Changes between Version 27 and Version 28 of u/erica/UniformCollapse
- Timestamp:
- 06/29/15 16:11:31 (10 years ago)
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u/erica/UniformCollapse
v27 v28 8 8 This acceleration is due to all of the mass ''inside'' of the sphere, i.e. the mass contained within a radius r, [[latex($M_r$)]]. Since this equation governs the local acceleration of the gas at any radius, we can consider how the acceleration of any shell within the sphere behaves over time, once we know r(t) of course. Initially (t=0), however, we do know the radius, so we can check the behavior of this equation at t=0 for various shells within the sphere. 9 9 10 Consider first the outer most radius. A sphere with radius [[latex($r=r_0$)]] has within it mass that goes like [[latex($M_{r0} \propto r_0^3$)]]. Therefore, we see that initially, the acceleration of the gas is proportional to [[latex($r_0$)]], that is, [[latex($\frac{dr^2}{dt^2}\propto r_0$)]]. The density of the sphere is uniform, and this means repeating this procedure for any smaller radii shows that the acceleration of interior shells decreases as r decreases. In other words, the furthest shells have the highest acceleration . They speed up from an initial velocity of 0 the fastest. This has to be the case, as10 Consider first the outer most radius. A sphere with radius [[latex($r=r_0$)]] has within it mass that goes like [[latex($M_{r0} \propto r_0^3$)]]. Therefore, we see that initially, the acceleration of the gas is proportional to [[latex($r_0$)]], that is, [[latex($\frac{dr^2}{dt^2}\propto r_0$)]]. The density of the sphere is uniform, and this means repeating this procedure for any smaller radii shows that the acceleration of interior shells decreases as r decreases. In other words, the furthest shells have the highest acceleration-- they speed up from an initial velocity of 0 the fastest. This has to be the case, as 11 11 12 12 - all shells reach the origin at the same time (so outer shells have furthest to go in the same amount of time, i.e. they must travel faster) 13 - shells do not cross over the collapse 13 - shells do not cross over the collapse, 14 14 - which means that the mass contained within each concentric sphere remains a constant over the collapse 15 15 16 This last point is useful for integrating the equation. Since the mass is constant in any shell ( i.e. shell by the way here is used to mean concentric sphere), we can replace [[latex($M_r$)]] in the equation by [[latex($M_r = \frac{4}{3} \pi r_0^3 \rho_0$)]]. Integrating this equation under the boundary condition: the initial velocity [[latex($\frac{dr}{dt}=0$)]] at the initial radius of the sphere [[latex($r=r_0$)]], leads to an equation that describes the radius of the outersphere over time (i.e. over the course of collapse), and the velocity at this radius over time. We will look at these equations next.16 This last point is useful for integrating the equation. Since the mass is constant in any shell (by the way, 'shell' here is analogous to a 'concentric sphere'), we can replace [[latex($M_r$)]] in the equation by [[latex($M_r = \frac{4}{3} \pi r_0^3 \rho_0$)]] when integrating this equation for the entire sphere. Integrating this equation under the boundary conditions: the initial velocity is zero, [[latex($\frac{dr}{dt}|_{t=0}=0$)]], and this occurs when [[latex($r=r_0$)]], leads to an equation that describes the position of the radius of the sphere over time (i.e. over the course of collapse), and the velocity at this radius over time. We will look at these equations next. 17 17 18 18 19 19 '''Position equation''' 20 20 21 Integrating the above equation twice with a change of variables yields the equation of motion :21 Integrating the above equation twice with a change of variables yields the equation of motion. 22 22 23 23 [[latex($\xi + \frac{1}{2} \sin(2\xi) = kt$)]] … … 31 31 [[latex($\frac{r}{r_0}=\cos^2(\xi)$)]] 32 32 33 Now, that equation of motion is a real hassle to deal with, so numerical solution is necessary. This, we can use mathematica for to find the roots of this equation in terms of [[latex($\xi,t$)]]. Once we have [[latex($\xi,t$)]] pairs, we can take the [[latex($\cos^2(\xi)*r_0$)]] to get a list of values for r fora list of discrete t.33 Now, that equation of motion is a real hassle to deal with, so numerical solution is necessary. This, we can use mathematica for to find the roots of this equation in terms of [[latex($\xi,t$)]]. Once we have [[latex($\xi,t$)]] pairs, we can take the [[latex($\cos^2(\xi)*r_0$)]] to get a list of values for r corresponding to a list of discrete t. 34 34 35 35 [[latex($\boxed{r(t) = r_0 \cos^2(\xi(t))}$)]] 36 36 37 These lists can then be plotted in mathematica :37 These lists can then be plotted in mathematica. For example, here is the position over time of some concentric spheres of the same initial density: 38 38 39 39 [[Image(position.png, 40%)]] 40 [[br]]'''''Radius of concentric spheres over time.''''' '''This plot shows the size of the outer radius of a collapsing sphere over time. The blue line shoes the outer most radius, [[latex($r_0=10$)]], the yellow is a sphere of same initial density but half the radius, and so on. From this plot, we can see that the shells do not cross over time, that the collapse speeds up over time, and that they all collapse to 0 radius in a free fall time.40 [[br]]'''''Radius of concentric spheres over time.''''' '''This plot shows the radius of a collapsing sphere over time. The blue line shows the radius of a sphere with initial radius [[latex($r_0=10$)]], the yellow line is a sphere of same initial density but half the radius (and thus can be envisioned as a smaller, inner, concentric sphere), and so on. From this plot, we can see that the shells do not cross over time, that the collapse speeds up over time, and that they all collapse to 0 radius in a free fall time. 41 41 42 Next, by using the values of r(t),t in the velocity equation, we can seestudy the velocity profiles of collapsing uniform spheres.42 Next, by using the discrete values of r(t) and t in the velocity equation, we can study the velocity profiles of collapsing uniform spheres. 43 43 44 44 '''Velocity equation''' … … 46 46 [[latex($\boxed{\frac{dr}{dt} = -\sqrt{\frac{8\pi}{3} G \rho_0 r_0^2(\frac{r_0}{r(t)}-1)}}$)]] 47 47 48 - describes the outer velocity of a sphere that contains mass M_ras a function of r(t), where r(t) is a list of discrete radii as described above48 - describes the outer velocity of a sphere that contains mass [[latex($M_r$)]] as a function of r(t), where r(t) is a list of discrete radii as described above 49 49 - at radius [[latex($r = r_0$)]], [[latex($\frac{dr} {dt} = 0$)]] by construction 50 50 - if r were decreasing linearly over time, then plotting v over time on a linear scale would show that v decreases as the square root. However, a look at r vs. t shows that r does not decrease linearly, so v does not strictly go at the negative square root.
