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?
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.
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)?