--------------------- #1514 - From Bubble Nebula to Rocky Planets
- Nebula are astronomy’s beautiful images. Google the “Bubble Nebula” for a great example. These are stellar nurseries where suns and planets are forming. Use calculus to calculate how fast a planet forms in the accretion disk of a new born star inside one of these nurseries.
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- The Bubble Nebula is located in the Constellation Cassiopeia the Queen. Cassiopeia is the “lazy w” of stars located in the opposite side of the night sky’s Northern Star from the Big Dipper. The south-west arm in the winter shy’s lazy “w” points to the Bubble Nebula (NGC7635).
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- The Bubble Nebula was first discovered in 1787 by William Herschel. He is the astronomer that discovered the planet Uranus in 1781. He was using a 6 inch telescope in his backyard.
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- Today the Nebula is classified as an emission nebula with 3 distinct expanding shells. What is visible as the “ Bubble” is the smallest of these shells. There is a massive star at the center of the Bubble emitting powerful ultraviolet light. The UV radiation is stripping away the electrons from the atoms in the interstellar medium. This ionization creates free electrons and positive ion nuclei.
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- When the atomic nuclei recapture the electrons their energy level falls and energy is released as photons of visible light. These light photons cause the expanding gas cloud to glow like a neon sign. Our Sun will have a similar fate as a “ planetary nebula” when it dies in 5 billion years from now. The Bubble’s star is 40 times more massive than our Sun. It has a shorter lifetime because it creates a massive stellar wind blowing outward at 4,000,000 miles per hour.
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- The Bubble has been pushed out to 10 lightyears diameter. It is 7,100 lightyears away. A similar show of ionized atomic hydrogen can be seen in the Orion Nebula (M42) See Review #1509.
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- The temperature inside the cloud where the stars and planets are forming is only 10 Kelvin (-442 degrees Fahrenheit). Electrostatic forces gather the cloud and gravity collapse it into clumps. The clumps collapse further into protostars. When densities and temperatures reach high enough level the star turns on with nuclear fusion.
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- These stars are typically massive stars 15 to 30 Solar Mass. Their Solar Winds expand in an irregular fashion as they smash into different densities of the interstellar medium. Over 5,000 of these space bubbles have been found in our Milky Way Galaxy. Google the Bubble Nebula to see an example of what our neighborhood was like 5 billion years ago.
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- The young stars have rotating accretion disks that are forming planets, asteroids and comets. How long does it take a planet the size of Earth to form? With some math and some assumptions we can make a first approximation. The answer is about ¾ of a million years for an Earth-size planet to form. Here is how the math works:
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------------------- The planet is a sphere, and the volume of a sphere is V = 4/3 *pi *r^3
------------------ The density of the Earth is D = 3,000 kilograms / meter^3
------------------ The mass of the Earth is M = 5.9*10^24 kilograms.
------------------- Mass = Volume * Density
------------------- M = V * D
------------------- M = 4/3* pi * r^3 * D
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- To express the change of mass as a function of time, we can use the change in radius as a function of time. The goal here is to see how fast the mass can accumulate in the accretion disk in order to form a planet. We use calculus that integrates collapsing sperical areas of the disk until the core reaches the density of the final Earth-size planet.
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---------------- M (t) = 4/3 * pi* r(t)^3 * D
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- Solving this equation to get the radius as a function of time.
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---------------- r(t) = ( 3M) / 4*pi*D) ^ 1/3
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- Next we need an expression for the rate of change of mass due to the collapsing accretion disk. Each shell is an area and these areas are adding up as they collapse and the radius gets smaller and smaller. Here is the rate of change of mass due to the change of radius with time.
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------------------------- dM / dt = 4*pi* [r (t)]^2 * (d * v)
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- In this case the lower case “ d” is the density of the accretion disk of in-falling material.
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------------------------ d = 0.000001 kilograms / meter^3
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- The velocity of the in-falling material is “v” = 2,200 miles per hour, ( 1,000 meters / second)
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------------------------ v = 1,000 meters / second
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--------------------- The velocity times the density = (v *d) = 10^3 meters / sec * 10^-6 kilograms / meter^3 = 10^-3 meters^2 * kilogram* second.
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---------------------- (v * d) = 10^-3 meters^2 * kilogram* seconds
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- Remember our goal is to determine the “time” it takes to form the planet. We use the expression for the rate of change of mass with time as a function of collapsing radius:
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------------------------- dM / dt = 4*pi* [r (t)]^2 * (d * v)
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- And, substitute the expression for radius as a function of time
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----------------------------- r(t) = [ 3M / 4*pi*D) ^ 1/3]
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-------------- dM / dt = 4*pi* [ 3M / 4*pi*D) ^ 2/3]* (d * v)
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- Moving mass as a constant out of the equation:
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--------------- dM / dt = 4*pi* [( 3 / 4*pi*D) ^ 2/3]* (d * v) * M^2/3
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- Dividing each side by M^2/3 and expressing change in mass as a function of time:
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---------------- dM /M^2/3 = 4*pi* ( 3 / 4*pi*D) ^ 2/3* (d * v) dt
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- We integrate both sides of this differential equation to find an expression for the mass as a function of time.
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----------------- 3 M^1/3 = 4*pi* ( 3 / 4*pi*D) ^ 2/3* (d * v) * t
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------------------ M = ( 4 *pi* d * v / 3)^3 * ( 3 / 4*pi*D) ^ 2 * t^3
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- Substituting the values in for mass and densities, then solving for time”
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------------------- 5.9*10^24 = ( 4.19 * 10^-3 )^3 * ( 0.239 / 3000) ^ 2 * t^3
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-------------------- 5.9*10^24 = ( 73.6*10^-9 ) * ( 6.33*10^-9) * t^3
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-------------------- 5.9*10^24 / 4.66 * 10^-16 = t^3
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------------------- t^3 = 1.27 * 10^40 seconds
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------------------- t = 2.33 *10^13
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- There are 31,600,000 seconds in one year:
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-------------------- t = 2.33*10^13 sec / 3.16*10^7 sec/year
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-------------------- t = 738,000 years
- The time it takes for an Earth -size planet to from is about ¾ million years.
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