LIVE INTERACTIVE LEARNING @ YOUR DESKTOP The Search for Earth-like Worlds - How a Little Bit of Math Goes a Long Way! Presented by: Dr. Sten Odenwald March 31, 2011
Exoplanet Exploration Dr. Sten Odenwald NASA March 30, 2011 Space Math @ NASA http://spacemath.gsfc.nasa.gov
Let s pause for questions from the audience
Bizarro solar system with cubical star and planet!
Bizarro solar system with cubical star and planet! From Earth, the transiting planet dims the starlight during its transit. The fraction of starlight dimming is just given by the ratio of their projected crosssections:
Bizarro solar system with cubical star and planet! From Earth, the transiting planet dims the starlight during its transit. The fraction of starlight dimming is just given by the ratio of their projected crosssections: Star = 16 Planet = 4 Ap 4 --- = ----- A* 16
A star dimmed its brightness by 4%. How big is the planet compared to the star? A) 0.4 B) 0.2 C) 0.02
The star HAT-P-7 Distance 320 parsecs Mass 1.5 suns Temperature 6350 K Radius 1.2 million km The graph shows that the star dimmed to 0.9930 from an initial brightness of 1.0000 What is the radius of the transiting planet?
Initial brightness 1.0000 Final brightness 0.9930 Dimming = 0.007 Rs = 1.2 million km
Our answer = 100,000 km. Jupiter R = 71,000 km So we got : 1.4 times Jupiter.
Let s pause for questions from the audience
Kepler 10b has a mass of 4 times Earth and a radius 1.4 times Earth. What is its surface gravity if its acceleration is approximately given by: M G = 9.8 ------ m / s 2 R 2 A) 20 C) 17.22 B) 5 D) 9.8
Water or Ice? Planet surface temperature range Water boils 212 F Water freezes 32 F Or in other words 273 < T < 373
Venus D = 0.7 AU T = 750 K D = 0.8 AU Earth D = 1.0 AU T = 184 K to 331 K D = 1.8 AU Europa D = 5.2 AU T = 50 K to 110 K
Venus D = 0.7 AU T = 750 K D = 0.8 AU Goldilocks Zone (Habitable Zone) D = 1.8 AU Europa D = 5.2 AU T = 50 K to 110 K
Planet surface temperature determined by: Luminosity of star L Distance from star D Reflectivity A
To melt ice, you must first raise it to a temperature of 32 F (0 C or 273 K) The energy input is called the Specific Heat = 2,000 Joules/kg/C Then you must actually melt the ice at 0C Called the Latent Heat of Fusion 333,000 Joules/kg Example. I have a 2 kg block of ice, T = 250 K, and I want to melt it. How much thermal energy do I need?
To melt ice, you must first raise it to a temperature of 32 F (0 C or 273 K) The energy input is called the Specific Heat = 2,000 Joules/kg/C Then you must actually melt the ice at 0C Called the Latent Heat of Fusion 333,000 Joules/kg Example. I have a 2 kg block of ice, T = 250 K, and I want to melt it. How much thermal energy do I need? E = 2,000 J/kgC x ( 273-250) x ( 2.0 kg) + 330,000 J/kg x 2 kg E = 92,000 J + 660,000 J so E = 752,000 Joules
Reflectivity A Also called the Albedo (0.0 < A < 1.0) The more light a body reflects, The less it absorbs, So the cooler it will be.
Reflectivity A Also called the Albedo (0.0 < A < 1.0) The more light a body reflects, A becomes larger The less it absorbs, ( 1 A) becomes smaller So the cooler it will be. Asphalt is hot to the touch A = 0.1 White clothing is cold to the touch A = 0.9
Distance from star D The farther you are from a star, the less energy will be deposited to a given surface to heat it. Ye Old Inverse Square Law Delivered energy per square meter Total energy from the star = ------------------------------------------------- 4 π D 2
Stefan-Boltzmann Law For any body that emits like a black body The temperature of a black body, T Is related to the emission of light energy per square meter, F F = σ T 4 where σ = 5.67x10-8 Watts meter -2 K -4 Example. On a hot day, a square-meter of asphalt emits 460 watts about what is its black body temperature?
Let s pause for questions from the audience
Now lets put it all together!
Now lets put it all together! The temperature of a planetary surface depends on σ T 4 =
Now lets put it all together! The temperature of a planetary surface depends on The light energy emitted by the star it orbits L σ T 4 = ---------------------------------------
Now lets put it all together! The temperature of a planetary surface depends on The light energy emitted by the star it orbits The distance of the planet from the star L σ T 4 = --------------------------------------- 4 π D 2
Now lets put it all together! The temperature of a planetary surface depends on The light energy emitted by the star it orbits The distance of the planet from the star And the amount of light the surface can absorb L ( 1 A ) σ T 4 = --------------------------------------- 4 π D 2
Now lets put it all together! The temperature of a planetary surface depends on The light energy emitted by the star it orbits The distance of the planet from the star And the amount of light the surface can absorb Oh there is also a factor of 4, to account for the planet absorbing light from π r 2 but emitting over 4 π r 2 L ( 1 A ) σ T 4 = --------------------------------------- 4 π D 2 1 ----- 4
Now lets put it all together! The temperature of a planetary surface depends on The light energy emitted by the star it orbits The distance of the planet from the star And the amount of light the surface can absorb There is also a factor of 4, to account for the planet absorbing light from π r 2 but emitting over 4 π r 2 So the final, fundamental, formula for planetary equilibrium temperatures is:
Example for Earth: D = 150 million kilometers L = 3.8 x 10 26 watts A = 0.30 So T = 254 K
Habitable Zones The Habitable Zone of every star is defined, mathematically, by determining the range for D where: Water is liquid: 273 < T < 373 For a star of a given luminosity L
Kepler search after 1 year: Total searched.. 156,453 Transit candidates. 1235 Earth-sized. 68 In Habitable Zone.. 5
If 5 Goldilocks Earth-sized planets were detected by Kepler in a search of 156,435 stars, how many are there in the entire Milky Way with 300 billion stars? A) Significantly more than 9,600,000 B) About 9,600,000 C) Much less than 9,600,000
By next fall, Space Math @ NASA Will produce a resource guide with other math problems related to the search for life in the universe Space Math @ NASA http://spacemath.gsfc.nasa.gov
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