Exact density-potential pairs from the holomorphic Coulomb field

0.36 MB·English

by  L. Ciotti

Mon.Not.R.Astron.Soc.000,000–000 (2007) Printed5February2008 (MNLATEXstylefilev2.2)
Exact density-potential pairs from the holomorphic
Coulomb field
Luca Ciotti1 and Giacomo Giampieri2
1Astronomy Department, University of Bologna, viaRanzani 1, 40127 Bologna, Italy
7 2Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA
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Accepted version,January12,2007

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ABSTRACT

2
1 We show how the complex-shift method introduced by Appell in gravity to the

case of a point mass (and applied among others in electrodynamics by Newman,
1
Carter, Lynden-Bell, and Kaiser to determine remarkable properties of the electro-
v
magneticfieldofrotatingchargedconfigurations),canbe extendedtoobtainnewand
8
explicitdensity-potentialpairsforself-gravitatingsystemsdepartingsignificantlyfrom
8
3 sphericalsymmetry.The rotationalpropertiesoftwoaxysimmetricbaroclinicgaseous
1 configurations derived with the proposed method are illustrated.
0
Key words: celestial mechanics – stellar dynamics – galaxies: kinematics and dy-
7
namics
0
/
h
p
-
o 1 INTRODUCTION with the Kutuzov-Osipkovmethod (1980, see also Kutuzov
r 1998).Anotherwaytoconstructinasystematicwayexplicit
t For the discussion of many astrophysical problems where
s density-potentialpairs with finite deviations from spherical
a gravity is important, a major difficulty is set by the po-
symmetry has been presented in Ciotti & Bertin (2005).
: tential theory. In general, to calculate the gravitational po-
v This technique is based on an elementary property of the
tential associated with a given density distribution one has
i asymptotic expansion of the homeoidal potential quadra-
X to evaluate a three-dimensional integral. Except for special tureformulaforsmallflattenings1,andrecently,ithasbeen
r circumstances,whereasolutioncanbefoundintermsofele-
applied to the modeling of gaseous halos in clusters (Lee &
a
mentaryfunctions,onehastoresorttonumericaltechniques
Suto 2003, 2004), to the study of the dynamics of elliptical
and sophisticated tools, such as expansions in orthogonal
galaxies (Muccione & Ciotti 2003, 2004), and finally to a
functions or integral transforms.
possible interpretation of the rotational field of the extra-
Under spherical symmetry, the density-potential rela-
planar gas in disk galaxies (Barnab`e et al. 2005, 2006).
tioncan bereducedtoaone-dimensional integral,while for
In the context of classical electrodynamics a truly re-
axisymmetric systems one in general is left with a (usually
markableand seamingly unrelatedresult hasbeenobtained
non-trivial) two-dimensional integral. As a result, the ma-
byNewman(1973,seealsoNewman&Janis1965,Newman
jority of the available explicit density-potential pairs refers
etal.1965), whoconsidered thecaseoftheelectromagnetic
tosphericalsymmetryandonlyfewaxiallysymmetricpairs
fieldofapointchargedisplacedontheimaginaryaxis.Suc-
areknown(e.g.,seeBinney&Tremaine1987,hereafterBT).
cessive analysis of Carter (1968), Lynden-Bell (2000, 2002,
Inspecialcases(inparticular,whenadensity-potentialpair
2004ab, and references therein; see also Teukolsky, 1973;
can be expressed in a suitable parametric form) there exist
Chandrasekhar, 1976; Page, 1976), and Kaiser (2004, and
systematic procedures to generate new non-trivial density-
referencestherein)revealedtheastonishingpropertiesofthe
potentialpairs(e.g.,seethecaseofMiyamoto&Nagai1975
resultingholomorphic”magic”field.Aspointedouttousby
andtherelatedSatoh1980disks;seealsoEvans&deZeeuw
the Referee, the complex-shift method was first introduced
1992;deZeeuw&Carollo 1996).Fornon-axisymmetricsys-
byAppell(1887, seealsoWhittaker& Watson1950) tothe
tems the situation is worse. One class of triaxial density
case of the gravitational field of a point mass, and succes-
distributions for which the potential can be expressed in a
sively used in General Relativity (e.g., see Gleiser & Pullin
tractable integral form is that of the stratified homeoids,
such as the Ferrers (1887) distributions (e.g., see Pfenniger
1984, Lanzoni & Ciotti 2003, and references therein) and
special cases of the family considered by de Zeeuw & Pfen- 1 Such expansion can be traced back to the treatise on geodesy
niger (1988). Additional explicit density-potential pairs are by Sir H. Jeffreys (1970, and references therein; see also Hunter
givenbytheEvans(1994) modelsandbythoseconstructed 1977).
