Download The Jeans instability in smoothed particle hydrodynamics

Mon. Not. R. Astron. Soc. 296, 442-444 (1998)
The Jeans instability in smoothed particle hydrodynamics
A. P. Whitworth *
Department of Physics & Astronomy, University of Wales, Cardiff CF2 3YB
Accepted 1998 January 15. Received 1997 November 14; in original form 1997 August 27
ABSTRACT
We derive analytically the Jeans criterion for a gas simulated using an SPH code in which the
number of neighbours 3\f
- neighb is held constant (approximately) and the gravity-softening
length, E, equals the smoothing length, h (approximately). We show that the Jeans criterion is
reproduced accurately for resolved structures, i.e. those represented by > — neighb particles.
Unresolved structures are stabilized, as long as (i) the smoothing kernel W(u) is sufficiently
centrally peaked, and (ii) the Jeans mass is resolved. Provided that these conditions are
satisfied, then, in simulations of the formation of stars and galaxies, any fragmentation that
occurs should be both physical and resolved. In particular there should be no creation of subJeans condensations owing to numerical instability.
-
Key words: hydrodynamics – methods: numerical – stars: formation – galaxies: formation –
large-scale structure of Universe.
1 INTRODUCTION
Smoothed particle hydrodynamics (SPH) is a numerical method
developed initially by Lucy (1977) and Gingold & Monaghan
(1977). It is Lagrangian, involves no imposed grids or symmetry
constraints, and is therefore well suited to problems involving
complex geometry and large ranges of density and/or linear scale.
In combination with tree-based methods for calculating the gravitational field, it provides a powerful tool for simulating problems in
self-gravitating gas dynamics, and this is its main application in
astrophysics.
In particular, it is used to study the growth of large-scale structure
in the Universe, and hence to predict the distribution of galaxies
resulting from different cosmological models, to study the formation and evolution of galaxies and clusters of galaxies, to study star
formation, and to study interactions between stars (both impulsive
interactions and mass exchange). In many of these problems, it is
essential that Jeans instability be simulated properly. In other
words, Jeans-unstable condensations should condense out on the
appropriate time-scale, and Jeans-stable condensations should not
condense out. This paper is concerned with evaluating these
requirements.
Detailed descriptions of SPH are given in reviews by Benz (1990)
and Monaghan (1992). Contemporary SPH codes used for problems
involving gravitational fragmentation usually incorporate the following features, and in what follows we shall presume that they are
standard. (i) The smoothing kernel has finite extent and the particles
all have the same mass m. (ii) Each particle i has an individual
smoothing length h i , which evolves with time so that the number of
neighbours within its smoothing kernel is approximately constant
*E mail: [email protected]
-
at a prescribed (and universal) value 5\f
- neighb • (iii) For threedimensional simulations most codes use 40 Nneighb 60 .
-Knei ghb is chosen to give an acceptable compromise between
suppressing stochastic fluctuations (thereby approximating to the
continuum limit) and excessive smoothing (which would erase
realistic structures and introduce a high effective viscosity). (iv)
Each particle also has an individual gravity softening length Ei
which evolves with time in step with h i so that e i (t)lh i (t) = K,
where K is a constant of order unity. This is to ensure that the
gravitational acceleration agrav and the hydrostatic acceleration
ahy dro are softened/smoothed on the same scale. (v) Gravity is
either kernel-softened or Plummer-softened. In order to make the
discussion more specific, we shall concentrate on the kernelsoftening case, but the arguments apply equally well to both
cases. This is because in both cases, if the smoothing kernels of
two particles i and j overlap (i.e. h i + hi > S, where S is their
separation), their mutual gravitational attraction is reduced by a
factor of order [S/(h i h3 )] 3 , in accordance with Gauss's gravitational theorem.
For an SPH code with these features, the mass resolution of the
code is Mmin = neighbM. That is to say, only features involving
more mass than M min are resolved properly. Bate & Burkert (1997)
have demonstrated the spurious fragmentation that can occur if an
SPH code that does not impose E h is used to simulate a
fragmentation problem. They also demonstrated that, even with
E h, fragmentation is suppressed artificially in regions where the
local Jeans mass is smaller than Mmin , i.e. unresolved Jeansunstable condensations are stabilized numerically. The main result
of the present paper is to confirm these findings, and to evaluate how
they are affected by different choices of kernel function.
It is also possible that SPH is prone to a numerical instability
which results in the formation of condensations that (1) should be
.
© 1998 RAS
Jeans instability in SPH 443
Jeans-stable and/or (2) contain fewer than 3\r
, ighb particles. We
show that such condensations will not form, as long as (i) the
standard features listed above are implemented, (ii) the dimensionless smoothing kernel, W(u), is sufficiently centrally condensed,
and (iii) the Jeans mass is resolved, Mmin M.
