Changes between Version 6 and Version 7 of u/erica/scratch

Timestamp:

01/27/16 17:14:38 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/scratch

@@ -1 +1 @@
 These plots (described in blog post), 
 
+[[Image(accretionluminosity2.png, 50%)]]
 [[Image(accretionluminosity1.png, 50%)]]
-[[Image(accretionluminosity2.png, 50%)]]
+
 
 give an indication of how good or bad using the equation, 
@@ -13 +14 @@
 
 
-However, as these plots show, using the previous equation for different values of the kinetic energy at r, ke(r) = 1/2 mv^2^(r), does not produce wildly differing accretion energies at the stellar surface. Therefore using the simpler equation (the first one listed) is a fine approximation. Instead of reading off the error from these plots, I made a few tables for the different curves. This gives us a sense of how small a zone-size would have to be before we would start to expect deviations. Note that the kinetic energy at r which would have been acquired from freefall alone is just GM/r. 
+However, as these plots show, using the previous equation for different values of the kinetic energy at r, i.e. 1/2 mv^2^(r) = ke(r), does not produce wildly differing accretion energies until you move to the far left on the x-axis. In all but the most extreme cases, in fact, the degree of error is negligible (<1%), even at a distance of 1/100,000 of a parsec (i.e. much less than a typical cell size) away from the stellar surface. At that distance, it doesn't matter whether the gas parcel is starting from rest, moving slowly (less than freefall speed), or moving fast (up to 10x freefall speed), the accretion energy that parcel will release at the stellar surface ''is the same''.  This is a statement that ''most'' of the energy gained from gravitational infall occurs in the final legs of the journey. Therefore under most circumstance using the simpler equation (the first one listed) should be a *great* approximation. 
 
-First, let's consider cases where the kinetic energy was ''greater'' than GM/r. This corresponds to the lower plot. In this plot, the accretion energy (per unit mass) for a freely falling particle from infinity, GM/R, is normalized to 1 and lies exactly on top of the x-axis. This plot shows that particles starting at r with kinetic energies > GM/r, would produce stronger accretion energy than those which would have fallen in from freefall alone. As the speed increases, the produce greater and greater accretion energies. As the distance to the star decreases, they  I am next going to solve the following equations for r given various ke(r),
+To get a handle of the resolution where this approximation may break down, I made a few tables of error for some different scenarios. In what follows recall that the kinetic energy at r which would have been acquired from freefall alone is just GM/r. 
+
+First, let's consider ke(r) > GM/r. This corresponds to the upper plot. In this plot, the accretion energy (per unit mass) for a freely falling particle from infinity, GM/R, is normalized to 1 and lies exactly on top of the x-axis. This plot shows that particles falling in from r with ke(r) > GM/r, would produce stronger accretion energy than those which would have fallen in from freefall alone. As the speed increases, they would produce greater and greater accretion energies. I am next going to solve the following equations for r given various ke(r)>GM/r at the .01% error level and the 30% level,
 
 
@@ -22 +25 @@
 [[latex($ke(r)-\frac{G*M}{r}+\frac{G*M}{R} = 1.3 (\frac{GM}{R})$)]]
 
-Note the LHS of these equations give the (specific) accretion energy at the surface of the star, ke(R), from gas that fell from a distance r away, i.e., 
-
-[[latex($ke(R)=ke(r)-\frac{G*M}{r}+\frac{G*M}{R} $)]]
-
-Here is a table of error and r for various ke(r),
+Note the RHS is our approximation, i.e. the accretion energy from infinity, whereas the LHS of these equations give the assumed more 'realistic' accretion energy at the surface of the star, ke(R), due to gas haven fallen in from a distance r away. Here is a table of error and r for various ke(r)>GM/r,
 
 
@@ -38 +37 @@
 
 
-This shows that you get the greatest error closer into the surface of the star, and even when you are going very fast starting from a distance r = .002 you still only have a .01% error. This means that for any cells larger than this r, wouldn't expect great deviation, even in the extreme 1000*freefall case.
+This shows that you get the greatest error closer into the surface of the star, and even when you are going very fast starting from a distance r = .002 you still only have a .01% error. This means that for any cells larger than this r, we shouldn't expect great deviation, even in the extreme 1000*freefall case. 
 
 Now, what if a gas parcel started from rest, a distance r away from the star surface? Now we are solving,