2 Ciotti & Giampieri
1989;Letelier&Oliveira1987,1998;D’Afonseca,Letelier& theentire space
Oliveira, 2005 and references therein).
1
In this paper we show how the complex-shift method W ρ Φ d3x
c≡ 2 c c
can be also applied in classical gravitation to obtain exact Z
and explicit density-potential pairs deviating significantly = 1 [ (ρ ) (Φ ) (ρ ) (Φ )]d3x (5)
fromsphericalsimmetry;wearenotawarethatthispossibil- 2 ℜ c ℜ c −ℑ c ℑ c
Z
ityhasbeennoticedintheliterature.Thepaperisorganized
coincides with the self-gravitational energy W =
asfollows.InSect.2wepresenttheideabehindthemethod,
0.5 ρΦd3x of the real unshifted seed density. Thus the
and we discuss its application to the case of spherical sys-
imaginary part of W is zero, i.e.
tems,describing twonewself-gravitating and axisymmetric R c
systemsobtainedfrom thePlummerandIsochronespheres. 1
[ (ρ ) (Φ )+ (ρ ) (Φ )]d3x=
In Sect. 3 we focus on the rotational fields of thenew pairs 2 ℜ c ℑ c ℑ c ℜ c
wthheemnaininterrepsruelttesdobastaginaseedo,uwshsiylestienmtsh.eSAecptpioennd4ixsuwmempraersieznest GZ ℜ[ρc(x)]ℑ[ρc(x′)]d3xd3x′=0, (6)
− x x′
the peculiar behavior of the singular isothermal sphere un- Z Z || − ||
derthe action of thecomplex shift. and W is the difference of the gravitational energies of the
c
realandtheimaginarypartsoftheshifteddensity.Thevan-
ishing of the double integral (6) shows that the integrand
necessarily changes sign, i.e., the complex shift cannot gen-
erate two physically acceptable densities. Additional infor-
2 GENERAL CONSIDERATIONS
mationsare providedbyconsidering that,forthesamerea-
Westartbyextendingthecomplexificationofapointcharge sonsjust illustrated, alsothetotalmass ofthecomplexified
Coulomb field discussed by Lynden-Bell (2004b), to the distribution M = ρ d3x coincides with the total (real)
c c
gravitational potential Φ(x) generated by a density distri- mass of theseed density distribution M = ρd3x, so that
bution ρ(x). From now on x = (x,y,z) will indicate the R
position vector, while <x,y>≡xiyi is the standard inner (ρc)d3x=0. R (7)
product over the reals (repeated index summation conven- ℑ
Z
tion implied).
Identities (6)-(7) then leave open the question whether at
Let assume that the (nowhere negative) density distri-
least (ρ ) can be characterized by a single sign over the
bution ρ(x) satisfies thePoisson equation ℜ c
wholespace.Itresultsthateithercasescanhappen,depend-
2Φ=4πGρ, (1) ingon thespecificseeddensityandshift vectoradopted.In
∇
fact,inthefollowing Sectionweshowthatsimpleseedden-
andthattheassociated complexifiedpotentialΦc withshift sities exist so that (ρc) is positive everywhere, while in
ia is defined as the Appendix we preℜsent a simple case in which also (ρ )
c
ℜ
changes sign.
Φ (x) Φ(x ia), (2)
c
≡ −
where i2 = 1 is the imaginary unit and a is a real vector.
