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    How many 10" balls will fill a sphere of diameter five miles?

  • #2
    Originally posted by philastro View Post
    How many 10" balls will fill a sphere of diameter five miles?
    If spheres are arranged in densest possible arrangement, the packing factor is pi*sqrt(2)/2, about 74%. random pack is 63.4% or less. I'll use 0.63.

    The volume of a sphere is (pi/6)D where D is the diameter. Five miles is 316800". If PF is packing factor, we need PF x (Da/Db)
    or 0.63 x (316800/10) = 2.00 x 1013, give or take.

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    • #3
      Fascinating stuff, I always loved the word problems like this.

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      • #4
        Thanks very much John. In case anyone wonders what this is about, it was inspired by an explanation of hydrogen atom structure in Atomic Physics, by J.M.Valentine, along with my study of Astrophysics in seeking an explanation as to how a small amount of material from White Dwarf or Neutron stars can weigh so much (on Earth that is). He explains that most of an atom is empty space, its appearance as a solid ball from the outside being given by the incredible speed at which electrons rotate around the nucleus. As an example, if the hydrogen proton and its electron were magnified to the size of a football (roughly 10" in one estimate, but not being sports-mad I wouldn't know!) then the entire atom would be 5 miles in diameter with nothing inside except its nucleus. Under intense stellar gravitational pressure the electron is squeezed out of its orbit, i.e. the atom is ionized. Exposed protons now get squeezed together until nuclear reactions begin as they are forced to fuse. The point is that the space originally occupied by ONE atom (now reduced to its 10" proton) is now occupied by as many protons as will fit into the original 5-mile diameter sphere of the atom prior to ionization. So by the time the protons are so crushed together that reactions occur, the space now contains your 2 x 10 to 13th power , or 20,000,000,000,000 (20 ENGLISH billion) protons. Assuming an arbitrary one ton as the weight of a single proton, any weight you like to make it more understandable than the actual weight, that means the original space occupied by a theoretical one-ton atom now weighs 20 billion tons. Quite logical, the principle is that it explains why stellar material is so heavy, and why the sun for example, contains such a colossal amount of fusible gas that it releases its energy at a prodigious but steady rate over a period of an estimated 10,000,000,000 years. Material in White Dwarf and Neutron stars is far more dense and correspondingly far heavier.
        Last edited by philastro; 02-10-2018, 04:33 AM.

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        • #5
          As a follow-up from the same book, ignoring the one-ton example I used, the principle of being able to have so much weight/many particles of mass, concentrated in a given volume through gravitational pressure having been established, it will help to have an idea of how many atoms, even before ionization, make up a sample of an element. Valentine uses a 1-cm. square of gold leaf 1 mm. thick. It weighs approx. 2 gm. and contains 6,000,000,000,000,000,000,000 ( 6 thousand trillion) atoms. This is an English trillion, not U.S. who would wrongly in my view, write it with fewer noughts. Added to my previous post this helps to show how and why, with so many atoms available, our sun and the stars produce so much energy over such long periods of time.

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