Changes between Version 28 and Version 29 of u/erica/UniformCollapse
- Timestamp:
- 06/29/15 16:34:47 (10 years ago)
Legend:
- Unmodified
- Added
- Removed
- Modified
-
u/erica/UniformCollapse
v28 v29 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 corresponding to a list of discrete t.33 Now, that equation of motion is a real hassle to deal with, and a numerical solution is necessary (i.e. a closed form solution DNE). Thus, we must find discrete solutions to this equation. For this, we can use mathematica 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($\boxed{r(t) = r_0 \cos^2(\xi(t))}$)]] … … 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 [[latex($M_r$)]] as a function of r(t), where r(t) is a list of discrete radii as described above48 - describes the velocity of a sphere at its outer most edge 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. … … 52 52 53 53 [[Image(velocityplot.png, 40%)]] 54 [[br]]'''''Velocity of outer radius over time.''''' '''This plot shows that for initial concentric spheres of the same density, by varying outer radius, spheres with larger radii have steeper velocity profiles over time. Note this velocity is the velocity out the outer edge of the sphere. The free-fall time here, as can be seen where all of the curves begin to asymptote, is about [[latex($t_{ff} \approx .24$)]]. The smallest sphere plotted has an initial radius of [[latex($\frac{1}{1000}r_0$)]] (this is the green line), whereas the velocity of the outer most sphere within initial radius [[latex($r_0$)]] is the blue line. Note that this shows that all shells continue to accelerate over time, that the furthest away shellshave the highest initial acceleration, all consistent with the discussion above.54 [[br]]'''''Velocity of outer radius over time.''''' '''This plot shows that for concentric spheres of the same initial density, spheres with larger radii have steeper velocity profiles over time (again, this is the velocity at the outer edge of the spheres). The free-fall time here, as can be seen where all of the curves begin to asymptote, is about [[latex($t_{ff} \approx .24$)]]. The smallest sphere plotted has an initial radius of [[latex($\frac{1}{1000}r_0$)]] (this is the green line), whereas the velocity of the sphere with initial radius [[latex($r_0$)]] is the blue line. Note this shows all shells continue to accelerate over time and that the shells that are furthest away have the highest initial acceleration, all consistent with the discussion above. 55 55 56 Now, instead of seeing this plot in terms of time, we can instead put i n terms of radius, since r is a function of time.56 Now, instead of seeing this plot in terms of time, we can instead put it in terms of radius, since r is a function of time. Here that is shown for the sphere with initial radius, [[latex($r=r_0$)]]: 57 57 58 58 [[Image(vofr.png, 40%)]] 59 [[br]]'''''Velocity of outer radius as a function of the outer radius.''''' '''That is to say, "when the outer radius has fell to a new position of "r outer", the velocity of that outer radius is "v outer" How can you check this plot? You would look at the r vs. t plot, see at what time does the radius reach X, and then read off the velocity at that time. These numbers will match what you are seeing in this plot.'''59 [[br]]'''''Velocity of outer radius as a function of the outer radius.''''' '''That is to say, "when the outer radius has fell to a new position of "r outer", the velocity of that outer radius is "v outer"''' 60 60 61 Now, all this beckons -- what is the radial velocity profile of the collapsing sphere over time? This should be roughly vertical lines in the velocity of outer radius over time plots, and after some fancy coding in mathematica (see attached notebook if interested), one can arrive at this beauty: 61 How can you check this plot? You would look at the r vs. t plot, see at what time does the radius reach X, and then read off the velocity at that time from the v vs. t plot. These numbers will match what you are seeing in this plot. 62 63 Now, all this beckons -- what is the ''radial velocity profile'' of a collapsing sphere over time? This should be roughly vertical lines in the velocity of outer radius over time plots, and after some fancy coding in mathematica (see attached notebook if interested), one can arrive at this beauty: 62 64 63 65 [[Image(radialvelocity.png, 40%)]] 64 [[br]]'''''Radial velocity of collapsing uniform density sphere over time''''' '''At t=0, the sphere is static, and the velocity of the sphere is everywhere 0 (blue line). At the next time, we see that the radius of the sphere has shrunk, and that the velocity of the gas in the sphere is fastest at the outer edge, but goes to 0 linearly as you approach the origin (orange line). Each of these lines are evenly spaced in time, which shows the growing acceleration of the sphere. The red line is approaching the free fall time. As you can see from the velocity equation, at this point r->0, and we are getting a singularity in the velocity. This is why the line is tending toward an infinite slope. '''66 [[br]]'''''Radial velocity of collapsing uniform density sphere ([[latex($r=r_0$)]]) over time.''''' '''At t=0, the sphere is static, and the velocity of the sphere is everywhere 0 (blue line). At the next time, we see that the radius of the sphere shrank, and that the velocity of the gas in the sphere is fastest at the outer edge, but goes to 0 linearly as you approach the origin (orange line). Each of these lines are evenly spaced in time, which shows the growing acceleration of the sphere. The red line is approaching the free fall time. As you can see from the velocity equation, at this point r->0, and we are getting a singularity in the velocity. This is why the line is tending toward an infinite slope. ''' 65 67 66 68 … … 71 73 [[latex($\rho(t) = \frac{{M_r}}{\frac{4}{3}\pi r(t)^3}$)]] 72 74 73 Plotting this over time for spheres of various initial radii shows ,75 Plotting this over time for spheres of various initial radii shows: 74 76 75 77 [[Image(density.png, 40%)]] 76 [[br]]'''''Density of collapsing sphere over time.''''' '''Shown here are 2 overlaid initial spheres, one with radius [[latex($r=r_0=10$)]] and one with [[latex($r=r_0/2=5$)]]. They are exactly the same, so you only see one line.78 [[br]]'''''Density of collapsing sphere over time.''''' '''Shown here are 2 overlaid spheres, one with initial radius [[latex($r=r_0=10$)]] and one with [[latex($r=r_0/2=5$)]]. They are exactly the same, so you only see one line. That is, the density is constant throughout a collapsing uniform density sphere. 77 79 78 This plot shows that the entire interior of the sphere experiences a homologous (uniform) increase in density. The density increases everywhere exactly the same! Furthermore, the density increases by many orders of magnitude over the period of collapse. Assuming the collapse is isothermal (which is a good approximation over much of the collapse of protostars), then thisimplies that the Jeans mass decreases drastically over the period of collapse. This means that local regions of the collapsing sphere themselves will begin collapsing, and the whole cloud can be expected to fragment.80 This plot shows that the entire interior of the sphere experiences a homologous (uniform) increase in density. The density increases everywhere exactly the same! Furthermore, the density increases by many orders of magnitude over the period of collapse. Assuming the collapse is isothermal (which is a good approximation over much of the collapse of protostars), implies that the Jeans mass decreases drastically over the period of collapse. This means that local regions of the collapsing sphere themselves will begin collapsing, and the whole cloud can be expected to fragment. 79 81 80 82 '''Profiles together of a collapsing sphere''' 81 83 82 Here is a plot showing the profiles of a collapsing uniform sphere:84 Here is a plot showing the key temporal profiles of a collapsing uniform sphere: 83 85 84 86 [[Image(collapse.png, 40%)]]