−
Theideabehindtheproposedmethodisbasedontherecog- 2.1 Spherically symmetric “seed” potential
nitionthat1)thePoissonequationisalinearPDE,andthat
We now restrict the previous general considerations to a
2) the complex shift is a linear coordinate transformation.
sphericallysymmetricrealpotentialΦ(r),wherer x =
From these two properties, and from equations (1) and (2) ≡|| ||
√<x,x> is thespherical radius, and ... is thestandard
it follows that || ||
Euclidean norm. After the complex shift the norm must be
2Φ =4πGρ , (3) stillinterpretedoverthereals2(e.g.,seeLynden-Bell2004b),
c c
∇
so that
where
x ia 2 =r2 2i<x,a> a2, (8)
ρ (x) ρ(x ia). (4) || − || − −
c
≡ − wherea2 a 2.Withoutlossofgeneralityweassumea=
Thus,byseparatingtherealandimaginary partsofΦc and (0,0,a),a≡nd||th||e shifted radial coordinate becomes
ρ obtainedfrom theshiftofaknownrealdensity-potential
pcair one obtains two real density-potentialpairs. rc2=r2−2iarµ−a2; µ≡cosθ, (9)
A distinction isin orderherebetween electrostatic and
where θ is the colatitude of the considered point, µr = z,
gravitational problems: in fact, while in the former case a
and
density (charge) distribution with negative and positive re-
gions can be (at least formally) accepted, in the gravita- Φ =Φ r2 2iaz a2 , (10)
c
− −
tional case the obtained density components have physical
(cid:16)p (cid:17)
meaning only if they do not change sign (see however Sect.
ρ =ρ r2 2iaz a2 . (11)
4). Quite interestingly, some general result about the sign c − −
of the real and imaginary parts of the shifted density can (cid:16)p (cid:17)
beobtainedbyconsideringthebehaviorofthecomplexified
self-gravitational energy and total mass. In fact, from the 2 Ifoneadoptthestandardinnerproductoverthecomplexfield,
linearityoftheshift,itfollowsthatthevolumeintegralover onewouldobtain||x−ia||2=r2+||a||2.
Exact density-potential pairs 3
Nsyostteemth,atth,ewrheaelnasntdaritminmgafgroinmaraysppahretsriocafltlhyessyhmifmteedtrdicensseietdy ℑ(ρc)= 3ℑ4(πΨc) 5ℜ(Ψc)4− 10ad42z2 + d8a4(zΨ4c)4 . (22)
(cid:20) ℜ (cid:21)
ρ canbeobtained1)fromevaluationoftheLaplaceopera-
c
We verified that the two new density-potential pairs (19)-
tor applied to thereal and imaginary partsof thepotential
(21) and (20)-(22) satisfy thePoisson equation (1).
Φ , 2) by expansion of the complexified density in equa-
c
Notethat,atvariancewith (Ψ ),thepotential (Ψ )
tion (11), or finally 3) by considering that for spherically ℜ c ℑ c
change sign crossing the equatorial plane of the system. In
symmetric systems ρ = ρ(Φ), and so the real and imagi-
accordance with this change of sign, also (ρ ) is negative
narypartsofρc(Φc)canbeexpressed(atleast inprinciple) ℑ c
for z < 0, as can be seen from equation (22), and thus
as functions of the real and imaginary part of the shifted
cannot be used to describe a gravitational system; we do
potential.
not discussthispair anyfurther,while we focus on thereal
components of the shifted density-potential pair. Near the
2.2 The shifted Plummer sphere centerd 1 a2+r2[1+2a2µ2/(1 a2)]+O(µ2r4),andthe
∼ − −
leadingtermsoftheasymptoticexpansionofequations(19)
Asafirstexample,inthisSectionweapplythecomplexshift
and (21) are
to the Plummer (1911) sphere. We start from the relative
potential Ψ= Φ, where 1 r2(1 a2+3a2µ2)
GM 1 − ℜ(Ψc)∼ √1 a2 − 2(−1 a2)5/2 +O(r4), (23)
Ψ= , ζ 1+r2, (12) − −
b ζ ≡ 3 15r2(1 a2+7a2µ2)
and r is normalizedptothemodelscale-lenght b,so that the ℜ(ρc)∼ 4π(1 a2)5/2− 8π(−1 a2)9/2 +O(r4);(24)
associated density distribution is − −
inparticular,theisodensesareoblateellipsoidswithminor-
ρ= 3M 1 (13) to-major squared axis ratio (1 a2)/(1+6a2). For r ,
4πb3ζ5 d r2+1 a2+2a2µ2+O(µ−2r−2),and →∞
∼ −
(neo.rgm.,asleiezeBdTd)e.nFsoitrye(atsoeMof/nbo3t)aatniodnp,ofrtoemntinaolw(toonGwMe/wb)il,launsde ℜ(Ψc)∼ 1r − 1−a22+r33a2µ2 +O(r−5), (25)
so
ρc = 43π[ℜ(Ψc)+iℑ(Ψc)]5. (14) ℜ(ρc)∼ 4π15r5 51 − 1−a22+r27a2µ2 +O(r−9), (26)
(cid:18) (cid:19)
From substitution (8) (where also the shift a is expressed
sothat (ρ )coincideswiththeunshiftedseed density(13)
c
in b units), the shifted potential Ψc =1/ζc depends on the and it iℜs spherically symmetric and positive3. Thus, near
square root of ζ2=1 a2+r2 2iaz deiϕ,with
c − − ≡ the center and in the far field (ρc) > 0 for 0 a < 1.
