis by plants for the sun.

Stars



Star       Mass    Luminosity   Color  Temp     Lifetime   Death    Remnant        Size of
type      (solar    (solar            (Kelvin)  (billions                          remnant
          masses) luminosities)                 of years)

Brown Dwarf  <0.08                       1000  Immortal
Red Dwarf     0.1        .0001   Red     2000   1000      Red giant  White dwarf   Earth-size
The Sun       1         1        White   5500     10      Red giant  White dwarf   Earth-size
Blue star     10    10000        Blue   10000      0.01   Supernova  Neutron star  Manhattan
Blue giant    20   100000        Blue   20000      0.01   Supernova  Black hole    Central Park

The minimum mass for hydrogen fusion is 0.08 solar masses.

      Mass < 9     -->  Ends as a red giant and then turns into a white dwarf.
  9 < Mass         -->  Ends as a supernova
  9 < Mass < 20    -->  Remnant is a neutron star.
 20 < Mass         -->  Remnant is a black hole.
130 < Mass < 250   -->  Pair-instability supernova (if the star has low metallicity)
250 < Mass         -->  Photodisintegration supernova, producing a black hole and relativistic jets.

Heat capacity of atmospheres

          Air        Air     Air Column  Gravity  Temperature   Rotation
        Density    Pressure     Mass
        kg/m^3    10^5 N/m^2   kg/m^2     m/s^2     Kelvin        days

Venus     67        92.1      1038000      8.87      735         243.0
Titan      5.3       1.46      108000      1.35       94          15.9
Earth      1.2       1.00       10200      9.78      287           1.00
Mars        .020      .0063       170      3.71      210           1.03

The "Air column mass" is the mass of air above a square meter of an object's surface.

Mass / Area = Pressure / GravitationalAcceleration

The atmosphere of the Earth is thick enough to block cosmic rays and the atmosphere of
Mars isn't.

An ideal blackbody radiates power according to the Stefan-Boltzmann law:
Radiation Power  =  Temperature  *  SurfaceArea^2  *  5.7*10^-8 Watts/K/m^2

Heat capacity of air = 1020 Joules/kg/Kelvin

For each of the above objects, how many days does it take for radiation to
decrease the temperature of the air by 10 Kelvin?


Atmospheric density and the transmission of light

I0 = Intensity of sunlight above the atmosphere
I  = Intensity of light that gets through the atmosphere and reaches the planet's surface
d  = Atmospheric thickness in kg/meter^2
D  = A constant

Light transmission can usually be modeled as an exponential

I/I0  =  exp(-d/D)

Based on data from the web, what is I/I0 for the Earth, and what is D?

By what factor would you have to increase the Earth's atmospheric thickness to
reduce I/I0 by a factor of 1/2?

Titan atmosphere thickness  =  10.6 * Earth atmosphere thickness
What would you estimate is I/I0 for Titan?


Blackbody radiation

i(f) = Intensity as a function of frequency
I    = Total intensity
I    = Integral i(f) df

h = Planck constant
k = Boltzmann constant
T = Temperature

i(f) = Constant * f^3 / (exp(-hf/kT) - 1)

The classical formula can often be obtained from the quantum formula by taking the limit
h --> 0

What is the limit as h->0 of i(f)?