Mon. Not. R. Astron. Soc. 000, 000{000 (1994) Printed August 3, 1995

On the deprojection of axisymmetric bodies

Ortwin Gerhard1 and James Binney21Astronomisches Institut, Universit�at Basel, Venusstrasse 7, CH-4102 Binningen, Switzerland2Theoretical Physics, Keble Road, Oxford OX1 3NP

ABSTRACT

Axisymmetric density distributions are constructed which are invisible when viewedfrom a range of inclination angles i. By adding such distributions to a model galaxy, itcan be made either disky or boxy without in any way a�ecting its projected image. Asthe inclination of a galaxy decreases from edge-on to face on, the range of `invisible'densities, the uncertainty in the deprojection, and the sensitivity of the deprojectionto noise all increase. The relation between these phenomena is clari�ed by an analysisof Palmer's deprojection algorithm.

These results imply that disk-to-bulge ratios are in principle ill-determined fromphotometry unless the disk is strong or the system is seen precisely edge-on. Theuncertain role of third integrals in galaxies makes it unclear to what degree this inde-terminacy can be resolved by kinematic studies.

Key words: Galaxies: fundamental parameters { galaxies: photometry { galaxies:kinematics and dynamics

1 INTRODUCTION

Since many galaxies are at least approximately axisymmet-ric and transparent, it is desirable to be able to estimate thethree-dimensional luminosity density �(r; �) of an axisym-metric galaxy from its projected surface brightness I(x;y),given an assumed inclination angle i between the galaxy'ssymmetry axis (the z-axis) and the line of sight to the ob-server. The Richardson{Lucy algorithm (Richardson 1972,Lucy 1974) has been successfully used to recover distribu-tions �(r; �) for a large number of galaxies with i assumeddi�erent from 90� (e.g., Binney Davies & Illingworth 1990;van der Marel 1991; Dehnen 1995).

The uniqueness of the resulting model galaxies is un-clear, however. On the one hand, Rybicki (1986) gave asimple argument based on the `Fourier slice theorem' that,for i 6= 90�, any given surface brightness distribution, I(x;y)must be the projection of an in�nite number of luminositydensities �(r; �). On the other hand, Palmer (1994) provedthat the relationship between I and � is unique for i 6= 0provided that the density �(R; �) is `band-limited': that isthat its expansion in Legendre polynomials Pl(cos �),

�(r; �) =

LXl=0

�l(r)Pl(cos �); (1)

contains only a �nite number of terms. Since any plausibleluminosity density e� can be approximated to su�cient ac-curacy by a band-limited density �, Palmer's result suggeststhat for practical purposes the relationship between I and �

is one-to-one for all i 6= 0�.If two space-densities can be found which, for �xed i,

project to the same surface brightness distribution, then thedi�erence between these densities projects to zero bright-

ness. We refer to such an `invisible' density distribution asa `konus' density. Here we demonstrate by explicit construc-tion of a family of konus densities that, for i 6= 90�, an in�-nite number of physically plausible luminosity densities are

compatible with a given surface-brightness distribution. Inparticular, we show that one may choose between disky andboxy luminosity densities for the same photometric data.

We describe how the range of possible luminosity densi-ties that are compatible with a given surface brightness dis-tribution grows continuously as the inclination varies fromedge-on to face-on, and how this change is re ected in thedivision of the space of possible luminosity densities �(r; �)into its three parts: konus densities, densities that can beobtained by an extension of Palmer's inversion algorithm,and other visible densities.

In Section 2, we construct a family of konus densitiesand show that by adding one of these to a given depro-jected brightness distribution we can turn a boxy distribu-tion into a disky one, or vice versa. In Section 3 we showthat Palmer's algorithm can provide a consistent deprojec-tion of data that do not derive from a band-limited densitydistribution provided certain conditions on the data are sat-is�ed. These conditions will be satis�ed by most data whenthe assumed inclination angle is i ' 90�, and become moreand more restrictive as i! 0.

Section 4 discusses the e�ects of noise in the data, thedivision of the space of possible surface brightness distribu-tions into ones that can be deprojected and ones that cannot,and the implications of the existence of konus densities forthe detectability of low-luminosity disks in elliptical galax-ies.

Section 5 sums up. In an appendix we give a group-theoretic proof of the result that resolves the apparent con-

2 O.E. Gerhard and J.J. Binney

ict between the papers of Rybicki and Palmer; namely thatthe Fourier transform of a band-limited distribution is itselfband-limited.

2 KONUS DENSITIES

We now show that there exist distinct three-dimensionallight distributions that are both physically plausible { i.e.,are smooth and su�ciently compact { and project to thesame surface brightness distribution. This we do by explic-itly calculating a family of axisymmetric luminosity distri-butions which are invisible when viewed at all inclinationangles smaller than some critical value.

We start with a review of Rybicki's (1986) applicationof the Fourier slice theorem. By writing the axisymmetricdensity �(x) in terms of its Fourier transform A(k),

�(x) =1

(2�)3

Zd3k A(k) exp(ik � x); (2)

and projecting this along z, say,

I(x; y) =

Zdz �(x)

=1

(2�)2

Zd3kA(kx; ky; 0) �(kz) exp(ik � x);

(3)

we see that the Fourier transform of the projected surfacedensity I(x;y) is the two-dimensional slice A(kx; ky; 0) ofA(k). This simple result is known as the Fourier slice theo-rem.

Consider now the projection of an axisymmetric den-sity distribution. From the observed image we can obtainthe two-dimensional slice of the density's Fourier transformwith k perpendicular to the direction of projection. How-ever, because of the assumed axial symmetry, all lines ofsight which are related to the actual line of sight by a sim-ple rotation around the symmetry axis must give identicalimages. Therefore by symmetry the Fourier transform isin fact known in all parts of Fourier space that are sweptby rotating this two-dimensional slice around the object'ssymmetry axis.

