Table Of ContentAstronomy & Astrophysics manuscript no. bhlflow February 2, 2008
(DOI: will be inserted by hand later)
High Mach-number Bondi–Hoyle–Lyttleton flow around a small
accretor
1
Richard G. Edgar
Stockholms Observatorium, AlbanovaUniversitetscentrum,SE-106 91, Stockholm, Sweden
5
Received / Accepted
0
0
2 Abstract. In this paper, we discuss a two-dimensional numerical study of isothermal high Mach number Bondi–
Hoyle–Lyttleton flow around a small accretor. The flow is found to beunstable at high Mach numbers, with the
n
instability appearing even for a larger accretor. The instability appears to be the unstable radial mode of the
a
accretion column predicted by earlier analytic work.
J
5
Key words.Accretion, accretion disks – Hydrodynamics– Instabilities
1
v
0 1. Introduction studied, due to the largecomputationalexpenseinvolved.
7 Edgar & Clarke (2004) observed a flow instability while
0 The Bondi–Hoyle–Lyttleton (BHL) accretion problem
studing the effect of radiative feedback on Bondi–Hoyle–
1 concerns the accretion rate of a point mass moving at
Lyttleton flow, and this paper looks more closely at this
0 constantvelocitythroughauniformgascloud.Thisprob-
5 instability.Thecomputationalmethodisdiscussedinsec-
lem was first considered by Hoyle & Lyttleton (1939).
0 tion2andtheresultsinsection3.Finally,theconclusions
ConsideringNewtoniantrajectoriesinapressurelessfluid,
/ are presented in section 4.
h
Hoyle & Lyttleton concluded that material with an im-
p
pact parameter (ζ) less than
-
o 2. Method
2GM
r
ζ <ζ = (1)
st HL v∞2 The Zeus-2d code of Stone & Norman (1992) was used
a
to perform the calculations presented here. The author
: would accrete onto the point mass, M (v is the velocity
v ∞ parallelised the code using OpenMP, and found that it
of the gas relative to the point mass at infinity). If the
i
X scaled well to up to eight processors. Extra routines to
density of the gas is ρ (at infinity) then the accretion
∞ monitor accretion rates and drag were added to the main
r rate onto the point mass will be
a code, and calculations were performed in spherical polar
4πG2M2ρ (r,θ) co-ordinates.
M˙ =πζ2 v ρ = ∞ (2)
HL HL ∞ ∞ v3 A linear grid of 100 cells was used in the θ direction,
∞
while the radialgrid was logarithmically spaced. The size
For a full discussion of the problem, see Edgar (2004).
of the innermost radial cells was chosen to make the in-
Numericalstudies haveshown(see Edgar(2004),andref-
ner portions of the grid as ‘square’ as possible (to avoid
erences therein) that, although the accretion rate pre-
possible truncation errors). For the inner boundary con-
dicted by equation 2 is fairly accurate, the flow geome- dition, three alternatives were tried: Zeus’ own ‘outflow’
try is somewhat more complicated, due to the effects of
boundary, a constant (small) density, and the density set
pressure.
toasmallfractionoftheinnermost‘active’cell.Theouter
Bondi–Hoyle–Lyttleton accretion is of particular in-
boundary lay at 10ζ (as defined by equation 1) with a
HL
terest in star formation studies, since protostars in clus-
uniforminflowimposedontheupstreamside,andoutflow
ters are thought to undergo competitive accretion, at
conditions on the downstream side.
rates similar to that given by equation 2 (Bonnell et al.
The gridwas initialised to a uniform flow,and the ac-
2001). However, under such conditions, v is low, and
∞ cretor introduced over a number of timesteps at the start
so ζ is very large compared to the radius of the ac-
HL of the run. The simulations were run for as long as possi-
cretor (R ). This region of parameter space is poorly
acc ble, to ensure that any ‘switch-on’ effects died away. The
Send offprint requests to: Richard Edgar, e-mail: goal was a crossing time of the whole grid, but this was
[email protected] not possible for the runs with the small accretor.
2 R. G. Edgar: High M BHL Flow around a small accretor
Fig.1.AccretionratesasafunctionoftimeforlowMach
number flow around a large accretor
Fig.2.Detailoffigure1toshowdifferencesduetobound-
Inthesimulations,dimensionlessunits wereused.The
ary conditions
incoming material had v = 1, and the sound speed was
∞
set according to the required Mach number (M). The
point mass was set to M = 1 (as was the gravitational
constant, G). Using equation 1 we see that ζ = 2 for
HL
these choices. A value of ρ =0.01 was chosen (arbitrar-
∞
ily,sincetheflowwasisothermalandnotself-gravitating),
giving M˙ =0.126 in these units.
