Changes between Version 21 and Version 22 of u/GasPhiBE
- Timestamp:
- 10/04/12 16:11:21 (12 years ago)
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u/GasPhiBE
v21 v22 55 55 We can arrive at this same equation by taking a slightly different conceptual route. We can consider phi in ambient medium to be due to the superposition of a uniform sphere on top of a point mass object. In pictures, this is like: 56 56 57 [[Image(PhiGraphic.png, 70%)]]57 [[Image(PhiGraphic.png, 40%)]] 58 58 59 59 In equations, this is like: … … 85 85 86 86 87 [[Image(LightPhi.png, 60%)]]88 [[Image(LineLightPhi.png, 50%)]]87 [[Image(LightPhi.png, 40%)]] 88 [[Image(LineLightPhi.png, 30%)]] 89 89 90 90 Ah, that looks about right. … … 92 92 Here is the function of phi as found above: 93 93 94 [[Image(MMaLight.png, 60%)]].94 [[Image(MMaLight.png, 40%)]]. 95 95 96 96 I see that for small r, there behavior is qualitatively similar, and on the same order. The theoretical curve, considering both phi from a point source and phi from a uniform sphere seems to fail at large r. This indicates that the ambient is not sufficiently important at this density to effect the point gravity potential beyond Rbe. That is, phi is well approximated by simply a point source object of mass on the order of Mbe for all r within simulation box. … … 100 100 From my sim: 101 101 102 [[Image(MatchedPhi.png, 60%)]]102 [[Image(MatchedPhi.png, 40%)]] 103 103 104 [[Image(LineMatchedPhi.png, 50%)]]104 [[Image(LineMatchedPhi.png, 30%)]] 105 105 106 106 Here is the function of phi as found above: 107 107 108 [[Image(MatchedMMa.png, 60%)]].108 [[Image(MatchedMMa.png, 40%)]]. 109 109 110 110 Now, this is where things get hard for me to understand. First, I see the range of phi, and magnitude of phi is drastically different for this matched ambient case. This may make sense, based on more mass in the ambient meaning stronger gravitational potential due to the ambient. However, I would expect that at very small radii, just above Rbe, the approximation of the ambient as a uniform sphere should be very good. This is because there is SOOOO much more mass in the ambient than in the sphere (see above calculation). Thus, very early on, I would expect the potential to be growing more and more negative as more and more mass is enclosed in this uniform sphere. Indeed, we see this behavior from the function derived above, but NOT in the simulation. Can someone please explain this discrepancy? … … 114 114 Ah, so the potential needs to be calculated up to the correct additive constant, which the above formulation was NOT doing. I see now that by using Gauss's law to get the force, and integrating from infinity, the potential matches that of my sims. 115 115 116 [[Image(ForWiki1.png, 60%)]] 117 [[Image(NewPhiMatched.png, 60%)]] Compare to:[[Image(LineMatchedPhi.png, 50%)]] 116 [[Image(ForWiki1.png, 40%)]] 118 117 119 [[Image(NewPhiLight.png, 60%)]] Compare to:[[Image(LineLightPhi.png, 50%)]] 118 The Matched case: 119 120 [[Image(NewPhiMatched.png, 40%)]] 121 122 Compare to: 123 124 [[Image(LineMatchedPhi.png, 30%)]] 125 126 and for the light case: 127 128 [[Image(NewPhiLight.png, 60%)]] ' 129 130 Compare to: 131 132 [[Image(LineLightPhi.png, 50%)]]