If the density distribution is observed edge-on, thissweeping process covers all of Fourier space, and the three-dimensional density can be uniquely recovered from the pro-jected image. In a face-on view, only a single Fourier planeis known even after the sweeping process, and, as is well-known, the deprojection is in this case highly non-unique.If, however, the direction of projection is inclined with re-spect to the symmetry axis by an angle i 6= 90�, then therotation of the Fourier slice leaves a cone around the sym-metry axis with half-angle (90��i) uncovered { the so-called`cone of ignorance'. The surface of this cone opens to aplane (half-angle 90�) in the face-on case, and closes aroundthe symmetry axis in an edge-on projection (half-angle 0�).Any density distribution derived from a Fourier transformwhich is non-zero only in the cone of ignorance will have zeroprojected brightness and thus be a konus distribution. Con-versely, a density distribution whose Fourier transform isnon-zero anywhere outside the cone of ignorance will havenon-zero projected brightness. Hence konus densities areprecisely those densities whose Fourier transforms are non-zero only in the cone of ignorance. Note that a konus density

for inclination i is also a konus density for any inclinationi0 < i, since the cone of ignorance for i lies within the conesof ignorance for all smaller inclinations.

In the following we use Cartesian coordinates (x;y; z)and cylindrical polar coordinates (R; �; z) in the galaxy-intrinsic frame, with z along the symmetry axis, and(x0; y0; z0) coordinates in the frame of the observer, suchthat projection is along ez0 and the x= x0-axis is the lineof nodes. The inclination angle i is measured between ezand ez0, the ez0 -axis having direction (0; sin i; cos i) in thegalaxy-intrinsic frame. The boundary of the cone of igno-rance in k-space is given by

jkzj=kR � jkzj=pk2x + k2y = cot(90� � i); (4)

or

jkzj = kR tan i: (5)

The physical density corresponding to a Fourier densityA(k) in the cone of ignorance is then

�K(x) =1

(2�)3

ZdkR kR

Zjkz j�kR tan i

dkz exp(ikzz)

�Z 2�

0

d�k exp[iRkR cos(�k � �)]A(k);

(6)

where k � (kR cos �k; kR sin �k; kz). By symmetry, A(k)must be independent of �k, so as not to generate a depen-dence of �K(x) on �. Thus the �k-integral simply evaluatesto 2�J0(kRR), where J0 is the usual Bessel function. Wewill, moreover, assume that A(k) is a symmetric functionwith respect to kz=0,

A(k) = 12[B(kR; kz) +B(kR;�kz)] ; (7)

so that �K(x) becomes the cosine-Bessel transform

�K(R; z) =1

2�2

Z1

0

dkR kRJ0(kRR)

�1Z

kRtan i

dkz cos(kzz)A(kR; kz):

(8)

2.1 A speci�c family of konus densities

From the discussion above it is clear that A(k) must be zeroon and outside the cone of ignorance. So that it lead to aphysically reasonable space density we also require it to besmooth, and to have certain characteristic scales. This lastrequirement derives from the fact that a konus density mustnecessarily be positive in some parts of space and negative inothers, and after adding a konus density to a visible one wewant the resulting density to be everywhere non-negative.This requires that the konus density remain �nite as r ! 0and fall o� su�ciently rapidly as r !1. Consequently, thekonus density must have certain characteristic scales, andthese will be inversely related to corresponding scales in itsFourier density A(k).

It is likely that a general investigation of Fourier den-sities that are con�ned to the cone of ignorance will be adi�cult numerical problem. We have therefore sought ananalytical example and, after some experimentation with

Deprojecting axisymmetric bodies 3

Gradshteyn & Ryzhik (1980; hereafter GR), have concen-trated on the following family of konus Fourier densities:

A(k) =

��2 exp(���) exp(��kR) for � > 0,0 otherwise,

(9)

where

� � jkzj � kR tan i: (10)

A(k) decreases smoothly to zero on the boundary of the cone(� = 0) and has its main contribution in a limited region ofFourier space around the kz-axis, as determined by the twocharacteristic length scales � and �.

With this A(kR; kz), the cosine-Bessel transform of (8)can be done simply. The kz-integral gives

1ZkRtan i

dkz cos(kzz)�2 exp(���)

=@2

@�2� cos(zkR tan i)� z sin(zkR tan i)

�2 + z2

= g1(z) cos(zkR tan i) + g2(z) sin(zkR tan i);

(11a)

where we have used formula 3.893.2 of GR and

g1(z) � 2�3�6�z2(�2 + z2)3

g2(z) � 2z3�6�2z(�2 + z2)3

:

(11b)

Note that for z � � the cosine and sine terms in this equa-tion decrease like / z�4 and / z�3, respectively.

Before proceeding further with the wave density ineq. (9), it is instructive to replace the term exp(��kR) by(2kR)

�1�(kR). Then the remaining integral can immedi-ately be done, and we obtain the density of a plane-parallelsheet

�s(z) =1

2�2�3 � 3�z2

(�2 + z2)3: (12)

The surface density obtained on integrating along any di-rection with inclination i0 6= 90� is

Is(i0) =

Zdz0�s(z) =

Zdz

cos i0�s(z)

=1

2�2 cos i0

1Z�1

dz�3 � 3�z2

(�2 + z2)3= 0:

(13)

Only when the sheet is seen edge-on (i0=90�) is the massivewire along kz part of the slice in Fourier space about whichinformation is available, and only in this case will the pro-jected density be non-zero, with positive or negative values,depending on the vertical coordinate y0=z.

We now return to the exponential wave density (9). GR,formula 6.751.3, give the integral

C0 =

Z1

0

dkR exp(��kR) cos(zkR tan i)J0(kRR)

=

�A +

pA2 +B

�1=2p2pA2 +B

;

(14)

where

A � R2 � z

2 tan2 i+ �2;

B � 4�2z2 tan2 i:(15)

From this we may obtain the two kR-integrals required in(8) as

C1 =

Z1

0

dkR kR exp(��kR) cos(zkR tan i)J0(kRR)

= �@C0

@�=

�D1 + 4�z2 tan2 iD2p2D3

(16)

and, with a � z tan i,

S1 =

Z1

0

dkR kR exp(��kR) sin(zkR tan i) J0(kRR)

= �@C0

@a=

z tan i (�D1 + 4�2D2)p2D3

;

(17)

where

D1 � A2 +A

pA2 + B � B;

D2 � A+ 12

pA2 + B;

D3 ��A+

pA2 +B

�1=2 �A2 +B

�3=2:

(18)

Thus, �nally, the density corresponding to the Fourierdensity (9) may be written as

�K(R; z) =1

2�2[g1(z)C1(R; z) + g2(z)S1(R; z)] ; (19)

with the functions gi(z) de�ned by eqs (11b), and the re-maining auxiliary quantities speci�ed in eqs. (14) { (18).