HL
3. Results
Three sets of simulations were performed. For eachof the
sets,onerunforeachoftheinnerboundaryconditionswas
made. First, a low Mach number (M = 3) flow around a
large(R =0.005ζ )accretorwasusedtotestthecode.
acc HL
The second set used the same size of accretor, but with
the Mach number increased to M=20. For the final set,
the accretor size was reduced to 0.0005ζHL. Fig.3. Gravitational drag force in units of M˙HLv∞ for
We will now discuss the resultsof eachof these sets in low Mach number flow around a large accretor
turn.
thissetofruns,wecanconcludethatthecodeisbehaving
3.1. Low Mach Number, Large Accretor
well.
The accretion rate as a function of time is presented in
figure1.Afteraninitialturn-onperiod,theaccretionrate
3.2. High Mach Number, Large Accretor
settles to a fairly steady value, slightly larger than that
predicted by equation 2. Changing the inner boundary The accretionrates for the three runs of high Mach num-
condition does not affect the results significantly, as is ber flow arounda large accretorare presented in figure 4.
shown in figure 2. The different boundary conditions are The behaviour is very different to that shown in figure 1,
givingslightlydifferentcurves,butareveryclosetogether. with a large oscillating accretion rate. We can see that
The gravitational drag force on the accretor for this changingtheboundaryconditiondoeschangetheflowhis-
setofrunsisplottedinfigure3.Thisisplottedinunitsof tory, but the change does not appear to be significant -
M˙ v whichis the naturalscalingfor the problem.The the instability is similar in all cases.
HL ∞
drag force rises smoothly to around 18M˙HLv∞. Figure 5 shows the corresponding gravitational drag
Inspection of the density field shows a wake with a for this set of runs. As might be expected from the accre-
fairlybroadopeningangle,butnobowshock-asistypical tion rate curves, it is quite noisy. Again, there are slight
for isothermalBHL flow.None of these results are partic- differencesduetotheinnerboundarycondition,butthese
ularly surprising, when compared to the work of Ruffert do not seem to be significant compared to the general in-
(1996)(model‘GS’isclosesttothatpresentedhere).From stability of the flow.
R. G. Edgar: High M BHL Flow around a small accretor 3
Fig.4.AccretionratesasafunctionoftimeforhighMach Fig.6. Sample radial velocity in the wake for high Mach
number flow around a large accretor number flow around a large accretor
trinsic to the flow along the line. Settling this dispute is
beyond the scope of the present work.
Looking at a single accretion rate history curve from
figure 4, around two to three oscillations occur per simu-
lationtime unit.Thisisinaccordancewiththeprediction
of Cowie, who pointed out that the instability must grow
faster than the characteristicaccretiontime (ζ /v =2
HL ∞
in this paper), or it will be swallowed by the accretor be-
fore it can grow. Examining the velocity structure of the
wake (figure 6 for a sample curve), oscillations in vr are
observed superimposed on the freefall curve, as predicted
byCowie.Theoscillationwavelengthdoesnotvarymuch,
as required by the WKB approximation. The stagnation
Fig.5.Gravitationaldrag(inunitsofM˙ v )asafunc- point is close to r =3.5,whichis unsurprising,giventhat
HL ∞ the accretion rates are slightly higher than M˙ (which
tion of time for high Mach number flow around a large HL
would place the stagnation point at r = 2 – cf Edgar
accretor
(2004)).
Unfortunately, computation of the power spectrum of
Thewake’sopeningangleismuchlessthanforthelow the accretionrates (figure 7) is rather unenlightening due
Mach number case, as one would expect. The oscillating to noise. The spectrum peaks at a frequency of around 2
accretionrateappearstobethe‘overstability’ofradialos- to3(itistoonoisytobemoreaccurate),andhasapower
cillationsdescribedbyCowie(1977)andSoker(1990).The lawfalloffathigherfrequencies.Theremayalsobe aside
analysiswas performed for a pressure-free(ballistic) flow, peak at a frequency of around 0.5, but this is only cov-
andthehighMachnumberisothermalnatureofthesecal- ered by a few points, and hence may just be noise. The
culationsisacloseapproximationofthiscondition.Cowie very intense bursts of accretion seen in figure 4 seem to
extendedtheanalysisofBondi & Hoyle(1944)(seeEdgar beassociatedwithlarger‘blobs’ofmatterbeingaccreted.
(2004) for a slightly more detailedexposition) to time de- However, these are relatively rare, with only a couple ap-
pendentflow.Perturbingthesteadystatesolutionandap- pearing over the entire duration of the simulation (and
plyingtheWKBapproximation(thatis,thewavelengthof one of those was probably a ‘switch-on’ transient).
the perturbation is assumed to be short compared to the
scale on which it varies), Cowie found that the accretion
3.3. High Mach Number, Small Accretor
column should be unstable. Soker showed that viscosity
couldsuppressthisinstability(whichhasnothappenedin Finally, we come to the simulations of high Mach num-
thepresentwork).However,theydisagreedonthephysical ber flow around a small (0.0005ζ ) accretor. These sim-
HL
natureoftheinstability:Cowieattributedittotheincom- ulations were painfully expensive computationally. Each
ingmaterial.Sokerassertedthattheequationshowingthe simulation in the first two sets required only about six
possibilty for instability (the dispersion relation from the days of wall time (on two processors). These final three
WKB approximation) did not contain terms relating to each occupied two processors for eight months, but still
the inflowing material, and hence the instability was in- did not achieve the same elapsed simulation time. As the
4 R. G. Edgar: High M BHL Flow around a small accretor
Fig.7. Sample accretion rate power spectrum for high Fig.9.Gravitationaldrag(inunitsofM˙ v )asafunc-
HL ∞
Mach number flow around a large accretor tion of time for high Mach number flow around a small
accretor
Table 1. Mean Accretion Rates in units of M˙ for high
HL
Machnumberflowfordifferentinnerboundaryconditions
Accretor Size Zeus Const. ρ Const. ρ Frac.