ℜ ≤
d ζ 2 = (1 a2+r2)2+4a2z2, (15) In addition, on the model equatorial plane z = 0 (where
≡| c| − d = 1 a2 +R2, and R is the cylindrical radius), (Ψ )
c
and p coincid−es with the potential of a Plummer sphere ofℜscale-
1 a2+r2 2az lenght √1 a2, and from equation (21) it follows that
cosϕ= − d , sinϕ=− d . (16) (ρc)=3 −(Ψc)5/(4π)>0 for 0 a<1.
ℜ ℜ ≤
However, for z =0 a negative term is present in equa-
Notethatcosϕ>0everywherefora<1,andinthefollow- 6
tion(21),andthepositivityof (ρ )isnotguaranteedfora
ing discussion we restrict to this case. Thesquare root ℜ c
generic valueof theshift parameter in the range 0 a<1.
ζc =√deπki+ϕi/2, (k=0,1), (17) Infact, anumericalexploration revealsthat (ρc)≤becomes
ℜ
negative on the symmetry axis R = 0 at z 0.81 for
is made a single-valued function of (r,z) by cutting the ≃
a = a 0.588; the negative density region then expands
complex plane along the negative real axis (which is never m ≃
touchedbyζ2)andassumingk=0,sothatthemodelequa- around this critical point for increasing a > am. The iso-
c
density contours of (ρ ) in a meridional plane are shown
torial planeismapped intotheline ϕ=0.With thischoice ℜ c
inthetoppanelsofFig.1,forshiftparametervaluesa=1/2
theprincipaldeterminationofΨ reducestoΨwhena=0,
c
(left) and a = 23/40 = 0.575 (right). The most salient
and simple algebra shows that
property is the resulting toroidal shape of the model with
ϕ √1+cosϕ ϕ √2az the large shift, which reminds similar structures known in
cos = , sin = . (18)
2 √2 2 −d√1+cosϕ the literature, as the Lynden-Bell (1962) flattened Plum-
mersphere,thedensitiesassociated withtheBinney(1981)
The real and imaginary parts of Ψ are then given by
c
logarithmic potential and the Evans (1994) scale-free po-
ℜ(Ψc)=ℜ ζζ¯cc2 = √d+1√−2da2+r2, (19) t(esenetiaallss,othCeioTttoio,mBreert(i1n9,8&2)LanodndCriilolotti2&004B)e,rtainnd(2t0h0e5e)xtaocrti
(cid:18)| | (cid:19)
MONDdensity-potentialpairsdiscussedinCiotti,Nipoti&
ζ¯ az
(Ψ )= c = , (20) Londrillo (2006). Unfortunately, we were not able to find
ℑ c ℑ(cid:18)|ζc|2(cid:19) d2ℜ(Ψc) theexplanation (if any) behind this similarity (see however
respectively, and from equation (14) we obtain the expres- Sect.4).
sions of the(normalized) axysimmetric densities:
ℜ(ρc)= 3ℜ4(πΨc) ℜ(Ψc)4− 10ad42z2 + d85a(4Ψz4c)4 , (21) 3beTsheeisnijsuastgbeyneerxaplapnrdoipnegrteyquoaftsihonift(e1d1)spfhorerric→al s∞ys.tems, as can
(cid:20) ℜ (cid:21)
4 Ciotti & Giampieri
3 3
2 2
1 1
z 0 z 0
-1 -1
-2 -2
-3 -3
-3-2-1 0 1 2 3 -3-2-1 0 1 2 3
R R
3 3
2 2
1 1
z 0 z 0
-1 -1
-2 -2
-3 -3
-3-2-1 0 1 2 3 -3-2-1 0 1 2 3
R R
Figure 1. Isodensity contours in the (R,z) plane of ℜ(ρc) of the shifted Plummer sphere for a = 1/2 (top left) and a = 23/40 (top
right),andoftheshiftedIsochronespherefora=1/2(bottom left)anda=4/5(bottom right).Thecoordinatesarenormalizedtothe
scale-lenghtbofthecorrespondingseedsphericalmodel.
