Changes between Version 9 and Version 10 of u/erica/scratch
- Timestamp:
- 01/27/16 21:28:51 (9 years ago)
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u/erica/scratch
v9 v10 1 1 These plots (described in blog post), 2 2 3 [[Image(accretionluminosity2.png, 50%)]] 4 [[Image(accretionluminosity1.png, 50%)]] 3 [[Image(accretionluminosity2.png, 35%)]] 4 5 [[Image(accretionluminosity1.png, 35%)]] 5 6 6 7 7 give an indication of how good or bad usingthe equation,8 give an indication of how good or bad of an approximation it is to use the equation, 8 9 9 10 [[latex($L= \frac{G\dot{m} M}{R}$)]] 10 11 11 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,12 for the expected accretion luminosity of cells surrounding a sink. Given gas in the box has not traveled from infinity in free-fall to the sink surface, one might be inclined to instead use, 12 13 13 [[latex($L= \frac{1}{2} mv^2(r) - \frac{GM \dot{m}}{r}+\frac{GM\dot{m}}{R}$)]]14 [[latex($L= \frac{1}{2}\dot{m} v^2(r) - \frac{GM \dot{m}}{r}+\frac{GM\dot{m}}{R}$)]] 14 15 15 16 16 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 circumstanceusing the simpler equation (the first one listed) should be a *great* approximation.17 which describes the luminosity due to gas falling in from a distance r away (instead of infinity). However, as these plots show, using this equation for different values of the kinetic energy at r (ke(r)), i.e. ke(r)=1/2 mv^2^(r), does not produce wildly differing accretion energies until you move to the far left on the x-axis. In fact, in all but the most extreme cases 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). 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 circumstances, using the simpler equation (the first one listed) should be a *great* approximation. 17 18 18 19 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. … … 37 38 38 39 39 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.40 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 pc you still only have a .01% error. This means that for any cells larger than this r, it virtually doesn't matter how fast the material is going (it can even be going 1000x freefall speed!!) -- it is going to end up with nearly the same accretion luminosity as if it traveled from infinity. Therefore, the approximation seems good over a wide range of potential speeds and down to very small distances away from the sink. 40 41 41 42 Now, what if a gas parcel started from rest, a distance r away from the star surface? Now we are solving, … … 51 52 || 0 || 30 || 7.5*10^-8^ || 52 53 53 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 .54 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 (<1%), assuming the material is moving slowly. 54 55 55 56 56 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?57 Lastly, what if the parcel was moving ''slower'' than freefall? In particular, what if ke(r) was 1/2, 1/5, 1/10 the freefall kinetic energy at r... at what distance would we see a 0.01% error? What about a 30% error? 57 58 58 59 [[latex($ke(r)-\frac{G*M}{r}+\frac{G*M}{R} = .99 (\frac{GM}{R})$)]] … … 68 69 || 1/10*freefall || 30 || 6.8*10^-8^ || 69 70 70 71 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 over many length scales (i.e. only breaks down once your cell size gets very small, because then the gas will not have traveled very far until it hits the stellar surface). As a caveat, the sink is so much less then a cell size and has no radius. Thus, these are all upper limits to the resolution requirements. 71 Again, we can expect gas falling in from something like 10^-6^ pc away from the sink to have enough time to gain a significant fraction of the accretion luminosity from infinity. Therefore, unless we are really modeling distances much smaller than this -- we have shown that really for a wide range of starting kinetic energies, the infinity luminosity should do just fine. As a caveat, the sink radius is so much less than a cell size (it is a sub-grid model, no way of telling its size other than it is small compared to a grid zone).. this implies that these are all ''upper'' limits to the resolution requirements. So basically, it makes sense to use L = G M m-dot/R.
