Can SPH correctly solve for hydrostatic

equilibrium? Molly Peeples The Ohio State University Center for Cosmology and Astro-Particle Physics June 16, 2010 at the 2010 Great Lakes Cosmology Workshop with David Weinberg, Neal Katz, Dušan Kereš, Romeel Davé, and Ben Oppenheimer

Two phase media & galaxy formation:

cold mode gas accretion (Kereš et al. 2005,2009)

galaxy winds (Oppenheimer et al.)

plus forming cold clouds via thermal instabilities, stripping gas from infalling satellites, star formation in cooling gas clouds... c.f. morning talks by Tremonti, Feldman, Leitner, Nickerson...

Can SPH correctly solve for hydrostatic

equilibrium? Molly Peeples The Ohio State University Center for Cosmology and Astro-Particle Physics June 16, 2010 at the 2010 Great Lakes Cosmology Workshop with David Weinberg, Neal Katz, Dušan Kereš, Romeel Davé, and Ben Oppenheimer

The setup: spherical cold blob initially in hydrostatic equilibrium with hot ambient medium

Ambient medium: •  T = 106K • 100ρamb=ρblob

Spherical blob: •  T = 104K

(ρT)amb=(ρT)blob

Let it evolve: blob oscillates, eventually heats up and contracts slightly

Let it evolve: blob oscillates, eventually heats up and contracts slightly blob becomes over-pressured

distance from blob center

log

[de

nsity

* te

mp

era

ture

]

Pressure difference as a function of time ... blob is oscillating but still over-pressured

Resolution: Blob is even more over-pressured at lower resolutions, but oscillations damp out more quickly

So what’s going on here?

 SPH force calculations assume smooth density gradients within smoothing kernels

 sharp boundary violates this!

 leads to numerical surface tension from low-density particles near boundary having their densities over-estimated

Agertz et al. (2007)

gap forms from sharp change in smoothing lengths

relative pressure SPH (rpSPH): possible solution?

€

d v d t

= −

∇ Pρ

€

d v d t

= − ∇

Pρ

⎛

⎝ ⎜

⎞

⎠ ⎟ +

Pρ2 ∇ ρ

⎡

⎣ ⎢

⎤

⎦ ⎥

versus

Gadget-2 (and most other SPH codes)

rpSPH (Abel 2010)

relative pressure SPH (rpSPH): possible solution?

€

d v id t

= − m j fiPi

ρi2 ∇ iWij (hi) + f j

Pj

ρ j2 ∇ iWij (h j )

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑

€

d v id t

= − m j fi

Pj − Pi

ρ j2 ∇ iWij (hi)

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑

versus Gadget-2

rpSPH

relative pressure SPH (rpSPH): possible solution?

€

d v id t

= − m j fiPi

ρi2 ∇ iWij (hi) + f j

Pj

ρ j2 ∇ iWij (h j )

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑

€

d v id t

= − m j fi

Pj − Pi

ρ j2 ∇ iWij (hi)

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑

versus Gadget-2

rpSPH

momentum: ✔ pressure: ✖

momentum: ✖ pressure: ✔

€

d v id t

= − m j fi

Pj − Pi

ρ j2 ∇ iWij (hi)

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑

relative pressure SPH (rpSPH): possible solution?

€

d v id t

= − m j fiPi

ρi2 ∇ iWij (hi) + f j

Pj

ρ j2 ∇ iWij (h j )

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑Gadget-2

rpSPH

versus

Bernoulli

Newton

momentum: ✔ pressure: ✖

momentum: ✖ pressure: ✔

€

d v id t

= − m j fi

Pj − Pi

ρ j2 ∇ iWij (hi)

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑

relative pressure SPH (rpSPH): possible solution?

€

d v id t

= − m j fiPi

ρi2 ∇ iWij (hi) + f j

Pj

ρ j2 ∇ iWij (h j )

⎡

⎣ ⎢

⎤

⎦ ⎥

j=1

N

∑Gadget-2

rpSPH

versus

momentum: ✔ pressure: ✖

momentum: ✖ pressure: ✔

rpSPH: Needs 128 neighbors to not crash... blob ends up slightly under-pressured... in regular SPH more neighbors means more over-pressured

rpSPH: Needs 128 neighbors to not crash... blob ends up slightly under-pressured... in regular SPH more neighbors means more over-pressured

rpSPH Gadget-2.0.4

Same initial conditions; Nngb = 128 ± 2