2.3 The shifted Isochrone sphere ρ = Ψ2c 3+2r2Ψ . (29)
c 4πζ3 c c
c
Following the treatment of the Plummer sphere, we now
consider the slightly more complicate case of the shifted The real an(cid:0)d imaginar(cid:1)y parts of Ψ = (1+ζ¯)/[1+ ζ 2+
c c c
| |
Isochronesphere.Itsrelativepotentialanddensityaregiven 2 (ζ )] are easily obtained from equations (15)-(18) as
c
ℜ
by
1+√d+1 a2+r2/√2
GM 1 (Ψc)= − , (30)
Ψ= , (27) ℜ 1+d+ 2(d+1 a2+r2)
b 1+ζ −
M 3(1+ζ)+2r2 (Ψ )= p √2az ,(31)
ρ= 4πb3 (1+ζ)3ζ3 , (28) ℑ c d(d+1 a2+r2)[1+d+ 2(d+1 a2+r2)]
− −
whereζ isdefinedinequation(12),andagainalllenghtsare while (ρp) and (ρ ) can be obtainped by expansion of
c c
ℜ ℑ
normalizedtothescaleb(H´enon1959,seealsoBT).Inanal- equation (29); however their expression is quite cumber-
ogy with equation (14), the normalized shifted density can some, and so not reported here. Again, (Ψ ) changes sign
c
ℑ
bewritten as a function of thenormalized shifted potential when crossing the model equatorial plane, revealing that
Exact density-potential pairs 5
also ℑ(ρc) changes sign in order to produce the resulting ρvϕ2 = ∞ ∂ρ ∂Φ ∂ρ ∂Φ dz′. (37)
vertical force field near z = 0, and the vanishing of the in- R ∂R∂z′ − ∂z′∂R
tegral in equation (7). In accordance with equation (11) on Zz (cid:16) (cid:17)
Weremark that this “commutator-like” relation is not new
themodelequatorialplane (ρ )isfunctionallyidenticalto
c
ℜ (e.g., see Rosseland 1926, Waxmann 1978, and, in the con-
anIsochronesphereofscale-lenght√1 a2 (andsopositive
− text of stellar dynamics, Hunter1977).
for 0 a<1), while for r
≤ →∞ Before solving equations (34) for the real parts of the
ℜ(Ψc)∼ rr−21 + 1+a22−r33a2µ2 +O(r−4), (32) dbeansiscityp-rpoopteerntytiaolfpraoirtsatoinfgSeflcut.id2,ciotnifisguusreaftuiolntso.rFecoarllexsoamme-
ℜ(ρc)∼ 4π1r5 2r−3+4a21−r6µ2 +O(r−7). (33) pinleit,iainl cseovnedriatlioansstrfooprhyhsyicdarlodaypnpalimcaitciaolns(ime.gu.l,attihoenss)e,t-euqpuao-f
(cid:18) (cid:19) tions (34) are solved for the density under the assump-
As for the shifted Plummer sphere, also in this case (ρ ) tion of a barotropic pressure distribution (and neglecting
c
ℜ
coincides in the far field with the seed density, and so it is the gas self-gravity), and for assigned ρ(R, ) = 0 or
∞
positive for 0 a < 1, while it becomes negative on the z ρ(R,0)=ρ0(R).
≤
axis at z 0.648 for a > a 0.804. In the bottom pan- As well known, this approach can lead only to hydro-
m
≃ ≃
els of Fig. 1 we show the isodensity contours in the merid- static (Ω = 0) or cylindrical rotation (Ω = Ω[R]) kinemati-
ionalplaneof (ρ )fortherepresentativevaluesoftheshift calfields.Infact, accordingtothePoincar´e-Wavretheorem
c
ℜ
parameter a = 1/2 (left) and a = 4/5 (right): again, the (e.g., see Lebovitz 1967, Tassoul 1980), cylindrical rotation
toroidalshapeandthecriticalregionsonthesymmetryaxis is equivalent to barotropicity, or to the fact that the accel-
are apparent in thecase of the larger shift. eration field at the r.h.s. of equations (34) derives from the
effectivepotential
R
3 ROTATIONAL FIELDS Φeff Φ Ω2(R′)R′dR′ (38)
≡ −
ZR0
A natural question to ask is about the nature of the kine-
matical fields that can be supported by the found density- (where R0 is an arbitrary but fixed radius), so that thegas
potential pairs (Ψ ) (ρ ) when considered as self- density and pressure are stratified on Φeff.