This density is (i) everywhere non-singular, (ii) com-pact, and (iii) by construction, invisible at all inclinationssmaller than i. To prove (i) we note from (15) that for � 6= 0A2 � 0 and B � 0, but both cannot be zero simultaneously.Thus the denominator D3 of both C1 and S1 is positivede�nite. Also, for � 6= 0 the denominator of both g1(z)and g2(z) is positive de�nite. To show (ii) we consider theasymptotic behaviour at large R and z. For R!1 at �xedz, we have A / R2, B / R0, C1 / A�3=2 / R�3, S1 / R�3

and, since gi are independent of R, also �K / R�3. Forz ! 1 at �xed R, A / z2, B / z2, C1 / z�3, S1 / z�2,and �K / g2(z)S1(R; z) / z�5. Finally, for the special di-rection given by R2=z2 tan2 i, we have A = �2, B = 4�2R2,and with r2 = R2 + z2 C1 / r�1=2, S1 / r�1=2, g1 / r�4,g2 / r�3 and so �K / r�7=2.

To con�rm that this �K(R; z) is invisible in projectionfor the inclination angle i used in its construction, or indeedin any more face-on projection i0 with i0 � i, we have in-tegrated the density (19) numerically along lines of sight,using a Romberg midpoint rule algorithm and eq. (18) ofBinney, Davies & Illingworth (1990). At all image pointstried the projected density was found to be vanishingly smallfor i0 � i, and then to rise relatively rapidly from zero fori0 � i.

The konus densities (19) tend at large R and small z toconstant-scale-height disks. To see this, we �rst note thatfor R!1 and �xed kR the Bessel function in eqs. (16) and(17) is asymptotically

J0(kRR) ��

2

�kRR

�1=2cos

�kRR� �

4

�+O

�1

kRR

�: (20)

Thus as R ! 1, only wavenumbers near kR=0 (very longwavelengths) contribute to the konus density (19), suggest-ing an asymptotic connection to the plane-parallel sheet so-lution (12). To make this more explicit, we rewrite the konus

4 O.E. Gerhard and J.J. Binney

Figure 1. Contour plots in the meridional plane of two konus densities: � = � = 1, i = 45� (left panel); � = 0:75, � = 4, i = 80� (right

panel). Contours are uniformly spaced in log(j�j) and for negative density are dotted.

density (19) as

�K(R; z) =1

2�2R2

Z1

0

dx xJ0(x) exp(��x=R)

�hg1(z) cos

�z

Rx tan i

�+ g2(z) sin

�z

Rx tan i

�i:

(21)

For R ! 1, the konus density takes its largest values nearthe R-axis (z � R); at z�R it is smaller by a factor of order(�=R)2. Thus we may approximate the square bracket inthe last equation by simply g1(z), after which the remainingintegral gives (GR, 6.623.2)

�K(R; z) ' g1(z)

2�2R2

Z1

0

dx xJ0(x) exp(��x=R)

=�

�2R3

�3 � 3�z2

(�2 + z2)3;

(22)

which is the plane-parallel sheet solution (12) multiplied by afactor proportional to R�3. Thus asymptotically for large R,the konus density (19) approaches a disk-like con�gurationthat consists of a thin positive-density disk surrounded by athicker layer of negative density.

2.2 Some illustrations

Fig. 1 shows contour plots in the meridional (R; z) planeof two members of this family of konus densities. The �rstis constructed with parameters � = � = 1 and i = 45� ineq. (9), the second with �=0:75, � = 4, and i=80�. Solidcontours in Fig. 1 delineate regions in which the konus den-sity is positive, dotted contours signify regions of negativedensity. The structure apparent in Fig. 1 is typical of allthe konus densities we have generated from eq. (9) { a re-gion of positive-density around the minor axis is followed bya region of negative density at intermediate latitudes, andthen by another region of positive density around the R-axis.The approach as R ! 1 of this last region to the plane-parallel sheet is apparent; at the largest radius plotted, the

asymptotic density in the mid-plane, �(R; 0) ' �=(�2R3�3)is accurate to better than a percent.

These density distributions project to zero surfacebrightness for all observers viewing them from inclinationsi0 � 45� and i0 � 80�, respectively { such lines of sight passthrough regions of both positive and negative density. Onthe other hand, in an edge-on view (i=90�) there are lines ofsight which pass through only the region of positive densitynear the R-axis, or only the region of negative density aboveit, depending on height z. The structures in the right-handpanel of Fig. 1 appear to be attened because for high in-clination the cone of ignorance is narrow, allowing only forfairly long-wavelength components in the radial direction.

By adding or subtracting such density components toa `visible' density distribution, such as those commonly in-ferred for elliptical galaxies, it is clearly possible to gen-erate intrinsic disk-like or peanut-like components withoutaltering the observed image. Ideally one would demonstratethis by choosing an underlying elliptical density distribution�E(R; z), such as a modi�ed Hubble pro�le, say, and addinga variety of possible konus densities. Since here we have�xed the functional form of the konus density (19) a pri-ori, we will instead vary the functional form of the ellipticaldensity distribution. Fig. 2 shows three examples of suchsuperpositions

�(R; z) = �j(R; z) + f �K(R; zj�;�; i): (23)

In the top panel of Fig. 2 a boxy density distribution isobtained by subtracting a multiple of the 45�-konus densityof Fig. 1 from the elliptical density distribution

�3(m) � �0m�1(m+mc)

�2: (24)

Here m is the usual spheroidal radius m2 � R2 + z2=q2.This boxy distribution and the elliptical model (24) projectto exactly the same elliptical isophotes for all inclinationangles i0 � 45�. The middle panel of Fig. 2 shows contoursof a strongly disky intrinsic distribution, obtained by adding

Deprojecting axisymmetric bodies 5

a multiple of the �=2, �=4, 45�-konus density to the quasi-halo density

�2(m) � �0(m2 +m

2c)�1: (25)

Again the isophotes of this disky model are precisely ellipti-cal when viewed under inclination angles i0 � 45�. Finally,the bottom panel of Fig. 2 shows the superposition of thequasi-halo model (25) with the 80�-konus density of Fig. 1;this shows that invisible disklike structures can be generatedeven for high inclination i0 � 80�.