Large 1.47 1.50 1.48
Small 1.46 1.46 1.47
ble 1. Shrinking the accretor does not seem to have made
anysignificantdifferencetotheresults(beyondincreasing
the requiredcomputing resourcesby well overan orderof
magnitude).
4. Conclusions
Fig.8.AccretionratesasafunctionoftimeforhighMach
number flow around a small accretor Based on the results presented above, we see that high
MachnumberBondi–Hoyle–Lyttletonflowaroundasmall
accretor is unstable. Although the average accretion
gridisextendedtowardsthe pointmass,thegravitational rate is generally similar to that originally predicted by
accelerationgetsstronger.Therearemorecells,theveloc- Hoyle & Lyttleton (1939), shortincreases to rates several
ities are higher, and the cell size is smaller. These three timesthisarepossible.Theinstabilityseemstoberelated
effects combine to cause the consumption of prodigious to the high Mach number of the flow - at lower Mach
computing resources. numbers,pressuremusthavea stabilisingeffect. So far as
The accretion rate onto the point mass as a function these simulations can determine, the instability is the ra-
of time is plotted in figure 8. The behaviour is generally dialoscillationoftheaccretioncolumndescribedbyCowie
similar to figure 4, with large fluctuations once the flow (1977) and Soker (1990).
becomes established. There are differences in detail be- Observationally, the instability may be visible for
tween the three boundary conditions, but the qualitative deeply embedded young protostars.For a 1M⊙ protostar
behaviourisobviouslythe same.Figure9showsthe grav- orbiting in a protocluster, the time unit in the present
itational drag for this set of runs. Note that the gravita- paper would correspond to approximately one year. The
tional force reaches larger values - as one might expect fluctuations in the accretion rate would correspond to a
given the smaller size of the point mass (cf the Coulomb brightening of a factor of a few over a period of days to
logarithm which appears in the calculation of dynamical weeks,anddimmingoverasimilartimescale.Thisisquite
friction.Seee.g.Binney & Tremaine(1987)).Iftheflowis different from the FU Orionis phenomenon which is at-
dense enough to make M˙HL interestingly large, then this tributedtoaccretiondiscactivity(cfBell et al.(2000)and
dragforcewillalterv significantly.However,thegeneral referencestherein),wheretheoutburstsriseonatimescale
∞
patternissimilartofigure5,showingfluctuationscompa- of months and last for years.
rable to those of the accretion rate. In the author’s opinion, there is little to be gained
The mean accretion rates for the two sets of simula- by the further study of the pure Bondi–Hoyle–Lyttleton
tions with high Mach number flow are presented in ta- problem. Shrinking the accretor size revealed no new be-
R. G. Edgar: High M BHL Flow around a small accretor 5
haviour, at a tremendous cost in computer time. Instead,
theBondi–Hoyle–Lyttletonmodelshouldberetainedasa
useful ‘benchmark’ and work focused on improving simu-
lations of more realistic scenarios.
References
Bell, K. R., Cassen, P. M., Wasson, J. T., & Woolum,
D. S. 2000, Protostars and Planets IV, 897
Binney, J. & Tremaine, S. 1987, Galactic dynamics
(Princeton, NJ, Princeton University Press, 1987, 747
p.)
Bondi, H. & Hoyle, F. 1944, MNRAS, 104, 273
Bonnell,I.A.,Bate,M.R.,Clarke,C.J.,&Pringle,J.E.
2001, MNRAS, 323, 785
Cowie, L. L. 1977, MNRAS, 180, 491
Edgar, R. 2004,New Astronomy Review, 48, 843
Edgar, R. & Clarke, C. 2004, MNRAS, 349, 678
Hoyle,F. & Lyttleton,R. A. 1939,Proc.Cam.Phil. Soc.,
35, 405
Ruffert, M. 1996, A&A, 311, 817
Soker, N. 1990, ApJ, 358, 545
Stone, J. M. & Norman, M. L. 1992, ApJS, 80, 753
Acknowledgements. The author is supported by EU-RTN
HPRN-CT-2002-00308,“PLANETS”andthecalculationspre-
sentedherewereperformedonthemachinesofHPC2N,Ume˚a
and NSC, Link¨oping. Comments from an anonymous referee
helped improve thediscussion of theinstability