c c
ℜ − ℜ However, barotropic equilibria are just a very special
gravitating stellar systems. In general, the associated two-
class of solutions of equations (34). For example, in the
integrals Jeans equations (e.g., BT, Ciotti 2000) can be
context of galactic dynamics isotropic axisymmetric galaxy
solved numerically after having fixed the relative amount
models often show streaming velocities dependenton z (for
ofordered streaming motionsandvelocity dispersion in the
simple examples see, e.g., Lanzoni & Ciotti 2003, Ciotti &
azimuthaldirection(e.g.,seeSatoh1980,Ciotti&Pellegrini
Bertin 2005). Such baroclinic configurations (i.e., fluid sys-
1996). Forsimplicity herewerestrict totheinvestigation of
temsinwhichP cannotbeexpressedasafunctionofρonly,
the isotropic case, when the resulting Jeans equations are
formallyidenticaltotheequationsdescribingaxysimmetric, and Φeff does not exist at all), have been studied in the
past for problemsranging from geophysics, tothetheoryof
self-gravitating gaseous systems in permanent rotation:
sunspotsandstellarrotation(e.g.,seeRosseland1926),and
1 ∂P ∂Φ
= , to the problem of modeling the decrease of rotational ve-
ρ ∂z −∂z locity of theextraplanargas in disk galaxies for increassing
 (34)
z (e.g., see Barnab´e et al. 2005, 2006). Note that a major
1 ∂P = ∂Φ +Ω2R, problem posed by the construction of baroclinic solutions
ρ ∂R −∂R
is the fact that the existence of physically acceptable solu-
(inordertosimplifythenotation,inthefollowingweintend tions (i.e. configurations for which vϕ2 ≥ 0 everywhere) is
ρ= [ρ ]andΦ= [Φ ]).Thequantitiesρ,P andΩdenote not guaranteed for arbitrary choices of ρ and Φ. However,
c c
ℜ ℜ
thefluiddensity,pressureandangularvelocity,respectively; the positivity of the integrand in equation (37) for z 0
the rotational (i.e., streaming) velocity is v = ΩR, while is a sufficient condition to have v2 0 everywhere. In f≥act,
ϕ ϕ ≥
v =v =0. inBarnab´eetal.(2006) severaltheoremsonbarocliniccon-
R z
In problems where ρ and φ are assigned, the standard figurations have been proved starting from equation (37):
approach for the solution of equations (34) is to integrate inparticulartoroidalgasdistributionsarestronglyfavoured
for the pressure with boundarycondition P(R, )=0, in order to have v2 0 in presence of a dominating disk
∞ ϕ ≥
∞ gravitational field. We note that also in the present cases
P = ρ∂Φ dz′, (35) the density is of toroidal shape, altough the distribution is
∂z′
Zz self-gravitating.
and then to obtain the rotational velocity field from the Inthecaseofthe (ρc) (Ψc)pairoftheshiftedPlum-
ℜ −ℜ
radial equation mer sphere, the (normalized) commutator in equation (37)
is a very simple function
R∂P ∂Φ
v2 = +R . (36)
ϕ ρ ∂R ∂R ∂ρ ∂Φ ∂ρ ∂Φ 15a2Rz(1 a2+r2)
= − , (39)
However, v2 can be also obtained without previous knowl- ∂R∂z′ − ∂z′∂R π d7
ϕ
edgeofP,becausebycombiningequations(35)-(36)andin- which is positive for z > 0 and 0 a < 1, so that v2 0
≤ ϕ ≥
tegratingbypartswiththeassumptionthatP =ρ∂Φ/∂R= as far as (ρ ) is positive everywhere, i.e., for 0 a<a .
c m
ℜ ≤
0 for z = it follows that Remarkably, the integration of equation (37) can be easily
∞
6 Ciotti & Giampieri
3 3
2 2
1 1
z 0 z 0
-1 -1
-2 -2
-3 -3
-3-2-1 0 1 2 3 -3-2-1 0 1 2 3
R R
3 3
2 2
1 1
z 0 z 0
-1 -1
-2 -2
-3 -3
-3-2-1 0 1 2 3 -3-2-1 0 1 2 3
R R
Figure 2.Isorotational contours inthemeridionalplaneofℜ(ρc)ofPlummerspherefora=1/2(topleft),anda=23/40(topright).