It is not entirely clear which of the features of the exam-ples we have shown is generic and which determined by thepeculiarities of the particular family of konus densities withwhich we have worked. Certainly, it would be straightfor-ward to generate konus densities with di�erent asymptoticpro�les from the / R�3; z�5 characteristic of our family:densities with steeper asymptotic pro�les could be generatedby either di�erentiating eq. (19) w.r.t. �, or by replacing theexponential in eq. (9) by a Gaussian, say. Also, it is clearthat any konus density must be made up of approximatelyconical shells of positive and negative density, and it seemsunlikely that any has fewer than the three cones characteris-tic of our family. Konus densities with more cones certainlyexist as one may demonstrate simply by adding konus densi-ties of the family (19) for di�erent values of i. For example,Fig. 3 shows the result of adding half the 45�-density ofFig. 1 to the 80�-density of Fig. 1. The combined konusdensity is invisible at i � 45�, and at r > 4 kpc it has threeconical regions of positive density and two regions of nega-tive density. Adding such a density to an ellipsoidal modelwould not merely e�ect the transition from disky to boxydistributions.

The examples of Fig. 2 are signi�cantly in uenced bya combination of the R�3 asymptotic pro�le and the factthat for R � � the region of highest density around theR-axis tends to a constant-scale-height disk. Between themthese limit the amplitude f of the konus density which canbe added to a spheroidal distribution without generatingimplausible isodensity surfaces.

Far from the origin, the structure of a konus density willbe dominated by the behaviour of its Fourier transform nearthe origin of k-space. It happens that the particular wave-density (9) has a pronounced peak at (kR = 0; kz = 2=�),with the result that waves that run parallel to the z-axisbecome dominant at large r. The tendency to a constantscale-height disk discussed above is a manifestation of thisfact. In principle there is no reason why the Fourier trans-form of a konus density should not be zero along the kz-axis.In this case the konus density would at large R be dominatedby waves with non-zero kR=kz, and would not tend to a con-stant scale-height disk. Some such Fourier transforms mightlead to analytically tractable k-space integrals, or the inte-grals could be done numerically. Hence it is clear that therange of physically reasonable konus densities is certainlylarger than that displayed by the family (19).

As the inclination increases towards edge-on, the angu-lar scales of the konus densities decrease and the form ofthe 80�-konus density of Fig. 1 suggests that the availablefreedom in ��=� is less at higher than at lower inclinations.However, because this statement is based on the detailedform of our konus densities, it is not well established. Weshall return to this point with di�erent arguments below.

Figure 2. Disky and boxy systems with elliptical surface bright-

ness contours. Each panel shows contours in the meridional plane

for two deprojections of the same surface brightness data. The

dotted elliptical contours correspond to one of the densities �j of

eqs (24) or (25). The full contours show the sum of these densities

and a konus density of plausible amplitude f . Top panel: j = 3,

q = 0:7,mc = 2; � = � = 1, i = 45�, f < 0. Middle panel: j = 2,

q = 0:7, mc = 2; � = 2, � = 4, i = 45�, f > 0. Bottom panel:

j = 2, q = 0:5, mc = 3; � = 0:75, � = 4, i = 80�, f > 0.

6 O.E. Gerhard and J.J. Binney

Figure 3. A linear combination of the two konus densities of

Fig. 1: half the left panel has been added to the right panel. The

resulting konus density has �ve conical regions and is invisible at

inclinations i � 45�.

We have illustrated the e�ects of deprojection degen-eracy by adding konus densities to intrinsically spheroidaldistributions, whose isophotes are elliptical. But we couldequally well have added konus densities to disky distribu-tions that had pointed isophotes. Then we would have foundthat the amplitude and thickness of the disk implied by givendata were highly ambiguous.

3 LEGENDRE EXPANSIONS

How do the results just presented relate to Palmer's (1994)proof that a band limited density distribution can beuniquely recovered from its projected surface brightness forall inclinations i 6= 0? To understand this we will �rst reviewPalmer's algorithm, and then return to the konus densitiesdescribed above. We employ galaxy-intrinsic spherical polarcoordinates (r; �; �).

3.1 Palmer's algorithm

The Legendre polynomial expansion of a typical, smoothluminosity density e�(r; �) will be the in�nite sum

e�(r; �) =1Xl=0

e�l(r)Pl(cos �): (26)

For simplicity we assume that e� is symmetric about theequatorial plane, with the result that only even l need beincluded in the sum. Let (s; ') be suitably oriented polar

coordinates for the plane of the sky. Then the projection eIof e� may be written

eI(s; ') = 12I0 +

1Xn=2;4:::

In(s) cos n': (27)

If e� were band-limited, the sum over n in (27) would run onlyup to n = L, the largest value of l involved in the Legendre

polynomial expansion of e�. Thus the Fourier decompositionof the projection of a band-limited density distribution isband-limited in the conventional sense.

We consider the band-limited approximations �(L) to e�that are obtained by setting to zero all the In with n > L.Let the Legendre polynomial expansion of �(L) be

�(L) =

LXl=0

�(L)

l (r)Pl(cos �): (28)

By expressing the (r; �; �) coordinates natural to the galaxyin terms of coordinates (s; '; z0) natural to the observer, andintegrating along z0, Palmer shows that

In(s) = 4

LXl=n

(l�n) even

pnl (cos i)

Z1

s

�(L)

l (r) pnl (�)r drpr2 � s2

; (29)

where pnl is related to the conventional associated Legendrefunction Pn

l by

pnl �

r(l � jnj)!(l + jnj)!P

nl and

� � cos�arcsin(s=r)

�=p1� s2=r2:

(30)

[With this de�nition,R 1

�1dx (pnl )

2 = 2=(2l + 1).] Palmer

shows that when i 6= 0, equations (29) can be converted

into expressions for the �(L)l in terms of the In, which maybe determined observationally. Speci�cally,

�(L)

l (r) = �2ll![(2l)!]�1=2

2�rpll(cos i)

Z1

1

d

d�

hI(L)

l (�r)

�l

id�p�2 � 1

; (31)

where

I(L)

l (s) � Il(s)� 4

LXl0=l+2

pll0 (cos i)

�Z

1

s

�(L)

l0(r)pll0 (�)

r drpr2 � s2

:

(32)

Palmer's proof that band-limited densities can be uniquelyinverted follows from the fact that for i 6= 0 this systemof equations can be straightforwardly solved: one �rst �nds�(L)

L , which depends only on IL. Then one solves for �(L)L�2,

which depends on IL�2 and �(L)

L , and so on down the series

of the �(L)l .

