1.2. FUNDAMENTACIÓN TEÓRICA
1.2.5. Optimización de Recursos
Dust grains are modelled using Lagrangian tracer particles, where each simulated particle represents a collection of dust grains with similar properties and motions. Trajectories of the particles are integrated using the fully implicit method of Bai & Stone (2010), which we have incorporated into the VL integrator in Athena. In a Cartesian coordinate system, Athenasolves an equation of motion for each particle given by
dvi
dt =−
vi−u tstop
, (4.2.9)
with vi the velocity vector of particle i,u the local gas velocity vector, andtstop the particle
stopping time due to gas drag. Neglecting grain charges and assuming only pure hydrogen gas, the (collisional) drag law is given by (Draine & Salpeter, 1979)
dvi dt ≈ − 2πa2nk BT G0(s) (4/3)πρda3 , (4.2.10) with G0(s)≈ 8s 3√π(1 + 9π 64s 2 )1/2 (4.2.11) and s ≡(mHv 2 rel 2kBT )1/2, (4.2.12)
where a is the dust grain radius, kB is the Boltzmann constant, T is the temperature of the
constant at ρd = 3.0 g cm−3), mH is the mass of hydrogen, and vrel ≡vi−u is the relative
velocity difference between the dust and gas. The stopping distance is evaluated as
tstop = √ π 2√2 aρd n√mHkBT (1 + 9πmH 128kBT v2rel)−1/2. (4.2.13)
The gas properties (n,T,u) at each particle’s location are calculated from nearby grid points using a triangular-shaped cloud (TSC) interpolation scheme (Hockney & Eastwood, 1988). There is no momentum feedback from the particles on the gas.
Dust grain sizes
The drag force and the sputtering rates depend on the dust grain radius a. Since the size distribution of grains formed in SN ejecta is still a matter of debate (Clayton & Nittler, 2004; Bianchi & Schneider, 2007; Nozawa et al., 2007; Sarangi & Cherchneff, 2015; Marassi et al., 2015), we follow the approach of Ouellette et al. (2010) and implement an initial ‘distribution’ of four radii: a= 10, 1, 0.1, and 0.01µm. Each radius group is initialized with the same number of particles (Np= 105), and the sputtered mass from each radius group is
tracked using a separate passive scalar field (ρd, see Section 4.2.3).
Sputtering
The dust grains will be eroded by both thermal and non-thermal (kinetic) sputtering. We use sputtering rates estimated from the results of Nozawa et al. (2006), neglecting the slight differences in sputtering rate due to dust composition.
Non-thermal sputtering results from high-speed collisions of a dust grain with gas molecules and depends on the magnitude of the relative velocity |vrel| between the gas and the dust.
For simplicity, we adopt the polynomial fit of Ouellette et al. (2010, eqs. 13,14)2 to the
2Note that Ouellette et al. (2010) contain a typographical error in the definition of x; cm s−1 should be km s−1.
−9 −8 −7 −6 −5 −4 −9 −8 −7 −6 −5 −4 log {n −1 (da/dt)} [ µ m yr −1 cm 3 ] 4 5 6 7 8 9 10 1 2 3 4 5
log vrel [ km s−1 ] (non−thermal sputtering)
log T [ K ] (thermal sputtering) thermal non−thermal
Figure 4.2: Polynomial fits to the thermal (solid red line) and non-thermal (dashed blue line) sputtering rates, estimated from fig. 2 of Nozawa et al. (2006). The non-thermal sputtering varies with the relative velocity |vrel| between the dust and the gas (bottom axis), and the
thermal sputtering varies with the gas temperatureT (top axis). Both rates depend on the gas number density n and are given in volumetric units (µm yr−1 cm3).
non-thermal (kinetic) sputtering rates of Nozawa et al. (2006, fig. 2b):
yk=−0.1084x4k+ 1.7382x3k−10.5818x2k+ 28.1292xk−32.7024 (4.2.14)
with xk= log10(|vrel|/1 km s−1) and
(da
dt)k=−10
yk ( n
1 cm−3) µm yr
−1, (4.2.15)
with the velocity difference between the dust and the gas|vrel|in km s−1, and the gas number densityn. Fig. 4.2 shows the volumetric non-thermal sputtering raten−1(da/dt) (solid blue
line) as a function of the relative velocity |vrel| (bottom axis).
