Changes between Version 5 and Version 6 of u/erica/scratch

Timestamp:

01/27/16 16:28:24 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/scratch

@@ -1 @@
-From these plots (described in blog post), 
+These plots (described in blog post), 
 
 [[Image(accretionluminosity1.png, 50%)]]
 [[Image(accretionluminosity2.png, 50%)]]
 
-I can read off the percent error for using various equations to calculate the accretion energy. To get a sense of the error, lets consider a few different initial kinetic energies, ke(r), -- some less than and some greater than the kinetic energy which would have been acquired from freefall alone (i.e. GM/r). 
+give an indication of how good or bad using the equation, 
 
-First, I will consider cases where the kinetic energy was greater than GM/r. This corresponds to the lower plot. I can solve the following equations for r, given various ke(r),
+[[latex($L= \frac{G\dot{m} M}{R}$)]]
+
+is for the accretion luminosity. Given gas in the box has not traveled from infinity in free-fall to the sink surface, one might be inclined to instead use,
+
+[[latex($L= \frac{1}{2}m v^2(r) - \frac{GM \dot{m}}{r}+\frac{GM\dot{m}}{R}$)]] 
+
+
+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. 
+
+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),
 
 
@@ -13 +22 @@
 [[latex($ke(r)-\frac{G*M}{r}+\frac{G*M}{R} = 1.3 (\frac{GM}{R})$)]]
 
-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., 
+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),
 
-Now, solving for r given various starting ke(r) and percents of error, we have,
 
 || '''ke(r)''' || '''Error (%)''' || '''Distance (pc)'''||