Changes between Version 4 and Version 5 of u/erica/scratch
- Timestamp:
- 01/27/16 16:05:54 (9 years ago)
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u/erica/scratch
v4 v5 1 From these plots (described in blog post), 2 1 3 [[Image(accretionluminosity1.png, 50%)]] 2 4 [[Image(accretionluminosity2.png, 50%)]] 3 5 6 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). 4 7 5 Having a kinetic energy (per unit mass) at r that is higher than would have been acquired from freefall alone (which would just be GM/r), i.e. solving the equations: 8 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), 9 6 10 7 11 [[latex($ke(r)-\frac{G*M}{r}+\frac{G*M}{R} = 1.01 (\frac{GM}{R})$)]] … … 9 13 [[latex($ke(r)-\frac{G*M}{r}+\frac{G*M}{R} = 1.3 (\frac{GM}{R})$)]] 10 14 11 again where the specific kinetic energy at the surface of the star (R) after having fallen from a distance r away from the center of the star is given by,15 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., 12 16 13 17 [[latex($ke(R)=ke(r)-\frac{G*M}{r}+\frac{G*M}{R} $)]] 14 18 15 19 20 Now, solving for r given various starting ke(r) and percents of error, we have, 16 21 17 22 || '''ke(r)''' || '''Error (%)''' || '''Distance (pc)'''|| … … 24 29 25 30 31 This shows that you get the greatest error closer into the surface of the star, and even when you are going very fast starting from a distance r = .002 you still only have a .01% error. This means that for any cells larger than this r, wouldn't expect great deviation, even in the extreme 1000*freefall case. 32 26 33 Now, what if a gas parcel started from rest, a distance r away from the star surface? Now we are solving, 27 34 … … 30 37 [[latex($-\frac{G*M}{r}+\frac{G*M}{R} = .70 (\frac{GM}{R})$)]] 31 38 39 for a .01 and 30% error. This gives the following distances, 40 32 41 || '''ke(r)''' || '''Error (%)''' || '''Distance (pc)'''|| 33 42 || 0 || .01 || 2.2*10^-6^ || 34 43 || 0 || 30 || 7.5*10^-8^ || 35 44 45 Again, the closer into the star we get, the worse the approximation gets -- but that a cell size of r~ 2.2*10^-6^ shouldn't produce much deviation. 36 46 37 Lastly, what if the parcel was moving, however, it was moving ''slower'' than freefall? Now ke(r) will be a fraction of the freefall energy in the table below. In particular, what if ke(r) was 1/2, 1/5, 1/10 the freefall kinetic energy... at what distance, r, would I see a 0.01% error? What about a 30% error? 47 48 Lastly, what if the parcel was moving ''slower'' than freefall? Now ke(r) will be a fraction of the freefall energy in the table below. In particular, what if ke(r) was 1/2, 1/5, 1/10 the freefall kinetic energy... at what distance, r, would we see a 0.01% error? What about a 30% error? 38 49 39 50 [[latex($ke(r)-\frac{G*M}{r}+\frac{G*M}{R} = .99 (\frac{GM}{R})$)]] … … 48 59 || 1/10*freefall || .01 || 2*10^-6 || 49 60 || 1/10*freefall || 30 || 6.8*10^-8^ || 61 62 63 This all shows us that '''the majority of energy gains from a stellar source occurs very close to the source''', and thus, using the equation of accretion luminosity for a particle starting from rest at infinity is a fine approximation.