3.2 Extension to non-band-limited data

We now investigate under what circumstances Palmer's al-gorithm can be used to deproject a surface-brightness distri-bution eI(s; �) that is not band-limited. Consider the di�er-ence between the band-limited approximations �(L+2) and�(L) to e� that are obtained with Palmer's method by trun-cating the Fourier series (27) for eI at n = L+ 2 and n = L,respectively. We have that

��l � �(L+2)

l � �(L)

l

= �2ll![(2l)!]�1=2

2�rpll(cos i)

Z1

1

d

d�

h�Il(�r)

�l

id�p�2 � 1

;(33)

where for l=L+ 2

�IL+2 = IL+2; (34)

Deprojecting axisymmetric bodies 7

and for l � L

�Il(s) = �4L+2X

l0=l+2

pll0 (cos i)

Z1

s

��l0(r)pll0 (�)

r drpr2 � s2

: (35)

Consider now the way in which ��l depends upon thesignal IL+2 under the assumption that the In are slowlyvarying functions of s. For large l, the integral in equation(33) may be estimated as follows. We have

d

d�

h�Il(�r)

�l

i' � l �Il(�r)

�l+1; (36)

and the dominant contribution to the integral comes from� ' 1, soZ

1

1

d

d�

h�Il(�r)

�l

id�p�2 � 1

' �l�l�Il(r)Z

1

1

d�

�l+1p�2 � 1

' �l (l � 1)!!

l!!

�

2�l�Il(r);

(37)

where �l is a number of order unity. Thus from equation(33)

j��lj ' l(l � 1)!!

l!!

r(2l)!!

(2l� 1)!!

�l�Il(r)

4rpll(cos i): (38)

Note that since l = l!!(l�1)!!=[(l�1)!!(l�2)!!)], we have forlarge l that l!!=(l � 1)!! '

pl. Hence for large l,

j��l(r)j � l3=4 �Il(r)

rpll(cos i): (39)

Similarly, the integral in equation (35) can be estimated asZ1

s

��l0(r) pll0 (�)

r drpr2 � s2

' �l0��l0 (s)

Z1

s

pll0

�p1� s2=r2

�r drpr2 � s2

= �l0 s ��l0 (s)(�1)l�2l+1l!

r(l0 + l)!

(l0 � l)!

(l� 3)!!

(l� 2)!!

� 3F2�l0+l+1

2;� l0�l

2; l�1

2; l+ 1; l

2; 1�:

(40)

where we have used equation 7.132.6 of GR. Here 3F2 is thegeneralized hypergeometric series, which in this case termi-nates after 1

2(l0 � l) + 1 terms. In general we �nd that a

reasonable approximation to the resulting scaling of �Il is

j�Il(s)j �L+2X

l0=l+2

pll0 (cos i) s ��l0 (s)

l0 (l=l0)2

�L+2X

l0=l+2

1

l01=4

�l0

l

�2pll0 (cos i)

pl0

l0(cos i)

�Il0 (s):

(41)

This equation allows us to estimate the e�ect on therecovered density distribution of adding one more term IL+2to the expansion of the surface brightness: every coe�cient�l depends through (31) on the corresponding `e�ective data'coe�cient Il, and all of these are modi�ed by the addition ofIL+2 in a way that we now estimate from (41). By equation(34), �IL+2 = IL+2, so from (41) we have

�IL � 1

(L+ 2)1=4

�L+ 2

L

�2 pLL+2(cos i)pL+2L+2(cos i)

IL+2: (42a)

Figure 4. Typical Legendre functions normalized according to

equation (30).

Using this to eliminate �IL from the corresponding expres-sion for �IL�2, we �nd

�IL�2 ��

1

(L+ 2)1=4L1=4

pLL+2(cos i)

pL+2L+2(cos i)

pL�2L (cos i)

pLL(cos i)

+1

(L+ 2)1=4pL�2L+2(cos i)

pL+2L+2(cos i)

��L+ 2

L� 2

�2IL+2:

(42b)

The pattern of coe�cients obtained by continuing this pro-cess down to arbitrary l is now apparent. The dependence of�Il on IL+2 is governed by the ratios of the type pl

0

�2k

l0=pl

0

l0 .Fig. 4 shows three typical Legendre functions. From this onesees that the functions pl

0

l0 / sinl0

i that occur in the denom-inators above fall monotonically as the inclination changesfrom edge-on to face-on, and become very small by a valueof x � cos i which diminishes as l0 increases. The func-tions pl

0

�2k

l0that appear in the numerators above oscillate

for small cos i, and tend to zero � sin(l0

�2k) i as cos i ! 1.Consequently, the ratios of Legendre functions in (42) willbe of order unity near i = 90� but be large for near face-on inclinations. In fact, in the limit i ! 0 the products ofLegendre functions in each term in the series of (42) will allbe comparable because they all scale as sin�(L+2�l) i. Formoderate inclinations and values of L, the ratio �Il=IL+2will be dominated by these products. Hence the additionof a small coe�cient IL+2 can profoundly modify the e�ec-tive data Il at all orders, and hence signi�cantly change therecovered density distribution.

As we have de�ned it, �Il is a function of the order Lof the last included term. If deprojection is to make sense,the di�erence between the deprojections that one obtainson truncating the data at either order L or order L + 2must tend to zero as L ! 1. Consequently, we requirelimL!1 �Il = 0 for all l � L. For i = 90�, the ratios ofLegendre functions in equation (42) will evaluate to less thanunity, so limL!1 �Il is zero provided the Fourier coe�cientsIL fall o� at least as fast as L�2, as they will because thesurface brightness is a continuous function of position angleon the sky. However, for small i, the limit will vanish only ifthe coe�cients IL fall o� extremely rapidly since they mustoverwhelm the very rapid growth with L of the products ofLegendre functions.

