Changes between Version 27 and Version 28 of u/GasPhiBE


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Timestamp:
01/15/16 15:03:15 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/GasPhiBE

    v27 v28  
    99* A) The BE sphere is 150 solar masses. The mass of the ambient medium then is given by the equation:
    1010
    11 {{{#!latex
    12 {4\over3} \pi \rho ({r^3-Rbe^3})
    13 
    14 }}}
     11[[latex(${4\over3} \pi \rho ({r^3-Rbe^3})$)]]
    1512
    1613Where 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.
    1714
    1815By 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($M = {0.0007 * {4 \over 3} \pi ({15.4^3 -1 })} = 10.7 * Mscale = 4.77 * 10^36 g = 2,248 ~Solar ~Masses$)]]
    2317
    2418That is, '''Mamb~14Mbe'''.
     
    3024* 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:
    3125
    32 {{{#!latex
    33 \phi = -{G*Mbe \over r } - {G*Mr \over r} \space (for \space r > Rbe)
    34 
    35 }}}
     26[[latex($\phi = -{G*Mbe \over r } - {G*Mr \over r} \space (for ~\space r > Rbe)$)]]
    3627
    3728Here we say
    3829
    39 {{{#!latex
    40 Mr = Mass \space enclosed = {4 \over 3} * \pi * \rho * {(r^3 - Rbe^3)}
    41 
    42 }}}
     30[[latex($Mr = Mass \space ~enclosed = {4 \over 3} * \pi * \rho * {(r^3 - Rbe^3)}$)]]
    4331
    4432
     
    4634=>
    4735
    48 {{{#!latex
    49 \phi = -{G \over r} [Mbe + {4 \over 3} * \pi *\rho ({r^3-Rbe^3})]
    50 
    51 }}}
     36[[latex($\phi = -{G \over r} [Mbe + {4 \over 3} * \pi *\rho ({r^3-Rbe^3})]$)]]
    5237
    5338From 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.
     
    6045
    6146Point:
    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 $)]]
    6748
    6849Total Phi:
    6950
    70 {{{#!latex
    71 \phi = - \space{ G {(Mbe - {4 \over 3} * \pi* \rho * Rbe^3)} \over r} + {-G * {4 \over 3} * \pi * r^3 * \rho \over r}
    72 
    73 }}}
     51[[latex($\phi = - \space{ G {(Mbe - {4 \over 3} * \pi* \rho * Rbe^3)} \over r} + {-G * {4 \over 3} * \pi * r^3 * \rho \over r} $)]]
    7452
    7553Some manipulation shows this to be the equation as above.