Section 2 describes a simple derivation of the Jeans criterion,
based on the time-dependent virial theorem. Section 3 applies this
2
derivation to condensations in an SPH simulation, treating first
resolved condensations, and then unresolved ones. This allows us to
formulate the requirement for condensations that should be Jeansstable to be stable. Section 4 summarizes our conclusions.
2 JEANS INSTABILITY: BASIC
FORMULATION
Consider an extended medium having uniform density P o and
uniform isothermal sound speed a 0 . Now suppose that, owing to
some unspecified fluctuation, a spherical portion of the medium
having mass M0 attempts to condense out. We make the following
assumptions. (i) The mass of the proto-condensation, M 0 , is
constant. (ii) The proto-condensation remains spherical with
radius R, and hence mean density 0. = 3M0 /4'rrR 3 . (iii) The gas in
the proto-condensation responds isothermally, so the mean pressure
in the proto-condensation is P int = 3M0 aFi/41rR 3 . (iv) Radial excursions of the proto-condensation are sufficiently subsonic that the
gas outside the proto-condensation exerts an approximately constant external pressure text = p o a (2) on the boundary of the protocondensation.
Then if we neglect all purely numerical factors, the radial
acceleration at a representative point in the proto-condensation
comprises a hydrostatic contribution,
(minin 1/3
1/3
(mmin)
h
R<R.
Mc.
P )
Because h < R, pressure gradients in the proto-condensation are
well resolved, and the radial hydrostatic acceleration experienced
by a typical particle is again given by
2
ao ao po R
I?) hydro R
2
Mo
to lowest order in (hIR). There is a systematic fractional error in
P) hydro of order - (hIR) - (Mmin/Mo) 1/3 .
Because E h < R, gravity softening is also a small perturbation, and the radial gravitational acceleration experienced by a
typical particle is given by
GMo
gray
R2
to lowest order in (hIR). The systematic fractional error in R) gray is
of order - (h/R) 3 - (Mmin/Mo
Hence, to lowest order, the net radial acceleration is again given
by
a o [
(M0)-1R 2 (Mc) R
tFF VJ)
/1//j
but with a systematic
- (hIR)
//j
V?j)
fractional
error of order
- (Mmin /MO) 1/3
It follows that, if the condensations are well resolved,
Mo >> Mmin , Jeans instability is accurately reproduced, but the
growth-time of the instability is underestimated, by a factor
1 - 0 [ (Vinlin ) 1/3 1
MO
dP/dR
(Pext — Pint) /R
p
M/R3
P ) hydro —
smoothing lengths inside the proto-condensation are given by
aFi a (2) p 0 R 2
R
M0
Bate & Burkert (1997) attest that acceptable results are obtained
provided that Mo > 2M„ in .
and a gravitational contribution,
P) gray —
3.2 Unresolved condensations
GM°
R2
If we introduce the freefall time (for the background medium),
) —1/2
-,, the Jeans mass, Mj (a o tFF ) 3 p o , and the Jeans
tFF
(GPO
radius, Rj a O tFF , then the net radial acceleration becomes
(R) 2 (M0 ( R —2]
ao R)
tFF RJ
Mir
RJ
Mj i?j)
If we continue to neglect purely numerical factors, we find that
proto-condensations with M0 > Mj have R < 0 for all R, and are
therefore unstable against collapse, i.e. Jeans-unstable. Protocondensations with M0 < Mj have a stable equilibrium state (R = 0
and d.k/dR < 0) in which the density is only slightly above the
background, so these proto-condensations are stable. (They have a
second equilibrium state at higher density, but that one is unstable.)
An unresolved proto-condensation has M 0 < Mmin , and so its
particles must necessarily trawl extensively outside the condensation itself to find sufficient neighbours. To lowest order, and again
neglecting purely numerical factors, the smoothing lengths inside
the proto-condensation are given by
(h 3 - R 3 ) p0 Mmin — Mo
whence
(Mmin)
[R3 ± lmin 43 )] 1/3
Po
h
1/3
PO
From this it follows that
2
Mj
aoh
GMmin Mmin
3/2
•
Next we define a blurring factor
3 JEANS INSTABILITY: SPH FORMULATION
3.1 Well-resolved condensations
A well-resolved proto-condensation has Mo >> Mmi n . Consequently,
its particles can - and usually do - find most of their neighbours
within the proto-condensation itself. This is particularly true if the
proto-condensation has developed a large density contrast relative
to the background. Therefore, to lowest order in (hIR), the
© 1998 RAS, MNRAS 296, 442-444
R
[1 + (Mm i n — Mo
Mj
_3 —1/3
R]
< 1.