8 O.E. Gerhard and J.J. Binney

3.3 An example

It is instructive to see how these ideas work out in prac-tice. Consider deprojecting a system whose noise-free sur-face brightness follows the modi�ed Hubble pro�le

I(x;y) =I0

�22where �

22 � 1 +

x2 + y2=q22r20

: (43)

The luminosity density of this system is

�(r; �) =�0

�33where �23 � 1 +

r2

r20

�sin2 � + cos2 �=q23

�(44)

with

�0 =q2I0

2q3r0;

q23 = q

22 csc

2i� cot2 i:

(45)

By straightforward contour integration one may show thatfor this system

IL(s) =2I0

12(q�22 + 1)s2 + r20

�p1� �2 � 1

�

�L=2

p1� �2

; (46)

where L is even and

�(s) � � q�22 � 1

(q�22 + 1) + 2(r0=s)2: (47)

Since�p1 � �2 � 1

�

�' 1� q2

1 + q2(r � r0)

=sin2 i(1� q23)�

1 +pq23 sin

2 i+ cos2 i�2 :

(48)

we have that IL tends to zero with L as

IL � sinL i(1 � q23)

L=2: (49)

When this is inserted into equations (42) for �Il, the factorsinL+2 i in IL+2 overwhelms the factors sin�(L+2�l) i implicitin the products of Legendre functions, with the result that�Il � sinl i just as Il � sinl i. Thus Palmer's algorithm whenused to deproject a noise-free modi�ed Hubble pro�le mightbe expected always to converge to the same density distri-bution, independent of the inclination at which the systemis viewed.

Figure 5 shows a numerical veri�cation of this propo-sition. The open symbols show for a typical radius valuesof log10 j�lj that were recovered by deprojecting a modi�ed-Hubble system of axis ratio q3 = 0:6 and inclination i = 10�.The open symbols give the values obtained for L = 4; 6 and8, while the full symbols give the values that one obtainsby directly expressing �(r; �) as a Legendre series. It canbe seen that as L is increased, good values are obtained formore and more of the �l. A virtually indistinguishable �gurecould be shown for any other inclination angle i > 0. Thusin this case, Palmer's algorithm does seem to recover thecorrect three-dimensional density from its projected surfacebrightness distribution.

3.4 Konus densities and Palmer's algorithm

We �rst demonstrate that the Legendre expansion of a konusdensity must contain an in�nite number of terms; i.e., it

Figure 5. Legendre coe�cients for a galaxy that has the modi-

�ed Hubble pro�le. An E4 galaxy is viewed at i = 10�. The full

symbols show the values of log10 j�lj for this density distributionat a typical radius. The open symbols show the values of the

same quantities that are recovered from the projected density for

L = 4;6;8. Triangles give values of �0, squares of �2, pentangles

of �4, and so forth.

cannot be band-limited. Indeed, if �l(r) =0 for l >L, thenby (29) it follows from the fact that IL(s) = 0 for all s,that �L(r) = 0 for all r. Repeating this argument withn = L � 1; : : : in (29) we see that a band-limited densitydistribution with zero surface brightness is identically zero.Fig. 6 illustrates this result by showing for one of the konusdensities of Fig. 1 the coe�cients �l(r) with l � 250 at sixvalues of r.

By adding a konus density to a band-limited distribu-tion, we see that a band-limited surface brightness I(s;�)does not imply that the underlying density �(r; �) is band-limited, while eq. (29) clearly shows that the reverse is true.

Since, by construction, a konus density projects to zerosurface brightness, one might think it impossible to recoversuch a density by Palmer's deprojection algorithm. Con-sider, however, the result of truncating the Legendre expan-sion of a konus density at order L. This band-limited densitydistribution projects to a band-limited surface brightnessdistribution I(L)(s; �). For any �nite L we may in princi-ple deproject this by Palmer's algorithm. The band-limitedresult of this deprojection must coincide with our originaltruncated konus density, for otherwise the di�erence be-tween these two densities would be a band-limited konusdensity, which we have just shown to be impossible. Re-peating this experiment for larger and larger values of L,we obtain more and more accurate approximations to theoriginal konus density, and in the limit L ! 1 we obtainthe konus density itself. Notice that this implies

limL!1

Pa�I(L)�6= Pa

�limL!1

I(L)�; (50)

where Pa stands for Palmer's operator. Indeed, as L in-creases, I(L) becomes smaller and smaller and vanishes inthe limit L!1. Hence Palmer's algorithm extracts betterand better approximations to the konus density's �l fromsmaller and smaller values of Il. Equations (42) give someinsight into how this is achieved, although they describe alimiting process that is di�erent from the one under consid-eration here: now with each increment in L, every Il varies,

Deprojecting axisymmetric bodies 9

Figure 6. The coe�cients �l(r) with l � 250 for the 80�-konus

density of Fig. 1. The larger the value of r, the more slowly �l de-

clines with r. This re ects the fact that at largeR these distribu-

tions approximate disks of constant scale-height. The 45�-konus

density yields a similar �gure.

rather than just IL+2. In other words, we obtain a konusdensity from a series of Fourier series rather than from asingle Fourier series.

4 DISCUSSION

4.1 Noise

We have seen that equations (42) impose constraints on thebehaviour of the Fourier coe�cients Il at large l that mustbe satis�ed if a meaningful deprojection is to be obtained byPalmer's algorithm. When the surface brightness distribu-tion I(s; �) is contaminated by noise, these conditions mustbe violated at su�ciently large l, since noise will tend tomake jIlj approximately independent of l for large l. More-over, as the assumed inclination i decreases, the conditionsimposed by equations (42) become ever more severe, withthe result that the value of l at which a given body of noisydata will �rst violate these conditions diminishes with i.

In practice galaxy images are deprojected using theRichardson{Lucy (R{L) algorithm rather than Palmer's.Experience shows that for a given image there is a small-est value of i for which the R{L algorithm yields a plau-sible density distribution. When using the R{L algorithm,both images and density distributions are usually �tted bylow-order functions of the angles � and �. This smooth-ing process will strongly attenuate the high-frequency powerin both data and model, but it will not amount to stricttruncation of the implicit Fourier and Legendre expansions.Consequently, the R{L algorithm will not be working withband-limited data or models. Equations (42) indicate thatat su�ciently small values of i, even the residual small high-l terms Il in the data can make important contributionsto the low-l coe�cients �l. Since the high-order Il will be

noise dominated, it follows that the inversion will be entirelynoise-dominated for su�ciently small assumed inclinations.Clearly, the deleterious e�ects of noise can only be increased

by increasing the angular resolution at which the data arerepresented.

We have seen that the low-amplitude brightness dis-tributions of truncated konus densities deproject to densi-ties of amplitude unity. Any component of residual noise insmoothed data that is equal to the surface brightness of atruncated konus density can be matched only by projectinga truncated konus density. Hence no matter how such dataare �tted, a model that �ts the data accurately must includea truncated konus density of signi�cant amplitude. More-over, the less the data have been smoothed, and thereforethe higher the value of L at which the underlying series aretruncated, the smaller is the amplitude of noise in the datathat is required to produce a konus density of unit ampli-tude.

