Where do galaxies get their spin from?

They got it from their mama

A story of angular momentum,

tracer particles and data analysis

$z ≥ 2$: some puzzle(s) of galactic disks

Alignment at large scales
→ common origin of gal. spins
Spin of DM halo ≠ spin of disk
→ diff. evol. of DM vs baryons
Virial shock → should prevent coherent accretion
Feedback → disk destruction?

Filamentary accretion is responsible for most of the mass and angular momentum acquisition (at z>3 ; Kereš+05, Pichon+11, Tillson+12, …)

Filamentary accretion: natural "bridge" between large scale structures (cosmic web) and galaxy formation

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

easy, $\sum m \mathbf{r}\times \mathbf{v}$
Lagrangian trajectories
temperature along Lagrangian traj.
easy, right? $\sum \mathbf{r}\times \mathbf{f}$
$\nabla P, \nabla \phi$
(grav. accel. provided by simulation)

Tracer particles

Eulerian method

Lagrangian history

Tracer particles!

Eulerian method

How to access Lagrangian history in Eulerian simulation?

Lagrangian history

Tracer particles!

$v$-based tracers

Using linear interpolation of velocity

MC-based tracers

\[p_{i,j} \propto \mathrm{flux}(i\rightarrow j)\]

Bonus: also work for SF, feedback, …

Eulerian method

How to access Lagrangian history in Eulerian simulation?

$v$-based tracers

Using linear interpolation of velocity

MC-based tracers

\[p_{i,j} \propto \mathrm{flux}(i\rightarrow j)\]

Bonus: also work for SF, feedback, …

Reference

Difference

$v$-based tracers

Using linear interpolation of velocity

MC-based tracers

\[p_{i,j} \propto \mathrm{flux}(i\rightarrow j)\]

Bonus: also work for SF, feedback, …

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

✅ easy, $\sum m \mathbf{r}\times \mathbf{v}$
❔ Lagrangian trajectories
❔ temperature along Lagrangian traj.
❔ easy, right? $\sum \mathbf{r}\times \mathbf{f}$
❔ $\nabla P, \nabla \phi$
(grav. accel. provided by simulation)

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

✅ easy, $\sum m \mathbf{r}\times \mathbf{v}$
✅ Lagrangian trajectories
❔ temperature along Lagrangian traj.
❔ easy, right? $\sum \mathbf{r}\times \mathbf{f}$
❔ $\nabla P, \nabla \phi$
(grav. accel. provided by simulation)

Grid data

Particle data

Easy!

Use cic/nearest interpolation

yt.add_deposited_particle_fields(
    ("tracer", "particle_mass"),
    method="cic"
)
p = yt.ProjectionPlot(
  ds,
  ("deposit", "tracer_cic_mass")
)

Grid data

Particle data

(Now it is) easy!

Use "mesh deposition"

ds.add_mesh_sampling_particle_field(
  ("gas", "density"),
  "tracer"
)
ds.r["tracer", "cell_gas_density"]
# Took 1.4s for 1 090 895 particles!

p = yt.ParticleProjectionPlot(
  ds, 
  ("tracer", "cell_gas_density")
)

pos = ds.r["tracer", "particle_position"][:1000]
new_data = ds.arr([
  ds.r[p]["density"] 
  for p in pos
])

# Argh! Took 3.5s for 1 000 particles!
# ETA: 1hr for all particles

Easy?

ds.add_mesh_sampling_particle_field(
  ("gas", "density"),
  "tracer"
)

# Precompute cell index
ds.r["tracer", "cell_index"]        # 1.3s

# Fast access!
ds.r["tracer", "cell_gas_density"]  # 300ms

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

✅ easy, $\sum m \mathbf{r}\times \mathbf{v}$
✅ Lagrangian trajectories
✅ temperature along Lagrangian traj.
❔ easy, right? $\sum \mathbf{r}\times \mathbf{f}$
❔ $\nabla P, \nabla \phi$
(grav. accel. provided by simulation)

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

✅ easy, $\sum m \mathbf{r}\times \mathbf{v}$
✅ Lagrangian trajectories
✅ temperature along Lagrangian traj.
✅ easy, right? $\sum \mathbf{r}\times \mathbf{f}$
❔ $\nabla P, \nabla \phi$
(grav. accel. provided by simulation)

$$ \nabla P =\frac{P_{i+1} - 2P_i + P_{i-1}}{2} $$

Gradient needs (at least) 3 cells in each direction,

but octrees only have 2!

Background: https://www.youtube.com/watch?v=Mf7P5hqD0eY

only 2 cells

$$ \nabla P =\frac{P_{i+1} - 2P_i + P_{i-1}}{2} $$

Gradient needs (at least) 3 cells in each direction,

but octrees only have 2!

Background: https://www.youtube.com/watch?v=Mf7P5hqD0eY

only 2 cells

Let us "extend" our octs with "ghost zones"

$$ \nabla P =\frac{P_{i+1} - 2P_i + P_{i-1}}{2} $$

Gradient needs (at least) 3 cells in each direction,

but octrees only have 2!

Background: https://www.youtube.com/watch?v=Mf7P5hqD0eY

now 4 cells wide!

Let us "extend" our octs with "ghost zones"

$P_{i+1}$

$P_{i-1}$

$P_{i}$

$$ \nabla P =\frac{P_{i+1} - 2P_i + P_{i-1}}{2} $$

Gradient needs (at least) 3 cells in each direction,

but octrees only have 2!

Background: https://www.youtube.com/watch?v=Mf7P5hqD0eY

now 4 cells wide!

Let us "extend" our octs with "ghost zones"

$P_{i+1}$

$P_{i-1}$

$P_{i}$

I am only showing 6 out of 56 ghost cells for clarity.
But trust me they are here.
You can also use more than 2-cells (for higher order derivatives).

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

What to do?

Study AM kinematics,
in accretion flows,
both cold and hot,
dynamics: compute torques,
here, pressure & gravitational torques.

✅ easy, $\sum m \mathbf{r}\times \mathbf{v}$
✅ Lagrangian trajectories
✅ temperature along Lagrangian traj.
✅ easy, right? $\sum \mathbf{r}\times \mathbf{f}$
✅ $\nabla P, \nabla \phi$
(grav. accel. provided by simulation)

Inspecting Kinematics

Plot of $ \frac{1}{\Delta r}\int_{r}^{r+\Delta r} \mathrm{d}r\,\mathbf{r} \times \mathbf{v}$

Large scale

Galaxy

Direction of accretion

drop by $\times 5$

❄️

🔥

So the AM drops by $\times 5$

(in one free-fall time)

(Local) dynamics

The AM dynamics is given by

$\displaystyle \phantom{\frac{\mathrm{d} \mathbf{l}}{\mathrm{d} t}} + \text{gravitational torques} $

$ \displaystyle \frac{\mathrm{d} \mathbf{l}}{\mathrm{d} t} = \textcolor{darksalmon}{\text{pressure torque}} $

$\displaystyle \phantom{\frac{\mathrm{d} \mathbf{l}}{\mathrm{d} t} + } \underbrace{\phantom{\text{gravitational torques}}}_{\text{\textcolor{darkgray}{DM} + \textcolor{orange}{Star} + \textcolor{deepskyblue}{Gas}}} $

Pressure dominates the ⊥ acceleration in CGM…

locally!

Mean of ⊥ force magnitude