Changes between Version 27 and Version 28 of u/GasPhiBE
- Timestamp:
- 01/15/16 15:03:15 (9 years ago)
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u/GasPhiBE
v27 v28 9 9 * A) The BE sphere is 150 solar masses. The mass of the ambient medium then is given by the equation: 10 10 11 {{{#!latex 12 {4\over3} \pi \rho ({r^3-Rbe^3}) 13 14 }}} 11 [[latex(${4\over3} \pi \rho ({r^3-Rbe^3})$)]] 15 12 16 13 Where rho is the uniform ambient density, and we are approximating the box as a sphere. Note, when these quantities are in computational units, one converts to mass in cgs, as defined in the BE problem module, by multiplying this equation by Mscale found in scales.data. 17 14 18 15 By this method, I found the mass in the light ambient medium to be: 19 {{{#!latex 20 M = {0.0007 * {4 \over 3} \pi ({15.4^3 -1 })} = 10.7 * Mscale = 4.77 * 10^36 g = 2,248 Solar Masses 21 22 }}} 16 [[latex($M = {0.0007 * {4 \over 3} \pi ({15.4^3 -1 })} = 10.7 * Mscale = 4.77 * 10^36 g = 2,248 ~Solar ~Masses$)]] 23 17 24 18 That is, '''Mamb~14Mbe'''. … … 30 24 * A) We can look at this problem in 2 ways that give the same answer. First we can add the potential of the BE sphere as though it were a point charge with M=Mbe to the potential of the ambient, approximated as a uniform sphere of r=l/2, where l is box length: 31 25 32 {{{#!latex 33 \phi = -{G*Mbe \over r } - {G*Mr \over r} \space (for \space r > Rbe) 34 35 }}} 26 [[latex($\phi = -{G*Mbe \over r } - {G*Mr \over r} \space (for ~\space r > Rbe)$)]] 36 27 37 28 Here we say 38 29 39 {{{#!latex 40 Mr = Mass \space enclosed = {4 \over 3} * \pi * \rho * {(r^3 - Rbe^3)} 41 42 }}} 30 [[latex($Mr = Mass \space ~enclosed = {4 \over 3} * \pi * \rho * {(r^3 - Rbe^3)}$)]] 43 31 44 32 … … 46 34 => 47 35 48 {{{#!latex 49 \phi = -{G \over r} [Mbe + {4 \over 3} * \pi *\rho ({r^3-Rbe^3})] 50 51 }}} 36 [[latex($\phi = -{G \over r} [Mbe + {4 \over 3} * \pi *\rho ({r^3-Rbe^3})]$)]] 52 37 53 38 From this we can see that when the ambient rho is SMALL, phi is dominated by the first term. That is, phi can be approximated by a point source of mass=Mbe for small r. However, there appears to be a turning point in phi. We can expect the r^2^ term to dominate at large r for non-negligible rho. This would make phi more and more negative. … … 60 45 61 46 Point: 62 63 {{{#!latex 64 m = Mbe - {4 \over 3} * \pi * Rbe^3 * \rho 65 66 }}} 47 [[latex($m = Mbe - {4 \over 3} * \pi * Rbe^3 * \rho $)]] 67 48 68 49 Total Phi: 69 50 70 {{{#!latex 71 \phi = - \space{ G {(Mbe - {4 \over 3} * \pi* \rho * Rbe^3)} \over r} + {-G * {4 \over 3} * \pi * r^3 * \rho \over r} 72 73 }}} 51 [[latex($\phi = - \space{ G {(Mbe - {4 \over 3} * \pi* \rho * Rbe^3)} \over r} + {-G * {4 \over 3} * \pi * r^3 * \rho \over r} $)]] 74 52 75 53 Some manipulation shows this to be the equation as above.