Inthebottom panelscontoursofconstantpressureforthesamemodels.
carried out in terms of elementary functions Note that v2 a2, and so it vanishes (as a consequence
ϕ ∝
of the imposed isotropy) when reducing to the spherically
3a2R2 d4(1+a2+r2) symmetricseeddensity:infact,itcanbeprovedthatthear-
ρv2 = 1 gumentinparenthesisinequation (40)isregularfor a 0.
ϕ 2πd5 ( − 12a4(1+R2)3 × The baroclinic nature of the equilibrium is apparent,→and
confirmedbyFig.2,whereweshowthecontoursofconstant
6a4(1d+4 R2)2 − 2a2(1d+2 R2) + 1+1+a2a+2+r2r−2 d , (40) vϕ2 for a =1/2 and a= 23/40 (top panels, correspondig to
(cid:20) (cid:21)) themodelsplottedinthetoppanelsofFig.1).Thenormal-
ized streaming velocity field in theequatorial plane is
while the direct integration of the pressure equation is
muchmorecomplicate(againshowingtherelevanceofequa- a2R26(1+R2)2 4a2(1+R2)+a4
v2(R,0)= − , (42)
tion [37] in applications), and we report its normalized ex- ϕ 3 (1+R2)3(1 a2+R2)3/2
pression on theequatorial plane only −
and has a maximum at R<1, while the normalized circular
2(1+R2) a2 velocity is the same as th∼at of a Plummer sphere of scale-
P(R,0)= 16π(1+R2)2(1 −a2+R2)2. (41) lenght √1 a2 (cf. thecomment after equation [26]), i.e.,
− −
Exact density-potential pairs 7
v2(R)=R∂Φ(R,0) = R2 . (43) Aninterestingissueraisedbythepresentanalysisisthe
c ∂R (1 a2+R2)3/2 toroidal shape of theobtained densities. While wewere not
−
abletofindageneralexplanationofthisphenomenon,some
Itiseasytoverifythatequations(41)and(42)satisfyequa-
hints can be obtained by considering the behavior of the
tion (36).
complex-shiftfor a 0.Infact,inthiscaseitcanbeeasily
Unfortunately,wewereunabletoobtaintheexplicitfor- →
provedthatwhiletheoddtermsoftheexpansioncorrespond
mulaefortheanalogous quantitiesfor theshifted Isochrone
totheimaginarypartfotheshifteddensity(andcontainthe
model. However, it is easy to prove that also in this case
thecommutatorisproportionaltoa2,andsoitvanishes(as factor z), the even terms are real and, to the second order
in a,
expected) for a 0. A comparison of the rotational fields
→
of the two models in the far field can be obtained by using a2dρ a2z2 dρ d2ρ
the asymptotic expressions (25)-(26) and (32)-(33) can be ℜ(ρc)∼ρ− 2rdr + 2r3 dr −rdr2 +O(a4). (46)
used.Inparticular,fortherealpartoftheshiftedPlummer (cid:18) (cid:19)
sphere we found This expansion is functionally similar to that obtained by
Ciotti &Bertin (2005) for thecase of oblatehomeoidal dis-
v2 = 2a2R2 5a2R2(3−a2−9a2µ2) +O(R2r−9), (44) tributions, where the term dependent on z2 is immediately
ϕ r5 − 3r7
identifiedwithasphericalharmonic(e.g.,Jackson 1999)re-
while for (ρc) of theIsochrone shifted sphere sponsibleofthetoroidalshape.Itisclearthatwhenrestrict-
ℜ
ing to the small-shift approximation (46), all the computa-
a2R2 4 13
v2 = + tions presented in this paper can be carried out explicitly
ϕ r5 3 − 5r
for simple seed distributions. Moreover, from equation (46)
a2R2((cid:16)3159+57(cid:17)80a2+21780a2µ2) +O(R2r−8). (45) it is easy to prove that negative values of density appears
r7 on the z axis - no matter how much the shift parameter is
Notethat,eventhoughtheradialasymptoticbehaviorofthe small - when the seed density has a central cusp ρ r−γ
∝
densityisdifferentinthetwoconsideredcases,theobtained with γ >0 (e.g., see theexplicit example in Appendix).