Thermal sputtering is due to the thermal motion of the gas and depends on the tem- perature T. Similar to the procedure used by Ouellette et al. (2010) for the non-thermal sputtering rate, we generate an average fit to the thermal sputtering rates of Nozawa et al. (2006, fig. 2a) with the polynomial
yt=−0.001911x4t + 0.12275x 3
t −2.4011x 2
with xt = log10(T /1 K) and (da dt)t =−10 yt ( n 1 cm−3)µm yr −1. (4.2.17)
Fig. 4.2 also shows the volumetric thermal sputtering rate n−1 (da/dt) (dashed red line) as a function of the temperature T (top axis).
We treat thermal and kinetic sputtering independently, adding the contributions to determine the erosion. However, the thermal motions of the gas will skew the relative velocity difference between the dust and the gas, particularly at high temperatures. We note that the more detailed treatment of Bocchio et al. (2014) leads to slightly lower sputtering rates in the high temperature regime, suggesting that our sputtering rates are overestimated and hence our injection efficiencies should be considered lower bounds in this regard.
The erosion rates (equations 4.2.15 and 4.2.17) are applied at first order via operator splitting. A particle is assumed to be completely destroyed when its radius decreases to 10−4 µm. As the particles are eroded, they release SLRs back into the gas phase. To
continue tracking the sputtered SLRs in gas phase, we deposit the sputtered dust mass into a passive density fieldρd3. This field is initially set to zero and is advected with the gas. Each
initial grain radius group has its own unique passive density field. The mass is distributed into nearby cell-centered field locations using the same TSC interpolation scheme used to determine gas properties (Hockney & Eastwood, 1988).
At each time-step, the mass lost by each particle is given by
∆Mp =
4πρp
3 [a
3−(a−∆a)3], (4.2.18)
where a is the current grain radius, ∆a = [(da/dt)k+ (da/dt)t] dt is the total change in
radius due to both non-thermal and thermal sputtering,Mp is the mass of each particle, and 3Note that this is a passive density rather than a concentration (i.e., color) field. The density is a conserved
ρp is the density of each particle. It is important to note that ρp 6=ρd, as each particle in
Athenarepresents a collection of many individual dust grains. As the density of each dust
grain is fixed at ρd = 3.0 g cm−3, the exact number of dust grains per particle depends on
the dust mass and the number of particles used; e.g. for 1 M of 1 µm dust distributed in 105 particles, each particle represents ∼1039 dust grains. For simplicity, we normalize such
that each particle has an initial mass of unity. Hence, if every particle of a given radius group is completely destroyed, the total mass of the passive density field is Np. With this
simplification, ρp = 3/(4πa30), where a0 is the original radius of the particle, and
∆Mp =
a3−(a−∆a)3 a3
0
. (4.2.19)
Verification of dust dynamics and sputtering
We have modified the Lagrangian tracer particles in Athena to include the drag force
given by Eq. 4.2.13 and the sputtering rates given by Eqs. 4.2.15 and 4.2.17. We verify the dynamics and destruction in a simple test case. A particle representing each radius group is initialized atx= 0 with vrel = 500 km s−1 in a uniform medium of densityn= 10 cm−3 and
temperature T = 105 K. Fora= 10, 1, 0.1, and 0.01 µm, the stopping times are 1.50×105, 1.5×104, 1.5×103, and 1.5×102 years, respectively. Figure 4.3 compares the evolution of the position, velocity, and radius of the four particles in Athena to the exact solution. Overall, the implementation in Athena performs well, with an rms error in the position
of 1.4×10−2 for the largest particles. The rms error increas with decreasing radius, up to
0.84 for the smallest grains, but these grains are less important for enrichment as they stop rapidly and carry the least SLR mass.
−5 0 5 10 15 20 −5 0 5 10 15 20 x (pc) 10 µm 1 µm 0.1 µm 0.01 µm 0 100 200 300 400 500 v (km s −1 ) 0.00 0.01 0.02 0.03 0.04 t (Myr) 10−7 10−6 10−5 10−4 10−3 a (cm)
Figure 4.3: Time evolution of the position x, velocity v, and radius a of dust particles in a simple test case. The dust radius is indicated by color. The exact solution is given by the solid line, and results from Athena are overlaid as open squares. Overall, our
implementation in Athenafollows the dynamics and sputtering to high accuracy; only the smallest (a = 0.001µm) grains show disagreement, but as these grains have little mass and are rapidly stopped they contribute very little to the enrichment.