It is interesting to study a practical example of a trun-cated konus density being introduced into a model by noisein the data. Fig. 7 shows a deprojection of the surfacebrightness distribution of NGC 2300 for an assumed incli-nation angle of i = 50�. This galaxy is approximately E2,and has slightly boxy (a4 �< 1%) isophotes. The deprojec-tion was done with the R{L algorithm of Binney, Davies& Illingworth (1990) as implemented by W. Dehnen. After10 iterations the rms deviation in surface brightness for sixradial rays was 0:006mag. Despite this very accurate rep-resentation of the nearly elliptical isophotes, the contoursof deprojected density in the meridional plane are stronglynon-elliptical and their shapes vary with radius. The shapesof these contours suggest that a truncated konus density hasbeen added to a smooth elliptical model.

Ambiguities in the result of the deprojection can beavoided by introducing additional constraints, such as thatthe deprojected density distribution should be as smoothas possible, or as elliptical as possible, etc. However, therealization that many elliptical galaxies display �ne struc-ture such as weak disks casts doubts on the wisdom of suchprescriptions. The successes of the R{L algorithm referredto in the introduction may in part be due to its e�ectivelysmoothing the angular density distribution during the de-projection, thus removing the higher Pl(cos �).

4.2 What can and cannot be seen

We have made it plausible that Palmer's algorithm can beapplied to some non-band-limited surface brightness distri-butions in addition to band-limited ones. For any assumedinclination i, let 2DP(i) denote the set of surface-brightnessdistributions I which yield a well-de�ned deprojected den-sity distribution in the limit L!1 as Palmer's algorithm issuccessively applied to the Fourier decomposition truncatedat each order L. Then 2DP(i) comprises nearly all surface-brightness distributions for i = 90� and shrinks continuouslywith i until at i = 0 it comprises only circularly-symmetricbrightness distributions. Thus, whereas in Palmer's originaldiscussion the case i = 0, in which Palmer's algorithm can-not be applied even to band-limited distributions, appearedanomalous, we now see that it is merely the endpoint of acontinuous evolution.

For any given value of i, there is a set 3DP(i) of density

10 O.E. Gerhard and J.J. Binney

0 2 4 6 8

0

2

4

6

8

R / kpc

deprojected density of the galaxy ngc2300

Figure 7. Contours of constant density in the meridional plane

of a model of NGC 2300 obtained by the R{L deprojection al-

gorithm. These are reminiscent of the sum of a konus density

and an elliptical density distribution. The assumed inclination is

i = 50� and the rms deviations of the projected model from the

data are 0:006mag. Reproduced from H�odtke (1995).

Figure 8. Schematic of function spaces

distributions that can be obtained by applying Palmer's al-gorithm to all the members of 2DP(i). One expects that theset 3DP(i) of functions of two variables will have essentiallythe same size as the set 2DP(i) of similar functions of whichit is the image. No konus density lies in 3DP(i) because torecover a konus density by Palmer's algorithm one has tochange all the Fourier coe�cients In at each stage of thelimiting process, rather than adding one more term to theFourier series. Hence the set of konus densities, 3DK(i), isdisjoint from 3DP(i). On adding a member of 3DK(i) toa member of 3DP(i) we obtain a density distribution thatlies in neither set. Hence the space of all possible densitydistributions falls into three parts as is schematically illus-trated by Fig. 8. That �gure also illustrates how 3DK(i)increases from the empty set to a large part of the space asi diminishes from 90� to zero.

4.3 Konus densities and disky distributions

Konus densities are astrophysically important because ad-dition of a konus density can change one's model of a given

galaxy from disky to boxy or vice versa. This strongly un-derlines the point made by Rix & White (1990) that manyelliptical galaxies might contain non-negligible disks. How-ever, the statement of Rix & White was that these diskswould be merely too faint to detect in currently availabledata, whereas disks contributed by konus densities could notbe photometrically detected, even in principle. The ampli-tude of any disk that could be masked either by the additionof a konus density or by noise in the observations, decreasesas the assumed inclination increases towards edge-on.

One cannot place an upper limit on the luminosity of an`invisible' disk that can be added to a boxy galaxy withoutexploring the set of possible konus densities more thoroughlythan we have been able to do. It seems clear that konusdensities will always comprise a nested sequence of roughlyconical regions of alternating positive and negative density.We have shown that some konus densities have more thanthe three conical regions that are characteristic of our basicfamily, but it seems unlikely that any have fewer. It is cer-tain that great variety is possible in the way in which thepeak value of j�j within a conical region declines with radiusin a konus density. For �xed inclination, our konus densitiesare characterized by two scale lengths � and �. The latterdetermines the radius beyond which the density in the equa-torial plane falls as a power law, � � R�3, while � sets thescale height of the disk that emerges at z � R as R ! 1.These features are certainly speci�c to our examples ratherthan characteristic of konus densities as a whole.

Galaxies are made up of stars moving collisionlessly onorbits in a given gravitational potential, and before one canfeel free to add or subtract any component it is importantto be assured that this can be constructed by placing starson orbits. Fortunately, Lynden-Bell's (1962) demonstrationthat for any �(r; �) there correspond in�nitely many distri-bution functions f(E;Lz) assures us that this is so.

In the light of the previous discussion, attempts to de-tect disks through their e�ect on kinematic observables as-sume greater importance. Unfortunately, uncertainty as tothe role of a third integral in galaxy distribution functionsmakes this approach also di�cult. If we knew that the dis-tribution function were of the simple form f(E;Lz), theneach of the possible density distributions that are consistentwith a given galaxy image, would predict di�erent velocitypro�les along the various lines of sight. Unfortunately, wehave no reason to expect f to be of any particular form,and the prospects for disentangling the e�ects of ambiguityin �(r; �) and ambiguity in phase-space structure seem slim.Said di�erently, for each of the in�nitely many densities thatare consistent with a given galaxy image, there correspondin�nitely many possible distribution functions f , and eachf predicts di�erent kinematics. It seems likely that for anytwo di�erent densities we can �nd corresponding distribu-tion function that predict identical kinematics.

5 CONCLUSIONS

We have shown that for i 6= 90�, the deprojection of axisym-metric density distributions is non-unique in practice as wellas in principle. That is, there are di�erent, astrophysicallyplausible intrinsic density distributions that project to thesame surface brightness.