Because h > R, pressure gradients in the proto-condensation are not
resolved. For a typical particle in the proto-condensation, the
pressure gradient is reduced (below the true value) by a factor of
order (Mo /Mmin ) B4 W( B), where W' -dW/du. The density is
reduced by a factor of order --- B3 . Hence P) hydro is moderated by a
factor of order (Mo /Mmin ) BW(B).
444 A. P. Whitworth
Because E h > R, gravity is softened inside the proto-condensation. With the assumption of kernel softening, the radial gravitational acceleration is moderated by a factor of order B 3 .
Thus the net radial acceleration becomes
R
(m0)-1 (R)
Rj
mj
ao [ (R Rj
tFF
X[
BW(B)] —
[ (
2]
mo)
Mmin
2
(kR )
„
(1)
X (B3)
The hydrostatic contributions [the contributions moderated by
the term (Mo /Mmin ) BW(B) act to stabilize the proto-condensation. On their own they lead to an equilibrium state in which there is
no density contrast between the proto-condensation and the background. It is the gravitational contribution (the contribution moderated by the term B3 ) that - if sufficiently large - leads to Jeans
instability. Therefore unresolved proto-condensations will be destabilized artificially if the moderating terms reduce the hydrostatic
contribution to R more than the gravitational contribution.
From equation (1), it is clear that the most critical case is the one
with the smallest (M o /Mmin ), i.e. just two SPH particles. For two
particles having separation S, their mutual (repulsive) hydrostatic
acceleration is
a o2 m
, S
A) hydro
W
hMminh)
(
and their mutual (attractive) gravitational acceleration is
Gm S
P) gray
h3 •
Neglecting rotation (which will be steadily damped by artificial
viscosity), the particles will cluster unless
(
s
)
—
1
W,
S'
> Gmmin
h
aFjh
Mmin
Mj
(2)
) 1/3
(Mmin
In SPH codes that (i) impose E h, (ii) adapt h to give an
approximately constant number of neighbours, and (iii) satisfy
inequality (4), unresolved proto-condensations are stabilized. Irrespective of whether they are Jeans-unstable or not, their contraction
is inhibited by the blurring of gravitational and hydrostatic forces.
Conversely, resolved proto-condensations reproduce the Jeans
instability acceptably - except in as much as the time-scale for
condensation may be underestimated in cases where the protocondensation is only just resolved. These results confirm the
findings of Bate & Burkert (1997). Given the properties of the
smoothing kernels employed in contemporary SPH codes, artificial
formation of condensations by numerical instability should be
effectively suppressed, as long as Mmin < Mj.
ACKNOWLEDGMENTS
I am grateful to Henri Boffin for drawing my attention to a flaw in an
earlier version of this paper, and to the referee, Matthew Bate, for
his useful comments.
< 1,
and, if they cluster, S/h will become smaller. Therefore, if we define
limit d In -(- 1
C
(3)
d ln(u)
u ►o
we can put
s
-W' (- ) h
Inequality (2) then becomes
i s\ (c-i)
iMmin \ 2/3
Mj
4 CONCLUSIONS
2/3
In a well-settled distribution, the distance S between nearest
neighbours satisfies
(h)
It follows that, if C > 1, artificial clustering of particles will be
endemic, unless Mmin < Mj. If C = 1, stability against artificial
clustering requires Mmin < Mj . If C < 1, artificial clustering of
particles will not occur as long as Mmin 3\111;giZ2MJ, or equivalently m Nne(ieib)/2*.
This is quite a stiff constraint. The commonly used M4 kernel
(Monaghan 1985) has C = 1, as do the Gaussian kernel (Gingold &
Monaghan 1977), the super-Gaussian kernel (Gingold & Monaghan
1982) and the quartic kernel used by Lucy (1977). Therefore, for
these kernels, our analysis suggests that stability against artificial
clustering can only be ensured if the Jeans mass remains resolved.
The exponential kernel (Wood 1981) and the kernel used by
Thomas & Couchman (1992) have C = 0, and our analysis suggests that these kernels should afford greater stability against
artificial clustering.
REFERENCES
Bate M. R., Burkert A., 1997, MNRAS, 288, 1060
Benz W., 1990, in Buchler J. R., ed., The numerical modelling of non-linear
stellar pulsations. Kluwer, Dordrecht, p. 269
Gingold R. A., Monaghan J. J., 1977, MNRAS, 181, 375
Gingold R. A., Monaghan J. J., 1982, J. Comput. Phys., 46, 429
Lucy L. B., 1977, AJ, 82, 1013
Monaghan J. J., 1985, Comput. Phys. Rep. 3, 71
Monaghan J. J., 1992, ARA&A, 30, 543
Thomas P. A., Couchman H. M. P., 1992, MNRAS, 257, 11
Wood D., 1981, MNRAS, 194, 201
(4)
This paper has been typeset from a TEX/L A TEX file prepared by the author.
© 1998 RAS, MNRAS 296, 442-444