velocityfieldhasthesameradialdependenceinthefarfield, In this paper we restricted for simplicity to spherically
and in both cases v2 decreases for increasing z and fixed simmetric seed systems, which produce axysimmetric sys-
ϕ
R. In addition, v2 0 for r , as expected from the tems.Anaturalextensionofthepresentinvestigationwould
ϕ → → ∞
spherical symmetry of the density and potential for r be the study of the density-potential pairs originated by
→
: in fact, it is apparent from equation (37) that spherical the complex shift of seed disk distributions, such as the
∞
(additive)componentsofρandΦ(e.g.,theleadingtermsin Miyamoto & Nagay (1975) and Satoh (1980) disks, or even
the asymptotic expansions [25]-[26] and [32]-[33]) mutually the generalization of homeoidal quadrature formulae (e.g.,
cancel in thecommutator evaluation. see Kellogg 1953, Chandrasekhar 1969, BT). Note that in
such cases, at variance with the spherical cases here dis-
cussed, the shift direction is important: for example, the
4 CONCLUSIONS complexshift of adiskalong thez axiswill lead toadiffer-
ent system than an equatorial shift, that would produce a
Inthispaperwehaveshownthatthecomplex-shiftmethod, triaxial object.
introduced in electrodynamics by Newman, and throughly Asecondinterestinglineofstudycouldbetheuseofthe
studied among others by Carter, Lynden-Bell and Kaiser, complex-shift to produce more elaborate density-potential
canbeextendedtoclassicalgravitationtoproducenewand pairsbyaddingmodelswithaweightfunctionw(a),i.e.,by
explicit density-potential pairs with finite deviation from considering the behaviorof thelinear operator
sphericalsymmetry.Inparticular,weshowedthataftersep-
aration of the real and imaginary parts of the complexified ̺ = ρ(x ia)w(a)d3a, (47)
c
density-potentialpair,theimaginarypartofthedensitycor- −
Z
respondstoasystem ofnulltotalmass,whiletherealcom-
wheretheintegrationisextendedoversomesuitablychosen
ponentcan bepositive everywhere(dependingon theorigi-
region. Wedonot pursuethisline of investigation, butit is
nalseeddensitydistributionandtheamountofthecomplex
obvious that also shifted density components with negative
shift).
regionscanbeacceptedinthiscontext,asfarasoneisable
Assimpleapplication oftheproposed method weillus-
toconstructaweightfunctionleadingtoafinaldistribution
trated theproperties of theaxysimmetric systems resulting
nowhere negative (as it happens, for example, in spherical
from the shift of the Plummer and Isochrone spheres. The
harmonics expansions).
analysis revealed that the obtained densities are nowhere
negative only for shift valuesin some restricted range. This
property is linked to the presence of a flat ”core” in the
adopted(sphericallysymmetric)seeddistributions.Itisthus
ACKNOWLEDGMENTS
expected that also other distributions, such as the King
(1972) model, and the Dehenen (1993) and Tremaine et al. The paper is dedicated to the memory of my best friend
(1994)γ =0model,willleadtophysicallyacceptableshifted andcolleagueDr.GiacomoGiampieri,whotragicallypassed
systems. For the two new exact density-potential pairs we away on September 3, 2006 while working on this project.
alsofoundthattheassociatedrotationalfields(obtainedun- L.C. thanks Donald Lynden-Bell for enligthening discus-
dertheassumptionofglobalisotropyofthevelocitydisper- sions,andTimdeZeeuw,ScottTremaineandananonymous
siontensor)correspondtobaroclinicgaseousconfigurations. Refereefor interesting comments. Thiswork was supported
8 Ciotti & Giampieri
bytheItalian MIUR grant CoFin2004 ”Collective phenom- H´enon, M. 1959, Ann.d’Astrophys., 22, 126
ena in the dynamicsof galaxies”. Hunter,C. 1977, AJ, 82, 271
Jeffreys,SirH.1970,TheEarth(fifthedition)(Cambridge
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Kutuzov,S.A.1998, Astronomy Letters, 24, n.5, 645
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ρc = (r2 a2)−2+4a2z2 +i(r2 a2)2+4a2z2, (A1) Lanzoni, B., & Ciotti, L. 2003, A&A,404, 819
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