Deprojecting axisymmetric bodies 11

The di�erence between two such densities is invisible atall su�ciently face-on inclinations. It arises from a Fourierdensity in the `cone of ignorance', the region of Fourier spaceabout which the observed image contains no information.Examples of such `konus' densities can be constructed thatare everywhere non-singular and decay rapidly with radius.

By adding a konus density to any given model of theluminosity distribution of a non-edge-on galaxy, one canchange that model from disky to boxy or vice versa without,even in principle, altering the �t of the model to the photo-metric data. At near edge-on inclination, any added konusdensity will correspond to a thin disk, but the amplitude ofthis disk could still be signi�cant. Thus, especially in thecase of a weak disk, the disk-to-bulge ratio is photometri-cally undetermined.

Kinematic data will in some cases enable one to choosebetween luminosity distributions with identical photometricappearance. But it seems unlikely that even the most com-plete kinematic data will always resolve all ambiguity sincethe existence of a third integral for axisymmetric systemsimplies that an in�nite number of distribution functions cangenerate each of the in�nite number of intrinsic density dis-tributions that are compatible with given photometric data.

The best prospect for making progress is to concentrateon studying the most edge-on systems, for which the pho-tometric uncertainty is least. In such systems a cold diskwould be unmistakable but an appropriate konus densitycould add to one's model a thin hot disk without a�ectingany observable.

We have used Palmer's deprojection algorithm to clarifythe way in which the ambiguity of the deprojection, whicharises from the konus densities, increases as the assumedinclination i decreases from edge-on. As i diminishes, thehigher-order terms in the Fourier decomposition of the sur-face brightness a�ect more and more strongly the lower-order terms in the expansion of �(r; �) in Legendre poly-nomials. This has two consequences. First, the range ofsurface brightness distributions that can be successfully de-projected diminishes as i decreases. Second, with decreasingi the deprojected density becomes more and more sensitiveto noise in the data.

Although Palmer's algorithm is helpful for understand-ing the deprojection problem in general terms, it is muchharder to program and use than the R{L algorithm. Also,it does not constrain � to be non-negative as does the R{Lalgorithm.

Konus densities have in�nitely many terms in their Leg-endre expansions. The density distribution obtained bytruncating such an expansion at high order, projects to afaint surface brightness distribution. When this pattern ofsurface brightness is present in smoothed noisy data, a trun-cated konus density of large amplitude is generated when thedata are deprojected. We have illustrated this phenomenonusing an image of NGC 2300.

The results presented here for the axisymmetric casealso have implications for general triaxial systems. For theystrongly suggest that the range of triaxial densities thatare consistent with given photometry is far greater thanis commonly assumed. In particular, our results suggestthat the three-dimensional distributions that are compatiblewith given photometry di�er not only in the orientation andlengths of their axes, as described by Stark (1977), but also

by the addition or subtraction of weaker, more local struc-tures such as disks or dumb-bells. It would be interesting todisplay explicitly triaxial analogues of konus densities.

ACKNOWLEDGMENTS

We thank R. Bender for making available the photometricdata for NGC 2300 and for a stimulating discussion, andM. H�odtke for computing Fig. 7. OEG thanks the Heisen-berg foundation and the Schweizerischer Nationalfonds for�nancial support.

REFERENCES

Binney J.J., Davies R.L., Illingworth G.D., 1990, ApJ, 361, 78

Dehnen W., 1995, MNRAS, 274, 919

Gradshteyn I.S., Ryzhik I.M., 1980, Table of integrals, series,

and products, Academic Press, New York (GR)

H�odtke M., 1995, Diploma thesis, University of Heidelberg

Lucy L.B., 1974, AJ, 79, 745

Lynden-Bell D., 1962, MNRAS, 123, 447

Palmer P.L., 1994, MNRAS, 266, 697

Richardson W.H., 1972, J. Opt. Soc. Am., 62 55

Rix H.-W., White S.D.M., 1990, ApJ, 362, 52

Rybicki, G.B., 1986, in Structure and dynamics of elliptical

galaxies, IAU Symp. 127, ed. de Zeeuw P.T., Kluwer,

Dordrecht, 397

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Stark A.A., 1977, ApJ, 213, 368

APPENDIX: BAND LIMITATION OF �(K)

Axisymmetric systems with band-limited densities �(x)evade the Fourier slice theorem because their Fourier trans-forms are also band-limited. That is, if �(k) �

Rd3x exp(ik�

x)�(x) is the Fourier transform of � and (k; �; �) are spher-ical polar coordinates in k-space such that the axis � = 0is aligned with the galaxy's symmetry axis, then �(k; �) =PL

l=0�l(k)Pl(cos �). As Palmer shows, knowledge of band-

limited �(k; �) in the complement of Rybicki's \cone of ig-norance" su�ces to determine �(k; �) for all k. In fact, �(k)is determined by its value at only a �nite number of valuesof � at each value of k.

Palmer e�ectively demonstrates that � is band-limitedby explicit calculation. Here we show that this result followseasily from a general group-theoretic argument.

Let the functions fm(x) for m = �l; : : : ; l form a basisfor the spin-l irreducible representation of SO(3), where theaction on fl of a rotation R 2 SO(3) is

R : f ! f0 with f

0(x) � f(RT � x): (A1)

(Here R is the rotation matrix associated with R.) Then bythe uniqueness of the spin-l representation, any fm can bewritten as

fm(x) =

lXm0=�l

fmm0(r)Y ml (�; �): (A2)

12 O.E. Gerhard and J.J. Binney

Fourier transforming equation (A1) we have

f0(k) =

Zd3x0 exp(ik � x0)f 0(x0)

=

Zd3x0 exp(ik � x0)f(RT � x0)

=

Zd3x exp[ik � (R � x)]f(x)

=

Zd3k exp[i(RT � k) � x]f(x)

= f(RT � k):

(A3)

Consequently, the Fourier transforms fm also form a basisfor a (2l+1)-dimensional irreducible representation of SO(3),and by the uniqueness of the spin-l representation we have

fm(k) =

lXm0=�l

fmm0 (k)Y ml (�; �): (A4)

A band-limited �(x) is a function that can be written as alinear combination of functions that are members of spin-lrepresentations of SO(3) for l � L. By the above it immedi-ately follows that �(k) has this last property and so is itselfband-limited.

This paper has been produced using the Blackwell Scienti�cPublications TEX macros.